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arXiv:1104.1919v1 [cond-mat.str-el] 11 Apr 2011

Phase diagram and criticality of the three dimensional Hubbard model

G. Rohringer,1A. Toschi,1A. Katanin,2and K. Held1

1Institute for Solid State Physics, Vienna University of Technology 1040 Vienna, Austria

2Institute of Metal Physics, Ekaterinburg, Russia

(Dated: April 12, 2011)

By means of the dynamical vertex approximation (DΓA) we include spatial correlations on all

length scales beyond the dynamical mean field theory (DMFT) for the three dimensional Hubbard

model. The most relevant changes to the phase-diagram are: i) a sizable reduction of the N´ eel

temperature by ∼ 30% for the onset of antiferromagnetic long-range order and (ii) a deviation

from the mean-field critical behavior for T ∼ TN with the same critical exponents as for the three

dimensional Heisenberg (anti)-ferromagnet. Finally, DΓA also allows for a quantitative estimate of

the errors of DMFT in different regions of the phase-diagram.

PACS numbers: 71.10.Fd, 71.27.+a

Almost 50 years after the invention of the Hubbard

model [1–3] and despite modern petaflop supercomput-

ers, a precise analysis of the criticality of this most basic

model for electronic correlations has not been achieved

so far, at least not in three dimensions. Dynamical mean

field theory (DMFT) [4–6] was a big step forward to

quantitatively calculate the three dimensional Hubbard

model since the major contribution of electronic corre-

lations, i.e., the local ones, are well captured within

this theory. Local correlations give rise to quasiparti-

cle renormalization, the Mott-Hubbard transition, mag-

netism, and even more subtle issues such as kinks in a

purely electronic models [7]. However, non-local spatial

correlations are also naturally generated by a purely lo-

cal Hubbard interaction, and, as it is well known, they

become of essential importance in the vicinity of second-

order phase transitions.As these correlations are ne-

glected in DMFT, this scheme can provide only for a

conventional mean-field (MF) description of the critical

properties.

To overcome this shortcoming cluster extensions to

DMFT such as the dynamical cluster approximation

(DCA) and cluster-DMFT have been proposed, see Ref.

8 for a review. In these approaches non-local correla-

tions beyond DMFT are taken into account, however

only within the range of the cluster size; and due to

computational limitations the actual size of the cluster

is severely restricted to about 3-5 sites for each direction.

Hence, short-range correlations are included by these ap-

proaches, whereas long-range ones are not.

less, Kent et al. [9] were able to extrapolate the cluster

size of so-called Betts clusters to infinity, albeit assum-

ing from the very beginning the critical exponents to be

those of the Heisenberg model. This way they extrap-

olated the critical temperature of the paramagnetic-to-

antiferromagnetic phase transition which was found to be

in agreement with earlier quantum Monte Carlo (QMC)

simulations of Staudt et al. [10] on finite lattices.

As an alternative to cluster extensions and, in partic-

ular, to include long-range correlations on an equal foot-

Nonethe-

ing, more recently diagrammatic expansions of DMFT

have been proposed: (i) the dynamical vertex approxi-

mation (DΓA) [11–13] which approximates the fully ir-

reducible n-particle vertex to be local [11] or that of a

DCA cluster [13]; and (ii) the dual Fermion approach

[14]. The latter basically considers the same kind of di-

agrammatic extension as DΓA, however not in terms of

the real fermions, i.e., the electrons, but in terms of dual

fermions. As for phase transitions, DΓA with Moriyasque

λ−corrections[15] fulfills - in contrast with dual fermion

calculations of [16]- the Mermin and Wagner theorem in

two-dimensions and, as we will discuss in the following,

corrects the MF behavior for the critical exponents in

three dimensions.

In this paper, we apply DΓA for studying the phase-

diagram of the three dimensional Hubbard model at half-

filling. In particular, we (i) determine the phase diagram

with a N´ eel temperature (TN) substantially reduced com-

pared to the DMFT one, (ii) calculate the critical expo-

nents, and (iii) define the region where non-local correla-

tions become too strong so that DMFT is not applicable

anymore.

We consider the Hubbard model on a cubic lattice

H = −t

?

?ij?σ

c†

iσcjσ+ U

?

i

ni↑ni↓

(1)

where t denotes the hopping amplitude between nearest-

neighbors, U the Coulomb interaction, and c†

ates (annihilates) an electron with spin σ on site i;

niσ=c†

iσciσ. In the following, we restrict ourselves to the

paramagnetic phase with n = 1 electron/site at a finite

temperature T. For the sake of clarity, and in accordance

with previous publications, we will define hereafter our

energies in term of a typical energy scale D representing

twice the variance of the cubic DOS, i.e. D = 2√6t.

The DΓA approach to the model (1) was derived in

Refs. 11, 15. The dynamic non-uniform susceptibility

reads

iσ(ciσ) cre-

χs(c)

qω= [(φs(c)

q,ω)−1∓ U + λs(c)]−1

(2)

Page 2

2

where

φs(c)

q,ω =

?

νν′

Φνν′ω

s(c),q,

Φνν′ω

s(c),q= [(χν′

0qω)−1δνν′ − Γνν′ω

s(c),ir± U]−1, (3)

χν′

0qω= −T?

ble, Gk,ν= [iν − ǫk+ µ − Σloc(ν)]−1is the Green func-

tion, and Σloc(ν) the local self-energy. The vertex Γνν′ω

is determined from the solution of the single-impurity

problem[11], λs(c)is the Moriya lambda correction for

the spin (charge) channel [15]. In fact, the complete in-

clusion of such non-local corrections in the irreducible

vertices in all channels can be achieved only via the fully

self-consistent DΓA equations. However, as discussed in

Ref. [15], when considering a situation where no com-

petition between different instabilities occurs (as it hap-

pens here, because the antiferromagentic(AF)-instability

surely dominates all others in the particle-hole symmet-

ric case), a restriction to specific channels and the evalu-

ation of the self-consistency effect via the corresponding

Moriya lambda-corrections is possible.

case, charge excitations are expected to be gapped at

half filling, we can hence put λc = 0 and determine λs

from the sum rule

kGk,ν′Gk+q,ν′+ωis the particle-hole bub-

s(c),ir

In our specific

−

?∞

−∞

dν

πImΣk,ν= U2n(1 − n/2)/2. (4)

where the non-local self-energy is given by

Σk,ν =

1

2Un +1

2TU

?

ω,q

?3γνω

c,qχc

s,q− γνω

c,q− 2

+3Uγνω

s,qχs

qω+ Uγνω

qω

+

?

ν′

χν′

0qω(Γνν′ω

c,loc− Γνν′ω

s,loc)

?

Gk+q,ν+ω

(5)

with

γνω

s(c),q= (χν

0qω)−1?

ν′

Φνν′ω

s(c),q, (6)

and Γνν′ω

determined from the single-impurity problem.

Starting point of our investigation of the critical prop-

erties of the antiferromagnetic (AF) instability is the cor-

responding (divergent) susceptibility, i.e., the AF-spin

susceptibility

s(c),locis the reducible local spin (charge) vertex,

χAF= χs

Q,0=

β

?

0

dτ?Sz,Q(τ)Sz,−Q(0)? (7)

with Q = (π,π,π). While the DΓA with Moriyasque

corrections well reproduces the textbook Mermin and

Wagner results in the case of 2d-Hubbard model yield-

ing finite, but exponentially large susceptibility at finite

0

0.03

0.01

0.02

0.03

0.04

0.05

0.04 0.05 0.06

T/D

0.07 0.08

χAF

-1

U=1.00D

U=1.25DU=1.50D

U=2.25D

U=2.50D

DΓA

χAF

-1

TN

DMFT

DMFT

FIG. 1:

(π,π,π)) susceptibility as a function of T for different U val-

ues: Note the evident enlargement of the region of deviation

from the MF/DMFT exponent (i.e., 2ν = 1 linear behavior,

see inset) with increasing U.

(Color online) Inverse spin-spin AF (q = Q =

T[15], the situation in 3d is even more intriguing, since

the AF-phase remains stable in a broad region at finite

T, allowing for a direct study of the critical properties.

Of particular interest is the analysis of the evolution of

the critical region as a function of the Coulomb repulsion.

In Fig. 1, we show the inverse susceptibility χ−1

function of T for different U values. The vanishing of

χ−1

AFmarks the onset of the AF long-range order, defining

the corresponding TN for a given U. More important is,

however, the examination of the critical behavior: While

in a MF (or DMFT) approach χ−1

close to TNin accordance with the MF (Gaussian) critical

exponent (γ = 2ν = 1, with χ−1

inset of Fig. 1), DΓA data clearly show a bending in

the region close to the AF transition, indicating a DΓA

critical exponent γ definitively larger than 1. The non-

perturbative nature of DΓA also allows for a treatment of

the critical properties, e.g. the size of the critical region,

as a function of U: From our data it emerges that, in the

U-range studied, the size of the region where the critical

behavior deviates from the MF predictions (here: from

linearity) becomes larger with increasing U.

This result can be understood physically in the small-

U (weak-coupling) limit. As it is known, in three dimen-

sions, the applicability of the standard Landau-Ginzburg

expansion is (conventionally) defined via the so-called

Ginzburg criterion: The critical region with non-MF be-

havior grows like ∆T = (T − TN) ∝ T2

coupling TN ∼ e−

the size of the critical region is expected to grow quickly

with increasing U. This expectation is, in fact, perfectly

matching with our DΓA results, showing that the MF lin-

ear behavior is gradually confined to higher and higher

temperatures with increasing U (and TN), and basically

disappearing in the interval of temperature considered

AFas a

AFis vanishing linearly

AF∝ (T − TN)γ, see

N[17]. At weak-

1

WU (with W ∝ 1/D), and therefore

Page 3

3

0

0.064

0.01

0.02

0.03

0.072 0.08

T/D

aχ = 6.8510

TN = 0.0662

ν = 0.7070

χAF

-1(T)=aχ(T-TN)2ν

0

0.064

0.01

0.02

0.03

0.072 0.08

T/D

aχ = 6.5118

TN = 0.0663

ν = 0.6996

χAF

-1(T)=aχ(T-TN)2ν

0

0.064

0.25

0.5

0.75

1

0.072 0.08

aξ = 13.2878

TN = 0.0665

ν = 0.7070

ξ-1(T)=aξ(T-TN)ν

0

0.064

0.25

0.5

0.75

1

0.072 0.08

aξ = 14.5699

TN = 0.0661

ν = 0.7333

ξ-1(T)=aξ(T-TN)ν

FIG. 2: (Color online) Fit of χ−1

est interaction value considered, i.e., U = 2.5D. Left: fit with

fixed ν = 0.707 (3d-Heisenberg-exponent[21]). Right: free

fit, showing the good compatibility with the 3d-Heisenberg

universality class.

AF(T) and ξ−1(T) for the high-

for U > 1.5D. On the other hand, the region of devia-

tion from MF is still appreciable for the smallest set of

parameter shown in Fig. 1, whereas the bending of χ−1

becomes hardly visible for U as small as U = 0.5D (not

shown)[18].

A more quantitative study of the critical behavior re-

quires naturally a precise evaluation of the critical expo-

nent(s). From the behavior of the spin-susceptibility, one

can extract the values of the critical exponents ν, which

controls the divergence of the AF-correlations length ξ

(defined as the square root of the inverse mass of the

spin-spin propagator at q = Q, ω = 0) when T → TN.

This can be computed[19] either from the divergence of

χAF∝ (T −TN)−γ(i.e., directly from the data shown in

Fig. 1), where the index γ is defined[20] as γ = 2ν, or by

extracting from χAF the value of ξ as function of T by

fitting its momentum dependence for different T.

The results of our analysis, reported in Fig. 2, demon-

strate that the DΓA description of the antiferromagnetic

criticality of the Hubbard model is rather accurate. For

the largest values of U = 2.5D, indeed, both divergences

of χAFand ξ observed in DΓA can be described (left pan-

els of Fig. 2) with high-accuracy by the critical exponent

ν = 0.707 of the 3d−Heisenberg AF. This is expected to

be the correct exponent, not only because the half-filled

Hubbard can be mapped onto the Heisenberg model but

also since dimension and symmetry of the order param-

eter suggest the same universality class. Similar results,

though with a lower degree of precision, can be found by

directly fitting the value of the ν exponent to χ−1

(right panels): our fits provide an estimate of ν ranging

from ∼ 0.69 ÷ 0.73 for U = 2.5D. Note that the over-

AF

AFand ξ

0

0.1

0.5 1 1.5

U/D

2 2.5 3

T/D

crossover

PM

PI

AF

(*) diff.=|ΣDMFT(iν1)-ΣDΓA(kF,iν1)|/|ΣDMFT(iν1)|

DΓA

DCA

DMFT

10% diff.(*)

FIG. 3: (Color online) N` eel Temperature in DΓA (red filled

circles), compared with the corresponding DMFT (small black

squares) and DCA ones (open diamonds; both reproduced

from Ref. [9]). Also shown is the region where corrections

to DMFT in form of non-local correlations become relevant

(open circles; defined as 10 % relative change of the self-energy

at the first Matsubara frequency).

all numerical precision in the determination of the critical

exponent is limited by the density of the mesh adopted in

the momentum and frequency integrations for computing

the Moriya lambda-corrections close to the critical point.

This poses a limit to the temperature region accessible

in the vicinity of TN. In particular, it reduces the accu-

racy of our fitting procedure in the weak-coupling regime,

where the region of non-MF behavior becomes very nar-

row, and hence, harder to be reached numerically. In any

case, while for U < 1.25D it is difficult to perform pre-

cise estimates, in the whole region 1.25D < U < 2.5D

our fits do not indicate a change in the value of the crit-

ical exponent ν. This shows the Heisenberg universality

is still valid also in a parameter region (i.e., at interme-

diate coupling), where the Hubbard model is not well

approximated by the Heisenberg model.

A natural by-product of the calculations of the crit-

ical exponents is an accurate determination of TN at

the DΓA level, whose values are reported in the phase-

diagram of Fig. 3, and compared with the corresponding

estimates obtained in DMFT and DCA[9] (extrapolation

with Betts clusters). The inclusion of non-local fluctua-

tions at the DΓA level induces a reasonable reduction of

the N´ eel temperature w.r.t. DMFT in the whole phase

diagram.The overall agreement with the most accu-

rate DCA calculations available[9] appears satisfactory

in the intermediate-U region, demonstrating the validity

of DΓA to describe non-perturbatively the critical region,

without resorting to any finite-size extrapolation scheme.

Finally, we investigate the effects of the non-local cor-

rections on the spectral properties of the three dimen-

sional Hubbard model. On general ground, the maximum

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4

impact of non-local corrections is to be expected close

to the second-order transition line (which is reduced in

comparison to DMFT). This is because the correspond-

ing spin susceptibility, which explicitly enters in the DΓA

equations for the self-energy, is diverging at the transi-

tion (red line in the phase diagram). For the same reason,

non-local corrections should become weaker with increas-

ing temperature since AF-fluctuations are reduced in in-

tensity and spatial extension , see, e.g., the temperature

behavior of ξ in Fig. 2. Our aim is, hence, to provide an

estimate of the region of the paramagnetic phase in which

DMFT works and in which it does not. For this scope, we

take as our criterion the relative change between the DΓA

self-energy evaluated for a k-vector at the Fermi surface

(e.g., kF = (0,π

2,π)[22] ) and the corresponding (param-

agnetic) DMFT one at the lowest Matsubara frequency.

The results of our analysis are summarized in Fig. 3

where the whole region in the phase-diagram below the

violet line is characterized by a relative change between

the DMFT and the DΓA self-energy exceeding 10 %. In

this region, visible changes in the spectral functions, such

as loss of coherence of the QP peaks for U < 2D or an

extra-deepening of the pseudogap for U > 2D, are ex-

pected due to non-local correlations, and indeed are al-

ways seen when performing the corresponding analytic

continuations, e.g., with the Pad´ e method. Our findings

validates (a posteriori) the success of DMFT in describ-

ing electronic correlations in d = 3, provided one is not

interested to parameter regions close to (second-order)

magnetic instabilities. Let us just note here that, as one

can infer from the DΓA spectral functions of Ref. [15], the

situation is radically different for the corresponding two-

dimensional case[23], where DMFT calculations can be

considered reliable only in the high-temperature regime.

In conclusion, we have analyzed non-perturbatively the

effect of non-local correlations in the three dimensional

Hubbard model by means of DΓA. Our study confirms

that the local description of DMFT can be generally con-

sidered to be rather accurate in d = 3, except in the prox-

imity of second order phase transitions. In such regions,

however, spatial correlations strongly modify the physics

both at the level of spectral functions as well as of the

critical properties of the system. In this respect, DΓA is

a very powerful tool for studying the critical properties

beyond the MF/DMFT level: critical exponents of the

Hubbard model are found to be -within the error bars-

identical to those of the 3d Heisenberg model, and DΓA

provides also for a proper reduction of TN w.r.t. the

incorrect DMFT prediction. Moreover, since the DΓA

scheme includes both spatial and temporal electronic cor-

relations in a non-perturbative way, it looks naturally

very promising also for future analysis of quantum phase

transitions beyond the weak-coupling regime.

We acknowledge financial support from the EU net-

work MONAMI (GR), Austrian-Russian joint project

FWF I 610-N16 (AT), RFBR grants no. 10-02-91003-

ANF a, 11-02-00937-a and Max-Planck associated part-

ner group (AK), and FWF SFB ViCoM F41 (KH). Cal-

culations have been performed on the Vienna Scientific

Cluster (VSC).

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[18] An analogous argument applies to the superconducting

transition for the case U < 0 with |U| << t (weak-

coupling), explaining also why the violation of the MF

critical exponents are hardly observable in experiments

on conventional superconductors.

[19] The value of ξ has been computed by fitting the DΓA

spin susceptibility χ(q,Ω = 0) with the fitting function

χfit= A/[4(sin2(qx−π

2

)+sin2(qy−π

[20] Note that within Moriyasque DΓA, the index η is not

changed from its MF value (i.e., 0), since the explicitly

q-dependent terms of the spin-spin propagator (but not

its mass!) is computed at the level of DMFT.

[21] M.F. Collins, “Magnetic Critical Scattering”, Oxford

University Press, New York, 1989.

[22] Note that for the selected k−vector, the largest deviation

of the DΓA self-energy, w.r.t. DMFT, was observed.

[23] In fact, while in d = 2 the AF-instability is confined

to T = 0, its effects are much stronger than in d = 3

and visible quite far away from the instability itself. This

means that in d = 2, we would have drawn a phase-

diagram where the AF-line (red) would be dropping at

T = 0, while the violet line would be enhanced to include

the whole portion of the phase-diagram shown in Fig. 3.

2

)+sin2(qz−π

2

))+ξ−2].