Serge Florens and Antoine Georges- Quantum impurity solvers using a slave rotor representation

8/3/2019 Serge Florens and Antoine Georges- Quantum impurity solvers using a slave rotor representation

1/18

arXiv:cond-mat/0206571v2

[cond-ma

t.str-el]7Aug200

2

Quantum impurity solvers using a slave rotor representation

Serge Florens1, 2 and Antoine Georges1, 2

1 Laboratoire de Physique Theorique, Ecole Normale Superieure,

24 rue Lhomond, 75231 Paris Cedex 05, France2 Laboratoire de Physique des Solides, Universite Paris-Sud, Bat. 510, 91405 Orsay, France

We introduce a representation of electron operators as a product of a spin-carrying fermion andof a phase variable dual to the total charge (slave quantum rotor). Based on this representation,a new method is proposed for solving multi-orbital Anderson quantum impurity models at finiteinteraction strength U. It consists in a set of coupled integral equations for the auxiliary field Greensfunctions, which can be derived from a controlled saddle-point in the limit of a large number of fieldcomponents. In contrast to some finite-U extensions of the non-crossing approximation, the newmethod provides a smooth interpolation between the atomic limit and the weak-coupling limit, anddoes not display violation of causality at low-frequency. We demonstrate that this impurity solvercan be applied in the context of Dynamical Mean-Field Theory, at or close to half-filling. Goodagreement with established results on the Mott transition is found, and large values of the orbitaldegeneracy can be investigated at low computational cost.

Contents

I. Introduction 1

II. Rotorization 3A. Atomic model 3B. Rotor representation of Anderson impurity

models 4C. Slave rotors, Hubbard-Stratonovich and gauge

transformations 4

III. Sigma-model representation and solution in

the limit of many components 5

A. From quantum rotors to a sigma model 5B. Integral equations 5C. Some remarks 6

IV. Application to the single-impurity

Anderson model 7A. Spectral functions 7B. Low-energy behaviour and Friedel sum-rule 9

V. Applications to Dynamical Mean-Field

Theory and the Mott transition 10A. One-orbital case: Mott transition, phase

diagram 10B. Comparison to other impurity solvers:

spectral functions 11C. Double occupancy 12D. Multi-orbital effects 13E. Effects of doping 13

VI. Conclusion 14

Acknowledgments 15

A. The Atomic limit 15

B. Numerical solution of the real timeequations 15

C. Friedels sum rule 16

References 17

I. INTRODUCTION

The Anderson quantum impurity model (AIM) and itsgeneralizations play a key role in several recent devel-opments in the field of strongly correlated electron sys-

tems. In single-electron devices, it has been widely usedas a simplified model for the competition between theCoulomb blockade and the effect of tunnelling [1]. Ina different context, the Dynamical Mean-Field Theory(DMFT) of strongly correlated electron systems replacesa spatially extended system by an Anderson impuritymodel with a self-consistently determined bath of con-duction electrons [2, 3]. Naturally, the AIM is also essen-tial to our understanding of local moment formation inmetals, and to that of heavy-fermion materials, particu-larly in the mixed valence regime [4].

It is therefore important to have at our disposal quanti-tative tools allowing for the calculation of physical quan-tities associated with the AIM. The quantity of interestdepends on the specific context. Many recent applica-tions require a calculation of the localized level Greensfunction (or spectral function), and possibly of some two-particle correlation functions.

Many such impurity solvers have been developedover the years. Broadly speaking, these methods fall intwo categories: numerical algorithms which attempt at adirect solution on the computer, and analytical approx-imation schemes (which may also require some numeri-cal implementation). Among the former, the Quantum
http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2http://arxiv.org/abs/cond-mat/0206571v2


2/18

2

Monte Carlo approach (based on the Hirsch-Fye algo-rithm) and the Numerical Renormalization Group play aprominent role. However, such methods become increas-ingly costly as the complexity of the model increases. Inparticular, the rapidly developing field of applications ofDMFT to electronic structure calculations [5] requiresmethods that can handle impurity models involving or-bital degeneracy. For example, materials involving f-

electrons are a real challenge to DMFT calculations whenusing Quantum Monte Carlo. Similarly, cluster exten-sions of DMFT require to handle a large number of cor-related local degrees of freedom. The dimension of theHilbert space grows exponentially with the number ofthese local degrees of freedom, and this is a severe limi-tation to all methods, particularly exact diagonalizationand numerical renormalization group.

The development of fast and accurate (although ap-proximate) impurity solvers is therefore essential. In theone-orbital case, the iterated perturbation theory (IPT)approximation [6] has played a key role in the develop-ment of DMFT, particularly in elucidating the nature ofthe Mott transition [7]. A key reason for the success ofthis method is that it becomes exact both in the limit ofweak interactions, and in the atomic limit. Unfortunatelyhowever, extensions of this approach to the multi-orbitalcase have not been as successful [8]. Another widelyemployed method is the non-crossing approximation, orNCA [9, 10]. This method takes the atomic limit as astarting point, and performs a self-consistent resumma-tion of the perturbation theory in the hybridization to theconduction bath. It is thus intrinsically a strong-couplingapproach, and indeed it is in the strong-coupling regimethat the NCA has been most successfully applied. TheNCA does suffer from some limitations however, whichcan become severe for some specific applications. These

limitations are of two kinds:

The low-energy behaviour of NCA integral equa-tions is well-known to display non-Fermi liquidpower laws. This can be better understood whenformulating the NCA approach in terms of slavebosons [11, 12]. It becomes clear then that theNCA actually describes accurately the overscreenedregime of multichannel models. This is in a sense aremarkable success of NCA, but also calls for somecare when applying the NCA to Fermi-liquid sys-tems in the screened regime. It must be noted thatrecent progresses have been made to improve the

low-energy behaviour in the Fermi liquid case [13]. A more important limitation for practical applica-

tions has to do with the finite-U extensions of NCAequations. Standard extensions do not reproducecorrectly the non-interacting U = 0 limit. In con-trast to IPT, they are not interpolative solversbetween the weak-coupling and the strong couplinglimit. Furthermore, the physical self-energy tendsto develop non-causal behaviour at low-frequency(i.e. d() > 0) below some temperature (the

NCA pathology) At half-filling and large U, thisonly happens at a rather low-energy scale, but awayfrom half-filling or for smaller U, this scale canbecome comparable to the bandwith, making thefinite-U NCA of limited applicability.

In this article, we introduce new impurity solvers whichovercome some of these difficulties. Our method is basedon a rather general representation of strongly correlatedelectron systems, which has potential applications to lat-tice models as well [27]. The general idea is to introduce anew slave-particle representation of physical electron op-erators, which emphasizes the phase variable dual to thetotal charge on the impurity. This should be contrasted toslave-boson approaches to a multi-orbital AIM: there, oneintroduces as many auxiliary bosons as there are fermionstates in the local Hilbert space. This is far from eco-nomical: when the local interaction depends on the totalcharge only, it should be possible to identify a collectivevariable which provides a minimal set of collective slavefields. We propose that the phase dual to the total chargeprecisely plays this role. This turns a correlated electron

model (at finite U) into a model of spin- (and orbital-)carrying fermions coupled to a quantum rotor degree offreedom. Various types of approximations can then bemade on this model. In this article, we emphasize an ap-proximate treatment based on a sigma-model representa-tion of the rotor degree of freedom, which is then solvedin the limit of a large number of components. This resultsin coupled integral equations which share some similar-ities to those of the NCA, but do provide the followingimprovements:

The non-interacting (U = 0) limit is reproduced ex-actly. For a fully symmetric multi-orbital model athalf-filling and in the low temperature range, theatomic limit is also captured exactly, so that theproposed impurity solver is an interpolative schemebetween weak and strong coupling. The sigma-model approximation does not treat as nicely theatomic limit far from half-filling however (thoughimprovements are possible, see section V E).

The physical self-energy does not display any viola-tion of causality (even at low temperature or smallU). This is guaranteed by the fact that our inte-gral equations become exact in the formal limit ofa large number of orbitals and components of thesigma-model field.

It should be emphasized, though, that the low-energybehaviour of our equations is similar to the (infinite-U)NCA, and characterized by non-Fermi liquid power lawsbelow some low-energy scale.

As a testing ground for the new solver, we apply it inthis paper to the DMFT treatment of the Mott tran-sition in the multi-orbital Hubbard model. We findan overall very good agreement with the general as-pects of this problem, as known from numerical workand from some recently derived exact results. We also


3/18

3

compare to other impurity solvers (IPT, Exact Diago-nalization, QMC, NCA), particularly regarding the one-electron spectral function.

This article is organized as follows. In section II, weintroduce a representation of fermion operators in termsof the phase variable dual to the total charge (taking thefinite U, multi-orbital Anderson impurity model as anexample). In section III, a sigma-model representation

of this phase variable is introduced, along with a gener-alization from O(2) to O(2M). In the limit of a largenumber of components, a set of coupled integral equa-tions is derived. In section IV, we test this dynamicalslave rotor (DSR) approach on the single-impurity An-derson model. The rest of the paper (Sec. V) is devotedto applications of this approach in the context of DMFT,which puts in perspective the advantages and limitationsof the new method.

II. ROTORIZATION

The present article emphasizes the role played by thetotal electron charge, and its conjugate phase variable.We introduce a representation of the physical electronin terms of two auxiliary fields: a fermion field whichcarries spin (and orbital) degrees of freedom, and the(total) charge raising and lowering operators which werepresent in terms of a phase degree of freedom. Thelatter plays a role similar to a slave boson: here a slaveO(2) quantum rotor is used rather than a conventionalbosonic field.

A. Atomic model

We first explain this construction on the simple exam-ple of an atomic problem, consisting in an N-fold degen-erate atomic level subject to a local SU(N)-symmetricCoulomb repulsion:

Hlocal =

0 dd +

U

2

dd N

2

2(1)

We use here 0 d + U/2 as a convenient redefinitionof the impurity level, and we recast the spin and orbitaldegrees of freedom into a single index = 1 . . . N (forN = 2, we have a single orbital with two possible spinstates =, ). We note that 0 is zero at half filling,due to particle-hole symmetry.

The crucial point is that the spectrum of the atomichamiltonian (1) depends only on the total fermioniccharge Q = 0, , N and has a simple quadratic depen-dence on Q:

EQ = 0Q +U

2

Q N

2

2(2)

There are 2N states, but only N+ 1 different energy lev-els, with degeneracies

NQ

. In conventional slave boson

methods [14, 15], a bosonic field is introduced for eachatomic state |1 Q (along with spin-carrying auxil-iary fermions f). Hence, these methods are not describ-ing the atomic spectrum in a very economical manner.

The spectrum of (1) can actually be reproduced byintroducing, besides the set of auxiliary fermions f, asingle additional variable, namely the angular momen-tum L = i/ associated with a quantum O(2) rotor (i.e. an angular variable in [0, 2]. Indeed, the energylevels (2) can be obtained using the following hamiltonian

Hlocal =

0ff +

U

2L2 (3)

A constraint must be imposed, which insures that thetotal number of fermions is equal to the O(2) angularmomentum (up to a shift):

L =

ff

1

2

(4)

This restricts the allowed values of the angular momen-tum to be l = Q N/2 = N/2, N/2 + 1, , N/2 1, N/2, while in the absence of any constraint l can be anarbitrary (positive or negative) integer. The spectrum of(3) is 0Q + U l2/2, with l = Q N/2 thanks to (4), sothat it coincides with (2).

To be complete, we must show that each state in theHilbert space can be constructed in terms of these aux-iliary degrees of freedom, in a way compatible with thePauli principle. This is achieved by the following identi-fication:

|1 . . . Q

d =

|1 . . . Q

f

| = Q

N/2

(5)

in which |1 . . . Qd,f denotes the antisymmetric fermionstate built out ofd and f fermions, respectively, and| denotes the quantum rotor eigenstate with angularmomentum l, i.e. | = ei. The creation of a physi-cal electron with spin corresponds to acting on such astate with f as well as raising the total charge (angularmomentum) by one unit. Since the raising operator isei, this leads to the representation:

d f ei , d f ei (6)

Let us illustrate this for N = 2 by writing the four pos-

sible states in the form: |d = |f |0, |d = |f |0,|d = |f |+1 and |0d = |0f |1, and showingthat this structure is preserved by d = f

e

i. Indeed:

|d = d |d = f |f e+i |0 = |f |+1 (7)

The key advantage of the quantum rotor representationis that the original quartic interaction between fermionshas been replaced in (3) by a simple kinetic term (L2)for the phase field.


4/18

4

B. Rotor representation of Anderson impurity

models

We now turn to an SU(N)-symmetric Anderson im-purity model in which the atomic orbital is coupled to aconduction electron bath :

H =

0d

d +

U

2

d

d N

22

(8)

+k,

kck,ck, +

k,

Vk

ck,d + d

ck,

Using the representation (6), we can rewrite this hamil-tonian in terms of the (f, ) fields only:

H =

0ff +

U

2L2 +

k,

kck,ck, (9)

+k,

Vk

ck,fe

i + fck,ei

We then set up a functional integral formalism for the fand degrees of freedom, and derive the action associatedwith (10). This is simply done by switching from phase

and angular-momentum operators (, L) to fields (, )depending on imaginary time [0, ]. The action isconstructed from S 0 d[iL + H + ff], andan integration over L is performed. It is also necessaryto introduce a complex Lagrange multiplier h in orderto implement the constraint L =

f

f N/2. We

note that, because of the charge conservation on the localimpurity, h can be chosen to be independent of time, withih [0, 2/].

This leads to the following expression of the action:

S =

0

d

f( + 0 h)f +( + ih)

2

2U+

N

2h

+k,

ck,( + k)ck, + Vkc

k,fe

i + h.c.

(10)

We can recast this formula in a more compact form byintroducing the hybridization function:

(i)

k

|Vk|2i k (11)

and integrating out the conduction electron bath. Thisleads to the final form of the action of the SU(N) An-derson impurity model in terms of the auxiliary fermionsand phase field:

S =

0

d

f( + 0 h)f +( + ih)

2

2U+

N

2h

+

0

d

0

d()

f()f() ei()i(

) (12)

C. Slave rotors, Hubbard-Stratonovich and gauge

transformations

In this section, we present an alternative derivation ofthe expression (12) of the action which does not rely onthe concept of slave particles. This has the merit to give amore explicit interpretation of the phase variable intro-duced above, by relating it to a Hubbard-Stratonovich

decoupling field. This section is however not essentialto the rest of the paper, and can be skipped upon firstreading.

Let us start with the imaginary time action of the An-derson impurity model in terms of the physical electronfield for the impurity orbital:

S =

0

d

d( + 0)d +U

2

dd N

2

2

+

0

d

0

d()

d()d() (13)

Because we have chosen a SU(N)-symmetric form for theCoulomb interaction, we can decouple it with only onebosonic Hubbard-Stratonovitch field ():

S =

0

d

d( + 0 + i())d +2()

2U i N

2()

+

0

d

0

d()

d()d() (14)

Hence, a linear coupling of the field () to the fermionshas been introduced. The idea is now to eliminatethis linear coupling for all the Fourier modes of the -

field, except that corresponding to zero-frequency: 0 0

[2]. This can be achieved by performing the fol-lowing gauge transformation:

d() = f() e

i

0 ei0 / (15)

The reason for the second phase factor in this expressionis that it guarantees that the new fermion field f alsoobeys antiperiodic boundary conditions in the path in-tegral. It is easy to check that this change of variablesdoes not provide any Jacobian, so that the action simplyreads:

S =0

d

f( + 0 + i 0 )f +

2

()2U i N2 0

+

0

d

0

d()

f()f()ei

[

0

] (16)

We now set:

() =

+

1

0

with: 0

0

[2]

(17)


5/18

5

and notice that the field () has the boundary condition() = (0) [2]. It therefore corresponds to an O(2)quantum rotor, and the expression of the action finallyreads:

S =

0

d

f( + 0 + i0

)f +( +

0 )

2

2U i N

2

0

+

0

d

0

d()

f()f ()ei()i(

) (18)

This is exactly expression (12), with the identification:0/ ih. This, together with (17), provides an explicitrelation between the quantum rotor and Lagrange mul-tiplier fields on one side, and the Hubbard-Stratonovichfield conjugate to the total charge, on the other.

III. SIGMA-MODEL REPRESENTATION AND

SOLUTION IN THE LIMIT OF MANYCOMPONENTS

A. From quantum rotors to a sigma model

Instead of using a phase field to represent the O(2)degree of freedom, one can use a constrained (complex)bosonic field X ei with:

|X()|2 = 1 (19)

The action (12) can be rewritten in terms of this field,provided a Lagrange multiplier field () is used to im-plement this constraint:

S =

0

d

f( + 0 h)f +N

2h h

2

2U

+

0

d|X|2

2U+

h

2U(XX h.c.) + ()(|X|21)

+

0

d

0

d()

f()f ()X()X() (20)

Hence, the Anderson model has been written as a theoryof auxiliary fermions coupled to a non-linear O(2) sigma-model, with a constraint (implemented by h) relating thefermions and the sigma-model field X().

A widely-used limit in which sigma models becomesolvable, is the limit of a large number of componentsof the field. This motivates us to generalize (20) to amodel with an O(2M) symmetry. The bosonic field Xis thus extended to an M-component complex field X( = 1 . . . M ) with

|X|2 = M. The corresponding

action reads:

S =

0

d

f( + 0 h)f +N

2h M h

2

2U M

+

0

d

|X|22U

+h

2U(XXh.c.) + |X|2

+0d

0d

1

M(

),

f

()f (

)X()X

(

)

Let us note that this action corresponds to the hamilto-nian:

H =

0ff +

U

2M

,

L2, +k,

kck,ck,

+

k,,

VkM

ck,fX

+ f

ck,X

(21)

In this expression, L, denotes the angular momen-tum tensor associated with the X vector. The hamilto-

nian (21) is a generalization of the SU(N)O(2) = U(N)Anderson impurity model to an SU(N) O(2M) modelin which the total electronic charge is associated with aspecific component of L. It reduces to the usual Ander-son model for M = 1.

In the following, we consider the limit where both Nand M become large, while keeping a fixed ratio N/M.We shall demonstrate that exact coupled integral equa-tions can be derived in this limit, which determine theGreens functions of the fermionic and sigma-model fields(and the physical electron Greens function as well). Thefact that these coupled integral equations do correspondto the exact solution of a well-defined hamiltonian model

(Eq. (21)) guarantees that no unphysical features (like e.gviolation of causality) arise in the solution. Naturally, thegeneralized hamiltonian (21) is a formal extension of theAnderson impurity model of physical interest. Extendingthe charge symmetry from O(2) to O(2M) is not entirelyinocuous, even at the atomic level: as we shall see below,the energy levels of a single O(2M) quantum rotor havemultiple degeneracies, and depend on the charge (angularmomentum) quantum number in a way which does notfaithfully mimic the O(2) case. Nevertheless, the basicfeatures defining the generalized model (a localized or-bital subject to a Coulomb charging energy and coupledto an electron bath by hybridization) are similar to theoriginal model of physical interest.

B. Integral equations

In this section, we derive coupled integral equationswhich become exact in the limit where both M and Nare large with a fixed ratio:

N NM

(N, M ) (22)


6/18

6

Following [12] (see also [11]), the two body interaction be-tween the auxiliary fermions and the sigma-model fieldsis decoupled using (bosonic) bi-local fields Q(, ) andQ(, ) depending on two times. Hence, we consider theaction:

S =

0

d

f( + 0 h)f +

N

2h M h

2

2U M

+

0

d

|X|22U

+h

2U(XXh.c.) + |X|2

+

0

d

0

d MQ(, )Q(, )

( )

0

d

0

d Q(, )

X()X(

)

+

0

d

0

d Q(, )

f()f() (23)

In the limit (22), this action is controlled by a saddle-point, at which the Lagrange multipliers take static ex-pectation values h and , while the saddle point valuesof the Q and Q fields are translation invariant functionsof time Q( ) and Q( ).

Introducing the imaginary-time Greens functions ofthe auxiliary fermion and sigma-model fields as:

Gf()

Tf()f(0)

(24)

GX () + TX()X(0) (25)we differentiate the effective action (23) with respectto Q() and Q(), which leads to the following saddle-point equations: Q() = N()Gf() and Q() =()GX (). The functions Q(in) (= f) and Q

(in)

(= X ) define fermionic and bosonic self-energies:

G1f (in) = in 0 + h f(in) (26)

G1X (in) =2nU

+ 2ihnU

X (in) (27)

where n (resp. n) is a fermionic (resp. bosonic) Mat-subara frequency. The saddle point equations read:

X () = N()Gf() (28)f() = ()GX () (29)

together with the constraints associated with h and :

GX (= 0) = 1 (30)

Gf(= 0) =

1

2 2hNU (31)

+1

NU

GX (= 0) + GX (= 0

+)

There is a clear similarity between the structure of thesecoupled integral equations and the infinite-U NCA equa-tions [9]. We note also significant differences, such as theconstraint equations. Furthermore, the finite value of the

Coulomb repulsion U enters the bosonic propagator (27)in a quite novel manner.

The two key ingredients on which the present methodare based is the use of a slave rotor representation offermion operators, and the use of integral equations forthe frequency-dependent self-energies and Greens func-tions. For this reason, we shall denote the integral equa-tions above under the name of Dynamical Slave Rotor

method (DSR) in the following.

C. Some remarks

We make here some technical remarks concerning theseintegral equations.

First, we clarify how the interaction parameter U wasscaled in order to obtain the DSR equations above. Thisissue is related to the manner in which the atomic limit( = 0) is treated in this method. In the originalO(2) atomic hamiltonian (1), the charge gap betweenthe ground-state and the first excited state is U/2 at

half-filling (0 = 0). In the DSR method, the charge gapis associated with the gap in the slave rotor spectrum. Ifthe O(2M) generalization of (1) is written as in (21):

Hint =U

2M

,

L2, (32)

the spectrum reads: E = U ( + 2M 2)/(2M). As aresult, the energy difference from the ground state to thefirst excited state is E1 E0 = U(2M 1)/(2M) Uat large M, whereas it is U/2 at M = 1. In order to usethe DSR method in practice as an approximate impuritysolver, the parameter U should thus be normalized in a

different way than in (21), so that the gap is kept equalto U/2 in the large-M limit as well. Technically this canbe enforced by choosing the following normalization:

Hint =U

4M 2,

L2, (33)

instead of (32). Note that this scaling coincides with (32)for M = 1, but does yield E1 E0 = U/2 for large-M,as desired. This definition of U was actually used whenwriting the saddle-point integral equations (27), althoughwe postponed the discussion of this point to the presentsection for reasons of simplicity.

Let us elaborate further on the accuracy of the DSRintegral equations in the atomic limit. In Appendix A,we show that the physical electron spectral function ob-tained within DSR in the atomic limit coincides with theexact O(2) result at half-filling and at T = 0. This isa non-trivial result, given the fact that the constraint istreated on average and the above remark on the spectrumspectrum. In contrast to NCA, the DSR method (in itspresent form) is not based by construction on a strong-coupling expansion around the exact atomic spectrum,so that this is a crucial check for the applicability of this


7/18

7

method in practice. In the context of DMFT for exam-ple, it is essential in order to describe correctly the Mottinsulating state [7]. However, the DSR integral equationsfail to reproduce exactly the O(2) atomic limit off half-filling, as explained in Appendix A. Deviations becomesevere for too high dopings, as discussed in Sec. V E.This makes the present form of DSR applicable only forsystems in the vicinity of half-filling.

We now discuss some general spectral properties of theDSR solver. From the representation of the physical elec-tron field d = f

X, and from the convention chosen for

the pseudo-particules Greens functions (24-25), the one-electron physical Greens function is simply expressed as:

Gd() = Gf()GX () (34)Therefore eq. (30), combined with the fact that f hasa (1) discontinuity at = 0 (which is obvious from(26)), shows that d possesses also a (1) jump at zeroimaginary time. This ensures that the physical spec-tral weight is unity in our theory, and thus that phys-ical spectral function are correctly normalized. Because

the DSR integral equations result from a controlled largeN, M limit, it also insures that the physical self-energyalways have the correct sign (i.e. Im d( + i0+) < 0).This is not the case [16] for the finite U version of theNCA [10] which is constructed as a resummation of thestrong-coupling expansion in the hybridization function() (Strong coupling resummations generically sufferfrom non-causality, see also [17] for an illustration.)

Finally, we comment on the non-interacting limit U 0. This is a major failure of the usual NCA, which lim-its its applicability in the weakly correlated regime. Inthe DSR formalism, this limit is exact as can be noticedfrom equation (27). Indeed, as U vanishes, only the zero-

frequency component ofGX (in) survives, so that GX ()simply becomes a constant. Because of the constraint(30), we get correctly GX () = 1 at U = 0. From (29)and (34), this proves that Gd() is the non-interactingGreens function:

GU=0d (in) =1

in 0 (in) (35)

We finally acknowledge that an alternative dynamicalapproximation to the finite U Anderson model [13] wasrecently developed as an extension of NCA by Kroha,Wolfle and collaborators (a conventional slave boson rep-resentation was used in this work). Many progresses havebeen made following this method, but, to the authorsknowledge, this technique has not yet been implementedin the context of DMFT (one of the reasons is its com-putational cost). By developing the DSR approximation,we pursue a rather complementary goal: the aim here isnot to improve the low-energy singularities usually en-countered with integral equations, but rather to have afast and efficient solver which reproduces correctly themain features of the spectral functions and interpolatesbetween weak and strong coupling. In that sense, it isvery well adapted to the DMFT context.

IV. APPLICATION TO THE

SINGLE-IMPURITY ANDERSON MODEL

We now discuss the application of the DSR in the sim-plest setting: that of a single impurity hybridized with afixed bath of conduction electrons. For simplicity, we fo-cus on the half-filled, particle-hole symmetric case, whichimplies 0 = h = 0. The doped (or mixed valence) case

will be addressed in the next section, in the context ofDMFT.

As the strength of the Coulomb interaction U is in-creased from weak to strong coupling, two well-knowneffects are expected (see e.g [4]). First the width of thelow-energy resonance is reduced from its non-interactingvalue 0 |(0)|. As one enters the Kondo regime0 U (with the conduction electrons band-width), this width becomes a very small energy scale, ofthe order of the Kondo temperature:

TK =

2U0 exp(U/(80)) (36)A (local) Fermi liquid description applies, with quasi-

particles having a large effective mass and small weight:Z = m/m TK/0. The impurity spin is screened forT < TK.

Second, the corresponding spectral weight is trans-ferred to high energies, into Hubbard bands associatedto the atomic-like transitions (adding or removing anelectron into the half-filled impurity orbital), broadenedby the hybridization to the conduction electron bath.The suppression of the low-energy spectral weight cor-responds to the suppression of the charge fluctuations onthe local orbital. These satellites are already visible atmoderate values of the coupling U/0. As temperatureis increased from T < TK to T > TK, the Kondo quasi-

particle resonance is quickly destroyed, and the missingspectral weight is added to the Hubbard bands.The aim of this section is to investigate whether the in-

tegral equations introduced in this paper reproduce thesephysical effects in a satisfactory manner.

A. Spectral functions

We have solved numerically these integral equations byiteration, both on the imaginary axis and for real frequen-cies. Working on the imaginary axis is technically mucheasier. A discretization of the interval [0, ] is used(with typically 8192 points, and up to 32768 for reachingthe lowest temperatures), as well as Fast Fourier Trans-forms for the Greens functions. Searching by dichotomyfor the saddle-point value of the Lagrange multiplier (Eq. 30) is conveniently implemented at each step of theiterative procedure. Technical details about the analyticcontinuation of (28-31) to real frequencies, as well as theirnumerical solution are given in Appendix B.

On Fig. 1, we display our results for the impurity-orbital spectral function d(), at three different temper-atures (the density of states of the conduction electron


8/18

8

-3 -2 -1 0 1 2 30

0.5

1

1.5

2

FIG. 1: d-level spectral function d() for fixed U = 2 and in-verse temperatures = 4, 20, 2000, with the conduction elec-tron bath described in the text

bath is chosen as a semi-circle with half-width = 6, the

resonant level width is 0 = 0.16 and we take U = 2).The growth of the Kondo resonance as the temperature islowered is clearly seen. The temperatures in Fig. 1 havebeen chosen such as to illustrate three different regimes:for T TK , no resonance is seen and the spectral densitydisplays a pseudogap separating the two high-energybands; for T TK transfer of spectral weight to lowenergy is seen, resulting in a fully developed Kondo res-onance for T TK. We have not obtained an analyti-cal determination of the Kondo temperature within thepresent scheme. In the case of NCA equations, it is pos-sible to derive a set of differential equations in the limitof infinite bandwith which greatly facilitate this. Thisprocedure cannot be applied here, because of the form(27) of the boson propagator. Nevertheless, we checkedthat the numerical estimates of the width of the Kondopeak is indeed exponentially small in U as in formula(36), see Fig. 2. However, because U is normalized as in(33), (which gives the correct atomic limit), the prefactorinside the exponential appears to be twice too small.

On Fig. 3, we display the spectral function for a fixedlow temperature and increasing values of U. The strongreduction of the Kondo scale (resonance width) upon in-creasing U is clear on this figure. We note that the high-energy peaks have a width which remains of order 0,independently of U, which is satisfactory. However, wealso note that they are not peaked exactly at the atomic

value U/2, which might be an artefact of these integralequations. The shift is rather small however.

Fig. 4 illustrates how the high- and low- energy fea-tures in the d-level spectral functions are associated withcorresponding features in the auxiliary particle spectralfunctions f and X . In particular, the sigma-modelboson (slave rotor) is entirely responsible for the Hub-bard bands at high energy (as expected, since it describescharge fluctuations).

As stressed in the introduction, an advantage of our

1 1.5 2 2.5 3

U

0

0.01

0.02

0.03

0.04

0.05

0.06

Tk1 1.5 2 2.5 3

0.0001

0.001

0.01

FIG. 2: Kondo temperature TK from the exact formula (line)and from the numerical solution of the DSR equations (dots).Units of energy are such that 0 = 0.16.

-2 0 20

0.5

1

1.5

2

FIG. 3: d() at low temperature (= 600) and for increasingU = 1, 2, 3

-2 -1 0 1 2-2

-1

0

1

2

FIG. 4: Pseudo-particule spectral function at U = 2 and = 600. The Kondo resonance is visible in f(), brokencurve, whereas X() displays higher energy features, plaincurve


9/18

9

scheme is that the d-level self-energy is always causal,even for small U or large doping. This is definitely animprovement as compared to the usual U-NCA approx-imation. This is illustrated by Fig. 5, from which it isalso clear that d decreases (and eventually vanishes) asU goes to zero (for a more detailed discussion and com-parison to NCA, see Sec. III). However, it is also clearfrom this figure that the low-energy behavior of the self-

energy is not consistent with Fermi liquid theory. Thisis a generic drawback of NCA-like integral equation ap-proaches, that we now discuss in more details.

-2 -1 0 1 2-3

-2.5

-2

-1.5

-1

-0.5

0

FIG. 5: Imaginary part of the physical self-energy on the real-frequency axis, for U = 0.5 (upper curve) and U = 2 (lowercurve) at = 600

B. Low-energy behaviour and Friedel sum-rule

We discuss here the low-energy behavior of the integralequations (in the case of a featureless conduction electronbath), for both the d-level and auxiliary field Greensfunctions. As explained below, this low-energy behaviordepends sensitively on the ratioN= N/M which is keptfixed in the limit considered in this paper. The calcula-tions are detailed in Appendix C, where we establish thefollowing.

Friedel sum rule. The zero-frequency value of thed-level spectral function at T = 0 is independent ofU and reads:

d( = T = 0) = 1(0)

/2N+ 1 tan

2

NN+ 1

(37)

This is to be contrasted with the exact value forthe O(2) model (M = 1), which is independent ofN and reads:

exactd ( = T = 0) =1

(0)(M = 1, any N) (38)

This is the Friedel sum rule [4, 18], which in thisparticle-hole symmetric case simply follows fromthe Fermi-liquid requirement that the (inverse) life-time d ( = 0) should vanish at T = 0 (sinceG1d = i (i) d(i)). As a result, d(0)is pinned at its non-interacting value. The inte-gral equations discussed here yield a non-vanishingd ( = 0) (albeit always negative in order to sat-

isfy causality), and hence do not describe a Fermiliquid at low energy. The result (37) is identical tothat found in the NCA for U = , but holds herefor arbitrary U. There is actually no contradictionbetween this remark and the fact that our integralequations yield the exact spectral function in theU 0 limit. Indeed, the limit U 0 at finiteT, does not commute with , T 0 at finite U(in which (37) holds).

Low-frequency behavior. The auxiliary particlespectral functions have a low-frequency singu-larity characterized by exponents which dependcontinuously on

N(as in the U =

NCA):

f() 1/f() 1/||f , X () 1/X () Sign()/||X , with: f = 1 X = 1/(N+ 1).These behavior are characterized more precisely inAppendix C. A power-law behavior is also foundfor the physical self-energy d() d (0) at lowfrequency (as evident from Fig. 5).

Let us comment on the origin of these low-energy fea-tures, as well as on their consequences for the practicaluse of the present method.

First, it is clear from expression (21) that the Ander-son impurity hamiltonian generalized to SU(N)O(2M)actually involves M channels of conduction electrons.

Hence, the non-Fermi liquid behaviour found when solv-ing the integral equations associated with the N, M limit simply follows from the fact that multi-channelmodels lead to overscreening of the impurity spin, andcorrespond to a non-Fermi liquid fixed point. In thatsense, these integral equations reproduce very accuratelythe expected low-energy physics, as previously studiedfor the simplest case of the Kondo model in [11, 12].

Naturally, this means that the use of such integralequations to describe the one-channel (exactly screened)case becomes problematic in the low-energy region. Inparticular, the exact Friedel sum rule is violated, the d-level lifetime remains finite at low energy and non-Fermiliquid singularities are found. While the approach is rea-sonable in order to reproduce the overall features of theone-electron spectra, it should not be employed to calcu-late transport properties at low energy for example. Wenote however that the deviation from the exact Friedelsum rule vanishes in the N limit. This is expectedfrom the fact that in this limit the number of channels(M) is small as compared to orbital degeneracy (N). Theviolation of the sum rule remains rather small even forreasonable values of N. This parameter can actually beused as an adjustable parameter when using the present


10/18

10

method as an approximate impurity solver. There is nofundamental reason for which N = N should providethe best approximate description of the spectral func-tions of the one-channel case. We shall use this possi-bility when applying this method in the DMFT contextin the next section: there, we choose N = 3 in order toadjust to the known critical value of U for a single or-bital. We also used this value in the calculations reported

above. The zero-frequency value of the spectral densityis thus d(0) 1.85, while the Friedel sum rule wouldyield d(0) 1.95 (cf. Fig. 3). Hence the violation of thesum rule is a small effect (of the order of 5%), comparableto the one found with numerically exact solvers, dueto discretization errors [19]. Also, we point out that thepinning of d(0) at a value independent of U (albeit notthat of the Friedel sum-rule) is an important aspect ofthe present method, which will prove to be crucial in thecontext of DMFT in order to recover the correct scenariofor the Mott transition.

Finally, we emphasize that increasing the parameterN also corresponds to increasing the orbital degeneracyof the impurity level. This will be studied in more detailsin Sec. V D. In particular, we shall see that correlationeffects become weaker asNis increased (for a given valueof U), due to enhanced orbital fluctuations.

V. APPLICATIONS TO DYNAMICAL

MEAN-FIELD THEORY AND THE MOTT

TRANSITION

A. One-orbital case: Mott transition, phase

diagram

Dynamical Mean-Field Theory has led to significantprogress in our understanding of the physics of a corre-lated metal close to the Mott transition [2]. The detaileddescription of this transition itself within DMFT is nowwell established [2, 7, 15, 19, 20, 21, 22]. In this section,we use these established results as a benchmark and testthe applicability of the method introduced in this paperin the context of DMFT, with very encouraging results.As explained above, this is particularly relevant in viewof the recent applications of DMFT to electronic struc-ture calculations of correlated solids [5], which call forefficient multi-orbital impurity solvers.

As is well-known [2, 3, 6], DMFT maps a lattice hamil-

tonian onto a self-consistent quantum impurity model.We discuss first the half-filled Hubbard model, and ad-dress later the doped case. We then have to solve aparticle-hole symmetric Anderson impurity model :

S =

0

d

dd +U

2

dd 12

(39)

+

0

d

0

d ( )

d()d ()

subject to the self-consistency condition:

() = t2Gd() (40)

In this expression, a semi-circular density of states withhalf-bandwith D = 2t has been considered, correspond-ing to a infinite-connectivity Bethe lattice (z = ) withhopping tij = t/

z. In the following, we shall generally

express all energies in units of D (D = 1).

In practice, one must iterate numerically the DMFTloop: () Gd() ()new = t2Gd, using someimpurity solver. Here, we make use of the integralequations (28-31). The hybridization function () be-ing determined by the self-consistency condition (40),there are only two free parameters, the local Coulombrepulsion U and the temperature T (normalized by D).

We display in Fig. 6 the spectral functions obtained atlow temperature, for increasing values ofU, and in Fig. 7the corresponding phase diagram. The value of the pa-rameter N has been adapted to the description of theone-orbital case (see below). The most important pointis that we find a coexistence region at low-enough tem-

perature: for a range of couplings Uc1(T) U Uc2(T),both a metallic solution and an insulating solution of the(paramagnetic) DMFT equations exist. The Mott tran-sition is thus first-order at finite temperatures. This is inagreement with the results established for this problemby solving the DMFT equations with controlled numer-ical methods [19, 21], as well as with analytical results[20, 22]. In particular, the spectral functions that we ob-tain (Fig. 6) display the well-known separation of energyscales found within DMFT: there is a gradual narrowingof the quasiparticle peak, together with a preformed Mottgap at the transition. In the next section, we comparethese spectral functions to those obtained using other

approximate solvers. As pointed out there, despite someformal similarity in the method, it is well-known thatthe standard U-NCA does not reproduce correctly thisseparation of energy scales close to the transition [23].

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 6: Local spectral function at = 40 and U = 1, 2, 3for the half-filled Hubbard model within DMFT, as obtainedwith the DSR solver


11/18

11

0 1 2 3 4

U

0

0.025

0.05

0.075

0.1

T

FIG. 7: Single-orbital (paramagnetic) phase diagram at half-filling. Squares indicate the ED result, the DSR result is thesolid line, and IPT the broken line

Let us explain how the parameter N has been chosenin these calculations. As demonstrated below, the val-ues of the critical couplings Uc1 and Uc2 (and hence thewhole phase diagram and coexistence window) stronglydepends on the value of this parameter. This is ex-pected, sinceNis a measure of orbital degeneracy. Whathas been done in the calculations displayed above isto choose N in such a way that the known value [20]Uc2(T = 0) 2.9 of the critical coupling at which theT = 0 metallic solution disappears in the single-orbitalcase, is accurately reproduced. We found that this re-quires N 3 (note that N/M = 2 in the one-orbitalcase, so that the best agreement is not found by a naiveapplication of the large N, M limit). This value beingfixed, we find a critical coupling Uc1(T = 0)

2.3 in

good agreement with the value from (adaptative) exactdiagonalizations Uc1 2.4. The whole domain of coexis-tence in the (U, T) plane is also in good agreement withestablished results (in particular we find the critical end-point at Tc 1/30, while QMC yields Tc 1/40). Theseare very stringent tests of the applicability of the presentmethod, since we have allowed ourselves to use only oneadjustable parameter (N). In Sec.V D, we study how theMott transition depends on the number of orbitals, whichfurther validates the procedure followed here.

B. Comparison to other impurity solvers: spectral

functions

Let us now compare the spectral functions obtainedby the present method with other impurity solvers com-monly used for solving the DMFT equations. We startwith the iterated perturbation theory approximation(IPT) and the exact diagonalization method (ED). Bothmethods have played a major role in the early develop-ments of the DMFT approach to the Mott transition [ 7].A comparison of the spectral functions obtained by the

present method to those obtained with IPT and ED isdisplayed in Fig. 8, for a value of U corresponding to acorrelated metal close to the transition.

-3 -2 -1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 8: Comparison between DSR (dash), IPT (straight) andED (dot) at U = 2.4 and = 60

The overall shape and characteristic features of thespectral function are quite similar for the three meth-ods. A narrow quasiparticle peak is formed, togetherwith Hubbard bands, and there is a clear separation ofenergy scales between the width of the central peak (re-lated to the quasiparticle weight) and the preformedMott (pseudo-)gap associated with the Hubbard bands:this is a distinguishing aspect of DMFT. It is crucial foran approximate solver to reproduce this separation of en-ergy scales in order to yield a correct description of theMott transition and phase diagram.

There are of course some differences between the three

methods, on which we now comment. First, we note thatthe IPT approximation has a somewhat larger quasipar-ticle bandwith. This is because the transition point Uc2is overestimated within IPT (cf. Fig. 7), so that a morefair comparison should perhaps be made at fixed U/Uc2.It is true however, that the DSR method has a tendencyto underestimate the quasiparticle bandwith, and par-ticularly at smaller values of U. Accordingly, the Hub-bard bands have a somewhat too large spectral weight,but are correctly located in first approximation. Thedetailed shape of the Hubbard bands is not very accu-rately known, in any case. (The ED method involvesa broadening of the delta-function peaks obtained bydiagonalizing the impurity hamiltonian with a limitednumber of effective orbitals, so that the high-energy be-haviour is not very accurate on the real axis. This is alsotrue, actually, of the more sophisticated numerical renor-malization group). We emphasize that, since the DSRmethod does not have the correct low-frequency Fermiliquid behaviour, the quasiparticle bandwith should beinterpreted as the width of the central peak in d()(while the quasiparticle weight Z cannot be defined for-mally).

In Fig 9 and Fig. 10, we display the spectral functions


12/18

12

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

FIG. 9: U-NCA spectral function for U = 1.5 at low temper-ature.

obtained by using the NCA method (extended at finiteU in the simplest manner). It is clear that the U-NCA

underestimates considerably the quasiparticle bandwith(and thus yields a Mott transition at a rather low valueof the coupling). This is not very surprising, since thismethod is based on a strong coupling expansion aroundthe atomic limit, and one could think again of a com-parison for fixed U/Uc. More importantly, the U-NCAmisses the important separation of energy scales betweenthe central peak and the Mott gap. As a result, it doesnot reproduce correctly the phase diagram for the Motttransition within DMFT (in particular regarding coex-istence). A related key observation is that the U-NCAdoes not have the correct weak-coupling (small U) limit.To illustrate this point, we display in Fig. 10 the U-NCA

and DSR spectral functions for a tiny value of the inter-action U/D = 0.1: the DSR result is shown to approachcorrectly the semi-circular shape of the non-interactingdensity of states, while the U-NCA displays a charac-teristic inverted V-shape: in this regime the violationof the Friedel sum rule becomes large and the negativelifetime pathology is encountered within U-NCA. In con-trast, the DSR yields a pinning of the spectral densityat a U-independent value and does not lead to a viola-tion of causality. It should be emphasized however that,even though it yields the exact U = 0 density of statesat T = 0, the DSR method is not quantitatively veryaccurate in the weak-coupling regime.

C. Double occupancy

Finally, we demonstrate that two-particle correlatorsin the charge sector can also be reliably studied with theDSR method, taking the fraction of doubly occupied sitesd nn as an example.

Using the constraint (4), we calculate the connected

-4 -2 0 2 40

0.2

0.4

0.6

0.8

FIG. 10: Results for U = 0.1 showing how U-NCA overshootsthe Friedels sum rule. The Slave Rotor method is correctlyconverging towards the free density of states (semi-circular)

charge susceptibility in the following manner:

c()

n() 1

2

n(0) 1

2

=

L()L(0)

=1

U2()(0) (41)

=2

U2

GX ()2GX () + U () (GX ())2

The double occupancy is obtained by taking the equal-time value:

c( = 0) = 2 nn = 2d (42)

In Fig. 11, we plot the double occupancy obtained inthat manner, in comparison to the ED and IPT results.Within ED, d is calculated directly from the chargecorrelator. Within IPT, c() is not approximated veryreliably [24], but the double occupancy can be accuratelycalculated by taking a derivative of the internal energywith respect to U.

Fig. 11 demonstrates that the DSR method is very ac-curate in the Mott transition region and in the insulator.In particular, the hysteretic behaviour is well reproduced.In the weak coupling regime however, the approximationdeteriorates. This issue actually depends on the quan-tity: the physical Greens function has the correct limitU 0, as emphasized above, but this is not true for thecharge susceptibility (hence for d). The mathemati-cal reason is that the constraint (4) is crucial for writing(41), but this constraint is only treated on average withinour method. This shows the inherent limitations of slave-boson techniques for evaluating two-particle properties.The frequency dependence of two-particle correlators willbe dealt with in a forthcoming publication.


13/18

13

0 1 2 3 4

U

0

0.1

0.2

0.3

0.4

d

FIG. 11: Double occupancy at = 80 as a function of U.The ED result is indicated with squares. DSR overshoots theexact result on the left, whereas IPT is slightly displaced tothe right.

D. Multi-orbital effects

In this section, we apply the DSR method to study thedependence of the Mott transition on orbital degeneracyN. We emphasize that there are at this stage very fewnumerical methods which can reliably handle the multi-orbital case, specially when N becomes large. Multi-orbital extensions of IPT have been studied [8], but theresults are much less satisfactory than in the one-orbitalcase. The ED method is severely limited by the expo-nential growth of the size of the Hilbert space. The Motttransition has been studied in the multi-orbital case us-

ing QMC, with a recent study [25] going up to N = 8(4 orbitals with spin). Furthermore, we have recentlyobtained [15] some analytical results on the values of thecritical couplings in the limit of large-N. Those, togetherwith the QMC results, can be used as a benchmark of theDSR approximation presented here. As explained above,a value ofNN=2 3 was found to describe best the sin-gle orbital case (N = 2). Hence, upon increasing N, wechoose the parameter N such that NN/NN=2 = N/2.

In Fig. 12, we display the coexistence region in the(U, T) plane for increasing values of N, as obtained withDSR. The values of the critical interactions grow withN. A fit of the transition lines Uc1(T) (where the insula-tor disappears) and Uc2(T) (where the metal disappears)yields:

Uc1(N, T) = A1(T)

N (43)

Uc2(N, T) = A2(T)N (44)

These results are in good accordance with both the QMCdata [25], and the exact results established in [15]. Whenincreasing the number of orbitals, the coexistence regionwidens and the critical temperature associated with theendpoint of the Mott transition line also increases.

0 1 2 3 4 5 6 7 8 9

U

0

0.05

0.1

0.15

0.2

T

FIG. 12: Instability lines Uc1(T),Uc2(T), and coexistence re-gion for one, two, three and four orbitals (N = 2, 4, 6, 8) asobtained by DMFT (DSR solver)

We display on Fig. 13 the DSR result for the spectralfunction for N = 2 and N = 4, at a fixed value of U andT. Two main effects should be noted. First, correlationseffects in the metal become weaker as N is increased (fora fixed U), as clear e.g from the increase of the quasi-particle bandwith. This is due to increasing orbital fluc-tuations. Second, the Hubbard bands also shift towardslarger energies, an effect which can be understood in theway atomic states broadens in the insulator [26].

To conclude this section, we have found that the DSRyields quite satisfactory results when used in the DMFTcontext for multi-orbital models. In the future, we planto use this method in the context of realistic electronicstructure calculations combined with DMFT, in situa-tions where numerically exact solvers (e.g QMC) be-come prohibitively heavy. There are however some limi-tations to the use of DSR (at least in the present versionof the approach), which are encountered when the occu-pancy is not close to N/2. We examine this issue in thenext section.

E. Effects of doping

Up to now, we studied the half-filled problem (i.e. con-taining exactly N/2 electrons), with exact particle-holesymmetry. In that case, 0 d + U/2 = 0 and the La-grange multiplier h enforcing the constraint (31) could beset to h = 0. In realistic cases, particle-hole symmetrywill be broken (so that 0 = 0), and we need to considerfillings different from N/2. Hence, the Lagrange multi-plier h must be determined in order to fulfill (31), which


14/18

14

-2 0 20

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 13: Spectral function at U = 2, = 60 and for one (fullline) and two orbitals (broken line)

we rewrite more explicitly as:

nf =1

2 2h

NU (45)

+1

NU1

n

in

ein0+

+ ein0

2n/U + 2ihn/U X (in)

In this expression, nf =1

N

ff = 1N

dd

is the average occupancy per orbital flavor. (Note thatthe number of auxiliary fermions and physical fermionscoincide, as clear from (34)). nf is related to the d-levelposition (or chemical potential, in the DMFT context)by:

nf =

1

n

ein0+

in 0 + h f(in) (46)

-2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

FIG. 14: Doping the insulator (U = 3 and = 60) with0 = 0, 1 (corresponding to nf = 0.5, 0.4 respectively)

We display in Fig. 14 the spectral function obtainedwith the DSR solver when doping the Mott insulator

away from half-filling. A critical value of the chemicalpotential (i.e. of 0) is required to enter the metallicstate. The spectral function displays the three expectedfeatures: a lower and upper Hubbard bands, as well asa quasiparticle peak (which in this case where the dop-ing is rather large, is located almost at the top of thelower Hubbard band). However, it is immediately appar-ent from this figure that the DSR method in the doped

case overestimates the spectral weight of the upper Hub-bard band. This can be confirmed by a comparison toother solvers (e.g ED). Note that the energy scale be-low which a causality violation appears within U-NCAbecomes rapidly large as the system is doped, while nosuch violation occurs within DSR.

The DSR method encouters severe limitations howeveras the total occupancy becomes very different from N/2.This is best understood by studying the dependence ofthe occupancy upon 0, in the atomic limit. As explainedin Appendix. A, the use of sigma-model variables X (inthe large-M limit) results in a poor description of the nfvs. 0 dependence (Coulomb staircase). As a result,

it is not possible to describe the Mott transitions occur-ring in the multi-orbital model at integer fillings differ-ent from N/2 using the DSR approximation. It shouldbe emphasized however that this pathology is only dueto the approximation of the O(2) quantum rotor ei bya O(M) sigma-model field. A perfect description of theCoulomb staircase is found for all fillings when treatingthe constraint (4) on average while keeping a true quan-tum rotor [27]. Hence, it seems feasible to overcome thisproblem and extend the practical use of the (dynamical)slave rotor approach to all fillings. We intend to addressthis issue in a future work.

VI. CONCLUSION

We conclude by summarizing the strong points as wellas the limitations of the new quantum impurity solverintroduced in this paper, as well as possible extensionsand applications.

On the positive side, the DSR method provides aninterpolating scheme between the weak coupling andatomic limits (at half-filling). It is also free of some ofthe pathologies encountered in the simplest finite-U ex-tensions of NCA (negative lifetimes at low temperature).When applied in the context of DMFT, it is able to re-produce many of the qualitatively important features as-sociated with the Mott transition, such as coexisting in-sulating and metallic solutions and the existence of twoenergy scales in the DMFT description of a correlatedmetal (the quasi-particle coherence bandwith and thepreformed gap). Hence the DSR solver is quite usefulin the DMFT context, at a low computational cost, andmight be applicable to electronic structure calculationsfor systems close to half-filling when the orbital degen-eracy becomes large. To incorporate more realistic mod-elling, one can introduce different energy levels for each


15/18

15

correlated orbital, while the extension to non-symmetricCoulomb interactions (such as the Hunds coupling) mayrequire some additional work.

The DSR method does not reproduce Fermi-liquid be-haviour at low energy however, which makes it inad-equate to address physical properties in the very low-energy regime (as is also the case with NCA). The mainlimitation however is encountered when departing from

half-filling (i.e from N/2 electrons in an N-fold degener-ate orbital). While the DSR approximation can be usedat small dopings, it fails to reproduce the correct atomiclimit when the occupancy differs significantly from N/2(and in particular cannot deal with the Mott transitionat other integer fillings in the multi-orbital case). Wewould like to emphasize however that this results fromextending the slave rotor variable to a field with a largenumber of components. It is possible to improve thisfeature of the DSR method by dealing directly with anO(2) phase variable, which does reproduce accurately theatomic limit even when the constraint is treated at themean-field level. We intend to address this issue in afuture work. Another possible direction is to examinesystematic corrections beyond the saddle-point approxi-mation in the large N, M expansion.

Finally, we would like to outline some other possibleapplications of the slave rotor representation introducedin this paper (Sec. II). This representation is both phys-ically natural and economical. In systems with strongCoulomb interactions, the phase variable dual to the lo-cal charge is an important collective field. Promotingthis single field to the status of a slave particle avoidsthe redundancies of usual slave-boson representations.In forthcoming publications, we intend to use this rep-resentation for: i) constructing impurity solvers in thecontext of extended DMFT [29], in which the frequency-

dependent charge correlation function must be calculated[28] ii) constructing mean-field theories of lattice modelsof correlated electrons (e.g the Hubbard model) [27] andiii) dealing with quantum effects on the Coulomb block-ade in mesoscopic systems.

Acknowledgments

We are grateful to B. Amadon and S. Biermann forsharing with us their QMC results on the multiorbitalHubbard model, and to T. Pruschke and P. Lombardofor sharing with us their expertise of the NCA method.

We also acknowledge discussions with G. Kotliar (thanksto CNRS funding under contract PICS-1062) and withH. R. Krishnamurthy (thanks to an IFCPAR contractNo. 2404-1), as well as with M. Devoret and D. Esteve.

APPENDIX A: THE ATOMIC LIMIT

In this appendix we prove the claim that the atomiclimit of the model is exact at half-filling and at zero tem-

perature (at finite temperature, deviations from the exactresult are of order exp (U) and therefore negligeablefor pratical purposes). To do this, we first extract the val-ues of the mean-field parameters and h from the saddle-point equations at zero temperature and () 0.

1 = d

2

12

U + +

2ih

U

(A1)

(h 0) = 12

2hNU +4h

NU2

d

2

22

U + 2

+2h

U

2Performing the integrals shows that = (U24h2)/(4U)and (h 0) = 1/2, so that h = 0. If |0| > U/2, theequations lead actually to a solution with an empty orfull valence that we show on Fig. 15.

We can now compute the physical Greens functionGd() = Gf()GX () from the pseudo-propagatorsGf(in) = 1/(in) and:

GX (in) =1

2n/U + (U/4

20/U) 2i0n/U

(A2)

=1

in + 0 U/2 +1

in + 0 + U/2

Performing the convolution in imaginary frequencyand taking the limit T = 0 leads to:

Gd(in) =1

in

GX (in)Gf(in + in) (A3)

=1/2

in 0 + U/2 +1/2

in 0 U/2 (A4)

Because 0 = + U/2 this is the correct atomic limit ofthe single-band model (at half-filling). The result for theempty or full orbital is however not accurate, as shown infigure 15. This discrepancy with the correct result (evenfor one orbital) finds its root in the large M treatmentof the slave rotor X.

APPENDIX B: NUMERICAL SOLUTION OF

THE REAL TIME EQUATIONS

Here we show how the saddle-point equations (28-31)can be analytically continued along the real axis. Westart with X () = N()Gf(), which can be firstFourier transformed into:

X (in) =

0

d X ()ein (B1)

= N

d1

Gf(1)

d2

(2)

nF(1) nF(2)in + 1 2

where we used the spectral representation

G(z) =

d

G()

z (B2)


16/18

16

o

0

U/23U/2 U/2 3U/2

nf

1/4

1/2

3/4

1

FIG. 15: Impurity occupancy in function of the d-level posi-tion in the rotor description (full curve) and the exact result(dot curve), in the two-orbital case.

for each Greens function. The notations here are quite

standard: G() Im G( + i0+), nF() is the Fermifactor, and nB() denotes the Bose factor.

It is then immediate to continue in +i0+ in equa-tion (B1), and using again the spectral decomposition(B2), we derive an equation between retarded quantities:

X () = N

d

Gf()nF()( + ) (B3)

N

d

()nF()Gf( )

A calculation along the same lines for the fermionic self-energy f() = ()GX () leads to:

f() =

d

GX ()nB()( ) (B4)

d

()nF()GX ( )

The numerical implementation is then straightforwardbecause (B3) and (B4) can each be expressed as the con-volution product of two quantities, so that they can becalculated rapidly using FFT. The algorithm is loopedback using the Dyson equations (for real frequency):

G1f () = 0 + h f() (B5)

G1X () = 2

U+ + 2h

U X () (B6)

At each iteration, and h are determined using a bi-section on the equations (30-31), which can be properlyexpressed in terms of retarded Greens functions:

1 =

d

GX ()nB () (B7)

nf =1

2 2hNU

2

NU

d

GX ()nB() (B8)

where nf, the average number of physical fermions, is:

nf = Gf(= 0) =

d

Gf()nF() (B9)

We note here that solving these real-time integral equa-tions can be quite difficult deep in the Kondo regime ofthe Anderson model, or very close to the Mott transition

for the full DMFT equations. The reason is that Gf()and GX () develop low energy singularities (this is an-alytically shown in the next appendix), that make thenumerical resolution very unprecise if one uses FFT. Inthat case, it is necessary to introduce a logarithmic meshof frequency (loosing the benefit of the FFT speed, butincreasing the accuracy), or to perform a Pade extrapo-lation of the imaginary time solution.

APPENDIX C: FRIEDELS SUM RULE

We present here for completeness the derivation of the

Slave Rotor Friedels sum rule, equation (37), at half-filling. The idea, motivated by the numerical analysis aswell as theoretical arguments [12, 30], is that the pseudo-particles develop low frequency singularities at zero tem-perature:

Gf() = Af||f (C1)GX () = AX ||XSign() (C2)

Using the spectral representation

G() = +

0

d

eG() (C3)

we deduce the long time behavior of the Greens functions(we denote by (z) the gamma function):

Gf() =Af(1 f)

Sign()

||1f (C4)

GX () =AX (1 X )

1

||1X (C5)

We have similarly

() =(0)

(C6)

if one assumes a regular bath density of states at zerofrequency. The previous expressions allow to extract thelong time behavior of the pseudo-self-energies (using thesaddle-point equations (28-29)):

X () =NAf(1 f)

2(0)

||2f (C7)

f() =AX (1 X )

2(0) Sign()

||2X (C8)


17/18

17

The next step is to use (C3) the other way around to getfrom (C7-C8) the -dependance of the self-energies:

X () =N

Af(0)

1 f ||1f Sign() (C9)

f() =1

AX (0)

1 X ||1X (C10)

It is necessary at this point to calculate the real part ofboth self-energies. This can be done using the Kramers-Kronig relation, but analyticity provides a simpler route.Indeed, by noticing that (z) is an analytic function ofz and must be uni-valuated above the real axis , we find:

X (z) =N

Af(0)

1 fei(f1)/2

sin[(f 1)/2] |z|1f

f(z) =1

AX (0)

1 XeiX/2

sin[X /2]|z|1X

The same argument shows from (C1-C2) that:

Gf(z) = Af ei(f+1)/2

sin[(f + 1)/2]|z|f (C11)

GX (z) = AXeiX/2

sin[X /2]|z|X (C12)

We can therefore collect the previous expressions, usingDysons formula for complex argument:

G1f (z) = z f(z) (C13)

G1X (z) = z2

U+ X (z) (C14)

and this enables us to extract the leading exponents, aswell as the product of the amplitudes:

f =1

N+ 1 (C15)

X =N

N+ 1(C16)

AfAX =

N+ 11

(0)sin2

2

NN+ 1

(C17)

We finish by computing the long time behavior of thephysical Greens function Gd() = Gf()GX () to-gether with equations (C4-C5):

Gd() =

2(N

+ 1)tan

2

N

N+ 1

1

(0)

1

(C18)

This proves (37). In principle, next leading order cor-rections can be computed by the same line of arguments,although this is much more involved [12]. Non Fermi Liq-uid correlations in the physical Greens function wouldappear in this computation.

[1] For an introduction, see: L. Kouwenhoven and L. Glaz-man, Phys. World 14, 33 (2001)

[2] For a review, see: A. Georges, G. Kotliar, W. Krauthand M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996)

[3] For a review, see: T. Pruschke, M. Jarrell, and J.K. Fre-ericks, Adv. Phys. 42, 187 (1995)

[4] See e.g: A. Hewson, The Kondo problem to heavyfermions (Cambridge)

[5] For recent reviews, see: Strong Coulomb Correlationsin Electronic Structure Calculations (Advances in Con-densed Matter Science), Edited by V. Anisimov. Taylorand Francis (2001)K.Held et al. in Quantum Simulations of ComplexMany-Body Systems Edited by: J. Grotendorst, D.Marx and A. Muramatsu (NIC - Serie Band 11) cond-mat/0112079.

[6] A. Georges and G. Kotliar, Phys. Rev. B 45, 6479 (1992)[7] X.Y. Zhang, M. J. Rozenberg, and G. Kotliar, Phys. Rev.

Lett. 70, 1666, (1993); A. Georges and W. Krauth, Phys.Rev. B 48, 7167 (1993); M.J. Rozenberg, G. Kotliar andX.Y. Zhang, Phys. Rev. B 49, 10181 (1994).

[8] G. Kotliar and H. Kajueter, Phys. Rev. B 54, R14221(1996)

H. Kajueter and G. Kotliar, Int. J. Mod. Phys. 11, 729(1997)

[9] N. Bikers, Rev. Mod. Phys. 59, 845 (1993)[10] T. Pruschke and N. Grewe, Z. Phys. B 74, 439 (1989)[11] D.L. Cox and A.E. Ruckenstein, Phys. Rev. Lett. 71,

1613 (1993)[12] O. Parcollet, A. Georges, G. Kotliar and A. Sengupta,

Phys. Rev. B 58, 3794 (1998)O. Parcollet, PhD thesis, unpublished.

[13] J. Kroha and P. Wolfle, cond-mat/0105491[14] G. Kotliar and A.E. Ruckenstein, Phys. Rev. Lett. 57,

1362 (1986)[15] S. Florens, A. Georges, G. Kotliar and O. Parcollet,

cond-mat/0205263[16] T. Pruschke, private communication[17] S. Pairault, D. Senechal and A-M. Tremblay, Eur. Phys.

J. B 16, 85 (2000)[18] K. Yamada, Prog. Theo. Phys. 53, 970 (1975)[19] R. Bulla, A.C. Hewson and T. Pruschke, J. Phys: Con-

dens Matter 10 8365 (1998); R. Bulla, T.A. Costi and D.Vollhardt Phys. Rev. B 64, 045103 (2001)

[20] G. Moeller, Q. Si, G. Kotliar, M. Rozenberg and D.S.Fisher, Phys. Rev. Lett. 74, 2082 (1995)
http://arxiv.org/abs/cond-mat/0112079http://arxiv.org/abs/cond-mat/0112079http://arxiv.org/abs/cond-mat/0112079http://arxiv.org/abs/cond-mat/0105491http://arxiv.org/abs/cond-mat/0205263http://arxiv.org/abs/cond-mat/0205263http://arxiv.org/abs/cond-mat/0105491http://arxiv.org/abs/cond-mat/0112079http://arxiv.org/abs/cond-mat/0112079


18/18

18

[21] M.J. Rozenberg, R. Chitra and G. Kotliar, Phys. Rev.Lett 83, 3498 (1999). W. Krauth, Phys. Rev. B 62,6860 (2000); J. Joo and V. Oudovenko Phys. Rev. B 64,193102 (2001)

[22] G. Kotliar, Eur. Phys. J. B, 11, 27 (1999); G. Kotliar,E. Lange and M.J. Rozenberg, Phys. Rev. Lett. 84, 5180(2000).

[23] T. Pruschke, D.L. Cox and M. Jarrell, Phys. Rev. B 47,3553 (1993)

P. Lombardo and G. Albinet, Phys. Rev. B 65, 115110(2002)

[24] J.K Freericks and M. Jarrell, Phys. Rev. B 50, 6939

(1994)[25] B. Amadon and S. Biermann, unpublished[26] J.E. Han, M. Jarrell and D.L. Cox, Phys. Rev. B 58,

R4199 (1998)[27] S. Florens and A. Georges, in preparation[28] S. Florens, L. de Medici and A. Georges in preparation[29] Q. Si and J. L. Smith, Phys. Rev. Lett. 77, 3391 (1996);

H. Kajueter PhD thesis Rutgers University, 1996; J.L.Smith and Q. Si, Phys. Rev. B 61, 5184 (2000); A.M.

Sengupta and A. Georges Phys. Rev. B 52 10295 (1995).[30] E. Mueller-Hartmann, Z. Phys. B 57,281 (1984)

Documents

Serge Florens and Antoine Georges- Quantum impurity solvers using a slave rotor representation