monte carlo simulations of quantum systems with global...

Monte Carlo simulations of quantumsystems with global updates

Alejandro MuramatsuInstitut fur Theoretische Physik III

Universitat Stuttgart

Spinons, holons, and antiholons in one dimension

0 2 4 6 8 10

0

0

-6-4

-2 0

2 4

0

0

1

2

3

4

5

6

7

8

9

10

PSfragreplacements

n = 0:6 J=t = 0:5

!=t

k

A(k; !)

�

�

�

kF

kF

kF

EF

EF

5.1 The t-J model in one dimension


Phase diagramM. Ogata, M.U. Luchini, S. Sorella, and F.F. Assaad, Phys. Rev. Lett. 66, 2388 (1991)

1 2 3

separationgap

0.5(repulsive)

0

1

Spin

Luttinger liquid

Phase

J/t

n

ρ<1

Kρ>1

SC

ρK =1

K


Phase diagramM. Ogata, M.U. Luchini, S. Sorella, and F.F. Assaad, Phys. Rev. Lett. 66, 2388 (1991)

1 2 3

separationgap

0.5(repulsive)

0

1

Spin

Luttinger liquid

Phase

J/t

n

ρ<1

Kρ>1

SC

ρK =1

K

Phase diagram, with phases similar to those in high Tc superconductors

Exact results for the nearest neighbor t-J model


At J = 0One-particle spectral function from Ogata-Shiba wavefunctionK. Penc, K. Hallberg, F. Mila, and H. Shiba, Phys. Rev. Lett. 77, 1390 (1996)



2040

60

80

100

25

50

75

100

00.00050.001

0.00150.002

2040

60

80

100



2040

60

80

100

25

50

75

100

00.00050.001

0.00150.002

2040

60

80

100

Shadow bands



2040

60

80

100

25

50

75

100

00.00050.001

0.00150.002

2040

60

80

100

Shadow bands

At J = 2tBethe-Ansatz solutionP.A. Bares and G. Blatter, Phys. Rev. Lett. 64, 2567 (1990)No correlations functions

5.2 The 1/r2 (inverse square) t-J model in one dimension


HIS−t−J = −∑

i<j,σ

tij(c†i,σcj,σ + h.c.) +

∑

i<j

Jij

(

Si · Sj −14ninj

)

,

where at the supersymmetric (SuSy) point J = 2 t,

tij = Jij/2 = t(π/L)2/ sin2(π(i− j)/L)



i<j,σ


∑

i<j

Jij

(

Si · Sj −14ninj

)

,



Ground-state: Gutzwiller wavefunctionY. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67, 1338 (1991)



i<j,σ


∑

i<j

Jij

(

Si · Sj −14ninj

)

,




Compact support and excitation contentZ.N.C. Ha and F.D.M. Haldane, Phys. Rev. Lett. 73, 2887 (1994)

spinon Q=0, S=1/2 semionholon Q=+e, S=0 semionantiholon Q=–2e, S=0 boson



i<j,σ


∑

i<j

Jij

(

Si · Sj −14ninj

)

,




Compact support and excitation contentZ.N.C. Ha and F.D.M. Haldane, Phys. Rev. Lett. 73, 2887 (1994)

spinon Q=0, S=1/2 semionholon Q=+e, S=0 semionantiholon Q=–2e, S=0 boson

Semion −→ sjs` = is`sj −→ ‘half a fermion’

Spectral function for electron addition

M. Arikawa, Y. Saiga, and Y. Kuramoto, Phys. Rev. Lett. 86, 3096 (2001)

Spectral function for electron addition

M. Arikawa, Y. Saiga, and Y. Kuramoto, Phys. Rev. Lett. 86, 3096 (2001)

k

ωaU

kF

2kF

ωs

ωaL ωh

3

2

1

2π0

ω/t

π 2π−3kF

2π−2kF

3kF 2π−kF

Analytic expressions


Spectral function for electron addition (0 ≤ k < 2π)


Spectral function for electron addition (0 ≤ k < 2π)

A+(k, ω) = AR(k, ω) +AL(k, ω) +AU(k, ω)

where

AR(k, ω) =1

4π

∫ kF

0

dqh

∫ kF−qh

0

dqs

∫ 2π−4kF

0

dqa δ (k − kF − qs − qh − qa)

×δ [ω − εs(qs)− εh(qh)− εa(qa)]εgs−1s (qs) ε

gh−1h (qh) εga−1

a (qa)

(qh + qa/2)2 ,

with AL(k, ω) = AR(2π − k, ω)gs = 1/2, gh = 1/2, and ga = 2, statistical parameters, and

AU(k, ω) =

√

εa(k − 2kF )k(π − k/2)

δ{

ω − [εs(kF ) + εa(k − 2kF )]}

, (2kF ≤ k ≤ 2π − 2kF )

5.3 Spectral functions for the n.n. t-J model from QMCC. Lavalle, M. Arikawa, S. Capponi, F.F. Assaad, and A. Muramatsu,PRL 90, 216401, (2003)

0 2 4 6 8 10

0

-6-4

-2 0

2 4

0

0 5

10 15 20 25 30 35

PSfragreplacements

n = 0:6 J=t = 2

!=t

k

A(k; !)

�

�

kF

kF

EF

Spinon, holons, and antiholons in the n.n. t-J model at J=2t

0 KF 2 KF PI-8

-6

-4

-2

0

2

4

Y

ω

kF 2kF π

+ spinon: εsR(L)(qs) = tqs(

±v0s − qs

)

0 ≤ qs ≤ kF∗ holon: εhR(L)(qh) = tqh

(

qh ± v0c

)

0 ≤ qh ≤ kFantiholon: εh(qh) = tqh(2v0

c − qh)/2 0 ≤ qh ≤ 2π − 4kF

v0s = π, v0

c = π(1− n)

Excitation content of the hole spectrum at the supersymmetric pointM. Brunner, F.F. Assaad, and A. Muramatsu, Eur. Phys. J. B 16, 209 (2000)

Supersymmetric point J/t = 2 −→ exact holon and spinon dispersionsfrom Bethe-Ansatz

P.-A. Bares and G. Blatter, Phys. Rev. Lett. 64, 2567 (1990)P.-A. Bares, G. Blatter, and M. Ogata, Phys. Rev. B 44, 130 (1991)

38.018.125.96.255.655.495.585.464.863.843.492.941.891.471.421.261.021.020.980.880.860.780.740.660.640.630.680.851.071.201.491.451.45PSfrag replacements �

�2

00

�1�2�3�4�5

�6

�7

�8

1 2 3

4

5

!=t

Full line: One holon + one spinon withdispersions given by charge-spin separation Ansatz.

Crosses: One holon + one spinon withdispersions given by Bethe-Ansatz

.

Spectral function at J = 0.5 t

0 2 4 6 8 10

0

0

-6-4

-2 0

2 4

0

0

1

2

3

4

5

6

7

8

9

10

PSfragreplacements

n = 0:6 J=t = 0:5

!=t

k

A(k; !)

�

�

�

kF

kF

kF

EF

EF

Spinon, holons, and antiholons at J = 0.5 t and n = 0.6

0 KF 2 KF PI-8

-6

-4

-2

0

2

4

Y

ω

kF 2kF π

+ spinon: εsR(L)(qs) = (J/2) qs(

±v0s − qs

)

0 ≤ qs ≤ kF∗ holon: εhR(L)(qh) = tqh

(

qh ± v0c

)

0 ≤ qh ≤ kFantiholon: εh(qh) = tqh(2v0

c − qh)/2 0 ≤ qh ≤ 2π − 4kF

v0s = π, v0

c = π(1− n)

Charge-spin separation at J = 0.5 t

Photoemission I-Photoemission

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4

0-6 -4 -2 0 2 4 -4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6-4 -2 0 2 4 6

PSfrag

replacements

a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a)a) b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)b)

ω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/tω/t

πππππππππππππππππππππ

2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF2kF

kFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkFkF

EF

Antiholons at J = 0.5 t vs. doping

Antiholons at J = 0.5 t vs. doping

0

-6

-5

-4

-3

-2

-1

0

PSfrag

replacements

ρ = 1 J/t = 2ρ = 1 J/t = 2ρ = 1 J/t = 2

ω/tω/tω/t

πkF

2kFEF

0-6

-4

-2

0

2

4

PSfrag

replacements

ρ = 0.95 J/t = 2ρ = 0.95 J/t = 2ρ = 0.95 J/t = 2ρ = 0.95 J/t = 2ρ = 0.95 J/t = 2

ω/tω/tω/tω/tω/t

πkF

2kFEF

0

-6

-4

-2

0

2P

Sfragreplacem

ents



πkF 2kF

EF

0

-6

-4

-2

0

2

4

PSfrag

replacements



πkF 2kF

EF

0

-6

-4

-2

0

2

4

PSfrag

replacements



πkF 2kFE

F

0

-6

-4

-2

0

2

4

PSfrag

replacements



πkF 2kF

EF

Antiholons are generic to the nearest neighbor t-J model at finite J


• New excitation with Q = 2e, S = 0, and mah = 2mh

−→ not charge conjugated to holons




• Antiholons are bosons with Q = 2e −→ pre-formed pairs?





• Can they Bose-condensed for some J and n?





• Can they Bose-condensed for some J and n?

• Can they help to understand D = 2?

Photoemission and inverse photoemission with phase separation

0 2 4 6 8 10

0

-15 -10 -5 0 50 0

10

20

30

40

50

60

PSfragreplacements

n = 0:9 J=t = 4

!=t

k

A(k; !)�

�

kF

kF

EF

EF

Discontinuous spectrum on the photoemission side

5.4 Outlook of on-going and future work


• Spin and density correlation and spectral functions.−→ spin-gap and phase separation



• Correlation and spectral functions for superconductivity




• Feasibility of two-dimensional simulations




• Feasibility of two-dimensional simulations

• Extension to finite temperature

New representation and the minus-sign problem


Minus sign problem in one dimension: non-existent for N↑+N↓ = 4m+ 2,with m integer



Observation: < sign >> 0.95 for N↑ +N↓ = 4m




1 2 3 4 5 7 8 16




1 2 3 4 5 7 8 16

|↓> is a barrier for holes




1 2 3 4 5 7 8 16

|↓> is a barrier for holes

expect also less severe minus signproblem for J small, low doping, andlarge systems in two dimensions

Summary of Lecture 1


• Checkerboard decomposition −→ reduction to a 2-sites problem



• World-lines −→ useful in 1-D or with bosons in higher dimensions




• Local moves −→ inefficient (large autocorrelation times), possiblynon-ergodic.





• Fast Fourier transformation −→ alternative to checkerboarddecomposition.





• Fast Fourier transformation −→ alternative to checkerboarddecomposition.

• Jordan-Wigner transformation −→ from bosons to fermions in 1-DFor 2-D seeE. Frradkin, Phys. Rev. Lett. bf 63, 322 (1989)Y.R. Wang, Phys. Rev. B 43, 3786 (1991)


• Loop-algorithm −→ particular form of cluster algorithms.Statistical mechanics of graphsC. Fortuin and P. Kasteleyn, Physica 57, 536 (1972)



• Improved estimators −→ efficient reduction of noise




• Continuous imaginary time −→ no systematic error, very efficient forlarge systems

S-1/2 Heisenberg antiferromagnet up to 103 × 103 sitesJ.-K. Kim and M. Troyer, Phys. Rev. Lett. 80, 2705 (1998)




• Continuous imaginary time −→ no systematic error, very efficient forlarge systems

S-1/2 Heisenberg antiferromagnet up to 103 × 103 sitesJ.-K. Kim and M. Troyer, Phys. Rev. Lett. 80, 2705 (1998)

• Minus-sign problem −→ eliminated in special cases with the Meronmethod


• t-J model −→ strongly correlated fermions



• Canonical transformation −→ pseudospins + spinless holes




• Single hole dynamics and loop-algorithm −→ exact treatment of single holedynamics




• Single hole dynamics and loop-algorithm −→ exact treatment of single holedynamics

• Single hole in a 2-D quantum antiferromagnet−→ coherent quasiparticle with internal dynamics

holons and spinons confined by string potentialSelf-consistent Born approximation agrees very well with QMC


• Hybrid-loop algorithm −→ new algorithm for doped antiferromagnets



• Loop-algorithm −→ pseudospins




• Determinantal algorithm −→ holes





• Stabilization −→ like in simulations of the Hubbard model





• Stabilization −→ like in simulations of the Hubbard model

• Static and dynamical correlation functions −→ spectral functions


• t-J model in 1-D −→ rich phase diagram reminiscent of high Tcsuperconductors



• 1/r2 t-J model −→ free excitations with fractional statistics




• One-particle spectral functions −→ QMC for the nearest neighbort-J model vs. analytical resultsfor the 1/r2 t-J model




• One-particle spectral functions −→ QMC for the nearest neighbort-J model vs. analytical resultsfor the 1/r2 t-J model

• Antiholon −→ new generic excitation of the nearest neighbort-J model

Collaborators

• Dr. Michael Brunner (HypoVereinsbank-Munchen)

• Dr. Catia Lavalle (University of Stuttgart)

• Dr. Sylvain Capponi (Universite Paul Sabatier, Toulouse)

• Dr. Mitsuhiro Arikawa (University of Stuttgart)

• Prof. Dr. Fakher F. Assaad (University of Wurzburg)

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