Quantum transport in dispersionless electronic bands Alexander Kruchkov1 2 3 1Institute of Physics Swiss Federal Institute of Technology EPFL Lausanne CH 1015 Switzerland

2025-04-29 0 0 901.89KB 6 页 10玖币

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Quantum transport in dispersionless electronic bands

Alexander Kruchkov1, 2, 3

1Institute of Physics, Swiss Federal Institute of Technology (EPFL), Lausanne, CH 1015, Switzerland

2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

3Branco Weiss Society in Science, ETH Zurich, Zurich, CH 8092, Switzerland

Flat electronic bands are counterintuitive: with the electron velocity vanishing, our conventional

notions of quasiparticle transport are no longer valid. We here study the quantum transport in

the generalized families of perfectly ﬂat bands [PRB 105, L241102 (2022)], and ﬁnd that while the

conventional contributions indeed vanish, the quantum-geometric contribution gives rise to the en-

hanced electronic transport. This contribution is connected to the Wannier orbital quantization in

the perfectly ﬂat bands, and is present only for geometrically-nontrivial bands (for example, ﬂat

Chern bands). We ﬁnd structurally similar expressions for thermal conductance, thermoelectric re-

sponse, and superﬂuid weight in the ﬂat bands. In particular, we report the anomalous thermopower

associated with ﬂat topological bands reaching values as large as kB

eln 2≈60 µV/k, the quantum

unit of thermopower, which are not expected for the conventional dispersive bands.

Dispersionless electronic states (”ﬂat bands”) are

counterintuitive since the eﬀective electronic mass be-

comes inﬁnite, the quasiparticle velocity vanishes, and

the conventional notions of electron transport fail. The

perfectly ﬂat electronic bands is a change of paradigm

in condensed matter physics, but remained largely a hy-

pothetical object until the discovery of the twisted bi-

layer graphene [1, 2], where the dispersionless electronic

states are emerging at the magic angle 1.05◦[3, 4]. The

underlying ﬂat band is not just a lucky engineering of

material parameters, but it is of fundamental origin as

it can be tuned to perfectly ﬂat [5, 6]. It was further

understood that the magic-angle ﬂat bands are dual to

the lowest Landau level, and host a number of unconven-

tional phases, characterized by strange metallicity [7–9],

unconventional superconductivity [2, 10], fractional Hall

conductance [11], and giant thermopower [12]—untypical

for conventional electronic systems.

A recent interest in condensed matter physics is under-

standing the role quantum geometry of electronic states,

described by the quantum geometric tensor [13, 14]

G(n)

ij =h∂kiunk|[1 − |unkihunk|]|∂kjunki(1)

here unkis the associated Bloch state of the nth elec-

tronic band, which may or may not be ﬂat. The real

part of Gij =ReGij , is the Fubini-Study metrics describ-

ing the geometry of the bands, while the imaginary part

Fij =−2ImGij is Berry curvature reﬂecting the topol-

ogy of the Bloch states. In particular, it has been un-

derstood that the largely overlooked quantum metrics

plays important role in diﬀerent quantum transport phe-

nomena, ranging from quantum noise, optical conductiv-

ity, anomalous Hall eﬀect, and unconventional supercon-

ductivity, and adjacent topics [15–25]. The quantum-

geometric superconductivity [18] has a particular im-

portant role in twisted bilayer graphene, where it has

been shown that the quantum-geometric contribution to

the superﬂuid weight is key at the magic angle [26–29].

Moreover, it has recently been argued that the quantum-

geometric contribution to superﬂuid weight it TBG can

be probed in experiment with the ultraclean samples [30].

However, other anomalies in the topological ﬂat bands,—

such as e.g. giant thermoelectric power at the magic

angle [12]—remain to be revisited from the quantum ge-

ometric perspective as well.

In this direction, recent classiﬁcation of perfectly ﬂat

bands [31] presents a handy framework as it allows to

bridge the properties of Wannier orbitals, their quantum

geometry, and band ﬂatness limits. If we allow perfect

band ﬂatness, the quantum metrics of dispersionless elec-

tronic bands saturates the ”trace condition”

Tr Gij (k) = |Fxy(k)|.(2)

However, the nontrivial quantum geometry comes at cost

that the electronic Wannier orbitals will have a ﬁnite and

large cross-section [32]. In case of perfectly ﬂat Chern

bands (2), the Wannier orbital crossection r2

0experiences

Lifshitz-Onsager-like quantization [31], with

0=a2ZdkTrGij (k) = a2Zd2kFxy(k) = Ca2.(3)

In other words, the dispersionless electronic states are

spread over the atomic lattice, overlapping with Cneigh-

boring electronic orbitals, hence allowing quantum tun-

neling even in the absence of kinetic terms (Fig.1). We

show that this phenomenon promotes the unconventional

quantum transport in dispersionless electronic bands.

In this work we bring to the common denominator dif-

ferent quantum transport properties (electric conductiv-

ity, quantized Hall eﬀect, thermal conductivity, thermo-

electric response, and the superﬂuid weight), which for

the perfectly ﬂat band systems can be expressed through

their multiorbital quantum metrics:

Lij =X

nm,k

Inm(k) ReGnm

ij (k) + Jnm(k) ImGnm

ij (k),

where Lij is a quantum transport characteristic, and

Gnm

ij (k)≡ h∂kiunk|umkihumk|∂kjunkiis the general-

ized quantum-geometric tensor (deﬁned further in text),

arXiv:2210.00351v1 [cond-mat.str-el] 1 Oct 2022

Nontrivial flat band

Trivial flat band

FIG. 1. (Top) In trivial ﬂat bands, electrons are strongly lo-

calized (Wannier orbitals sharp) and electronic transport is

forbidden, the system is in the insulating phase. (Bottom) In

nontrivial ﬂat bands (e.g. Chern bands), the Wannier orbitals

cannot be exponentially localized; electrons do not move in

the classical sense. Under application of external ﬁelds, elec-

trons tunnel between overlapping Wannier orbitals, resulting

into unconventional conductivity without electron velocities.

and Inm(k) and Jnm(k) are system-dependent struc-

tural tensors expressed through quasiparticle propaga-

tors. Note that neither the quasiparticle velocities nor

bandwidth enter this expression; this ﬂat-band transport

is purely quantum, with its origin in Wannier functions

overlap (Fig.1).

Quantum transport formalism. In what follows

below we consider a weakly-dispersive Chern band, and

then set the bandwidth (and hence the Fermi veloc-

ity) to exact zero. The main result is illustrated for

the perfectly ﬂat Chern bands, however it also applies

to all geometrically-nontrivial ﬂat bands (Wannier func-

tion are not exponentially localized), thus including those

in twisted bilayer graphene and similar materials. For

the moment, we omit the explicit dependence on mag-

netic ﬁelds, however such generalization can be done.

Up to this moment, the derivation of polarization ten-

sor is rather conventional and can be obtained in several

ways. A disciplined way to derive it is through using the

current-current correlators in Matsubara framework [33].

The quantum transport properties are computed through

imaginary-time Matsubara correlators

Lαβ

ij (τ, τ 0) = −1

~hTτJα

i(τ)Qβ

j(τ0)i,(4)

where Qαis the generalized ”charge” operator (we denote

α= 1 for electric charge, α0= 2 for heat transfer), i, j =

x, y and Jαis the generalized current. For example, for

the electric current of charge eone writes [33]

J=e

c†

∂Hk

∂kck.(5)

To calculate the response functions Lij , we introduced

the auxiliary current-current correlators

Παβ

ij (τ) = −hTτJα

i(τ)Jβ

j(0)i,(6)

so in the frequency representation one has Lij (iωn) =

iωn[Πij (iωn)−Πij (0)] .To calculate the transport prop-

erties (such as conductance σij =L11

ij ), we further pro-

ceed to analytical continuation of Tij (iωn) and then take

the DC limit:

Lαβ

ij ≡lim

ω→0

Παβ

ij (ω)−Παβ

ij (0)

iω .(7)

The Onsager coeﬃcients Lαβ

ij fully describe the trans-

port properties of an electronic system. Experimentally,

the transport measurements across the sample are per-

formed by Ie=L11∆V+L12∆T(electric measurement)

and Ih=L21∆V+L22∆T(heat measurement) [34, 35].

We further compute the electric conductivity σij =L11,

thermal conductivity κij =β(L22 −L2

12/T L11), and ther-

moelectric response (Seebeck coeﬃcient Θ = β

L12/L11) in

the dispersionless electronic bands (here β=1/T ;eis in-

cluded in deﬁnition of Eq.(5)). A similar response struc-

ture to vector potential Awill imply the ﬁnite superﬂuid

weight DSin the dispersionless bands [19].

We start from the electric conductivity, for which we

calculate the electric polarization tensor Πij (ω) (we here

drop the superscripts αβ for brevity); for convenience,

below we use Πij (ω) = e2

~2˜

Πij (ω). Evaluating the current-

current correlator (6) with (5) in Matsubara representa-

tion gives [36]

Π±

ij (iω0) = 1

βX

iω0

Tr Gk(iω0

n)∂Hk

∂ki

Gk(iω0

n±iω0)∂Hk

∂kj

where Gk(iω0) is the Matsubara transform of the (renor-

malized) Green function Gk(τ, τ 0) = −hTτc†

k(τ)c)k(τ0)i,

where expectation value is taken over the interacting vac-

cum at temperature T. Here Matsubara frequencies iω0

are fermionic and iω0is bosonic.

The inﬂuence of the quantum-geometric tensor can be

demonstrated in the following way. The position operator

in the Bloch basis is [37]

rmn =i∂kδnm +hunk|i∂kumki.(8)

It follows that the generalized velocity operator in this

basis is given by

∂Hk

∂k=˙

r=vnkδnm +ωnm,khunk|∂kumki,(9)

where vnk=∂εnk

∂~kis the quasiparticle velocity in the

band (”Fermi velocity”) and ~ωnm,k=εnk−εmkare

transition frequencies of the multiorbital system (the sec-

ond term is called ”anomalous velocity” [16]). With the

velocity operator (9), the polarization tensor (8) have

two terms: the ﬁrst proportional to the Fermi velocities

O(v2

nk), and the second being independent of the band

dispersion itself. For illustrational purpose, it is useful to

write down the ﬁrst contribution which has a generic form

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QuantumtransportindispersionlesselectronicbandsAlexanderKruchkov1,2,31InstituteofPhysics,SwissFederalInstituteofTechnology(EPFL),Lausanne,CH1015,Switzerland2DepartmentofPhysics,HarvardUniversity,Cambridge,Massachusetts02138,USA3BrancoWeissSocietyinScience,ETHZurich,Zurich,CH8092,SwitzerlandFlatelect...

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Quantum transport in dispersionless electronic bands Alexander Kruchkov1 2 3 1Institute of Physics Swiss Federal Institute of Technology EPFL Lausanne CH 1015 Switzerland

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