The variational cluster method

David Senechal

Universite de SherbrookeDepartement de physique

CRM workshop on Quantum Information in QuantumMany-Body Physics, Oct. 18 2011

Outline

I Clusters and Cluster Perturbation Theory (CPT)I The Self-Energy Functional ApproachI The Variational Cluster Approximation (VCA)

Part I

Cluster Perturbation Theory

How do you solve a problem like the Hubbard Model?

H = Âi,j,s

tijc†iscjs + U Â

ini"ni# � µ Â

ini

I Wanted: GS + one-particle propertiesI The Green function Gµn(w):

Gµn(w) = Gµn,e(w) + Gµn,h(w)

Gµn,e(w) = hW|cµ1

w� H + E0c†

n|Wi

Gµn,h(w) = hW|c†n

1w + H � E0

cµ|Wi

I Susceptibilities (2-particle properties) areinteresting too. CuO2 plane in high-Tc materials

Clusters methods

Based on a tiling of the lattice (= superlattice)

e1

e2

10-site cluster making up a superlattice

(�⇡,�⇡)

(⇡, ⇡)

(0, 0)

K

˜k

k

Reduced Brillouin zone

Partial Fourier transforms

i, j : lattice site index k : full wavevector

m, n : superlattice site index k : reduced wavevectora, b : cluster site index K : cluster wavevector

f j = Âk

eik·rj f (k) f (k) =1N Â

je�ik·rj f j

fm = Âk

eik·rm f (k) f (k) =LN Â

me�ik·rm fm

fa =1pL

ÂK

eiK·ra fK fK =1pL

Âa

e�iK·ra fa

N : number of sites in the lattice (! •)L : number of sites in the cluster (small, e.g. 16)

CPT: Basic Idea

H

lattice Hamiltonian = H0! cluster Hamiltonian

+ V! inter-cluster hopping terms

t

hopping matrix = t

0! cluster hopping matrix

+ V

! inter-cluster hoppingI Treat V at lowest order in Perturbation theoryI At this order, the Green function is

G

�1(w) = G

0

! cluster Green function

�1(w)� V

C. Gros and R. Valenti, Phys. Rev. B 48, 418 (1993)

D. Senechal, D. Perez, and M. Pioro-Ladriere. Phys. Rev. Lett. 84, 522 (2000)

Basic Idea (cont.)

I In a mixed representation (reduced wavevector + cluster indices):

G

�1(k, w) = G

0�1(w)� V(k) .

I But

G

0�1 = w� t

0 �⌃

G0�1 = w� t

0 � V ,

I Thus : lattice self-energy is approximated as the cluster self-energy

G

�1(k, w) = G0�1(k, w)�⌃(w) ,

I Example : 2-site cluster (1D):

t

0 = �t

0 11 0

!

V(k) = �t

0 e�2ik

e2ik 0

!

Periodization

I CPT breaks translation invariance, which can be “restored” bycompleting the Fourier transform:

Gcpt(k, w) =1L Â

a,be�ik·(ra�rb)Gab(k, w) .

I Periodizing the Green function is the correct choice, from what weknow of the 1D Hubbard model:

�6 �3 0 3 6

!

(0)

(⇡/2)

(⇡)

Green function periodization

�6 �3 0 3 6

!

(0)

(⇡/2)

(⇡)

Self-energy periodization

One-dimensional example

Evolution of spectral function with increasing U/t:

Application to high-Tc superconductors

10%20% 20%10%

R2-xCexCuO4La2-xSrxCuO4

dopage aux électrons / conc. de Ce (x)dopage aux trous / conc. de Sr (x)

SC SC

AFAF

~ 300K

~ 30K

0

CPT : features

I Exact at U = 0I Exact at tij = 0I Exact short-range correlationsI Allows a continuum of wavevectorsI But : No long-range order or broken symmetryI Controlled by the size of the cluster

I Finite-size effects are importantI No self-consistency (unlike dynamical mean field theory)

Part II

The self-energy functional approach

Motivation

I CPT cannot describe broken symmetry states, because of the finitecluster size

I Idea : Add a Weiss field to the cluster Hamiltonian H0, e.g., forantiferromagnetism:

H0M = M Âa

eiQ

! (p, p)

·ra(na" � na#)

I This term favors AF order, but does not appear in H, and must besubtracted from V

I Need a principle to set the value of M : energy minimization?I Better : Potthoff’s self-energy functional approach

A variational principle for the Green function

I Define a functional of G:

W

! Grand potential functional

t[G] = F[G]� Tr ((G�10t � G

�1)G) + Tr

! Tr (A) = Âw,a Aaa(w)

ln(�G).I F[G] is the Luttinger-Ward functional (2PI diagrams):

� =

+ + + · · ·

I . . . with the propertydF[G]

dG= ⌃

M. Pottho↵, Eur. Phys. J. B 32, 429-436 (2003)

A variational principle for the Green function (cont.)

I The functional is stationary at the physical Green function (Euler eq.):

dWt[G]dG

= ⌃� G

�10t + G

�1 = 0.

I Its value at the stationary point is the grand potential:

W = E� µN

I Approximation schemes:I Type I : Simplify the Euler equationI Type II : Approximate the functional (Hartree-Fock, FLEX)I Type III : Restrict the variational space, but keep the functional exact

The Potthoff variational method

I Potthoff : Use the self-energy rather than the Green function

Wt[⌃] = F[⌃]� Tr ln(�G�10t +⌃)

F[⌃] = F[G]� Tr (⌃G)

I F is the Legendre transform of F:

dF[⌃]d⌃

=dF[G]

dG

dG[⌃]d⌃

�⌃

dG[⌃]d⌃

� G = �G

I New Euler equation:

dWt[⌃]d⌃

= �G+ (G�10t �⌃)�1 = 0

The Potthoff variational method: Reference System

I To evaluate F, use its universal character : its functional formdepends only on the interaction part.

I Introduce a reference rystem H0, which differs from H by one-bodyterms only (example : the cluster Hamiltonian)

I Suppose H0 can be solved exactly. Then, at the physical self-energy ⌃

of H0,W0 = F[⌃]� Tr ln(�G0)

I by eliminating F:

Wt[⌃] = W0 + Tr ln(�G0)� Tr ln(�G�10t +⌃)

= W0 + Tr ln(�G0)� Tr ln(�G)= W0 � Tr ln(1� VG

0)

The Potthoff functional

I Making the trace explicit, one finds

Wt[⌃] = W0 � T Âw

Âk

tr ln⇥

1� V(k)G0(k, w)⇤

= W0 � T Âw

Âk

ln det⇥

1� V(k)G0(k, w)⇤

I The sum over frequencies is to be performed over Matsubarafrequencies (or an integral along the imaginary axis at T = 0).

I The variation is done over one-body parameters of the clusterHamiltonian H0

I In particular, the Weiss field M is to be varied until W is stationary

Part III

The Variational ClusterApproximation

Basic Idea

I Set up a superlattice of clustersI Choose a set of variational parameters, e.g. Weiss fields for broken

symmetriesI Set up the calculation of the Potthoff functional:

Wt[⌃] = W0 � TLN Â

wÂk

ln det⇥

1� V(k)G0(k, w)⇤

I Use an optimization method to find the stationary pointsI Adopt the cluster self-energy associated with the stationary point

with the lowest W and use it as in CPT

Example : Neel Antiferromagnetism

I Used the Weiss field

H0M = M Âa

eiQ

! (p, p)

·ra(na" � na#)

I Profile of W for the half-filled, square lattice Hubbard model:

�0.3 �0.2 �0.1 0.0 0.1 0.2 0.3M

-0.1

-0.08

-0.06

-0.04

-0.02

0

⌦

U = 2

U = 4

U = 8

U = 16

0.0 0.1 0.2 0.3M

-4.51

-4.5

-4.49

-4.48

-4.47

-4.46

-4.45

-4.44

⌦

L = 4

L = 12

Example : Neel Antiferromagnetism (2)

0 5 10 15 20

U

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M (4⇥)

order parameter

ordering energy

Best scaling factor :

q =number of links

2⇥ number of sites2x1

2x2

2x3

2x4

B10

3x4

4x4

linksê2L

2x1

2x2

2x3

2x4

B10

3x4

4x4

1-1êL

0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

M

2x1

2x2

2x3

2x4

B10

3x4

4x4

linksê2L

2x1

2x2

2x3

2x4

B10

3x4

4x4

1-1êL

0.2 0.4 0.6 0.8 1.00.75

0.80

0.85

0.90

0.95

1.00

scaling factor

OrderParameter

Superconductivity

I Need to add a pairing field

Osc = Âij

Dijci"cj# + H.c

I s-wave pairing: Dij = dijI dx2�y2 pairing:

Dij =

(

1 if ri � rj = ±x

�1 if ri � rj = ±y

I dxy pairing:

Dij =

(

1 if ri � rj = ±(x + y)

�1 if ri � rj = ±(x� y)

0.0 0.1 0.2 0.3�

-1.82

-1.81

-1.8

-1.79

-1.78

-1.77

-1.76

⌦

dx

2�y

2

extended s-wave

s-wave

dxy

2⇥ 2 cluster

U = 8, µ = 1.2

Superconductivity (2)

I Pairing fields violate particle number conservationI The Hilbert space is enlarged to encompass all particle numbers with

a given spinI In practice, on uses the Nambu formalism, i.e., particle-hole

transformation on the spin-down sector :

ca = ca" and da = c†a#

Then the Hamiltonian looks like it conserves particle number, but notspin.

Superconductivity and Antiferromagnetism in thecuprates

I One-band Hubbard model for the cuprates: t0 = �0.3, t00 = 0.2,U = 8:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

orderparameter

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6n

pure dx

2�y

2

pure Neel AF

coex. Neel AF

coex. dx

2�y

2

M. Guillot, MSc thesis, Univ. de Sherbrooke (2007)

Thermodynamic consistency

I The electron density n may be calculated either as

n = TrG or n = �∂W∂µ

I The two methods give different results, except if the cluster chemicalpotential µ0 is treated like a variational parameter:

2⇥ 2 clusterU = 8normal state

0.5

0.6

0.7

0.8

0.9

1

n

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8µ

�@⌦

@µ

�

�

�

�

cons.

TrGcons.

TrG

�@⌦

@µ

Optimization procedure

I Need to find the saddle points of W(x) with the least possibleevaluations of W(x)

I Use the Newton-Raphson algorithm:I Evaluate W at a number of points at and around x0 that just fits a quadratic

formI Move to the stationary point x1 of that quadratic form and repeatI Stop when |xi � xi�1|, or the numerical gradient |rW|, converges

I The NR method is not robust : it converges fast when started closeenough to the solution

I Proceed adiabatically through external parameter space (e.g. asfunction of U or µ)

Example: Homogeneous coexistence of dSC and AForders

SFA and CDMFT : the Mott transition

cU

cU

M. Balzer et al., Europhys. Lett. 85 (2009),17002.

The SFA is also applied to systemsusually treated by DMFT orcluster DMFT.

t’ t’

t’

t’

V V

V V

t’’

t’’

t’’

t’’V

V6¡

<6¡

(A) (B)

2x2 cluster with bath

VCA vs Mean-Field Theory

I Differs from Mean-Field Theory:I Interaction is left intact, it is not factorizedI Retains exact short-range correlationsI Weiss field 6= order parameterI More stringent that MFTI Controlled by the cluster size

I Similarities with MFT:I No long-range fluctuations (no disorder from Goldstone modes)I Yet : no LRO for Neel AF in one dimensionI Need to compare different orderings or broken symmetry statesI yet : they may be placed in competition / coexistence

QUESTIONS ?