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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 ?