Some models and mechanisms for non-fermi liquidsweb.mit.edu/~senthil/www/aspen0118.pdfmodels, we formulate conjectures on coarse grained descriptions of a class of intermediate scale

Some models and mechanisms for non-fermi liquids

T. Senthil (MIT)

Collaborators

Debanjan Chowdhury(MIT)

Yochai Werman(Weizmann/Berkeley)

Erez Berg(Chicago)

Chowdhury, Werman, Berg, TS, arXiv: 1801.06178 (appeared today!)

Cuprate strange metal regime

Strange non-Fermi liquid

Power laws in many physical quantities distinct from that expected in a Fermi liquid.

Fermi surface but no Landau quasiparticles.

Compared to conventional textbook wisdom (= Landau fermi liquid theory) , the strange metal is a wild beast.

Cuprate strange metal regime

Power laws in many physical quantities distinct from that expected in a Fermi liquid.

Fermi surface but no Landau quasiparticles.

Compared to conventional textbook wisdom (= Landau fermi liquid theory) , the strange metal is a wild beast.

Strange non-Fermi liquid

Many interesting properties

Bi-2201: Martin et al, PR B, 1990 Nd-LSCO, Daou, ..., Taillefer, Nat. Phys., 2008

Most famous: linear resistivity

Many other anomalies in other properties, eg, optical transport, ARPES, ……

Well-known phenomenological description: Marginal Fermi Liquid (MFL) (Varma et al, 1989)?? Microscopic basis??

Q: What is the theory of the cuprate strange metal?

Q: What is the theory of the cuprate strange metal?

A: I do not know and will not provide an answer in this talk.

Other analogous metals

Various kinds of strange/bad/non-fermi liquids show up in a wide variety of correlated metals.

Eg: Many examples in heavy fermion systems, Ruthenates, pnictides,……..

(Various sessions at this meeting).

Others: Cobaltate Na0.7CoO2 (Cava, Ong, 02; Taillefer et al, 04)

Somewhat but not exactly similar phenomenology.

These likely realize distinct ways in which Fermi liquid theory breaks down.

The wild world of strange metalsStrange non-

All strange metals likely not exactly the same.

Can we tame some of these beasts?

Big questions slide from James Analytis

GTQ NUS Q U Y

b P bQ PQ O UNQ M XQ MW bU T YM U M UOWQ 4

9aQY UY M Y Y :Q XU ?U UP A:? bTd P bQ SQWUYQM G MYP T WP bQ Qc QO YUaQ MWN YP 4

b P Q PU PQ MRRQO TQ A:?4 9aQY TM PQWaQ MY MW QMPd aQ d PURRUO W NWQXl

Strategies for progress in theoretical physics

Ideal world: Solve the right model rightly

If not possible then…….


Solve the right model wrongly (and pray)

or

Solve the wrong model rightly (and see what you can learn).





or




Today: I will talk about an advance in building solvable translation invariant lattice models for marginal Fermi liquids and non-Fermi liquids.



or







or







or





Translationally invariant non-Fermi liquid metals with critical Fermi-surfaces:

Solvable models

Debanjan Chowdhury,1, ⇤ Yochai Werman,2 Erez Berg,3 and T. Senthil1

1Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139, USA.

2Department of Condensed Matter Physics,

Weizmann Institute of Science, Rehovot-76100, Israel.

3Department of Physics, University of Chicago, Chicago IL 60637, USA.

We construct examples of translationally invariant solvable models of strongly-correlated

metals, composed of lattices of Sachdev-Ye-Kitaev dots with identical local interactions.

These models display crossovers as a function of temperature into regimes with local quan-

tum criticality and marginal-Fermi liquid behavior. In the marginal Fermi liquid regime,

the dc resistivity increases linearly with temperature over a broad range of temperatures.

By generalizing the form of interactions, we also construct examples of non-Fermi liquids

with critical Fermi-surfaces. The self-energy has a singular frequency dependence, but lacks

momentum dependence, reminiscent of a dynamical mean field theory-like behavior but in

dimensions d < 1. In the low temperature and strong-coupling limit, a heavy Fermi liquid

is formed. The critical Fermi-surface in the non-Fermi liquid regime gives rise to quantum

oscillations in the magnetization as a function of an external magnetic field in the absence of

quasiparticle excitations. We discuss the implications of these results for local quantum crit-

icality and for fundamental bounds on relaxation rates. Drawing on the lessons from these

models, we formulate conjectures on coarse grained descriptions of a class of intermediate

scale non-fermi liquid behavior in generic correlated metals.

⇤ [email protected]

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Today’s listing: arXiv: 1801.06178

Two solvable models

1. One band model

- ``locally critical” NFL high T regime (T >> Tcoh)

- standard Landau Fermi liquid with a sharp Fermi surface at low T (<< Tcoh)

2. Two band model (a light weakly correlated band coupled to a different heavy strongly correlated band)

- broad intermediate temperature Marginal Fermi Liquid with a sharp Fermi surface- standard Landau Fermi liquid at low T

The one-band model10

…. ……..

….

….

.

.

.

.

…. ….………...

.

.

.

..

..

i j

kl

(a) (b)

FIG. 1: (a) A two-dimensional lattice where each site contains N orbitals (represented by di↵erent

colors). The hoppings, tcrr0 , between any neighboring sites (colored arrows) are diagonal in orbital-index.

Each site is identical and the system is translationally invariant. (b) The internal structure of a single site

with N orbitals. The on-site interactions, U c

ijk`

, are quartic in the fermion operators, with all orbital

indices unequal.

functions in our model are suppressed by powers of 1/N . We therefore expect that the correlation

functions in our model are self-averaging in the large N limit, as in the single-site SYK model.

A. Fermion Green’s Function

The fermion Green’s function can be analyzed diagrammatically, such that the large-N saddle-

point solution reduces to studying the following set of equations self-consistently,

Gc(k, i!) =1

i! � "k

� ⌃c(k, i!), (2a)

⌃c(k, i!) = �U2

c

ˆk

1

ˆ!1

Gc(k1

, i!1

) ⇧c(k + k

1

, i! + i!1

), (2b)

⇧c(q, i⌦) =

ˆk

ˆ!Gc(k, i!) Gc(k + q, i! + i⌦), (2c)

where´k

⌘´ddk/(2⇡)d and "

k

is the dispersion for the c�band. Formally, the above set of equa-

tions corresponds to resumming an infinite class of ‘watermelon-diagrams’, as shown in Fig. 2. One

Lattice of strongly correlated sites coupled by electron hopping.

Each site has N orbitals coupled together by frustrated interactions.

Solvable in large-N limit.

The one band model

Hc = �X

~r,~r0

X

`

tc~r,~r0c†~r`c~r0` +

1

N3/2

X

~r

X

ijk`

U cijk`c

†~ric

†~rjc~rkc~r`

10

…. ……..

….

….

.

.

.

.

…. ….………...

.

.

.

..

..





ijk`


indices unequal.




Gc(k, i!) =1

i! � "k

� ⌃c(k, i!), (2a)

⌃c(k, i!) = �U2

c

ˆk

1

ˆ!1

Gc(k1

, i!1

) ⇧c(k + k

1

, i! + i!1

), (2b)

⇧c(q, i⌦) =

ˆk

ˆ!Gc(k, i!) Gc(k + q, i! + i⌦), (2c)

where´k


k



can arrive at the same set of saddle-point equations by starting from the path-integral formulation,

as described in Appendix C. In Sec. III, we provide a simple alternate derivation of the results for

the one band model using scaling-type arguments which provide much physical insight.

Translation invariance: tc, U cijkl are the same at each lattice site.

Assume generic on-site inter-orbital interaction U cijkl which depends on (ijkl).

The one-band model (cont’d)

Choose interactions from a random distribution where U cijkl are independent,

uncorrelated variables with

U cijkl = 0, (U c

ijkl)2= U2

c

Hc = �X

~r,~r0

X

`

tc~r,~r0c†~r`c~r0` +

1

N3/2

X

~r

X

ijk`

U cijk`c

†~ric

†~rjc~rkc~r`

Large-N limit: Properties for any generic interaction can be calculated by averaging over this random probability distribution.

Note: each realization is exactly translation invariant.

The basic ingredient: a single siteThe Sachdev-Ye-Kitaev (SYK) model

10

i j

kl





ijk`


indices unequal.

functions in our model are suppressed by powers of 1/N . We therefore expect that the correlation

functions in our model are self-averaging in the large N limit, as in the single-site SYK model.




Gc(k, i!) =1

i! � "k

� ⌃c(k, i!), (2a)

⌃c(k, i!) = �U2

c

ˆk

1

ˆ!1

Gc(k1

, i!1

) ⇧c(k + k

1

, i! + i!1

), (2b)

⇧c(q, i⌦) =

ˆk

ˆ!Gc(k, i!) Gc(k + q, i! + i⌦), (2c)

where´k


k



Hint =1

N3/2

X

~r

X

ijk`

U cijk`c

†~ric

†~rjc~rkc~r`

All-to-all interactions taken from a random probability distribution.

Sachdev, Ye 1992Parcollet, Georges, 1998Kitaev, 2015……..

Model is self-averaging in large-N limit

Many novel features, for instance,

Power law Green’s function G(!) ⇠ 1p!F (

!T )

T = 0 entropy S = N(S0 + o(T ))

Comments

1. Our models build up a translation invariant lattice of SYK ``dots” coupled together by hopping.

2. Earlier work: Lattice of random SYK dots coupled by random hopping (Song, Jian, Balents, 17; Patel, McGreevy, Arovas, Sachdev 17).

Some common features (implying translation breaking is not essential).

Allowing translation invariance further enables us to address presence/absence of ``critical Fermi surfaces” (sharp Fermi surface without sharp quasiparticles).

Large-N solution

11

i i

j

k

l

q=0,Ω=0

k1,ω1

k+k1+k2, ω+ω1+ω2

k,ω

k2 ,ω2=

FIG. 2: The self-energy diagram, ⌃c

, for c�fermions with orbital index i in the single-band model due to

Uc

. The solid black lines represent fully dressed Green’s functions, Gc

(k,!); see Eq. (2a). The dashed line

corresponds to U2c

contraction and carries no frequency/momentum.

As we shall now show, the fermionic spectral function has qualitatively di↵erent behavior at

di↵erent temperatures. When the temperature is much lower than the characteristic crossover scale

⌦⇤c ⌘ W 2

c /Uc, the spectral function has a Fermi-liquid like form. In the interesting case Uc � Wc,

there is a second regime defined by ⌦⇤c ⌧ T ⌧ Uc, where the spectral function has an incoherent,

local form without any remnant of a Fermi-surface. To make this statement more precise, we can

take the limit of Uc ! 1 keeping Wc finite (such that ⌦⇤c collapses to zero), and then take the

limit of T ! 0, thus obtaining a compressible phase of electronic matter without quasiparticle-

excitations in a clean system, lacking any sharp momentum-space structure. We refer to this state

as a local incoherent critical metal (LICM).

To analyze the equations (2a-2c), we focus on the two extreme limits of T (or !) that are either

much larger or much smaller than ⌦⇤c . In the limit T ⌧ ⌦⇤

c , we find that the system follows Fermi

liquid behavior at su�ciently low frequencies. To show this, let us use a Fermi liquid-like ansatz

for the fermionic self energy. At low frequencies we assume that ⌃c has the following form near

the Fermi surface:

⌃c(k, i!) = �i(Z�1 � 1)! + (vF � vF )k + . . . , (3)

where Z is the quasiparticle residue, to be determined self-consistently, k = |k � kF | (kF is the

Fermi momentum), vF (vF ) are the renormalized (bare) Fermi-velocities with the renormalization

vF /vF = A to be determined self-consistently, and the . . . denote higher power terms in an expan-

sion in !, k. We stress that vF is di↵erent from the e↵ective Fermi velocity v⇤F = ZvF , which is the

physical speed with which quasi-particles propagate. For simplicity, we have dropped the constant

Exact lattice Green’s function given by solving a self-consistency equation for self-energy

Gc(~k, i!) =1

i! � ✏k � ⌃c(~k, i!)

⌃c(k, i!) = �U2c

Z

~k1,~k2

Z

!1,!2

Gc(~k1, i!1)Gc(~k2, i!2)Gc(~k + ~k1 + ~k2, i(! + !1 + !2))

Some similarity to DMFT equations but self-energy is a priori allowed to be momentum-dependent.

Results

Emergence of a coherence scale Tcoh

⇠ t

2c

Uc:

T � Tcoh

: Gc

(

~k, i!) inherits locally critical behavior of single dot (with

weak k-dependance).

T ⌧ Tcoh

: Landau Fermi liquid with sharp Fermi surface and heavy quasi-

particles.

For T � Tcoh

, resistivity ⇢ / T .

Physical picture

S/N

TTcoh

High T : Treat hopping tc

in perturbation theory.Scale invariant Green’s function in each site G(!) ⇠ 1p

!

=> Hopping becomes scale dependent.

E↵ective hopping at a temperature T

tRc

(T ) ⇠ tc

�U

c

T

� 12 .

Perturbation theory fails at Tcoh

when e↵ective hopping ⇡ Uc

.Natural to get FL at lower T .

Residual entropy at high-T which is relieved at low-T=> e↵ective mass ⇠ 1

T

coh

.

High-T NFL transport: Physical understanding

High-T NFL regime:Perturbation theory in e↵ective hopping => conductivity / (tR

c

(T ))2:

� ⇠ Ne2

h

✓tRc

(T )

Uc

◆2

⇠ Ne2

h

Tcoh

T

Resistivity ⇢ = h

Ne

2T

T

coh

(Linear resistivity with values exceeding Mott-Io↵e-Regal limit h

Ne

2 .)

Low-T Fermi liquid: ⇢(T ) = AT 2with A ⇠ �2

Note: similar to results of Song et al (17) but in a translation invariant model.As a matter of principle, disorder not necessary to get this kind of NFL.

Two band model

Weakly correlated c-band coupled to strongly correlated f-band.

20

IV. TWO-BAND MODEL — MARGINAL FERMI LIQUID

In the previous section, we saw an example of a crossover from a Fermi-liquid to an incoherent

metal, without any remnant of a Fermi-surface, in a one-band model. It is interesting to ask if

a critical Fermi-surface [19] can emerge in the general class of translationally invariant models

that are being considered here. Before proceeding further, it is useful to define precisely what

we mean by a critical Fermi-surface. Within our definition, the criticality is associated with the

gapless single-particle excitations of physical electrons over the entire Fermi surface, which remains

sharply defined14. However there are no Landau quasiparticles across the critical Fermi surface

and the quasiparticle residue Z is zero. We describe two classes of models in the next two sections

that host such a critical Fermi surface.

Let us begin with a model where we introduce an additional band of f�fermions with operators

f †r,`, fr,` (` = 1, ..., N) and an associated conserved U(1) charge density, Qf that may be tuned by

a chemical potential µf , which we set to zero. The modified Hamiltonian (with a U(1)c ⇥ U(1)f

symmetry) is

H = Hc +Hf +Hcf , (26)

where Hc is as described in Eq. (18), and Hf is defined in an identical fashion with translationally

invariant hoppings tfrr

0 and on-site interactions Ufijk`. The form of the inter-band interaction is

chosen to be

Hcf =1

N3/2

X

r

X

ijk`

Vijk`c†rif

†rjcrkfr`, (27)

where the coe�cients, Vijkl, are chosen to be identical at every site with Ufijkl = Vijkl = 0, and

where the distribution of the couplings satisfy (Ufijk`)

2 = U2

f , (Vijk`)2 = U2

cf . We now assume

that tfr,r0 ⌧ tc

r,r0 , i.e. the bandwidth for the f�fermions is much smaller than the bandwidth

for the c�fermions (Wf ⌧ Wc). The model described by (26) therefore has some similarity to

models for ‘heavy-Fermion’ systems, with a specific form of interaction terms, and where the direct

hybridization term, Hhyb

=P

r,ij Mij c†rifrj has been set to zero.

To leading order in 1/N , the saddle point equations for the Hamiltonian defined in Eq. (26) are

14 In contrast sometimes in the literature the phrase ‘critical Fermi surface’ is used to denote a Fermi surface of

emergent fermions (not locally related to microscopic degrees of freedom) which themselves may be critical.

Hf : translation invariant lattice of SYK dots (as before) with hopping tf ,interaction Uf

.

Hc = �P

~r,~r0P

i tc~r,~r0

⇣c†ricr0i + h.c

⌘

Hcf =

1N3/2

Pr

Pijkl Vijklc

†ricrkf

†rjfrl

Vijkl = 0, (Vijkl)2= U2

cf

Full model is translation invariant; solve as before by solving self-consistency equations for c and f self energies.

Results

Low coherence scale: Tcoh

⇠ t

2f

Uff,

Low-T (T ⌧ Tcoh

): Heavy Landau Fermi liquid with separate c and f Fermisurfaces

Broad intermediate temperatures Tcoh,f

⌧ T ⌧ min(tc

, Uf

):c-electrons form a Marginal Fermi Liquid with a sharp Fermi surface.(f are locally critical).

(Specialize to limit Ucf ⌧ tc and tf ⌧ other parameters.)

Focus on MFL .

Marginal Fermi Liquid solution (intermediate-T)

Physics: Scattering of c-electrons off fluctuations of critical f-electrons.

(+ weak k-dependent corrections)

Sharp c-Fermi surface satisfying Luttinger’s theorem

Heat capacity C(T ) ⇠ T ln(1/T )

Finite c-electron compressibility (non-interacting value + weak corrections).

⌃cf = � ⌫0U2cf

2⇡2Ufi! ln

⇣Uf

|! |⌘

Marginal Fermi Liquid (MFL) physical properties: transport

Linear dc resistivity

27

FIG. 6: Correction to the current vertex in Fig. 5(b) at the same order in 1/N .

in Fig. 5(a) without any rungs, which reduces the expression for the conductivity to

�0xx(⌦, T ) =

1

⌦

ˆd!

ˆk

(vxk

)2 Ak

(!) Ak

(⌦+ !) [f(!)� f(⌦+ !)], (37)

where Ak

(!) is the spectral function for the c fermions and f(...) represents the Fermi-Dirac

distribution function. At frequencies much higher than the temperature (⌦ � T ), this leads to:

�0xx(⌦) /

Nv2 Uf

U2

cf

1

⌦ ln(1/⌦)2, (38)

where v2 is the average of (vxk

)2 over the Fermi surface.

Let us now focus on the dc limit. We find that the scattering rate determined from the dc

resistivity is determined by the single-particle scattering rate of the c�fermions. This result is not

surprising, since in the regime that is being considered here where the f�fermions provide a momen-

tum independent scattering channel, providing an e↵ective ‘momentum sink’ for the c�fermions18.

Therefore, the resistivity (in units of h/e2) is given by,

⇢dc(T ) /U2

cf

Nv2 UfT. (39)

We can now estimate the frequency scale at which the high-frequency form of the optical conduc-

tivity matches the low-frequency dc result. A simple analysis immediately reveals the crossover

scale (in units of kB/~) to be

⌧�1

opt

⇠ T/ ln2(1/T ). (40)

18 One might be tempted to associate the momentum relaxing scattering in the clean system above with ‘umklapp’

scattering. However, we note that in the regime of interest here, there is no restriction on the respective c or

f�fermion densities.

High frequency conductivity

27



�0xx(⌦, T ) =

1

⌦

ˆd!

ˆk

(vxk

)2 Ak

(!) Ak

(⌦+ !) [f(!)� f(⌦+ !)], (37)

where Ak



�0xx(⌦) /

Nv2 Uf

U2

cf

1

⌦ ln(1/⌦)2, (38)








⇢dc(T ) /U2

cf

Nv2 UfT. (39)




⌧�1

opt

⇠ T/ ln2(1/T ). (40)



f�fermion densities.

Crossover scale = ``optical scattering rate”

27



�0xx(⌦, T ) =

1

⌦

ˆd!

ˆk

(vxk

)2 Ak

(!) Ak

(⌦+ !) [f(!)� f(⌦+ !)], (37)

where Ak



�0xx(⌦) /

Nv2 Uf

U2

cf

1

⌦ ln(1/⌦)2, (38)








⇢dc(T ) /U2

cf

Nv2 UfT. (39)




⌧�1

opt

⇠ T/ ln2(1/T ). (40)



f�fermion densities.Note: this rate is parametrically smaller than ``Planckian limit” despite linear resistivity.

Generalizations: Non-fermi liquids with ``critical Fermi surfaces”

Models discussed so far have 2 body interactions.

Generalization: q-body interactions with q > 2.

Two band model - Low-T: Landau Fermi liquid.

Broad intermediate-T: Non-fermi liquid with sharp ``critical Fermi surfacesatisfying Luttinger theorem.

Many interesting thermodynamic and transport properties.

Example: ⇢dc ⇠ T 4/q

=> sublinear power law

Signatures of critical Fermi surface: Quantum oscillations (with calculable

non-LK T-dependence), 2Kf singularities.

What have we learnt?

Concrete testing ground for many ideas that have been proposed about NFLs.

Example: Transport bounds

1. Need care in defining transport ``scattering rate” in a strange metal.

Procedure in Bruin,….MacKenzie et al, 2013 leads to scattering rate parametrically smaller than Planckian bound even with linear resistivity.

Not clear which, if any, transport rate satisfies a bound that is saturated in a strange metal.

2. Can get (sub)linear resistivity in a clean metal: disorder need not play a crucial role

Where do we go?

Models studied here encouraging but not directly applicable to non-fermi liquids in generic correlated models/materials.

Inspiration to develop a coarse grained picture of generic intermediate-T non-fermi liquids.

A key feature of these models: maximal many body chaos within each dot

Conjecture: Coarse graining of generic correlated model for a strange metal- Maximally chaotic bubbles of spatial size l >> a (microscopic length scale) but l << L (= system size).

Universal coarse grained theory: coupled maximally chaotic bubbles.

Many interesting questions + routes to progress ahead!!

Documents

Some models and mechanisms for non-fermi liquidsweb.mit.edu/~senthil/www/aspen0118.pdfmodels, we formulate conjectures on coarse grained descriptions of a class of intermediate scale