Extended Dynamical Mean Field Theory for Correlated Electron Models Haoyu Hu12 Lei Chen1 Qimiao Si1

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Extended Dynamical Mean Field Theory

for Correlated Electron Models

Haoyu Hu1,2, Lei Chen1, Qimiao Si1

1Department of Physics and Astronomy, Rice Center for Quantum Materials, Rice

University, Houston, TX 77005, USA

2Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San

Sebastian, Spain

An overarching question in strongly correlated electron systems is how the

landscape of quantum phases emerges from electron correlations. The method

of extended dynamical mean ﬁeld theory (EDMFT) has been developed for

clean lattice models of the correlated electrons. For such models, not only on-

site Hubbard-like interactions are important, but so are intersite interactions.

Importantly, the EDMFT method treats the interplay between the onsite and

intersite interactions dynamically. It was initially formulated for models of the

two-band Anderson-lattice type with intersite interactions, as well as for the one-

band Hubbard type with intersite Heisenberg-like terms that are often called

Hubbard-Heisenberg models. In the case of Kondo lattice models, the EDMFT

method incorporates a dynamical competition between the local Kondo and in-

tersite Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions. In these models,

the EDMFT-based analyses led to the notion of Kondo destruction, which has

played a central role in the understanding of quantum critical heavy fermion

metals. In this article, we summarize the EDMFT method, and survey its ap-

plications, particularly for Kondo/Anderson lattice models. We also discuss the

prospect for further developing the EDMFT method, as well as for applying it to

address the correlation physics in a variety of new settings. Among the latter are

the orbital-selective Mott physics that arises both in iron-based superconductors

and in frustrated bulk systems with topological ﬂat bands.

arXiv:2210.14197v1 [cond-mat.str-el] 25 Oct 2022

I. INTRODUCTION

A wide variety of bad metals form the bedrock of strongly correlated electron systems.

Often, a bad metal is operationally deﬁned in terms of the electrical resistivity at room

temperature reaches or exceeds the Mott-Ioﬀe-Regel limit [1, 2]. This limit is deﬁned in

terms of the mean free path `, corresponding to kF`/2π≈1 for each Fermi surface, which

signiﬁes a proximity to a Mott localized state [2]. A complementary criterion is based on a

sizeable reduction of the optical Drude weight from its non-interacting-electron expectation

[3–5]. Heavy fermion metals and metals that lie in the vicinity of a Mott insulator exemplify

bad metals.

Quantum many-electron systems are traditionally studied in terms of a perturbative ex-

pansion of the interaction [6]. The assumption is that the noninteracting electrons are

the building blocks for low-energy physics. A standard model is the one-band Hubbard

model, which contains an electron band of width Dand an onsite interaction – the Hub-

bard interaction – of strength U. The Fermi liquid description applies in an order-by-order

perturbative treatment of U/D to inﬁnite orders. Here, coherent quasiparticles appear as

a pole in the single-particle Green’s function G(k, ω). When U/D becomes of order unity,

non-perturbative eﬀects may develop. The most famous case is the development of a Mott

insulator for a half-ﬁlled band [7], in which the electron correlations drive the quasiparticle

weight to zero. Concurrently, a gap develops in the single-particle and charge excitation

spectrum; the ground state is no longer a metal and is no longer adiabatically connected to

its noninteracting counterpart.

In general, when the electron correlations reach and exceed the electron bandwidth, as is

the case in bad metals, other degrees of freedom emerge as a part of the building blocks for

the low-energy physics. A typical case involves spins, as seen from the notion of local moment

formation of a correlated and deep orbital in a metallic matrix [8]. As a result, models of

strongly correlated electrons may involve both local moments and itinerant electrons, as

exempliﬁed by heavy fermion metals [1, 9–12]. A similar set of building blocks develop once

a Mott insulator [7] is coupled to a metallic band.

When expressed in terms of the electron degrees of freedom, the correlation physics de-

scribes the eﬀect of Coulomb repulsion between the electrons [13, 14]. However, if represented

in terms of the emergent building blocks such as local moments, the competition between

diﬀerent types of eﬀective interactions becomes particularly explicit. A case in point is a

Kondo lattice Hamiltonian, which contains an antiferromagnetic spin-exchange (Kondo) in-

teraction between the local moments and spins of the conduction electrons on the one hand,

and the RKKY interaction between the local moments on the other.

The EDMFT method was introduced to dynamically study this type of competition [15–

17]. Consider the case of the Kondo lattice model. Here, the local Kondo interaction favors

the formation of a Kondo singlet between the local moments and spins of the conduction

electrons. Whereas the RKKY interaction, which often is also antiferromagnetic, promotes

singlet formation among the local moments. To properly account for the ampliﬁed quantum

ﬂuctuations that develop when the two types of eﬀects would produce comparable energy

scales, the dynamical interplay between the two interactions is crucial; this interplay is

captured by the EDMFT method. This is to be contrasted with the standard dynamical

mean ﬁeld theory [13, 14], in which the local interactions such as the Kondo couplings are

treated in a dynamical way, while the intersite interactions such as the RKKY couplings are

handled in terms of a static Hartree-Fock approximation.

The EDMFT method has played a central role in the elucidation of quantum phase

transitions of heavy fermion metals [18]. Studies based on such a method have led to the

advancement of the concept of Kondo destruction [19–23]. Qualitatively, the Kondo physics

underlies the formation of (heavy) quasiparticles in such systems. The quasiparticles develop

through the formation of the Kondo singlet, and appear in the form of a Kondo-driven

composite fermion. These processes are captured by the local self-energy of the conduction

electrons. As such, studying the dynamical competition between the RKKY and Kondo

interactions provides a way of describing the reduction – and eventual destruction – of the

Kondo singlet in the ground state and, by extension, the quasiparticles. In other words, the

EDMFT approach is ideally suited to access how (heavy) quasiparticles are lost.

The concept of Kondo destruction has inﬂuenced the development of quantum critical

metals in a profound way, particularly on the destruction of quasiparticles and dynamical

Planckian (~ω/kBT) scaling at the quantum critical point (QCP), and a sudden jump of a

“large” to “small” Fermi surface across the QCP [18].

The present article is devoted to three topics:

•We describe the dynamical equations associated with the EDMFT method.

•We illustrate the application of the EDMFT approach by focusing on the Kondo lattice

models and summarizing i) the methodological aspect of the approach and ii) the

results as pertaining to the heavy fermion quantum criticality. Further analyses that

have been motivated by the result of such calculations are touched upon, especially in

the form of a global phase diagram.

•The prospect for further progress along this general direction is discussed.

II. EDMFT APPROACH

We illustrate the EDMFT approach in terms of the single-band Hubbard-Heisenberg-type

model [15–17, 24]:

HU-v =X

Uni↑ni↓+X

hiji,σ

tijc†

iσcjσ +1

hiji

vij :ni:: nj: (1)

The ﬁrst two terms specify the Hubbard model for a spin-1

2band. The onsite Hubbard

interaction is Uand the hopping matrix is tij , whose Fourier transform corresponds to the

band dispersion εk. The third term describes an intersite density-density (vij ) interaction

(a spin-exchange interaction, Jij, can also be added, as originally done; see below), with ni

being the density operator of the c−electrons and : n:≡n− hnirepresenting its normal-

ordered form. For concreteness, we limit the intersite interactions to the nearest-neighbor

(hiji) contributions, but this can be readily generalized.

The EDMFT approach amounts to the summation of an inﬁnite series of diagrams as

outlined in Fig. 1. The approach is systematic and conserving. Moreover, the EDMFT

equations are generated by an eﬀective action function of the Baym-Kadanoﬀ type.

The EDMFT approach incorporates a local self-energy, Σ, for the single-electron Green’s

function Gand a related irreducible quantity in the density channel, M, which is deﬁned

in terms of a cumulant that is vij-irreducible [15]; for notational convenience, it has been

referred to as a (density) self-energy. The self-energies determine the dynamical density

susceptibility and single-particle Green’s function as follows:

χ(q, ω) = 1

M(ω) + vq

,(2)

j l

+...

FIG. 1. The single-site, two-site, and three-site diagrams for the Luttinger Ward potential in

the extended DMFT [16, 20]. A shaded circle contains all the on-site diagrams fully-dressed by

the fermion Green’s function (solid lines) as shown in the ﬁrst diagram. The dashed lines further

represent the intersite interactions; we have denoted them by Jij, but they could also be interactions

in other channels, such as vij of Eq. (1).

together with

G(k, ) = 1

+µ−k−Σ().(3)

These self-energies can be calculated from a local action, which can equivalently be ex-

pressed in terms of a local Hamiltonian:

Hloc,U-v =Un↑n↓+X

k,σ

Epc†

kσckσ+gX

:n:φp+φ†

−p(4)

Here the dispersion of the bosonic bath, along with its fermionic bath counterpart, are

self-consistently determined. The set of nonlinear equations can be expressed as follows.

The bath dispersion deﬁnes Weiss ﬁelds, χ0and G0, as follows:

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ExtendedDynamicalMeanFieldTheoryforCorrelatedElectronModelsHaoyuHu1;2,LeiChen1,QimiaoSi11DepartmentofPhysicsandAstronomy,RiceCenterforQuantumMaterials,RiceUniversity,Houston,TX77005,USA2DonostiaInternationalPhysicsCenter,P.ManueldeLardizabal4,20018Donostia-SanSebastian,SpainAnoverarchingquestioninst...

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