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

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|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

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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.

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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

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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

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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.

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