Understanding correlations and localization in the Hubbard model ...

Understanding correlations and
localization in the Hubbard model
from variational calculations
Balázs Hetényi
Institut für Theoretische Physik, Technische Universität Graz, Austria
Understanding correlations and localization in the Hubbard model from variational calculations – p. 1

Co-workers
H. G. Evertz: Institut für theoretische Physik, TU Graz
W. von der Linden: Institut für theoretische Physik, TU Graz
Understanding correlations and localization in the Hubbard model from variational calculations – p. 2

Outline
Introduction: the Hubbard model, variational wavefunctions
The role of the exchange hole in the Gutzwiller approximation
Combinatorial approximation for the Baeriswyl wavefunction
The question of the total position in periodic systems
Implications for the theory of polarization
Understanding correlations and localization in the Hubbard model from variational calculations – p. 3

The Hubbard model
Breakdown of band theory: correlation
Hamiltonian:
H =
hopping H t
{ }} {
−t ∑
{c † i,σ c j,σ + H.c.} +
〈i,j〉σ
interaction H
{ }} { U
U ∑ n i↑ n i↓
i
Limiting cases: Mott metal-insulator transition
Exact solution known in 1D via Bethe ansatz (Lieb
and Wu, 1968): insulator for finite U.
High-T c superconductivity
Understanding correlations and localization in the Hubbard model from variational calculations – p. 4

Variational approach
Gutzwiller wavefunction (GWF):
|Ψ G (γ)〉 = exp(−γ ∑ i
n i↑ n i↓ )|FS〉
Baeriswyl wavefunction (BWF):
|Ψ B (α)〉 = exp
⎧
⎨
⎩ −α ⎛
⎝ ∑
〈i,j〉σ
⎞⎫
⎬
c † iσ c jσ + H.c. ⎠
⎭ |Ψ G(γ → ∞)〉
Understanding correlations and localization in the Hubbard model from variational calculations – p. 5

Gutzwiller wavefunction
Approximate combinatorial solution:
Brinkman-Rice transition (U c = 8|¯ǫ|)
Exact solution in 1D for GWF metallic behaviour
for finite U (Metzner and Vollhardt, 1987)
GWF is metallic due to the lack of phase
dependence of the projector operator (Millis and
Coppersmith, 1991)
Antiferromagnetic correlations are well reproduced
by GWF (Horst et al., 1982)
Understanding correlations and localization in the Hubbard model from variational calculations – p. 6

Gutzwiller approximation
Brinkman-Rice transition: transition from a
paramagnetic metal to a paramagnetic insulator.
The Gutzwiller approximation (GA) corresponds to
the exact GWF in the ∞-D limit (Metzner and
Vollhardt, PRB, 1988; Kotliar and Ruckenstein, PRL
1986)
GA neglects explicit spin correlation
(configurational averages of the non-interacting
reference system enter as combinatorial factors), i.e.
the exchange hole is
〈FS|n iσ n jσ ′|FS〉 ≈ 1 − δ σσ ′δ ij .
Understanding correlations and localization in the Hubbard model from variational calculations – p. 7

Exchange hole
1
0,8
g(r)
0,6
0,4
0,2
1D
2D
3D
Infinite D
0
0 2 4
r/lattice parameter
Figure 1:
Two-body density for a system of spinless
fermions of various dimensions.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 8

Classification of hoppings
〈Ψ| |Ψ〉 ∆D
↑ 0 → 0 ↑ 0
↑ ↓ → 0 ↑↓ +1
↑↓ 0 → ↓ ↑ −1
↑↓ ↓ → ↓ ↑↓ 0
Allows writing the hopping term as a sum of different
classes of hoppings:
H t = H 0 t + H + t + H − t
Understanding correlations and localization in the Hubbard model from variational calculations – p. 9

Gutzwiller wavefunction
The GWF can be solved by Monte Carlo sampling (Yokoyama and
Shiba, JPSJ, 1986).
Distribution sampled:
P GWF [Ω] = |Det[k;g Ω ]| 2 |Det[l;h Ω ]| 2 exp[−2γD(Ω)].
Hopping estimator (〈Ψ|c † i↑ c j↑|Ψ〉):
χ ij↑
GWF (Ω) = −tDet∗ [k;g ′ ij
Ω]/Det ∗ [k;g Ω ] ×
exp[−γ∆D(g ′ ij
Ω,g Ω ;h Ω )].
Understanding correlations and localization in the Hubbard model from variational calculations – p. 10

Gutzwiller approximation
Two step approximation:
Replace Det[k;g Ω ]| 2 and Det[l;h Ω ]| 2 by configurational
averages, i.e.
P GA [Ω] = exp[−2γD(Ω)],
but average over configurations with one or zero particles of a
particular spin at each site (Pauli principle).
Replace the hopping estimator with
χ ij↑
GA (Ω) = Aexp[−γ∆D(g′ ij
Ω,g Ω ;h Ω )],
and scale A so that the result is exact at U = 0.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 11

GA-X
We have investigated implementing the exact exchange hole into the
GA.
Distribution from exact GWF:
P GWF [Ω] = |Det[k;g Ω ]| 2 |Det[l;h Ω ]| 2 exp[−2γD(Ω)].
Hopping estimator from GA:
χ ij↑
GA (Ω) = Ãexp[−γ∆D(g′ (ij)
Ω ,g Ω;h Ω )],
and scale
à so that the result is exact at U = 0.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 12

GA: Brinkman-Rice transition
|Energy|/(t*N)
1
0,1
0,01
0,001
N=36
N=60
N=120
N=180
GA (N=∞)
|Hopping|/(t*N)
1
0,1
0,01
0 5 10 15 20
U/t
0 5 10 15 20
U/t
Figure 2: Total energy and hopping of the GA for different
sizes including the thermodynamic limit.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 13

Comparison: GWF, GA, GA-X
Hopping energy/t
0
-10
-20
-30
-40
-50
-60
-70
Exact GWF
GA-X
GA
-80
0 1 2 3 4 5
γ
Figure 3: Comparison of the hopping energy for the exact
GWF, GA-X, and GA calculations for 60 lattice sites.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 14

GA-X vs. GWF with ∞-sites
0
Energy/t
-0,5
-1
-1,5
N=12
N=24
N=36
N=48
N=60
Exact
|Energy|/t
1
0,2
0,04
0 5 10 15 20
U/t
0 5 10 15 20
U/t
Figure 4: Comparison of the GA-X calculations with the
exact solution for the GWF in the thermodynamic limit
(Metzner and Vollhardt, PRL, 1990).
Understanding correlations and localization in the Hubbard model from variational calculations – p. 15

Hopping for GA-X
1
|Hopping energy|/(N*t)
0,1
0,01
N=36
N=48
N=60
0 5 10 15 20
U/t
Figure 5: Comparison of the hopping energy based on
GA-X calculations for 36, 48, and 60 lattice sites.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 16

Antiferromagnetism
0,03
0,02
Exact GWF
GA-X
GA
0,01
0
0 5 10 15 20
U/t
Figure 6: Comparison of the average of the square of the
staggered magnetism for GWF,GA-X,GA.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 17

Questions
We know that
in infinite dimensions the exact GWF and the GA coincide.
in infinite dimensions the exchange hole implicit in the GA is
exact.
We found in that
in 1D implementing the exact hole into the GA results in
quantitative agreement with the exact GWF results.
Question: can one generalize this conclusion?
Understanding correlations and localization in the Hubbard model from variational calculations – p. 18

Uses of the GA
Time-dependent GA (Seibold and Lorenzana,
(2001))
Multi-orbital Hubbard model (Bünemann et al. )
Varying particle number (Edegger et al. 2005)
Resonating valence bond method (Fazekas and
Anderson, 1974)
...applied to high-T c (Anderson et al. (2003))
GA was the first infinite dimensional solution which
inspired the DMFT method
Understanding correlations and localization in the Hubbard model from variational calculations – p. 19

GA applied to BWF
Assumption:
In the U = ∞ state the momentum states are evenly
distributed, but otherwise uncorrelated
Combinatorial approximation:
Include configurations with only one particle in each
momentum state kσ and assume probability
P GA−B [Γ] = exp[−2α ∑ k Γ l Γ
(ǫ kΓ + ǫ lΓ )].
Understanding correlations and localization in the Hubbard model from variational calculations – p. 20

GA applied to BWF: hopping
〈 ∑
k
ǫ k ñ k,σ
〉
≈
∑
Γ P GA−B[Γ]{ ∑ k Γ ,l Γ
(ǫ kΓ + ǫ lΓ )}
∑
Γ P ,
GA−B[Γ]
Understanding correlations and localization in the Hubbard model from variational calculations – p. 21

GA applied to BWF: interaction
Rewrite hopping in k space
(
∑
n i↑ n i↓ = 1 ∑
ñ k↑ ñ k
L
′ ↓ − ∑
k,k ′
i
kk ′ q≠0
˜c † k↑˜c† k ′ ↓˜c k+q↑˜c k ′ −q↓
)
.
〈
〉
˜c † k↑˜c† k ↓˜c k+q↑˜c ′ k ′ −q↓
=
B
∑ ′′
Γ P GA−B[Γ]{e −α(ǫ k Γ
+ǫ k ′ −ǫ Γ kΓ +q Γ
−ǫ k ′ Γ −q ) Γ }
∑
Γ P ,
GA−B[Γ]
Understanding correlations and localization in the Hubbard model from variational calculations – p. 22

GA-GWF vs. GA-BWF
GA-GWF
GA-BWF
Space sampled: r-space k-space
Configurations: one particle per rσ one particle per kσ
Distribution: e −2γD(Ω) e −2α P k Γ l Γ
(ǫ(k Γ )+ǫ(l Γ ))
Hopping:
Ae −γ∆D(g′ ij
Ω ,g Ω;h Ω )
∑k Γ l Γ
(ǫ kΓ + ǫ lΓ )
Interaction: UD(Ω) Be −α(ǫ k Γ
+ǫ k ′ Γ
−ǫ kΓ +q Γ
−ǫ k ′ Γ −q Γ
)
Understanding correlations and localization in the Hubbard model from variational calculations – p. 23

12 site system
0
Kinetic energy
-5
-10
-15
-20
Double occ.
3
2.5
2
1.5
1
0.5
Exact
Approx.
0
0 1 2 3
α
Figure 7: Exact and GA-B kinetic and interaction energies
for a 12 site system as a function of the variational
parameter.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 24

Energy of GA-B
0
-0.1
U=4.5
U=4.05
U=3.5
Energy/t
-0.2
-0.3
-0.4
0 1 2 3 4 5
α
Figure 8: Energy as a function of the variational parameter
for different values of the interaction strength.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 25

Energy of GA-B
Energy/t
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
GA
Exact Gutzwiller
Exact (Lieb & Wu)
GA-B
0 2 4 6 8 10 12 14 16 18 20
U/t
Figure 9: Energy as a function of the interaction strength.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 26

Localization
U 2
0.667
0.666
0.665
0.664
0.663
0.662
L=120
L=160
0.25
U/t
0 5 10 15 20
0.2
0.15
0.1
0.05
0
5 10
U/t
15 20
D
Figure 10: Second order Binder cumulant and double occupation
as a function of interaction strength.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 27

Metal-insulator transition
GA-GWF
GA-BWF
Order: continuous discontinuous
Metallic state: D ≤ 0.25 D = 0.25
Insulator: D = 0 D ≥ 0
Understanding correlations and localization in the Hubbard model from variational calculations – p. 28

Questions
Characteristics of the exchange hole in k-space
Solution for D = ∞?
Combined GA type approximation for the
Baeriswyl-Gutzwiller wavefunction?
Lattice quantum Monte Carlo method for the BWF
(Yokoyama and Shiba type)?
Understanding correlations and localization in the Hubbard model from variational calculations – p. 29

Localization: criterion for insulator
W. Kohn, Theory of the Insulating State, Phys. Rev.
133 A171 (1964).
Metallicity and insulating behavior can be
distinguished by the extent of localization in the
system in the many-body configuration space
How do we quantify the extent of localization in
practical situations?
Most often periodic boundary conditions are
used.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 30

Position operator in periodic systems
In a periodic system the position operator x ill-defined:
The space on which x is defined is not the same as
the space on which a periodic wavefunction φ(x) is
defined. ∫
dxφ(x) 2 x 2 = ∞
even if φ(x) is normalizable.
For single particle states one solution (Blount, 1962)
is to write the wave-function as a linear combination
of Bloch functions and obtain an equivalent
operator:
xf n = i ∂ f n + ∑ Understanding correlations and localization χ nn in the
′fHubbard n ′. model from variational calculations – p. 31

Position operator in periodic systems
Resta (1998) suggests evaluating
〈
Z L = exp
(i 2π )〉
L x
.
and then obtain the expectation value of the position
operator as
and the spread as
〈x〉 = L 2π Im lnZ L,
〈x 2 〉 − 〈x〉 2 ≈ −
( L
2π
) 2
ln|Z L |.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 32

Position operator in periodic systems
Compound momentum space shift operator U:
(
Uc jσ = exp i 2πx )
j
c jσ U.
L
Total position operator can be defined as:
X =
∑L−1
m=1
( 1
2 + U m
e −i2πm L − 1
)
.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 33

Position operator in periodic systems
Operator X is a many-body operator satisfying three
conditions:
X is Hermitian
X is the generator of compound momentum shifts:
(
U = exp i 2πX )
L
The time derivative of X corresponds to the total
current:
Ẋ = i[H, X] = −it ∑
(c † iσ c jσ − c † jσ c iσ).
〈i,j〉σ
Understanding correlations and localization in the Hubbard model from variational calculations – p. 34

Calculations
Brinkman-Rice
Metallic Gutzwiller:
|Ψ G 〉 = exp (−α ∑ i n i↑n i↓ ) |FS〉
Insulating Gutzwiller:
|Ψ G 〉 = exp (−α ∑ i n i↑n i↓ ) |AFM〉
Quantities calculated:
χ 4 = √ 〈X 4 〉 − 〈X 2 〉 2 /L 2
U 4 = 1 − 〈X4 〉
3〈X 2 〉 2
Understanding correlations and localization in the Hubbard model from variational calculations – p. 35

Brinkman-Rice
0.3
0.2
L=60
L=120
L=180
χ 4
0.1
0
0 5 10 15 20
U/t
Figure 11: χ 4 as a function of the variational parameter γ
for the Gutzwiller approximation (Brinkman-Rice transition).
Understanding correlations and localization in the Hubbard model from variational calculations – p. 36

Metallic Gutzwiller
0.4
χ 4
0.3
0.2
U 4
0.65
0.6
0.55
0.1
L=36
L=48
L=60
L=72
0 2 4 6
γ
0
0 2 4 6 8
γ
Figure 12: χ 4 and U 4 as a function of the variational parameter
γ for a metallic wavefunction.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 37

Insulating Gutzwiller
χ 4
0.2
0.15
0.1
0.05
0
L=36
L=48
L=60
L=72
U 4
0.67
0.66
0.65
0.64
0.63
0.62
0 2 4
γ
0 2 4
γ
Figure 13: χ 4 and U 4 as a function of the variational parameter
γ for an insulating wavefunction.
Understanding correlations and localization in the Hubbard model from variational calculations – p. 38

Conclusions
Metallicity and antiferromagnetism are introduced
into the Gutzwiller approximation via the exchange
hole.
Approximation scheme gives metal-insulator
transition for the Baeriswyl wavefunction.
Total position operator (for a many body system)
can be derived such that all of its moments can be
evaluated
Understanding correlations and localization in the Hubbard model from variational calculations – p. 39

Acknowledgments
FWF
HPC-Europa2 Grant
Understanding correlations and localization in the Hubbard model from variational calculations – p. 40