PROPOSAL SECTION C – PROPOSED RESEARCH
October 17, 2005
1 Introduction
Interacting electron systems have stimulated much of the work in condensed matter physics for
several decades. Usually the system organizes itself into a ground state with nearly independent
low-energy excitations. Examples are the Landau Fermi liquid[1] and incompressible fractional
quantum Hall states[2, 3].
At the same time, it has been realized that quenched disorder plays a central role in electron
systems. Early investigations[4] (with the exception of refs. [5]) concentrated on the effects of
disorder in noninteracting electron systems. Recently it has become clear that disorder-induced
spatial organization may lie at the heart of many strongly correlated systems[6].
This proposal focuses on systems in which the interplay of disorder and interactions conspires to
create novel states which exist solely by virtue of this interplay. The theme of much of the research
proposed below is to answer the following broad questions: (i) How does one create, understand,
and characterize strongly correlated/quantum fluctuating states in mesoscopic systems? (ii) How
does one understand the crossover from a mesoscopic to a bulk system near a bulk quantum
phase transition? In particular, what are the signatures of quantum criticality in transport in a
mesoscopic system? (iii) What are the possible ground states of a bosonic system in a strongly
disordered environment? Is a metallic state[7] possible?(iv) Does quenched disorder qualitatively
modify confinement/deconfinement in gauge theories as applied to condensed matter systems?
The following subsections motivate and sharpen these questions, and explain how they pertain
to the three major thrusts of this proposal: Mesoscopic systems, ν = 1 bilayer systems, and
deconfinement in antiferromagnets and quantum dimer models.
1.1 Open questions for interacting electrons in mesoscopics
The role of interelectron interactions in quantum dots[8, 9] has been clarified with the Universal
Hamiltonian[10, 11] emerging as the low-energy effective theory for diffusive and ballistic/chaotic
quantum dots in the three classical random matrix symmetry classes (the gaussian orthogonal–
GOE, unitary–GUE, and symplectic–GSE[12]). The Universal Hamiltonian[10, 11] (henceforth
called HU ) describes the physics of weak-coupling systems within the Thouless band (the energy
window within the Thouless energy ET of EF ). States separated by ε ¿ ET are correlated with
each other[4, 8, 9] by Random Matrix Theory (RMT)[12]. ET is the inverse ergodicization time of
an electron in the dot. For a diffusive dot ET ≈ hD/L2, where D is the bulk diffusion constant and
L is the linear size of the dot, while for a ballistic/chaotic dot ET ≈ hvF /L, where vF is the Fermi
velocity. In 2D, δ ≈ h2/mL2 is the single particle mean level spacing on the dot. The dimensionless
Thouless number g = ET /δ is an important parameter of the dot. For large g and low energies
1
|ε − εF | ¿ ET , it can be shown that the following HU [10] (for the GOE case) contains all the
relevant interactions in the renormalization group (RG) sense[13] at weak-coupling
HU =∑
α,s
εαc†αscαs +U
2N2 − J ~S2 + λ
∑
α
c†α↑c†α↓
∑
β
cβ↓cβ↑ (1)
Here N is the total number of electrons on the dot and ~S is the total spin. The first term is the
kinetic energy, the second is the charging energy, and the third is the Stoner energy. The final,
superconducting, coupling is missing in the GUE (a dot with an orbital magnetic field but no spin-
orbit couplings), while both the last two terms are missing in the GSE (a dot with both an orbital
magnetic field and spin-orbit couplings). An important prediction of HU is the mesoscopic Stoner
effect[10, 11], which pertains to the submacroscopic magnetization induced by the Stoner term long
before the bulk Stoner transition (occuring at J = δ). This has been verified experimentally[14].
For λ = 0 HU has no quantum fluctuations and is tractable.
A single-particle perturbation (such as a small orbital magnetic field, or a small spin-orbit cou-
pling) which moves the system from one symmetry class to another[12, 15] enhances mesoscopic
(sample-to-sample) fluctuations of the interaction matrix elements and renders the Universal Hamil-
tonian inapplicable during the crossover[16]. Furthermore, quantum fluctuations in a single sample
are enhanced in the RMT crossover[17], effectively moving the system into the quantum critical
regime[18]. This regime can be easily accessed experimentally by tuning external control parame-
ters. Since this regime is dominated by many-body quantum fluctuations, it is qualitatively different
from the Universal Hamiltonian regime. Very little is known about such mesoscopic regimes.
The simplest illustration of such regimes occurs in a superconducting grain (with order param-
eter ∆ and mean level spacing δ), which has been extensively studied [19, 20, 21, 22, 23, 24, 25].
However, many open questions remain in the regime ∆ ' δ, which arises naturally for ultrasmall
grains[26, 27, 28, 29], and also when an orbital magnetic field suppressed ∆ (in the GOE→GUE
crossover). Comparison with Richardson’s exact solution[30, 31] shows that the mean-field solu-
tion becomes poor for ∆ ' δ[29], implying that quantum fluctuations, both of amplitude[32] and
phase[33, 34] become important.
In addition to characterizing fluctuating ground states, one needs to find experimental signatures
for such states. Coulomb Blockade (CB) is experimentally precise[35] and theoretically rich[36],
with many dimensionless ratios (U/δ, U/T , Γ/δ where U is the charging energy, and Γ is the
level width of a single dot level coupled to the leads) allowing for detailed tests of theory. The
simplest case of only a charging interaction[36] (the “orthodox” model) has been investigated by
RG[37] instanton methods[38, 39], phase functional[40] methods (for large number of channels),
bosonization (for a single channel)[41, 42, 43], numerics[44], and most recently a self-consistent
“slave rotor” method[47], which can smoothly interpolate between high-T and the low-T Kondo[48]
regime in quantum dots[49, 50, 51, 52, 53, 54, 55, 56]. Recently mesoscopic fluctuations[45, 46] of
CB have been explored.
In the orthodox model there are no quantum fluctuations in an isolated dot. Coupling to the
leads results in quantum fluctuations, and entangled Kondo-like states emerge[41, 49, 50, 51, 52,
53, 54, 55, 56]. Even richer states might emerge when dots with strong quantum fluctuations (even
when isolated) are coupled to the leads in CB.
Finally, the approach to the bulk limits of various mesoscopic phenomena remains unclear in
many cases. For example: (i) Preliminary calculations by the PI and Y. Gefen of the average
susceptibility for the Pomeranchuk transition[57, 58, 59] in the diffusive bulk indicate that the
critical coupling is identical to the clean limit, u∗ = −2 (spinless). However, in the standard replica
2
treatment of disordered interacting electrons[5], these non-s-wave Landau channels are neglected in
favor of s-wave diffuson/cooperon modes. The interplay between s-wave and non-s-wave slow modes
(near the Pomeranchuk transition) needs to be understood. (ii) The PI’s treatment[60] of Stoner +
Kondo interactions in a quantum dot, with no spatially dependent spin waves, suggests that as the
Stoner coupling J increases the Kondo scale ∆K is initially enhanced before being suppressed. An
enhanced ∆K is also found in the quantum-critical regime near the bulk Stoner transition[61, 62, 63].
A complementary treatment on the bulk Stoner side[64, 65] shows a suppression of the Kondo scale
due to spin waves. Both these behaviors are seen in the PI’s zero-dimensional treatment[60]. It
would be very useful to understand the nontrivial crossover between the finite system/bulk on the
one hand, and paramagnetic/ferromagnetic states on the other.
Summarizing, the following broad questions in mesoscopics appear to be presently open: (i)
How does one create, control, and characterize the ground states of strongly correlated/fluctuating
mesoscopic systems? (ii) Can one develop a generalized theory of Coulomb Blockade for such states
and find their signatures? (iii) Can one characterize the crossovers between mesoscopic systems
and the bulk near a bulk phase transition?
1.2 Open questions for the ν = 1 bilayer quantum Hall system
Experiments on bilayer quantum Hall systems have established a number of properties akin to
superfluidity[66]. Early theoretical investigations of these systems concluded that[67, 68, 121] when
the interlayer separation is small: (i) The system should be a superfluid at low temperatures. (ii)
There should be a finite-T Kosterlitz-Thouless transition when superfluidity is lost. (iii) The charge
carriers are “merons” which are half-skyrmions and have charges of ±e/2.
In the tunnelling geometry, current flows into the top layer and flows out the bottom layer, and
the interlayer voltage Vint shows a narrow (but not infinitely narrow[69, 70]) peak at zero bias. As
T decreases, the height of the peak increases and its width decreases. For a true superfluid one
expects an infinitely narrow Josephson-like peak at zero bias for T below the Kosterlitz-Thouless
transition temperature TKT . In the counterflow geometry currents flow in opposite directions in
the two layers (a neutral excitonic current). Both the longitudinal and Hall voltages on one layer
are activated (' e−∆/T )[71] and current flows throughout the sample. In a true superfluid the
counterflow current would decay within a Josephson length, and the bulk currents and voltages
would be strictly zero.
The presence of dissipation at the lowest measurable T in a bosonic system makes this state
extremely unusual. Despite much theoretical work[125, 126, 127, 128, 129, 130, 131, 132], a complete
explanation remains elusive. Disorder is believed to be central to a full understanding of this state.
Some of the many questions remain unanswered are: (i) What is the nature of the true ground
state at T = 0? Is it a superfluid state or a vortex liquid[131, 132], or equivalently, a vortex
metal[7]? (ii) What is the role, if any, of real spin[122, 123] as opposed to pseudospin? (iii) What
is the best description of the quantum phase transition[126, 127, 128, 129] between two widely
separated ν = 12 systems and the interlayer coherent state? What is the effect of disorder on this
transition?
1.3 Open questions for deconfined criticality with randomness
Confinement and deconfinement have proven to be fruitful concepts in the physics of correlated
electron systems. Many strongly interacting systems can be recast as gauge theories[72], and
phenomena such as fractionalization[73] and topological order[74] can be simply understood.
3
The study of two-dimensional quantum spin- 12 models[75, 76], and their descendants, the quan-
tum dimer models[77, 73, 79, 80, 81], has been very fertile: The nearest-neighbor hard-core quantum
dimer model on a triangular lattice supports an entire spin liquid deconfined phase[78], with nearly
free monomer (spin- 12) excitations. Other such models have been constructed since[82], and related
to the quantum Lifshitz theory[83]. Coming back to spin models, it has been conjectured[84, 85]
that the Neel to Valance Bond Solid (VBS) transition[75, 76] for the square lattice Heisenberg
antiferromagnet could be generically second order, contrary to the usual Landau rules. Further,
in this picture the critical region is best described in terms of spinons which are confined in either
phase. The root cause[84, 85] of these phenomena are the Haldane-Berry phase factors[86] associ-
ated with the tunnelling of a skyrmion (a “hedgehog” event) in an antiferromagnet. The spatial
dependence of these phase factors, together with the translation invariance of the clean lattice
leads to a paramagnetic phase breaking translation invariance, and the hedgehogs having to appear
in quadruplets[86]. This quadrupling makes hedgehogs irrelevant in the RG sense exactly at the
transition[84, 85], leading to spinon deconfinement.
Since real samples always have quenched randomness, one can ask whether randomness qual-
itatively changes the physics. There are strong reasons to believe that it does. Looking first at
critical points, by a quantum extention[87] of the Harris criterion[88], it is seen that disorder is
relevant in the absence of hedgehogs both at the dimer transitions described by the quantum Lif-
shitz theory[82], and at the conjectured deconfined critical point of the antiferromagnet[84, 85].
The important questions are: (i) What is the nature of the transition with quenched disorder?
In classical models, by mapping the random-bond disorder to an effective random-field disorder,
it is known that disorder renders first-order transitions second-order[89, 90, 91]. The opposite
might well occur here. (ii) If the critical point remains second-order, are hedgehogs irrelevant at
the disordered critical point? Recall that the lattice symmetries[86] that forced the hedgehogs to
occur quadrupled (on the square lattice) are broken by disorder. (iii) Away from the critical point,
quantum Griffiths singularities[92, 93, 94, 95] are expected. What form do these singularities take
near the disordered quantum Lifshitz critical point (assuming it exists)?
Coming now to the triangular lattice dimer spin liquid[78], it is known that the topological
degeneracy of the ground state is stable against weak disorder[97]. How about excitations? Disorder
will induce local VBS order, and spinons are confined in the presence of VBS order. So, one expects
disorder to induce a long-range potential between the excitations. The important questions here
are: (i) What is the nature of this disorder-induced interaction between excitations? (ii) Is there
a critical strength of disorder beyond which they “reconfine”? If so, does topological order get
destroyed at this critical disorder?
2 Results of previous grant-supported research
During the grant period July 2003 – present the PI and co-workers have made progress on meso-
scopic systems with disorder and interactions[98, 99, 100, 17, 60, 101, 102], aspects of fractional
quantum Hall edge states[103], and developed a coherent network model[104] of the bilayer quan-
tum Hall “superfluid” at ν = 1. Several other projects are in progress and will be described briefly
below. Six papers have been published, four are in press, and several are in preparation.
Results on mesoscopic systems: Before the current grant period the PI and co-workers
have shown that[13, 105] (i) The most natural starting point to describe ballistic/chaotic dots is
a Landau Fermi liquid with the disorder represented by scattering off the walls. (ii) HU is the
weak-coupling low-energy effective theory of the finite-size fermionic RG in the g → ∞ limit.(iii) In
4
ballistic/chaotic quantum dots a phase transition to a bulk Pomeranchuk phase[57, 58, 59] (with a
distorted Fermi surface) is possible as g → ∞.
During the current grant period the research carried out by the PI and co-workers on meso-
scopics can be broadly divided into; (i) Characterizing the mesoscopic Pomeranchuk transition
in ballistic/chaotic dots[98, 99, 100], (ii) Establishing[17] a generic connection between single-
particle crossovers between different symmetry classes by an external perturbation[12] and the
weak-coupling to quantum critical many-body crossover[18] in interacting quantum dots. (iii)
Treating[60] the competition between the Stoner interaction between electrons on a large dot and
the Kondo interaction[48] of these electrons with the “impurity” spin on a small dot[49, 50, 51].
This leads to a counterintuitive enhancement of the Kondo scale under Zeeman fields which may
have been seen in experiments[108]. (iv) Constructing a supersymmetric nonlinear sigma model to
investigate ballistic systems with random boundary transmission[101, 102].
(i). In ballistic/chaotic dots with Landau Fermi liquid interactions, by using large-N methods
the PI and co-workers showed[99] that there is a crossover between the weak-coupling regime
dominated by the Universal Hamiltonian and a many-body quantum critical regime[18]. We
computed[99] the behavior of the quasiparticle decay rate in the weak-coupling to quantum critical
crossover. For symmetry-breaking in an odd angular momentum Landau channel, each isloated
sample has a two-fold ground-state degeneracy. This results in a linear dependence of the Coulomb
Blockade peak position as a function of an external orbital magnetic field. If the dot is coupled
weakly to leads, via a Kondo-like effect[109, 110] it can spontaneously break time-reversal symme-
try and develop a large persistent current[99]. Separately, we carried out a numerical analysis of
the persistent current in the RMT limit with Fermi-liquid interactions and found a diamagnetic
persistent current[98] in the presence of even channel Landau interactions. Our numerical analysis
on the Robnik-Berry billiard[111] found[100] that most of the assumptions of our previous work
are qualitatively and semi-quantitatively correct. However, we do not find any window between
the mesoscopic strong-coupling phase and the bulk Pomeranchuk phase[57, 58, 59]. The absence of
this window was pointed out in ref. [112] in a related model. Also[100], the mesoscopic fluctuations
of the effective potential are larger than expected from RMT for symmetry-breaking in an even
Landau channel. Finally[100], significant symmetry-breaking can occur even for weak-coupling in
the even Landau channel case, enhancing experimental visibility.
(ii) Crossovers between different symmetry classes of RMT[12, 15] occur ubiquitously in quan-
tum dots, and can be tuned, for example, by an external orbital magnetic field (GOE→GUE), or
by changing the size of a 2D GaAs dot (GOE→GSE). Single-particle RMT crossovers are charac-
terized by an energy scale EX : States separated by ε ¿ EX exhibit correlations which are fully
crossed over, while for ε À EX they exhibit the correlations of the original symmetry class. Dur-
ing the crossover, extra correlations develop[15], mesoscopic fluctuations of the interaction matrix
elements become large, and HU is inapplicable[16]. Consider an order parameter Q consistent with
symmetry class I, but not with class II (e.g. a spin polarization is consistent with GOE but not
GSE). Label the relevant dimensionless coupling J with the critical point J ∗ at which a transition
to a macroscopic value of 〈Q〉 occurs. Using correlations in the RMT crossover[113], the PI has
shown[17] a generic connection between the single-particle scale EX and the energy scale EQCX
to cross over to the quantum critical regime of Q: EQCX = |J − J∗|EX . All physical correlators
in the quantum critical regime are given in terms of explicit scaling functions[17] by a nonpertur-
bative calculation. The PI considered two examples: (a) The Stoner transition in the crossover
between the GOE and a new symmetry class discovered by Aleiner and Fal’ko[114], and (b) A
superconducting grain in the GOE→GUE crossover. The experimental implication is that one can
5
tune access to the quantum critical regime externally. These results hold for diffusive as well as
ballistic/chaotic dots.
(iii). In a recent Kondo-like experiment conduction electrons live in a large dot[108], while the
Kondo “impurity” spin is a small dot with an odd number of electrons. The electrons on the large
dot are described by HU . The PI has investigated[60] the competition between Kondo screening
and the Stoner interaction which tries to polarize the dot electrons. There is a regime[60] in
which the polarization of the dot electrons is suppressed, while the Kondo energy scale is enhanced.
Most strikingly, a Zeeman coupling leads to a huge enhancement of the Kondo scale[60], which is
the opposite of what occurs when the conduction electrons are noninteracting[106]. Under a large
Zeeman field, the Kondo state collapses into a mesoscopically polarized dot with the impurity slaved
to the dot polarization. These effects have been seen in experiments[108], but the dot there is open
while the theoretical model assumes a closed dot. Other explanations[107] of the enhancement of
the Kondo scale rely on the presence of two impurity spins, whereas the enhancement occurs[108]
for a single impurity as well.
(iv). In many ballistic mesoscopic systems (such as quantum corrals[115], quantum dots with
many leads, and optical/microwave cavities with leaky walls[116, 117]) the coupling to the outer
world is confined to the boundary. When the average coupling is weak and is gaussian distributed
around its mean, the PI’s postdoc, I. Rozhkov, and the PI have constructed a supersymmetric non-
linear sigma model (for integrable[101], and chaotic billiards[102]) which can be used to study any
physical correlator. The random coupling to the boundary acts as a natural regulator, and allows
us to avoid the technical difficulties which have dogged previous attempts at constructing ballistic
nonlinear sigma models[118]. The result is a set of modes confined to the boundary and diffusing in
angle, representing whispering gallery modes[119] interfering and diffracting out of the dot through
its leaky walls. This describes a situation complementary to “dynamic localization”[120] in which
the modes diffuse in angular momentum in a closed billiard.
Results in the ν = 1 bilayer quantum Hall system: As described in the introduction, ex-
periments have shown that an unusual dissipative state exists at the lowest measurable T in inter-
layer coherent systems at ν = 1. Based on the known effects of disorder on incompressible states,
Herb Fertig and the PI have constructed[104] a model which explains some of these observations.
In the Efros picture[134] (supported by imaging experiments[135]), the ν = 1 system cannot screen
disorder due to the dopant layer linearly, since it is incompressible. The system breaks up into
compressible regions (with ν < 1 or ν > 1) and incompressible strips (with ν = 1) of typical width
a few magnetic lengths, and screens the disorder nonlinearly. This is an example of the generic
phenomenon of pattern formation by disorder[6].
Our model treats the spatial structure induced by disorder explicitly. Along an Efros strip
disorder induces many solitons[104] each of which ends in weakly localized merons or antimerons.
There are two main sources of dissipation. At a node, where two or more incompressible strips
meet, the effects of disorder are the least, and therefore the tunnelling term with characteristic
energy h is the most effective. Thermal/quantum fluctuations induce the solitons to pass through
the node, disrupting the tunnelling at the node and produce a viscosity for the tunnelling current,
which translates into an interlayer voltage. For T À h a perturbative classical treatment leads to
an interlayer voltage Vint ' h2T−3/2, and a tunnelling conductance proportional to the area of the
sample[104].
In the counterflow geometry dissipation is produced by merons/antimerons thermally hopping
across the Efros strips. In order for a nonzero Hall resistance to exist, the merons must be in a
liquid state[131, 132]. We show by a replica-RG analysis of the classical model that the liquid state
6
can occur even as T → 0 for sufficiently strong disorder[104]. In this liquid state, the barrier to
these meron-antimeron tunnelling events is of the same order as the charge gap. Since merons and
antimerons are vortices and antivortices, their tunnelling across the sample creates a longitudinal
gradient in the interlayer voltage Vint, which can be measured as a longitudinal voltage on a single
layer. The merons carry a dipole moment which ultimately[104] leads to the activated Hall resis-
tance. In our model RH has the same activation gap as the longitudinal resistance (experimentally
true in electron but not in hole samples[71]).
Miscellaneous and ongoing work: The projects below either do not fit neatly into a par-
ticular category or are in progress. (i) In collaboration with his former postdocs, Y. Joglekar and
H. Nguyen, the PI has carried out a calculation of collective edge modes[103] in ν = 13 and 2
5 ,
using the Hamiltonian formulation of the FQHE developed by R. Shankar and the PI[136]. (ii)
R. Shankar and the PI have revisited deconfinement[137] in 1 + 1 dimensions[138, 139, 141], and
we find that often the particles usually considered to be “free” are only half-asymptotic, which
means that they must appear with particles and antiparticles alternating[139]. The confinement-
deconfinement transition in the massive Schwinger model[139, 140] is in the 2D Ising universality
class[137], and there are truly free Majorana fermions which have a nonlocal connection to the
original Dirac fermions. Similar results obtain in the spin chain with frustrated antiferromagnetic
interactions[141, 142]. (iii) A graduate student, O. Zelyak, and the PI are analyzing the problem of
a ballistic/chaotic system penetrated by a point flux, and the persistent current resulting from this
flux[143]. We find that this problem seems to be quite different from that of a uniform magnetic
field pervading the dot (which has been noticed before[144]). We also find that there is a large
diamagnetic persistent current which is proportional to the number of electrons in the dot (which
has not been noticed before). We are also investigating chaotic annular geometries, using an annu-
lar generalization of the Robnik-Berry billiard[111]. (iv) Y. Gefen and the PI are considering the
Pomeranchuk transition in a diffusive system. Our goal is to compute mesoscopic fluctuations of
the order parameter, and the behavior of the quasiparticle decay rate as the system crosses over
into the quantum critical regime on the weak-coupling side. This work is in progress.
3 Proposed projects
3.1 Proposed Projects in Mesoscopics
Let us recall the broad issues to be addressed in mesoscopics: (i) How does one create, control, and
characterize the ground states of strongly correlated/fluctuating mesoscopic systems? (ii) Can one
develop a generalized theory of Coulomb Blockade for such states? (iii) Can one characterize the
crossovers between mesoscopic systems and the bulk near a bulk phase transition?
3.1.1 Competition between Kondo and Stoner interactions
The PI’s work on Stoner + Kondo[60] has several limitations: (i) It is carried out for a single impu-
rity, whereas the experiments have been carried out for the conduction electrons interacting with
both a single impurity and with two impurities. The enhancement of ∆K with a Zeeman coupling
EZ is also seen in the two-impurity case[108]. (ii) The PI’s calculation assumes equally spaced
energy levels and an equal coupling of each level to the impurity spin, thus ignoring mesoscopic
fluctuations. (iii) The calculation is carried out at T = 0. (iv) The most important limitation is
that the PI’s work is carried out in a closed dot. The real experiments are carried out with dots
with a reasonably strong coupling to the leads (∆K is inferred by conductance measurements[108]).
7
Research is proposed to remove these limitations. The PI’s calculation will be repeated for
T > 0. The result will be interesting for large total spin, since[10, 11] for small r = 1 − J/δ (J
is the Stoner coupling of Eq. (1)) the ground state spin is of order 1/r, while there are many
energetically close values of total spin which are populated at nonzero T . This is expected to
enhance the Kondo scale even more than at T = 0, until T ' δ at which point particle-hole
excitations will start frustrating the Kondo state. The T, EZ dependence of ∆K is expected to be
a good experimental signature of the nature of the state. Note that the large-N approach is not
accurate for T comparable to ∆K [106], but we will restrict ourselves to T of order δ ¿ ∆K
The bulk limit (L → ∞) of the Stoner + Kondo model in an isolated dot is extremely interesting
near the Stoner transition: In the seminal work of Larkin and Melnikov[61], as the bulk system
approaches the Stoner transition, the spin susceptibility diverges, and the impurity polarizes a
huge droplet of electrons[62]. Since the effective size of the impurity is now large, the conduction
electrons can couple to it via many channels with nontrivial spatial dependence. The number of
channels tends to ∞ as the Stoner transition is approached[62]. This physics is a special case of
the bose-Kondo problem[63, 64].
It is proposed to extend the PI’s calculation to the crossover to the bulk in a diffusive dot
beyond ET . This can be accomplished with exactly the same methods as used by the PI, except
that now the wavefunction correlators of the dot states are no longer controlled by RMT, but
become wavevector and energy dependent[150]. The emergence of Landau-damped spin waves and
their interaction with the impurity[61, 62], as well as the emergence of many channels coupling to
the impurity will be investigated. Since the calculation will be carried out for a finite system with a
spin magnetization (albeit submacroscopic), it may shed light on the physics of the Kondo impurity
in the bulk Stoner phase[64, 65], where ∆K is suppressed. In this context, recall that even in the
PI’s zero-dimensional calculation[60], ∆K is suppressed for large enough J . On the other hand, as
J → 0 one expects to recover the usual disordered Kondo treatments in the bulk limit[151]. This
calculation will serve to clarify diverse crossovers and how they tend in the bulk limit to phase
transitions.
The quantum dot interacting with two magnetic impurities[108, 107] is interesting for a variety
of reasons. In the two-impurity case, the experiment has been explained by assuming that an
antiferromagnetic RKKY interaction I exists between the two impurities[107], leading to a singlet.
The Zeeman field closes the singlet-triplet gap leading to an enhancement of the (singlet-triplet)
Kondo scale[107]. This latter explanation does not include the Stoner interaction, and is inoperative
in the single-impurity case, in which an enhancement of the Kondo scale is also observed[108]. The
two-impurity case is theoretically interesting even in the bulk[152, 153, 154, 155], where a phase
transition (as I decreases) is found between a phase in which the two impurities form a singlet with
no Kondo effect, and one with a Kondo effect of both impurities[152, 155]. Finally, the two-impurity
Kondo problem in the dot, when Stoner interactions are included, offers a rich, yet experimentally
realized[108] example of competing interactions.
It is proposed to address the two-impurity Kondo problem with Stoner interacting conduction
electrons by using the large-N [156, 157, 158, 159] (or slave boson[160]) approach. At mean-field
level in the bulk, this approach fails to recover the unstable fixed point, giving instead a first-order
transition[154]. A potential way to overcome this limitation is to look at the effective theory of
phase fluctuations of the Kondo order parameter[158, 159] around the mean field solution. The
usual mean-field will be used to identify the phases and describe the low-energy physics deep in
a particular phase, while a careful analysis of the phase fluctuations will be performed near the
transition. In order to distinguish the different ground states, the most convenient signatures occur
8
in transport[107], which will be addressed in subsection 3. 1. 4 on Coulomb Blockade.
3.1.2 GOE to GUE crossover in superconducting grains
The PI’s analysis[17] of the relation between the single-particle RMT GOE→GUE crossover and
the weak-coupling to quantum critical interacting crossover in superconducting grains is limited to
weak-coupling (when the magnetic flux has killed the order parameter) and depends solely on the
noninteracting 4-point ensemble averages in the single-particle crossover computed recently[113].
When a superconducting order parameter ∆ is present, even considering only thermodynamic
quantities (leaving the transport properties for the Coulomb Blockade subsection), a number of
issues need to be faced: (i) For an isolated grain, charge quantization means that the phase of
∆ must be fluctuating[33, 34]. (ii) Comparing the BCS mean-field solution with Richardson’s
exact solution[30, 31] shows that different values of ∆BCS are inferred depending on what one
measures[29]. These values can be very different as ∆BCS approaches the mean level spacing δ.
This means that as the orbital field increases and ∆ becomes small, both amplitude[32] and phase
fluctuations[33, 34] of ∆ become important. (iii) Finally, mesoscopic fluctuations[34] need to be
considered in the crossover regime in order to compare to experiments.
Some of the ingredients for a nonperturbative calculation of the thermodynamic properties in
the crossover regime exist in the literature cited above, but other important ones are proposed
below.
The first step is to compute the spectrum and wavefunction correlators of the mean-field Hamil-
tonian, which includes the weak magnetic flux, and ∆.
HX(φ, ∆) = HO + ∆∑
i
(c†i↑c†i↓ + ci↓ci↑) + αHA (2)
where HO is the noninteracting Hamiltonian with Orthogonal symmetry with eigenstates labelled
by i, j and HA is a normalized antisymmetric Hamiltonian[12] induced by the external magnetic flux
Φ and α ' Φ/Φ0. This is a double-crossover and the ensemble averages of the 4-point correlators of
wavefunctions will determine the effective action[17] of the fluctuations of ∆. It is proposed to use
numerical[113] and supersymmetry methods[15] to determine the wavefunction correlations. The
known results in the limits[113] will provide a test of our results.
The next step is to include phase[33, 34] and amplitude[32] fluctuations of ∆ in the effective
theory. The best way, presaging the CB proposed research below, is to introduce a very large
charging energy to fix the number of particles. Since the total charge consists of electrons which
are part of a Cooper pair as well as electrons which are not, there are two phases involved, the
phase φ conjugate to the total charge, and the phase φ∆ of the order parameter ∆ = |∆| exp iφ∆.
When ∆ À δ, one expects the two phases to be locked at low energies, but we wish to investigate
the quantum critical regime when ∆ has no expectation value, but only slow fluctuations of both
amplitude and phase. As in the treatment of Coulomb Blockade, instantons[38] in both φ and φ∆
are expected to play an important role in determining the physics.
The resulting effective theory will be analysed using RG[37] instanton[38, 39], and slave rotor[47]
methods. The PI’s postdoc, I. Rozhkov, is an expert in supersymmetry techniques and will assist
the PI in this project. The graduate student, O. Zelyak, will also participate in this project.
3.1.3 The mesoscopic Stoner effect with weak spin-orbit interactions in GaAs
In GaAs there are “intermediate” spin-orbit RMT universality classes[114]. In the simplest one, to-
tal spin is not conserved but Sz is. Deep into this crossover, one recovers the mesoscopic Ising-Stoner
9
effect[161]. In the PI’s previous work[17], the analysis was restricted to Sz = 0 and mesoscopic
fluctuations were neglected.
It is proposed to examine the distribution of the ground state Sz as a function of the Stoner J
in the above crossover. However, Sz is not directly measurable. To make contact with measurable
quantities, one needs to include an in-plane field[162]. Due to the spin-orbit coupling, this in-plane
field breaks time-reversal symmetry. One is again led to a double crossover from the GOE with
perturbations taking the system to the Aleiner-Falko intermediate symplectic class[114] and the
GUE. The dependence of CB peak position on in-plane field will be calculated, and the connection
with ground state Sz found. Since experiments can tune through this crossover by changing the size
of the dot[114], the results will be directly relevant for dots constructed from GaAs heterostructures.
Both the postdoc and the student will participate in this project.
3.1.4 Coulomb Blockade in strongly correlated and fluctuating states
The treatment of CB in the simplest case of the orthodox model reduces in the δ/T → 0 limit
(after integrating out fermions) to the analysis of the effective action of the phase φ conjugate to
the total dot charge[40]. For nonzero δ/T the recently developed slave rotor[47] formulation which
treats the fermionic degrees of freedom on the same level as the bosonic ones, appears promising.
The new feature in the research proposed below is the existence of other collective variables, which
possess their own bosonic degrees of freedom, which can often be reduced to a phase (or an SU(2)
nonabelian phase for spin-rotation invariant systems). The theme of this subsection is the interplay
of the charge degree of freedom with the other collective variables, and with the fermions.
Coulomb Blockade in the Mesoscopic Pomeranchuk Regime: As the PI and co-
workers have shown[99, 100], mesoscopic systems have ground state expectation values for the Fermi
surface distortion, characterized by the vector σ (which determines the single-particle states), even
in the weak-coupling regime (|u| < |u∗|). This is a result of mesoscopic fluctuations and explicit
rotational symmetry breaking[99, 100]. Typically, the ground states for N and N + 1 particles will
have different values of σ, and the quantum fluctuations of σ will be small for large g. When an ad-
ditional electron enters the dot all the single-particle states change in a ground-state to ground-state
transition, implying that the transition amplitude will suffer orthogonality catastrophe effects[163]
and be very small. The typical barrier EB between the two minima is expected to be of order ET .
The distribution of effective tunnelling strengths between the two minima will be computed using
the large-g mean field theory[99] developed by the PI and co-workers and RMT. This is related to
the Caldeira-Leggett problem of tunnelling between two nearly degenerate states in the presence
of a dissipative bath[109, 110]. The new feature is that the charge degrees of freedom (phase φ)
are coupled to the two minima of σ for T ¿ EB, and to large fluctuations of σ in the quantum
critical regime. Therefore one expects a generalization of the charge Kondo effect[41] at low T , and
nontrivial T dependence of the CB.
Coulomb Blockade for the Universal Hamiltonian with large S: While there has been
progress[146, 147, 148, 149] over the last few years in treating the CB problem for the Universal
Hamiltonian[10, 11], our knowledge is by no means complete. For an isolated dot, as the Stoner
coupling approaches δ (or r = (1−J/δ) ¿ 1), the ground state spin becomes large[10, 11] (' 1/r),
and there are many (of order 1/√
r) spin states within an energy δ. When the dot is coupled to
leads the spin has to change when particle number (charge) changes. However, there are strong
constraints on transitions between spin states because they have to be connected by a single-particle
operator (for weak tunnelling). This will couple the charge and spin degrees of freedom.
Since the spin of a dot coupled to leads is not conserved, one can decouple the Stoner interaction
10
by a Hubbard-Stratanovich exchange field h, and the charging interaction[38, 39] by a scalar V .
The imaginary-time action (suppressing the dot-lead coupling) is
S =
β∫
0
dt
(
h2
4J+
V 2
2U+ iV N0 +
∑
kss′
cks((∂t + εk − iV )δss′ −h
2· ~τss′)cks′
)
(3)
where N0 is related to the gate voltage, the single-particle dot states are (k, s), and ~τ is a Pauli spin
matrix. Now, one represents[38, 39] the non-constant part of V in terms of a phase V (t) = V0 + φ,
and constructs “neutral” d-fermions by the unitary transformation cks = eiφ(t)(R(t))ss′dks′ , where
the unitary SU(2) rotation operator R(t) is defined by the requirement |h(t)|τz = h(t) ·R(t)†~τR(t).
All the spin dynamics is now contained in the SU(2) nonabelian phase. Since the ground state
spin of the decoupled dot is large, h will have a nonzero expectation value, with the d fermions
subject to a Zeeman field in the z-direction. Integrating out the lead fermions, one obtains an
effective action which couples φ, R, and the dot fermions. Assuming equal coupling of all lead and
dot modes the key term is of the form
Γ∑
k,k′
β∫
0
dtdt′
(t − t′)ei(φ(t)−φ(t′))d(t′)R†(t′)R(t)d(t) (4)
In phase-only models[40] (valid in the limit δ/T → 0[39]) the dot fermions are also integrated out
with their free action to leading order. Since we want to access all the temperature regimes as well
as Stoner physics, we need to work at nonzero δ/T . The dynamics of this model will be investigated
by RG[37], instanton[38, 39], and slave-rotor[47] methods, suitably generalized. For example, in
the slave-rotor approach, one would decouple the integrand of Eq. (5) by
〈ei(φ(t)−φ(t′))〉〈ds′(t′)ds(t)〉R†
s′s1(t′)Rs1s(t) + 〈ei(φ(t)−φ(t′))〉ds′(t
′)ds(t)〈R†s′s1
(t′)Rs1s(t)〉+ei(φ(t)−φ(t′))〈d(t′)R†(t′)R(t)d(t)〉 (5)
and self-consistently determine the different correlators.
Coulomb Blockade for Stoner + Kondo interactions in a quantum dot: This prob-
lem is relevant to measuring the correlated state formed by the dot electrons with an impurity
spin, previously investigated by the PI[60] when the dot and impurity are isolated from the leads.
This will involve an additional level of complexity to the previous subsection, since now the fluctu-
ating phase θ of the hybridization parameter between the dot electrons and the Abrikosov fermion
used to represent the impurity spin has to be taken into account[158, 159]. It is known that these
fluctuations have the action (to order 1/N in the large-N expansion, at T = 0, neglecting the
level spacing of the dot)[158, 159] πN
∫
dtdt′sin2 θ(t)−θ(t′)
2(t−t′)2
. Since they alter the phase of the fermion
wavefunctions, they will also couple with the Coulomb phase. In addition, due to the spin there
will be an SU(2) phase as well. Once again, RG, instanton, and large-N methods (all appropriately
generalized) will be used to analyze this model.
Coulomb Blockade in RMT crossover ensembles: An isolated superconducting grain
penetrated by a sufficiently strong orbital magnetic flux is in a state dominated by collective
quantum fluctuations[17]. As mentioned previously, there are two relevant phases, the phase φ
conjugate to the total charge, and the phase φ∆ of the superconducting order parameter. If there
are strong amplitude fluctuations, then the amplitude of ∆ is also a fluctuating field. As opposed
to subsection 3.1.2, where the focus was on the thermodynamics of the isolated grain penetrated
11
by a flux, here the focus is on CB. The T , charge asymmetry, and magnetic flux dependence of the
CB peak will be computed in all the three regimes, strong-coupling, quantum-critical and weak-
coupling. A clear signature of quantum criticality is expected in the T -dependence of CB peak
widths, which would be useful to identify this regime in transport experiments. Similarly, in the
Aleiner-Falko intermediate spin-orbit class[114], even a small Stoner interaction is enough to put
the dot in the quantum critical regime[17]. The CB of this system in a parallel field (the double
crossover) is expected to bear the signatures of quantum criticality. Both the postdot and the
student will participate in this project.
Coulomb Blockade in granular materials with chaotic/ballistic grains: A number of
investigations[39, 164, 165] have focused on granular materials recently. When the grains comprising
the system are chaotic, an extention of the recent supersymmetric formulation of the problem of a
single dot randomly coupled to the environment at its boundary is possible[101, 102]. Whispering
gallery modes, represented in the model by a superfield living on the boundary, are identified as the
important modes at long times. It is proposed to couple the boundary modes for neighboring grains.
If the coordination number z of the grain in the material is large, then it is appropriate to start
with the z → ∞ limit, consistent with the PI’s work with I. Rozhkov. In this limit, recent large-d
formulations (also called dynamical mean field theory (DMFT)) of correlated electron systems[166],
which have been extended to disordered systems[167] offer a fruitful way to proceed. Briefly, it is
proposed to self-consistently solve an “impurity” problem, which represents a single grain in its
random environment. Non-Fermi-liquid regimes are easily identifiable in this approach[166]. The
PI’s postdoc will participate in this project.
3.2 Proposed Projects in the ν = 1 bilayer system
Some of the unanswered questions for the ν = 1 bilayer system are: (i) What is the nature of the
true ground state at T = 0? Is it a superfluid state or a vortex liquid[131, 132], or equivalently,
a vortex metal[7]? (ii) What is the role, if any, of real spin[122, 123] as opposed to pseudospin?
(iii) What is the best description of the quantum phase transition[126, 127, 128, 129] between two
widely separated ν = 12 systems and the interlayer coherent state? What is the effect of disorder
on this transition?
The nature of the spinless ground state at T = 0: The classical analysis carried out by
Herb Fertig and the PI[104], based on the nonperturbative effects of disorder[134], decomposes the
system into two components, links and nodes where interlayer coherence is intact, and puddles,
where it is not. One key question is whether the quantum dynamics of this bosonic system is
dissipative[7, 131, 132]. Results from the gauge glass problem[168] are not directly applicable to
the ν = 1 bilayer because interlayer tunnelling and dynamical bond fluctuations (arising from the
quantum tunnelling of merons/antimerons across the strip) are absent in gauge glass models.
Two ways are proposed to address the issue of dissipation at T = 0 and the low T crossover.
The following Hamiltonian density incorporates all the relevant physics
∑
bonds
1
2(a2 + (∇× a − B0(~r))
2 − t∑
~r,ei
(φ†(~r + ei)eia·eiφ(~r) + h.c.) + λ
∑
~r
(φ†φ − φ20)
2)) (6)
where φ is the exciton destruction operator on a node (its phase is the phase of the “superfluid”),
and a denotes a dynamical gauge field living on the links with a preferred site-dependent flux B0
induced by disorder. The kinetic term of a denotes the quantum tunnelling of merons/antimerons
across the Efros strip. It is proposed to treat a node/dual site in the surrounding medium as a self-
consistent “impurity” problem and solve for the local Green’s function, in a large-d spirit[166, 167].
12
In this system, both baths are bosonic[63]. States where the rotors on the nodes and the fluxes on
the dual lattice sites have excitations at arbitrarily low energies will be looked for, since they can
self-consistently produce dissipation. In such a state the merons/antimerons/vortices will also be
delocalized across the entire system, with the potential of being in a liquid state[131, 132, 7].
A complementary approach is to disorder-average the above model using replicas. The resulting
theory will be analysed to see what kinds of saddle points exist, with the focus being on the density
of states of the low-lying excitations. This way has the disadvantage that it averages over the
spatial structure which, in the PI’s opinion, plays an important role in choosing the ground state of
the system. However, it will provide an independent check on the density of low-energy excitations.
The role of real spin: Most theoretical investigations assume that real spin plays no role
in the ν = 1 bilayer system. A clean 2DEG at ν = 1 is spontaneously spin polarized due to
exchange[169] even in the absence of a Zeeman coupling. However, for sufficiently strong disorder
the electron gas becomes spin disordered[170]. The PI[171] has previously shown that for single-
layer samples of similar quality to the bilayer samples, all the experimental data is consistent
with the system having no spontaneous spin polarization. In this context, recent experiments[122]
indicate an important role for spin in the compressible to bilayer coherent transition and in the
bilayer coherent state which is formed.
The effects of spin-pseudospin mixing will be seen in the response to a parallel field. When
interlayer tunnelling is present, a parallel field also couples to the pseudospin by introducing a
wavevector in the tunnelling, which is used to uncover the Goldstone mode[67, 70].
It is proposed to start with a theory including real spin[123] and disorder-average it along the
lines of the PI’s work on single-layer ν = 1 systems[171], and then compute physical quantities
such as the spectral density of spin excitations which can distangle the effects of parallel field on
pseudospin and real spin.
The quantum phase transition as a function of layer separation: As the layers are
separated, a quantum phase transition[126, 127, 128, 129] from the interlayer coherent superfluid
state to a state with two decoupled ν = 12 liquids occurs. The effect of disorder on this transition
is expected to be important in real materials.
In the Efros picture[134], as the layer separation increases the Coulomb disorder in the two layers
becomes less correlated. This leads to the nodes and incompressible strips getting “pinched off”,
with the couplings between the nodes becoming weaker. It is proposed to study this using the self-
consistent impurity formulation[166, 167] discussed earlier. When the couplings between the nodes
becomes weak enough, the collective Kondo-like state is expected to collapse into two critical ν = 12
liquids adiabatically connected to the noninteracting 0 → 1 plateau transition point. Note that
such a transition is qualitatively different from the transition in a clean system[126, 127, 128, 129],
where the compressible state has two decoupled Composite Fermion fermi liquids. This will provide
a disorder-dominated perspective of the transition.
3.3 Proposed Projects for deconfined criticality + quenched randomness
The broad issues here are: (i) What is the effect of disorder on deconfined critical points in the
absence of hedgehogs? Do the transitions remain second-order? (ii) If they do, how do single
hedgehogs (allowed for relevant disorder by the explicit breaking of lattice symmetries) modify
this? (iii) Can disorder “reconfine” excitations in a deconfined phase?
Quenched disorder at the Neel-VBS transition for square lattice S = 12 : The
Haldane-Berry phases[86] of the quantum tunnelling of skyrmions play a crucial role in selecting the
ground states for S = 12 systems. In a model with competing antiferromagnetic couplings[75, 76],
13
the paramagnetic phase breaks translation symmetry due to the proliferation of “hedgehogs” or
tunnelling events in which the skyrmion number of the system changes by 1. However, due to the
Haldane-Berry phases and lattice symmetries, the hedgehogs can only appear in multiples of four
on the square lattice[86]. In the recent picture of Senthil and co-workers[84], this renders them
irrelevant at the transition, leading to an emergent conservation of skyrmion number, which in
turn leads to the deconfinement of spinons at the critical point[84].
This transition is identical to that in a classical model in which hedgehogs are suppressed
by hand. Following inconclusive earlier work[173], M. Kamal and the PI in 1993[174] found a
disordering transition with exponents different from that of the usual 3D Heisenberg model. This
has been confirmed by Motrunich and Vishwanath[85] recently, who also mapped it to a non-
compact CP 1 model, and obtained a self-dual theory for the case when there is an additional
easy-plane anisotropy. They also showed that there is a gapless “photon” in the paramagnetic
phase[85], whose “field-strength” is the chirality operator in the spin language.
It is proposed to examine the role of quenched disorder at this classical critical[173, 85] point
numerically, with the disorder in 2D only. By a quantum extention[87] of the Harris criterion[88],
from the values of the exponents one expects that disorder is slightly relevant for the O(3) case and
strongly relevant for the easy-plane case. It is proposed to numerically find the critical exponents
for the critical point with quenched disorder (provided the transition is second-order). The behavior
of the “photon” correlator near the critical point will be determined.
The relevance or otherwise of single hedgehogs is also an issue at and near the critical point with
quenched disorder, since the lattice symmetries which forced them to occur quadrupled[86] are no
longer present. This will be investigated numerically on a disordered lattice by large-N methods,
similar to those carried out by the PI and S. Sachdev in just such a calculation in the clean system
in 1990[176]
Quenched disorder in the square lattice Valence Bond Solid: The VBS phase[75, 76]
has a 4-fold degenerate ground state[86] which breaks translation invariance. Quenched disorder
will favor a particular ordering in a given region. At any T 6= 0, the Imry-Ma[89, 90] as well as
RG[91] show that the order parameter will vanish for any nonzero disorder. The quantum nature of
the underlying spin theory emerges when four domains meet at a Z4 vortex and generate a localized
spinon[177] impurity. Deep in the VBS phase the bosonic excitations are massive triplets[75, 76].
As one approaches the critical point, dynamical Z4 vortices appear[177]. This is a self-generated
boson Kondo model[63, 64] near the critical point, and will be examined near the Heisenberg and
the easy-plane critical points by techniques developed for the bose-Kondo model[63, 64]. Given
a density of impurities (determined by the strength of the disorder) the bath and the localized
spinons will self-consistently affect each other. It is proposed to examine the resulting theory by
the slave rotor method[47].
Quenched disorder in quantum dimer models: As mentioned in the introduction, short-
range quantum dimer models[73, 77, 79, 81] have proven to be fruitful in exploring deconfined
states[78] and critical points[82]. The low-energy dynamics of the quantum dimer models near the
gapless RK point is also known to be the quantum Lifshitz theory[82, 83], a free field theory with
dynamical exponent z = 2. From the scaling dimensions it is seen that quenched bond disorder is
strongly relevant at all the quantum Lifshitz critial points.
It is proposed to investigate the following issues: (i) What is the nature of the transition with
quenched disorder? In classical models, it is known that disorder renders first-order transitions
second-order[90, 91]. The opposite might well occur here. (ii) If the critical point remains second-
order, what is the nature of the excitations? Are hedgehogs irrelevant at the disordered critical
14
point? (iii) Away from the critical point, quantum Griffiths singularities[92, 93, 94, 95] are expected.
What form do these singularities take near the disordered critical point (if it exists)? Real-space
RG[179] (known to be exact in one-dimensional quantum models for a broad enough distribution of
randomness[93]), direct scaling in 2 + 1D, and ε-expansions will be used to answer these questions.
Finally, the the effect of quenched disorder on the deconfined excitations on the triangular lattice
dimer spin liquid[78] will addressed by real-space RG. Since spinons are confined in VBS states,
and local VBS order is induced by disorder, one expects disorder to induce a long-range potential
between the excitations. This leads to the possibility of “reconfinement” of the excitations beyond
a critical disorder strength. One expects that such a reconfinement will go hand-in-hand with the
destruction of topological order.
4 Summary
The main theme of the research proposed here is the effect of disorder in strongly correlated systems
or those with strong quantum fluctuations. A large part of the research concerns mesoscopics, with
emphasis on correlated Kondo-like states, or states in a single-particle crossover which have strong
quantum fluctuations, and the signatures of such states in Coulomb Blockade. A major goal is
to determine the signatures of quantum criticality via Coulomb Blockade. If successful, this will
uncover a qualitatively new mesoscopic regime and have a significant impact on experiments. An
important secondary goal is to study in detail the crossover from mesoscopic to bulk behavior near
a bulk quantum phase transition. The intersection of these projects with other studies of disorder
and interactions in the bulk, and with the bose-Kondo and bose-fermi-Kondo models is expected
to shed light on both mesoscopic and bulk states.
The second set of projects concern the bilayer quantum Hall system, where disorder is known to
be central the the understanding of a highly unusual dissipative state in a bosonic system, but the
details of the explanation remain mysterious. Fundamental questions about the effects of quenched
disorder on the nature of the T = 0 ground state (possibly metallic[7]), the role of real spin, and
the compressible to interlayer coherent phase transition will be addressed.
The third set of projects addresses deconfined spin liquid states and deconfined criticality, where
the effects of disorder have not yet been probed. It is evident from Harris-criterion-like consider-
ations that disorder is relevant. The possibility exists that disorder could have the qualitatively
new effect of turning a second-order quantum phase transition first-order, and that disorder could
“reconfine” deconfined excitations in the spin liquid states.
In addition, the education of a postdoc and a graduate student in the current techniques of
mesoscopic and many-body physics is an integral part of this proposal.
References
[1] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory
in Statistical Physics, Dover Publications, New York, 1963.
[2] R.B.Laughlin, Phys. Rev. Lett. 50, 1395, (1983), “Anomalous Quantum Hall Effect: An
Incompressible Quantum Fluid with Fractionally Charged Excitations”.
[3] J. K. Jain, Phys. Rev. Lett. 63, 199, (1989), “Composite Fermion Approach for the Fractional
Quantum Hall Effect”; Phys. Rev. B 41, 7653 (1990), “Theory of the Fractional Quantum
Hall Effect”; Science 266, 1199 (1994), “Composite Fermions in the Quantum Hall Regime”;
15
For the latest review, see, J. K. Jain and R. K. Kamilla, “Composite Fermions: Particles of
the Lowest Landau Level”, in “Composite Fermions”, Olle Heinonen, Editor (World Scientific,
1998).
[4] K. B. Efetov, Adv. Phys. 32, 53 (1983), “Supersymmetry and the theory of disordered met-
als”; B. L. Al’tshuler ad B. I. Shklovskii, Sov. Phys. JETP 64, 127 (1986), “Repulsion of
energy levels and conductivity of small metal samples”.
[5] A. M. Finkel’shtein, Sov. Phys. JETP 57, 97 (1983), “Influence of Coulomb interaction on the
properties of disordered materials”; C. Castellani, C. Di Castro, P. A. Lee, and M. Ma, Phys.
Rev. B30, 527 (1984), “Interaction-driven metal-insulator transitions in disordered fermion
systems”; For a review, see, D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys.66, 261 (1994),
“The Anderson-Mott transition”.
[6] E. Dagotto, Science 309, 257 (2005), “Complexity in strongly correlated systems”.
[7] P. Phillips and D. Dalidovich, Science 302, 243 (2003), “The elusive Bose metal”.
[8] For reviews see, Y. Imry, Introduction to Mesoscopic Physics, Oxford University Press, 1997;
K. B. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, 1997.
[9] For recent reviews, see, T. Guhr, A. Muller-Groeling, and H. A. Weidenmuller, Phys. Rep.
299, 189 (1998), “Random-matrix theories in quantum physics: common concepts”; Y. Al-
hassid, Rev. Mod. Phys. 72, 895 (2000), “The statistical theory of quantum dots”; A. D.
Mirlin, Phys. Rep. 326, 259 (2000), “Statistics of energy levels and eigenfunctions in disor-
dered systems”.
[10] A. V. Andreev and A. Kamenev, Phys. Rev. Lett.81, 3199 (1998), “Itinerant ferromagnetism
in disordered metals: A mean-field theory”; P. W. Brouwer, Y. Oreg, and B. I. Halperin,
Phys. Rev. B60, R13977 (1999), “Mesoscopic fluctuations of the ground-state spin of a small
metal particle”; H. U. Baranger, D. Ullmo, and L. I. Glazman, Phys. Rev. B61, R2425 (2000),
“Interactions and interference in quantum dots: Kinks in Coulomb-blockade peak positions”;
I. L. Kurland, I. L. Aleiner, and B. L. Al’tshuler, Phys. Rev. B 62, 14886 (2000), “Mesoscopic
magnetization fluctuations for metallic grains close to the Stoner instability”
[11] I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep. 358, 309 (2002), “Quantum
effects in Coulomb Blockade”, and references therein; Y. Oreg, P. W. Brouwer, X. Waintal,
and B. I. Halperin, cond-mat/0109541 (to appear in “Nano-Physics and Bio-electronics”,
edited by T. Chakraborty, F. Peeters, and U. Sivan, Elsevier), and references therein, “Spin,
spin-orbit, and electron-electron interactions in mesoscopic systems”.
[12] M. L. Mehta, Random Matrices, Academic Press, San Diego, 1991.
[13] G. Murthy and H. Mathur, Phys. Rev. Lett. 89, 126804 (2002), “Interactions and disorder
in quantum dots: Instabilities and phase transitions”.
[14] D. S. Duncan, D. Goldhaber-Gordon, R. M. Westervelt, K. D. Maranowski, and A. C. Gos-
sard, Appl. Phys. Lett 77, 2183 (2000), “Coulomb-blockade spectroscopy on a small quantum
dot in a parallel magnetic field”; S. Lindemann, T. Ihn, T. Heinzel, W. Zwerger, K. Ensslin,
K. Maranowski, and A. C. Gossard, Phys. Rev. B66, 195314 (2002), “Stability of spin states
16
in quantum dots”; R. M. Potok, J. A. Folk, C. M. Marcus, V. Umansky, M. Hanson, and A.
C. Gossard, Phys. Rev. Lett.91, 016802 (2003), “Spin and polarized current from Coulomb
blockaded quantum dots”.
[15] H.-J. Sommers and S. Iida, Phys. Rev. E 49, 2513 (1994), “Eigenvector statistics in the
crossover region between Gaussian orthogonal and unitary ensembles”; Z. Pluhar, H. A. Wei-
denmuller, J. A. Zuk, C. H. Lewenkopf, and F. J. Wegner, Annals of Phys. 243, 1 (1995),
“Crossover from Orthogonal to Unitary Symmetry for Ballistic Electron Transport in Chaotic
Microstructures”; E. Kanzieper and V. Freilikher, Phys. Rev. B54, 8737 (1996), “Eigenfunc-
tions of electrons in weakly disordered quantum dots: Crossover between orthogonal and
unitary symmetry”; V. I. Fal’ko and K. B. Efetov, Phys. Rev. B50, 11267 (2994), “Statistics
of fluctuations of wave functions of chaotic electrons in a quantum dot in an arbitrary mag-
netic field”; Phys. Rev. Lett.77, 912 (1996), “Long-Range Correlations in the Wave Functions
of Chaotic Systems”; S. A. van Langen, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev.
E 55, 1 (1997), “Fluctuating phase rigidity for a quantum chaotic system with partially bro-
ken time-reversal symmetry”; Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers, Phys. Rev.
E55, R4857 (1997), “Parametric correlations of phase shifts and statistics of time delays in
quantum chaotic scattering: Crossover between unitary and orthogonal symmetries”; P. W.
Brouwer, X. Waintal, and B. I. Halperin, Phys. Rev. Lett.85, 369 (2000), “Fluctuating Spin
g-Tensor in Small Metal Grains”; A. Tschersich and K. B. Efetov, Phys. Rev. E 62, 2042
(2000), “Statistics of wave functions in coupled chaotic systems”.
[16] D. A. Gorokhov and P. W. Brouwer, Phys. Rev. Lett.91, 186602 (2003), “Fluctuations of g
Factors in Metal Nanoparticles: Effects of Electron-Electron Interaction and Spin-Orbit Scat-
tering”; Phys. Rev. B69, 155417 (2004), “Combined effect of electron-electron interactions
and spin-orbit scattering in metal nanoparticles”.
[17] G. Murthy, Phys. Rev. B 70, 153304 (2004): “Random matrix crossovers and quantum critical
crossovers for interacting electrons in quantum dots”.
[18] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. Lett. 60, 1057 (1988), “Low-
temperature behavior of two-dimensional quantum antiferromagnets”; Phys. Rev. B 39, 2344
(1989), “Two-dimensional quantum Heisenberg antiferromagnet at low temperatures”; For
a detailed treatment, see, S. Sachdev, Quantum Phase Transitions, Cambridge University
Press, Cambridge 1999.
[19] P. Lafarge, P. Joyez, D. Esteve, C. Urbina, and M. H. Devoret, Phys. Rev. Lett.70, 994
(1993), “Measurement of the even-odd free-energy difference of an isolated superconductor”.
[20] B. Janko, A. Smith, and V. Ambegaokar, Phys. Rev. B50, 1152 (1994), “BCS superconduc-
tivity with fixed number parity”.
[21] L. I. Glazman, F. W. J. Hekking, K. A. Matveev, and R. I. Shekhter, Physica B 203, 316
(1994), “Charge parity in Josephson tunnelling through a superconducting grain”; D. V.
Averin and Y. V. Nazarov, Physica B 203, 310 (1994), “Parity effect in a small supercon-
ducting island”; G. Schon, J. Siewert, and A. D. Zaikin, Physica B 203, 340 (1994), “Parity
effects in superconducting SET transistors”.
[22] K. A. Matveev and A. I. Larkin, Phys. Rev. Lett.78, 3749 (1997), “Parity effect in ground
state energies of ultrasmall superconducting grains”; A. Mastellon, G. Falci, and R. Fazio,
17
Phys. Rev. Lett.80, 4542 (1998), “Small superconducting grain in the canonical ensemble”;
S. D. Berger and B. I. Halperin, Phys. Rev. B58, 5213 (1998), “Parity effect in a small
superconducting particle”.
[23] C. Bruder, R. Fazio, and G. Schon, Physica B 203, 240 (1994), “Mesoscopic normal metal-
superconductor junction systems: The effective action approach”; R. Fazio, C. Bruder, A. van
Otterlo, and G. Schon, Physica B 203, 247 (1994), “The interplay of proximity and charging
effects”; A. D. Zaikin, Physica B 203, 255 (1994), “Influence of Coulomb and proximity
effects on electron tunnelling through normal metal-superconductor interfaces”.
[24] F. Guinea and G. Schon, J. Low Temp. Phys.69, 219 (1987), “Dynamics and phase transitions
of Josephson junctions with dissipation due to quasiparticle tunnelling”.
[25] M. V. Feigelman, A. Kamenev, A. I. Larkin, and M. A. Skvortsov, Phys. Rev. B66, 054502
(2002), “Weak charge quantization on a superconducting island”.
[26] D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett.74, 3241 (1995), “Spectroscopic
Measurements of Discrete Electronic States in Single Metal Particles”; C. T. Black, D. C.
Ralph, and M. Tinkham, Phys. Rev. Lett.76, 688 (1996), “Spectroscopy of the Supercon-
ducting Gap in Individual Nanometer-Scale Aluminum Particles”.
[27] F. Braun and J. von Delft, Phys. Rev. B59, 9527 (1999), “Superconductivity in ultrasmall
metallic grains”.
[28] J. von Delft, Ann. Phys. (Leipzig), 10, 1 (2001), “Superconductivity in ultrasmall metallic
grains”.
[29] M. Schechter, Y. Imry, Y. Levinson, and J. von Delft, Phys. Rev. B63, 214518 (2001),
“Thermodynamic properties of a small superconducting grain”.
[30] R. W. Richardson, Phys. Lett. 3, 277 (1963), “A restricted class of exact eigenstates of the
pairing-force Hamiltonisn”; J. Math. Phys. 18, 1802 (1977), “Pairing in the limit of a large
number of particles” and references therein.
[31] J. M. Roman, G. Sierra, and J. Dukelsky, Nucl. Phys. B 634[FS], 483 (2002), “Large-N limit
of the exactly solvable BCS model:analytics versus numerics”; G. Sierra, J. M. Roman, and
J. Dukelsky, Int. J. Mod. Phys. A 19S2, 381 (2004), “The elementary excitations of the BCS
model in the canonical ensemble?”.
[32] A. M. Finkelshtein, JETP Lett. 45, 46 (1987), “Superconducting transition temperature in
amorphous films”; Physica B 197, 636 (1994), .
[33] A. Larkin, Ann. Phys. (Leipzig) 8, 785 (1999), “Superconductor-insulator transitions in films
and bulk materials”; S. Kos, A. J. Millis, and A. I. Larkin, Phys. Rev. B70, 214531 (2004),
“Gaussian fluctuation corrections to the BCS mean-field gap amplitude at zero temperature”.
[34] M. A. Skvortsov and M. V. Feigelman, cond-mat/0504002 (2005), “Superconductivity in
disordered thin films; giant mesoscopic fluctuations”.
[35] U. Sivan, R. Berkovits, Y. Aloni, O. Prus, A. Auerbach, and G. Ben-Yoseph, Phys. Rev.
Lett. 77, 1123 (1996), “Mesoscopic Fluctuations in the Ground State Energy of Disordered
18
Quantum Dots”; S. R. Patel, S. M. Cronenwett, D. R. Stewart, A. G. Huibers, C. M. Marcus
, C. I. Duruoz, J. S. Harris, Jr. , K. Campman, and A. C. Gossard, Phys. Rev. Lett. 80, 4522
(1998), “Statistics of Coulomb blockade peak spacings”; ; F. Simmel, T. Heinzel, and D. A.
Wharam, Eur. Lett. 38, 123 (1997); S. Luscher, T. Heinzel, K. Ensslin, W. Wegscheider, and
M. Bichler, Phys. Rev. Lett. 86, 2118 (2001), “Signatures of Spin Pairing in Chaotic Quantum
Dots”; F. Simmel, D. Abusch-Magder, D. A. Wharam, M. A. Kastner, J. P. Kotthaus, Phys.
Rev. B 59, 10441 (1999), “Statistics of the Coulomb-blockade peak spacings of a silicon
quantum dot”; D. Abusch-Magder, F. Simmel, D. A. Wharam, M. A. Kastner, and J. P.
Kotthaus, Physica E 6, 382 (2000), “Spacing and width of Coulomb blockade peaks in a
silicon quantum dot”.
[36] K. K. Likharev, IBM J. Res. Dev. 32, 144 (1988), “Correlated discrete transfer of electrons in
ultrasmall tunnel junctions”; H. van Houten and C. W. J. Beenakker, Phys. Rev. Lett.63, 1893
(1989, “Comment on “Conductance oscillations periodic in thedensity of a one-dimensional
electron gas”; Y. Meir, N. S. Wingreen, and P. A. Lee, Phys. Rev. Lett.66, 3048 (1991),
“Transport through a strongly interacting electron system: Theory of periodic conductance
oscillations”; C. W. J. Beenakker, Phys. Rev. B44, 1646 (1991), “Theory of Coulomb-blockade
oscillations in the conductance of a quantum dot”.
[37] G. Falci, J. Heins, G. Schon, and G. T. Zimanyi, Physica B 203, 409 (1994), “Tunnelling in
the electron box in the nonperturbative regime”; G. Falci, G. Schon, and G. T. Zimanyi, Phys.
Rev. Lett. 74, 3257 (1995), “Unified scaling theory of the electron box for arbitrary tunnelling
strength”; W. Hofstetter and W. Zwerger, Phys. Rev. Lett.78, 3737 (1997), “Single-electron
box and the helicity modulus of an inverse square XY model”; J. Konig and H. Schoeller, Phys.
Rev. Lett.81, 3511 (1998), “Strong tunnelling in the single-electron box”. ; I. S. Beloborodov,
A. V. Andreev, and A. I. Larkin, Phys. Rev. B68, 024204 (2003), “Two-loop approximation
in the Coulomb blockade problem”.
[38] X. Wang and H. Grabert, Phys. Rev. B 53, 12621 (1996), “Coulomb charging at large con-
duction”; A. Kamenev and Y. Gefen, Phys. Rev. B 54, 5428 (1996), “Zero-bias anomaly in
finite-size systems”; A. Kamenev, Phys. Rev. Lett. 85, 4160 (2000), “Weak charge quantiza-
tion as an instanton of the interacting σ model”; A. Kamenev and A. I. Larkin, Phys. Rev.
Lett.89, 236801 (2002), “Coulomb blockade with dispersive interfaces”.
[39] K. B. Efetov and A. Tschersich, Phys. Rev. B 67, 174205 (2003), “Coulomb effects in granular
materials at not very low temperatures”.
[40] S. A. Bulgadaev, JETP Lett. 45, 622 (1987), “Phase diagram of a Josephson junction with
a “periodic” dissipation”; F. Guinea and G. Schon, Europhys. Lett.1, 585 (1986), “Coher-
ent charge oscillations in tunnel junctions”; S. E. Korshunov, JETP Lett. 45, 434 (1987),
“Coherent and incoherent tunnelling in a Josephson junction with a “periodic” dissipation”;
A. D. Zaikin, D. S. Golubev, and S. V. Panyukov, Physica B 203, 417 (1994), “Single elec-
tron tunnelling near the Coulomb blockade threshold”; Y. V. Nazarov, Phys. Rev. Lett.82,
1245 (1999), “Coulomb blockade without tunnel junctions”; H. Schoeller and G. Schon, Phys-
ica B 203, 423 (1994), “Resonant tunnelling and charge fluctuations in mesoscopic tunnel
junctions”;
[41] L. I. Glazman and K. A. Matveev, Sov. Phys. JETP 71, 1031 (1990), “Lifting of the Coulomb
blockade of one-electron tunneling by quantum fluctuations”; K. A. Matveev, Sov. Phys.
19
JETP 72, 892 (1991), “Quantum fluctuations of the charge of a metal particle under the
Coulomb blockade conditions”.
[42] K. A. Matveev, Physica B 203, 404 (1994), “Charge fluctuations under the Coulomb blockade
conditions”; K. A. Matveev, Phys. Rev. B51, 1743 (1995), “Coulomb blockade at almost per-
fect transmission; A. Furusaki and K. A. Matveev, Phys. Rev. Lett.75, 709 (1995), “Coulomb
blockade oscillations of conductance in the regime of strong tunnelling”.
[43] K. Flensberg, Physica B 203, 432 (1994), “Capacitance and conductance of dots connected
by point contacts”.
[44] X. Wang, R. Egger, and H. Grabert, Europhys. Lett. 38, 545 (1997), “Coulomb charging
energy for arbitrary tunnelling strength”; G. Goppert , H. Grabert, N. V. Prokof’ev, and
B. V. Svistunov, Phys. Rev. Lett.81, 2324 (1998), “Effect of tunnelling conductance on the
Coulpmb staircase”; C. P. Herrero, G. Schon, and A. D. Zaikin, Phys. Rev. B59, 5728 (1999),
“Strong charge fluctuations in the single-electron box: A quantum Monte Carlo analysis”.
[45] I. L. Aleiner and L. I. Glazman, Phys. Rev. B57, 9608 (1998), “Mesoscopic charge quan-
tization”; P. W. Brouwer and I. L. Aleiner, Phys. Rev. Lett.82, 390 (1999), “Effects of
electron-electron interaction on the conductance of open quantum dots”.
[46] I. S. Beloborodov and A. V. Andreev, Phys. Rev. B65, 195311 (2002), “Coulomb blockade in
metallic grains at large conductance”.
[47] S. Florens, P. San Jose, F. Guinea, and A. Georges, Phys. Rev. B68, 245311 (2003), “Co-
herence and Coulomb Blockade in single electron devices: a unified treatment of interaction
effects”; See also, S. Florens and A. Georges, Phys. Rev. B66, 165111 (2002), “Quantum
impurity solvers using a slave rotor representation”.
[48] J. Kondo, Prog. Theor. Phys. 32, 37 (1964).
[49] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 452 (1988), “Resonant Kondo transparency
of a barrier with quasilocal impurity states”; T. K. Ng and P. A. Lee, Phys. Rev. Lett.61, 1768
(1988), “On-site Coulomb repulsion and resonant tunnelling”; Y. Meir, N. S. Wingreen, and
P. A. Lee, Phys. Rev. Lett.70, 2601 (1993), “Low-temperature transport through a quantum
dot: The Anderson model out of equilibrium”.
[50] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, U. Meirav, and M. A.
Kastner, Nature 391, 156 (1998), “Kondo effect in a single-electron transistor”.
[51] Y. Meir and N. S. Wingreen, Phys. Rev. Lett.68, 2512 (1992), “Landauer formula for the
current through an interacting electron region”.
[52] W. B. Thimm, J. Kroha, and J. von Delft, Phys. Rev. Lett.82, 2143 (1999), “Kondo-box:
A magnetic impurity in an ultrasmall metallic grain”; I. Affleck and P. Simon, Phys. Rev.
Lett.86, 2854 (2001); P. Simon and I. Affleck, Phys. Rev. Lett.89, 206602 (2002).
[53] P. Simon and I. Affleck, Phys. Rev. B68, 115304 (2003), and references therein, “Kondo
screening cloud in mesoscopic devices”; P. S. Cornaglia and C. A. Balseiro, Phys. Rev. Lett.90,
216801 (2003), “Transport through quantum dots in mesoscopic circuits”.
20
[54] Y. Oreg and D. Goldhaber-Gordon, Phys. Rev. Lett.90, 136602 (2003), “Two-channel Kondo
effect in a modified single-electron transistor”.
[55] K. Le Hur, Phys. Rev. Lett.92, 196804 (2004), “ Coulomb Blockade of a Noisy Metallic Box:
A Realization of Bose-Fermi Kondo Models”; M.-R. Li and K. Le Hur, Phys. Rev. Lett.93,
176802 (2004), “ Double-Dot Charge Qubit and Transport via Dissipative Cotunneling”.
[56] L. I. Glazman and M. Pustilnik, J. Phys. Condens. Matter 16, R513 (2004), “Kondo effect
in quantum dots”.
[57] I. I. Pomeranchuk, Sov. Phys. JETP 8, 361 (1958), “On the stability of a Fermi liquid”.
[58] C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999), “Pseudogap Phase and the Quantum-
Critical Point in Copper-Oxide Metals”; cond-mat/0311145 (2003), “Cure to the Landau-
Pomeranchuk and Associated Long-Wavelength Fermi-surface Instabilities on Lattices”.
[59] V. Oganesyan, S. A. Kivelson, and E. Fradkin, Phys. Rev. B 64, 195109 (2001), “Quantum
theory of a nematic Fermi fluid”.
[60] G. Murthy, Phys. Rev. Lett. 94, 126803 (2005): “Interplay between the mesoscopic Stoner
and Kondo effects in quantum dots”.
[61] A. I. Larkin and V. I. Melnikov, Sov. Phys. JETP, 61, 1231 (1971), “Magnetic impurities in
an almost magnetic metal”.
[62] H. Maebashi, K. Miyake, and C. M. Varma, Phys. Rev. Lett.88, 226403 (2002), “Singular
effects of impurities near the ferromagnetic quantum critical point”; A. H. Castro-Neto, E.
Novais, L. Borda, G. Zarand, and I. Affleck, Phys. Rev. Lett.91, 096401 (2003), “Quan-
tum magnetic impurities in magnetically ordered systems”; Y. H. Loh, V. Tripathi, and M.
Turkalov, cond-mat/0405618 (2004), “Magnetic droplets in a metal close to a ferromagnetic
quantum critical point”.
[63] M. Vojta, C. Buragohain, and S. Sachdev, Phys. Rev. B61, 15152 (2000), “Quantum impurity
dynamics in two-dimensional antiferromagnets and superconductors”; S. Sachdev, J. Stat.
Phys. 115, 47 (2004), “Quantum impurity in a magnetic environment”; M. Vojta, cond-
mat/0412208 (2004), “Impurity quantum phase transitions”.
[64] A. M. Sengupta, Phys. Rev. B61, 4041 (2000), “Spin in a fluctuating field: The Bose (+Fermi)
Kondo models”.
[65] L. Zhu, S. Kirchner, Q. Si, and A. Georges, Phys. Rev. Lett.93, 267201 (2004), “Quantum
critical properties of the Bose-Fermi Kondo model in a large-N limit”; S. Kirchner, L. Zhu,
and Q. Si, cond-mat/0407307 (2004), “Destruction of the Kondo effect in a multi-channel
Bose-Fermi Kondo model”.
[66] See J.P. Eisenstein and A.H. MacDonald, Nature 432, 691 (2004), “Bose-Einstein condensa-
tion of excitons in bilayer electron systems”, and references therein.
[67] H.A. Fertig, Phys. Rev. B 40, 1087 (1989), “Energy spectrum of a layered system in a strong
magnetic field”.
21
[68] X.G. Wen and A. Zee, Phys. Rev. Lett. 69, 1811 (1992), “Neutral superfluid modes and
“magnetic” monopoles in multilayered quantum Hall systems”; Phys. Rev. B. 47, 2265 (1993),
“Tunneling in double-layered quantum Hall systems”; Z.F. Ezawa and A. Iwazaki, Phys. Rev.
B 47, 7295 (1993), “Quantum Hall liquid, Josephson effect, and hierarchy in a double-layer
electron system”.
[69] I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 84, 5808
(2000), “Resonantly Enhanced Tunneling in a Double Layer Quantum Hall Ferromagnet”; M.
Kellogg, I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett.88,
126804 (2002), “Observation of quantized Hall drag in a strongly correlated bilayer electron
system”.
[70] I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 87, 036803
(2001), “Observation of a Linearly Dispersing Collective Mode in a Quantum Hall Ferromag-
net”.
[71] M. Kellogg, J.P. Eisenstein, L.N. Pfeiffer, and K.W. West, Phys. Rev. Lett. 93, 036801 (2004),
“Vanishing Hall Resistance at High Magnetic Field in a Double-Layer Two-Dimensional
Electron System”; E. Tutuc, M. Shayegan, and D. Huse, Phys. Rev. Lett. 93, 036802
(2004), “Counterflow Measurements in Strongly Correlated GaAs Hole Bilayers: Evidence
for Electron-Hole Pairing”.
[72] E. Fradkin, Field Theories of Condensed Matter Systems, Westview Press, Oxford, 1991.
[73] P. W. Anderson, Mat. Res. Bull. 8, 153 (1973), “Resonating valence bonds: a new kind of
insulator?”; P. Fazekas and P. W. Anderson, Phil. Mag. 30, 423 (1974), “On the ground state
properties of the anisotropic triangular antiferromagnet”; G. Baskaran, Z. Zou, and P. W.
Anderson, Sol. State. Commun. 63, 973 (1987), “The resonating valence bond state and high-
T/sub c/ superconductivity-a mean field theory”; P. W. Anderson, Science 235, 1196 (1987),
“RESONATING VALENCE BOND STATE IN La2CuO4 AND SUPERCONDUCTIVITY”.
[74] X.-G. Wen and Q. Niu, Phys. Rev. B41, 9377 (1991), “Ground-state degeneracy of the
fractional quantum Hall states in the presence of a random potential and on high-genus
Riemann surfaces”; X.-G. Wen, Phys. Rev. B41, 12838 (1991), “Chiral Luttinger liquid and
the edge excitations in the fractional quantum Hall states”.
[75] N. Read and S. Sachdev, Phys. Rev. Lett.62, 1694 (1989), “Valence-bond and spin-Peierls
ground states of low-dimensional quantum antiferromagnets”; Phys. Rev. B42, 4568 (1990),
“Spin-Peierls, valence-bond solid, and Neel ground states of low-dimensional quantum anti-
ferromagnets”; W. Zheng and S. Sachdev, Phys. Rev. B40, 2704 (1989), “Sine-Gordon theory
of the non-Neel phase of two-dimensional quantum antiferromagnets”.
[76] S. Sachdev and K. Park, Annals of Phys. 298, 58 (2002), “Ground states of quantum anti-
ferromagnets in two dimensions”.
[77] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett.61, 2376 (1988), “Superconductivity and
the quantum hard-core dimer gas”.
[78] R. Moessner and S. L. Sondhi, Phys. Rev. Lett.86, 1881 (2001), “Resonating valence bond
phase in the triangular lattice quantum dimer model”; Phys. Rev. B68, 054405 (2003), “Ising
22
and dimer models in two and three dimensions”; E. Fradkin, D. A. Huse, R. Moessner, and S.
L. Sondhi, Phys. Rev. B69, 224415 (2004), “Bipartite Rokhsar-Kivelson points and Cantor
deconfinement”.
[79] S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, Phys. Rev. B35, 8865 (1987), “Topology of
the resonating valence-bond state: Solitons and highTc superconductivity”.
[80] S. Sachdev, Phys. Rev. B40, 5204 (1989), “Spin-Peierls ground states of the quantum dimer
model: A finite-size study”; L. B. Ioffe and A. I. Larkin, Phys. Rev. B40, 6941 (1989), “Su-
perconductivity in the liquid-dimer valence-bond state”; L. Levitov, Phys. Rev. Lett.64, 92
(1990), “Equivalence of the dimer resonating-valence-bond problem to the quantum rough-
ening problem”.
[81] E. Fradkin and S. Kivelson, Mod. Phys. Lett. B 4, 225 (1990), “Short range resonating valence
bond theories and superconductivity”.
[82] A. Ardonne, P. Fendley, and A. Fradkin, Ann. Phys. (N.Y.), 310, 493 (2004), “Topological
order and conformal quantum critical points”; E. Fradkin, D. A. Huse, R. Moessner, and S.
L. Sondhi, Phys. Rev. B69, 224415 (2004), “Bipartite Rokhsar-Kivelson points and Cantor
deconfinement”; A. Vishwanath, L. Balents, and T. Senthil, Phys. Rev. B69, 224416 (2004),
“Quantum criticality and deconfinement in phase transitions between valence bond states”.
[83] C. L. Henley, Jour. Stat. Phys. 89, 483 (1997), “Relaxation time for a dimer covering with
height representation”.
[84] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490
(2004), “”Deconfined” quantum critical points”; T. Senthil, L. Balents, S. Sachdev, A. Vish-
wanath, and M. P. A. Fisher, Phys. Rev. B 70, 14407 (2004), “Quantum criticality beyond
the Landau-Ginzburg-Wilson paradigm”.
[85] O. I. Motrunich and A. Vishwanath, Phys. Rev. B70, 075104 (2004), “Emergent photons
and transitions in the O(3) sigma model with hedgehog suppression”; O. I. Motrunich and
T. Senthil, Phys. Rev. B71, 125102 (2005), “Origin of artificial electrodynamics in three-
dimensional bosonic models”.
[86] F. D. M. Haldane, Phys. Rev. Lett.61, 1029 (1988), “O(3) nonlinear σ model and the topo-
logical distinction between integer and half-integer-spinantiferromagnets in two dimensions”.
[87] C. A. Doty and D. S. Fisher, Phys. Rev. B45, 2167 (1992), “Effects of quenched disorder on
the spin-12 quantum XXZ chain”.
[88] A. B. Harris, J. Phys. C7, 1671 (1974), “Effect of random defects on the critical behavior of
Ising models”.
[89] Y. Imry and S.-K. Ma, Phys. Rev. Lett.35, 1399 (1975), “Random-field instability of the
ordered state of continuous symmetry”.
[90] J. Z. Imbrie, Phys. Rev. Lett.53, 1747 (1984), “Lower Critical Dimension of the Random-
Field Ising Model”; J. Bricmont and A. Kupiainen, Phys. Rev. Lett.59, 1829 (1987), “Lower
Critical Dimension of the Random-Field Ising Model “; M. Aizenman and J. Wehr, Phys. Rev.
Lett.62, 2503 (1989), “Rounding of first-order phase transitions in systems with quenched
disorder”.
23
[91] K. Hui and A. N. Berker, Phys. Rev. Lett.62, 2507 (1989), “Random-field mechanism in
random-bond multicritical systems”.
[92] R. B. Griffiths, Phys. Rev. Lett.23, 17 (1969), “Nonanalytic behavior above the critical point
in a random Ising ferromagnet”.
[93] D. S. Fisher, Phys. Rev. B51, 6411 (1995), “Critical behavior of random transverse-field Ising
spin chains”; R. A. Hyman, K. Yang, R¿ N. Bhatt, and S. M. Girvin, Phys. Rev. Lett.76,
839 (1996), “Random bonds and topological stability in gapped quantum spin chains”; R. A.
Hyman and K. Yang, Phys. Rev. Lett.78, 1783 (1997), “Impurity-driven phase transition in
the antiferromagnetic spin-1 chain”; O. Motrunich, S.-C. Mau, D. A. Huse, adn D. S. Fisher,
Phys. Rev. B61, 1160 (2000), “Infinite-randomness quantum Ising critical points”.
[94] A. H. Castro Neto, G. E. Castilla, and B. A. Jones, Phys. Rev. Lett.81, 3531 (1998), “Non-
Fermi Liquid Behavior and Griffiths Phase in f-Electron Compounds”; A. H. Castro Neto and
B. A. Jones, Phys. Rev. B62, 14975 (2000), “Non-Fermi-liquid behavior in U and Ce alloys’:
Criticality, disorder, dissipation, and Griffiths-McCoy singularities”; M. B. Silva Neto and A.
H. Castro Neto, Europhys. Lett. 62, 890 (2003), “Metallic continuum quantum ferromagnets
at finite temperature”.
[95] A. J. Millis, D. K. Morr, and J. Schmalian, Phys. Rev. Lett.87, 167202 (2001), “Local defect
in a metallic quantum critical system”; Phys. Rev. B66, 174433 (2002), “Quantum Griffiths
phases in metallic systems”; N. Shah and A. J. Millis, Phys. Rev. Lett.91, 147204 (2003),
“Dissipative dynamics of an extended magnetic nanostructure: Spin necklace in a metallic
environment”;
[96] T. Vojta and J. Schmalian, cond-mat/0405609, (2005), “Quantum Griffiths phases in itinerant
Heisenberg magnets”.
[97] L. B. Ioffe, M. V. Feigel’man, A. Iosevich, D. Ivanov, M. Troyer, and G. Blatter, Nature 415,
503 (2002), “Topologically protected quantum bits using Josephson junction arrays”.
[98] D. Herman, H. Mathur, and G. Murthy, Phys. Rev. B 69, 041301 (2004): “Diamagnetic Per-
sistent Currents and Spontaneous Time-Reversal Symmetry Breaking in Mesoscopic Struc-
tures”.
[99] G. Murthy, R. Shankar, D. Herman, and H. Mathur, Phys. Rev. B69, 075321 (2004), “Solvable
regime of disorder and interactions in ballistic nanostructures: Consequences for Coulomb
blockade”.
[100] G. Murthy, R. Shankar and H. Mathur, Phys. Rev. B72, 075364 (2005), “Ballistic chaotic
quantum dots with interactions: A numerical study of the Robnik-Berry billiard”.
[101] I. Rozhkov and G. Murthy, cond-mat/0504653, submitted to Phys. Rev. B: “A nearly closed
ballistic billiard with random boundary transmission”
[102] I. Rozhkov and G. Murthy, cond-mat/0507186, submitted to Jour. Phys. A: “Ballistic dy-
namics of a convex smooth-wall billiard with finite escape rate along the boundary”.
[103] H. K. Nguyen, Y. N. Joglekar, and G. Murthy, Phys. Rev. B70, 035324 (2004), “Collective
edge modes in fractional quantum Hall systems”.
24
[104] H. A. Fertig and G. Murthy, cond-mat/0506292 to appear in Phys. Rev. Lett.: “A Coherence
Network in the Quantum Hall Bilayer”.
[105] G. Murthy and R. Shankar, Phys. Rev. Lett.90, 066801 (2003), “Quantum dots with disorder
and interactions: A solvable large-g limit”.
[106] A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, New
York, 1993.
[107] P. Simon, R. Lopez, and Y. Oreg, Phys. Rev. Lett.94, 086602 (2005), “Ruderman-Kittel-
Kasuya-Yosida and magnetic field interactions in coupled quantum dots”; M. G. Vavilov
and L. I. Glazman, Phys. Rev. Lett.94, 086805 (2005), “Transport spectroscopy of Kondo
quantum dots coupled by RKKY interaction”.
[108] N. J. Craig, J. M. Taylor, E. A. Lester, C. M. Marcus, M. P. Hanson, and A. C. Gossard,
Science 304, 565 (2004), “Tunable nonlocal spin control in a coupled-quantum dot system”.
[109] A. O. Caldeira and A. J. Leggett, Phys. Rev. Lett.46, 211 (1981), “Influence of dissipation on
quantum tunnelling in macroscopic systems”; V. Ambegaokar, U. Eckern, and G. Schon, Phys.
Rev. Lett.48, 1745 (1982), “Quantum dynamics of tunnelling between superconductors”.
[110] S. Chakravarty, Phys. Rev. Lett.49, 681 (1982), “Quantum fluctuations in the tunnelling
between superconductors”; A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher,
A. Garg, and W. Zwerger, Rev. Mod. Phys.59, 1 (1987), “Dynamics of the dissipative two-
state system”; A. J. Bray and M. A. Moore, Phys. Rev. Lett.49, 1545 (1983), “Influence of
dissipation on quantum coherence”.
[111] M. Robnik, J. Phys. A 17, 1049 (1984), “Quantizing a generic family of billiards with analytic
boundaries”; M. V. Berry and M. Robnik, J. Phys. A 19, 649 (1986), “Statistics of energy
levels without time-reversal symmetry: Aharonov-Bohm chaotic billiards”.
[112] S. Adam, P. W. Brouwer, and P. Sharma,Phys. Rev. B68, 241311 (2003), “Scaling approach
to electron-electron interactions in a chaotic quantum dot”.
[113] S. Adam, P. W. Brouwer, J. P. Sethna, and X. Waintal, Phys. Rev. B66, 165310 (2002),
“Enhanced mesoscopic fluctuations in the crossover between random matrix ensembles”, and
references therein.
[114] I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett.87, 256801 (2001), “Spin-Orbit Coupling Effects
on Quantum Transport in Lateral Semiconductor Dots”.
[115] G.A. Fiete and E.J. Heller, Rev. Mod. Phys. 75, 933 (2003), ‘Colloquium: Theory of quantum
corrals and quantum mirages”‘.
[116] G.C. des Francs, C. Girard, J.-C. Weeber, C. Chicane, T. David A. Dereux, and D. Peyrade,
Phys. Rev. Lett. 86, 4950 (2001), “Optical Analogy to Electronic Quantum Corrals”.
[117] J.U. Nockel and A. D. Stone, Nature 385, 45 (1997), “Ray and wave chaos in asymmetric
resonant optical cavities”.
25
[118] B. A. Muzykantskii and D. E. Khmelnitskii, JETP Lett. 62, 76 (1995), “Effective action in
the theory of quasi-ballistic disordered conductors”; A. V. Andreev, O. Agam, B. D. Simons,
and B. L. Altshuler, Phys. Rev. Lett.76, 3947 (1996), “Quantum Chaos, Irreversible Classical
Dynamics, and Random Matrix Theory”; Nucl. Phys. B 482, 536 (1996), “Semiclassical field
theory approach to quantum chaos”;Ya.M. Blanter, A.D. Mirlin, and B.A. Muzykantskii,
Phys. Rev. B 63, 235315 (2001), “Level and eigenfunction statistics in billiards with sur-
face disorder”; K.B. Efetov and V.R. Kogan, Phys. Rev. B 67, 245312 (2003), “Nonlinear
sigma model for long-range disorder and quantum chaos”; K.B. Efetov, G. Schwiete, and K.
Takahashi, Phys. Rev. Lett. 92, 026807 (2004), “Bosonization for Disordered and Chaotic
Systems”.
[119] C.D. Schwieters, J.A. Alford, and J.B. Delos, Phys. Rev. B 54, 10652 (1996), “Semiclassical
scattering in a circular semiconductor microstructure”; R.G Nazmitdinov, K. N. Pichugin,
L. Rotter, and P. Seba, Phys. Rev. E 64, 056214 (2001), “Whispering gallery modes in open
quantum billiards”; Phys. Rev. B 66, 085322 (2002), “Conductance of open quantum billiards
and classical trajectories”.
[120] G. Casati, B. V. Chirikov, J. Ford, and F. M. Izrailev, in Stochastic Behavior in Classical and
Quantum Hamiltonian Systems’, edited by G. Casati and J. Ford, Lecture Notes in Physics,
vol. 93 (Springer, Berlin, 1979); K. Frahm and D. L. Shepelyansky, Phys. Rev. Lett.78, 1440
(1997), “Quantum localization in rough billiards”; K. M. Frahm, Phys. Rev. B55, R8626
(1997), “Localization in a rough billiard: A σ-model formulation”.
[121] K.Moon, H.Mori, K.Yang, S.M.Girvin, A.H.MacDonald, L.Zheng, D.Yoshioka, and S.-
C.Zhang, Phys. Rev. B51, 5138 (1995), “Spontaneous interlayer coherence in double-layer
quantum Hall systems: Charged vortices and Kosterlitz-Thouless phase transitions”; K.Yang,
K.Moon, L.Belkhir, H.Mori, S.M.Girvin, A.H.MacDonald, L.Zheng, and D.Yoshioka, Phys.
Rev. B54, 11644 (1996), “Spontaneous interlayer coherence in double-layer quantum Hall
systems: Symmetry-breaking interactions, in-plane fields, and phase solitons”; See also
S.M.Girvin and A.H.MacDonald, Chapter 5 of Perspectives in Quantum Hall Effects, Das
Sarma and Pinczuk, Editors, 1997.
[122] I. B. Spielman, L. A. Tracy, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett.bf
94, 076803 (2005), “Spin transition in strongly correlated bilayer two-dimensional electron
systems”; N. Kumada, K. Muraki, K. Hashimoto, and Y. Hirayama, Phys. Rev. Lett.94,
096802 (2005), “Spin degree of freedom in the ν = 1 bilayer electron system investigated by
nuclear spin relaxation”; S. Luin, V. Pellegrini, A. Pinczuk, B. S. Dennis, L. N. Pfeiffer, and
K. W. West, Phys. Rev. Lett.94, 146804 (2005), “Observation of collapse of pseudospin order
in bilayer quantum Hall ferromagnets”.
[123] A. A. Burkov and A. H. MacDonald, Phys. Rev. B66, 115320 (2002), “Lattice pseudospin
model for ν = 1 quantum Hall bilayers”.
[124] E. Tutuc, S. Melinte, E. P. De Poortere, R. Pillarisetty, and M. Shayegan, Phys. Rev. Lett.91,
076802 (2003), “Role of density imbalance in an interacting bilayer hole system”; I. B. Spiel-
man, M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. B70, 081303
(2004), “Onset of interlayer phase coherence in a bilayer two-dimensional electron system:
Effect of layer density imbalance”.
26
[125] L. Balents and L. Radzihovsky, Phys. Rev. Lett. 86, 1825 (2001), “Interlayer Tunneling in
Double-Layer Quantum Hall Pseudoferromagnets”; A. Stern, S.M. Girvin, A.H. MacDonald,
and Ning Ma, Phys. Rev. Lett. 86, 1829 (2001), “Theory of Interlayer Tunneling in Bilayer
Quantum Hall Ferromagnets”; M. M. Fogler and F. Wilczek, Phys. Rev. Lett.86, 1833 (2001),
“Josephson effect without superconductivity: Realization in quantum hall bilayers”; Y. N.
Joglekar and A. H. MacDonald, Phys. Rev. Lett.87, 196802 (2001), “Is there a dc Josephson
effect in bilayer quantum Hall systems?”; R. L. Jack, D. K. K. Lee, adn N. R. Cooper, Phys.
Rev. B71, 085310 (2005), “Quantum and classical dissipative effects on tunneling in quantum
Hall bilayers”.
[126] J. Schliemann, S. M. Girvin, and A. H. MacDonald, Phys. Rev. Lett.86, 1849 (2001), “Strong
correlation to weak correlation phase transition in bilayer quantum Hall systems”; K. Yang,
Phys. Rev. Lett.87, 056802 (2001), “Dipolar excitons, spontaneous phase coherence, and
superfluid-insulator transition in bilayer quantum Hall systems at ν = 1”; M. Abolfath, L.
Radzihovsky and A. H. MacDonald, Phys. Rev. B65, 233306 (2002), “Global phase diagram
of bilayer quantum Hall ferromagnets”; K. Nomura and D. Yoshioka, Phys. Rev. B66, 153310
(2002), “Evolution of the ν = 1 bilayer quantum Hall ferromagnet”; D. Sheng, L. Balents,
and Z. Wang, Phys. Rev. Lett. 91, 116802 (2003), “Phase diagram for quantum Hall bilayers
at ν = 1”; J. Schliemann, Phys. Rev. B67, 035328 (2003), “Correlation energy, quantum
phase transition, and bias potential effects in quantum Hall bilayers at ν = 1”.
[127] E. Demler, C. Nayak, and S. Das Sarma, Phys. Rev. Lett.86, 1853 (2001), “Bilayer Coherent
and Quantum Hall Phases: Duality and Quantum Disorder”; Y.-B. Kim, C. Nayak, E. Dem-
ler, N. Read, and S. Das Sarma, Phys. Rev. B63, 205315 (2001), “Bilayer paired quantum
Hall states and Coulomb drag”.
[128] N. E. Bonesteel, I. A. McDonald, and C. Nayak, Phys. Rev. Lett.77, 3009 (1996), “Gauge
Fields and Pairing in Double-Layer Composite Fermion Metals”; S. H. Simon, E. H. Rezayi,
and M. V. Milovanovic, Phys. Rev. Lett.91, 046803 (2003), “Coexistence of composite bosons
and composite fermions in ν = 12 + 1
2 quantum Hall bilayers”.
[129] R. Rajaraman, Phys. Rev. B56, 6788 (1997), “Generalized Chern-Simons theory of composite
fermions in bilayer Hall systems”; T. Morinari, Phys. Rev. B59, 7320 (1999), “Composite-
fermion pairing in bilayer quantum Hall systems”; J. Ye, cond-mat/0310512, (2003), “Mutual
Composite Fermion and Composte Boson approaches to im-balanced bilayer quantum Hall
systems”; G. S. Jeon and J. Ye, Phys. Rev. B71, 035348 (2005), “Trial wavefunction approach
to bilayer quantum Hall systems”.
[130] H.A. Fertig and J.P. Straley, Phys. Rev. Lett. 91, 046806 (2003), “Deconfinement and Dissi-
pation in Quantum Hall Josephson Tunneling”.
[131] Z. Wang, Phys. Rev. Lett. 92, 136803 (2004), “Tunnelling, dissipation, and superfluid transi-
tion in quantum hall bilayers”; Phys. Rev. Lett.94, 176804 (2005), “Vorties, tunnelling, and
deconfinement in quantum Hall bilayers”.
[132] D.A. Huse, cond-mat/0407452 (2004), “Resistance due to vortex motion in the ν = 1 bilayer
quantum Hall superfluid”.
[133] Y. N. Joglekar, A. V. Balatsky, and A. H. MacDonald, Phys. Rev. Lett.92, 086803 (2004),
“Noise spectroscopy and interlayer phase coherence in bilayer quantum Hall systems”.
27
[134] A. L. Efros, Sol. St. Commun. 65, 1281 (1988), “Density of states of 2D electron gas and
width of the plateau of IQHE”; Sol. St. Commun. 70, 253 (1989), “Metal-nonmetal transition
in heterostructures with thick spacer layers”; Phys. Rev. B 45, 11354 (1992), “Homogeneous
and inhomogeneous states of a two-dimensional electron liquid in a strong magnetic field”; A.
L. Efros, F. G. Pikus, and V. G. Burnett, Phys. Rev. B 47, 2233 (1993), “Density of states of a
two-dimensinoal electron gas in a long-range random potential”; F. G. Pikus and A. L. Efros,
Phys. Rev. B 47, 16395 (1993), “Distribution of electron density and magnetocapacitance in
the regime of the fractional quantum Hall effect”.
[135] S. Ilani, J. Martin, E. Teitelbaum, J. H. Smet, D. Mahalu, V. Umansky, and A. Yacoby,
Nature 427, 328 (2004), “The microscopic nature of localization in the quantum Hall effect”.
[136] G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101 (2003), “Hamiltonian theory of the
fractional quantum Hall effects”.
[137] R. Shankar and G. Murthy, cond-mat/0508242, submitted to Phys. Rev. B: “Deconfinement
in d = 1: A closer look”.
[138] J. Schwinger, Phys. Rev.128, 2425 (1962), “Gauge invariance and mass. II”.
[139] S. Coleman, R. Jackiw, and L. Susskind, Annals of Phys. 93, 267 (1975), “Charge shielding
and quark confinement in the massive Schwinger model”; S. Coleman, Annals of Phys. 101,
239 (1976), “More about the massive Schwinger model”.
[140] G. Delfino and G. Mussardo, Nucl. Phys. B 516 675 (1998), “Non-integrable aspects of the
multi-frequency Sine-Gordon model”.
[141] F. D. M. Haldane, Phys. Rev. B25, 4925 (1982), “Spontaneous dimerization in the S = 12
Heisenberg antiferromagnetic chain with competing interactions”.
[142] C. Mudry and E. Fradkin, Phys. Rev. B50, 11409 (1994), “Mechanism of spin and charge
separation in one-dimensional quantum antiferromagnets”.
[143] For a review of persistent currents in ballistic structures, see, K. Richter, D. Ullmo, and R. A.
Jalabert, Phys. Rep. 276, 1 (1996), “Orbital magnetism in the ballistic regime: Geometrical
effects”.
[144] J. Desbois, S. Ouvry, and C. Texier, Nucl. Phys. B 528, 727 (1998), “Persistent currents and
magnetization in two-dimensional magnetic quantum systems”; R. Narevich, R. E. Prange,
and Q. Zaitsev, Phys. Rev. E62, 2046 (2000), “Square billiard with a magnetic flux”; G.
Chiappe and M. J. Sanchez, Phys. Rev. B65, 153409 (2002), “Persistent currents in diffusive
metallic cavities: Large values and anomalous scaling with disorder”.
[145] For recent examples, see, W. Hofstetter and H. Schoeller, Phys. Rev. Lett.88, 016803 (2002),
“Quantum phase transition in a multilevel dot”; K. Le Hur, P. Simon and L. Borda, Phys.
Rev. B69, 045326 (2004), “Maximized orbital and spin Kondo effects in a single-electron
transistor”.
[146] G. Usaj and H. U. Baranger, Phys. Rev. B64, 201319 (2001), “Coulomb blockade peak-
spacing distribution: Interplay of temperature and spin”; Phys. Rev. B66, 155333 (2002),
“Spin and e-e interactions in quantum dots: Leading order corrections to universality and
28
temperature effects “; Phys. Rev. B67, 121308 (2003), “Exchange and the Coulomb Blockade:
Peak height statistics in quantum dots”
[147] Y. Alhassid and T. Rupp, Phys. Rev. Lett.91, 156801 (2003), “Effects of spin and exchange
interaction on the Coulomb-blockade peak statistics in quantum dots”; Y. Alhassid, T. Rupp,
A. Kaminski, and L. I. Glazman, Phys. Rev. B69, 115331 (2004), “Linear conductance in
Coulomb-blockade quantum dots in the presence of interactions and spin”.
[148] D. P. Arovas, F. Guinea, C. P. Herrero, and P. San-Jose, Phys. Rev. B68, 085306 (2003),
“Granular systems in the Coulomb Blockade regime”; P. San-Jose, C. P. Herrero, F. Guniea,
and D. P. Arovas, cond-mat/0401557 (2004), “Interplay between exchange interactions and
charging effects in metallic grains”.
[149] M. N. Kiselev and Y. Gefen, arXiv:cond-mat/0504751, “The interplay of spin and charge
channels in zero-dimensional systems”.
[150] E. Abrahams, P. W. Anderson, P. A. Lee, and T. V. Ramakrishnan, Phys. Rev. B24, 6783
(1981), “Quasiparticle lifetime in disordered two-dimensional metals”; A. D. Mirlin, Phys.
Rep. 326, 259 (2000), “Statistics of energy levels and eigenfunctions in disordered systems”.
[151] F. J. Ohkawa, “Effects of localization on Kondo effect”, Prog. Theor. Phys. Suppl. 84, 166
(1985); I. Martin, Y. Wan, and P. Phillips, “Size Dependence in the Disordered Kondo
Problem”, Phys. Rev. Lett. 78, 114 (1997); E. Miranda, V. Dobrosavljevic, and G. Kotliar,
Phys. Rev. Lett.78, 290 (1997), “Disorder-driven non-fermi liquid behavior in Kondo alloys”.
[152] C. Jayaprakash, H. R. Krishnamurthy and J. W. Wilkins, Phys. Rev. Lett.47, 737 (1981),
“Two-impurity Kondo problem”; B. A. Jones and C. M. Varma, Phys. Rev. Lett.58, 843
(1987), “Study of two magnetic impurities in a Fermi gas”; Phys. Rev. B40, 324 (1989),
“Critical point in the solution of the two magnetic impurity problem”.
[153] A. Georges and Y. Meir, Phys. Rev. Lett.82, 3508 (1999); “Electronic correlations in transport
through coupled quantum dots”
[154] B. A. Jones, B. G. Kotliar, and A. J. Millis, Phys. Rev. B39, 3415 (1989), “Mean-field analysis
of two antiferromagnetically coupled Anderson impurities”.
[155] I. Affleck, A. W. W. Ludwig, and B. A. Jones, Phys. Rev. B52, 9528 (1995), “Conformal field
theory approach to the two-impurity Kondo problem: Comparison with numerical renormal-
ization group results”.
[156] S. Chakravarty, Bull. Am. Phys. Soc. March (1982).
[157] N. Read and D. M. Newns, J. Phys. C 16, 3273 (1983), “On the solution of the Coqblin-
Scrieffer hamiltonian by the large-N expansion technique”; ibid 16, L1055 (1983), “A new
functional integral formalism for the degenerate Anderson model”.
[158] N. Read, J. Phys. C 18, 2651 (1985) and references therein, “Role of infrared divergences in
the 1/N expansion of the U = ∞ Anderson model”. See also chapters 7 and 8 of [106].
[159] S. Chakravarty and C. Nayak, Int. J. Mod. Phys. B14, 1421 (2000), “Kondo impurity in a
disordered metal: Anderson’s theorem revisited”.
29
[160] P. Coleman, Phys. Rev. B29, 3035 (1984), “New approach to the mixed-valence problem”.
[161] Y. Alhassid and T. Rupp, cond-mat/0312691 (2003), “A universal Hamiltonian for a quantum
dot in the presence of spin-orbit interaction”.
[162] D. M. Zumbuhl, J. B. Miller, C. M. Marcus, D. Goldhaber-Gordon, J. S. Harris, K. Campman,
and A. C. Gossard, Phys. Rev. B72, 081305 (2005), “Conductance fluctuations and partially
broken spin symmetries in quantum dots”.
[163] R. O. Vallejos, C. H. Lewenkopf, and Y. Gefen, Phys. Rev. B65, 085309 (2002), “Orthogo-
nality catastrophe in parametric random matrices”.
[164] A. Altland, L. I. Glazman, and A. Kamenev, Phys. Rev. Lett.92, 026801 (2004), “Electron
transport in granular materials’; J. S. Meyer, A. Kamenev, and L. I. Glazman, Phys. Rev.
B70, 045310 (2004), “Electron transport in two-dimensional arrays”;
[165] I. S. Beloborodov, K. B. Efetov, A. V. Lopatin, and V. M. Vinokur, Phys. Rev. Lett.91,
246801 (2003), “Transport properties of granular metals at low temperatures”; A. V. Andreev
and I. S. Beloborodov, Phys. Rev. B69, 081406 (2004), “Mesoscopic charging effects in the
sigma model for granular materials”; I. S. Beloborodov, A. V. Lopatin, and V. M. Vinokur,
Phys. Rev. Lett.92, 207002 (2004), “Suppression of superconductivity in granular metals”;
I. S. Beloborodov, A. V. Lopatin, G. Schwiete, and V. M. Vinokur, Phys. Rev. B70, 073404
(2004), “Tunnelling density of states of granular metals”; I. S. Beloborodov, A. V. Lopatin,
and V. M. Vinokur, Phys. Rev. B70, 205120 (2004), “Universal description of granular metals
at low temperatures: Granular Fermi liquid”; I. S. Beloborodov, K. B. Efetov, A. V. Lopatin,
and V. M. Vinokur, Phys. Rev. B71, 184501 (2005), “Effects of fluctuations and Coulomb
interaction on the transition temperature of granular superconductors”.
[166] W. Metzner and D. Vollhardt, Phys. Rev. Lett.62, 324 (1989), “Correlated lattice fermions
in d = ∞ dimensions”; A. Georges and G. Kotliar, Phys. Rev. B45, 6479 (1992), “Hubbard
model in infinite dimensions”; For a review see, A. Georges, G. Kotliar, W. Krauth, and
M. J. Rozenberg, Rev. Mod. Phys.68, 13 (1996), “Dynamical mean field theory of strongly
correlated fermion systems and the limit of infinite dimensions”.
[167] ; V. Dobrosavljevic and G. Kotliar, Phys. Rev. Lett.71, 3218 (1993), “Hubbard models with
random hopping in d = ∞”; Phys. Rev. B50, 1430 (1994), “Strong correlations and disorder
in d = ∞ and beyond”;M. Ulmke, V. Janis, and D. Vollhardt, Phys. Rev. B51, 10411 (1995),
“Anderson-Hubbard model in infinite dimensions”.
[168] M. Nikolaou and M. Wallin, Phys. Rev. B69, 184512 (2004), “Zero temperature glass transi-
tion in teh two-dimensional gauge glass model”; L.-H. Tang and P. Tong, cond-mat/0412415
(2004), “Zero-temperature criticality in the two-dimensional gauge glass model”.
[169] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Phys. Rev. B47, 16419 (1993),
“Skyrmions and the crossover from the integer to fractional quantum Hall effect at small
Zeeman energies”; H. A. Fertig, L. Brey, R. Cote, A. H. MacDonald, A. Karlhede, S. L.
Sondhi, Phys. Rev. B55, 10671 (1997), “Hartree-Fock theory of Skyrmions in quantum Hall
ferromagnets”.
30
[170] M.M.Fogler and B.I.Shklovskii, Phys. Rev. B 52, 17366 (1995), ‘Collapse of spin splitting in
the quantum Hall effect”‘; see also, J.Dempsey, B.Y.Gelfand, and B.I.Halperin, Phys. Rev.
Lett. 70, 3639 (1993), “Electron-electron interactions and spontaneous spin polarization in
quantum Hall edge states”.
[171] G. Murthy, Phys. Rev. B64, 24130 (2001), “Effets of disorder on the ν = 1 quantum Hall
state”.
[172] G. Murthy, Phys. Rev. B 64, 241309 (2001), “Effects of Disorder on the ν = 1 Quantum
Hall State”.
[173] M. Lau and C. Dasgupta, Phys. Rev. B39, 7212 (1989), “Numerical investigation of the role
of topological defects in the three-dimensional Heisenberg transition”.
[174] M. Kamal and G. Murthy, Phys. Rev. Lett.71, 1911 (1993), “New O(3) transition in three
dimensions”.
[175] X. G. Wen, Phys. Rev. B44, 2664 (1991), “Mean-field theory of spin-liquid states with finite
energy gap and topological orders”.
[176] G. Murthy and Subir Sachdev, Nucl. Phys. B344, 557 (1990); “Action of hedgehog-instantons
in the disordered phase of the 2+1 dimensional CP N−1 model”.
[177] M. Levin and T. Senthil, Phys. Rev. B70, 220403 (R) (2004), “Deconfined quantum criticality
and Neel order via dimer disorder”.
[178] T. Senthil and M. P. A. Fisher, Phys. Rev. B62, 7850 (2000), “Z2 gauge theory of electron
fractionalization in strongly correlated systems”; R. Moessner, S. L. Sondhi, and E. Fradkin,
Phys. Rev. B65, 024504 (2002), “Short-ranged resonating valence bond physics, quantum
dimer models, and Ising gauge theories”; G. Misguich, D. Serban, and V. Pasquier, Phys.
Rev. Lett.89, 137202 (2002), “Quantum Dimer Model on the Kagome Lattice: Solvable
Dimer-Liquid and Ising Gauge Theory”.
[179] S.-K. Ma, C. Dasgupta, and C.-K. Hu, Phys. Rev. Lett.43, 1434 (1979), “Random antiferro-
magnetic chain”; C. Dasgupta and S.-K. Ma, Phys. Rev. B22, 1305 (1980), “Low-temperature
properties of the random Heisenberg antiferromagnetic chain”.
31