Monte Carlo simulations of quantum systems with global...
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Monte Carlo simulations of quantumsystems with global updates
Alejandro MuramatsuInstitut fur Theoretische Physik III
Universitat Stuttgart
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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
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5.1 The t-J model in one dimension
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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
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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 diagram, with phases similar to those in high Tc superconductors
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Exact results for the nearest neighbor t-J model
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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)
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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
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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
Shadow bands
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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
Shadow bands
At J = 2tBethe-Ansatz solutionP.A. Bares and G. Blatter, Phys. Rev. Lett. 64, 2567 (1990)No correlations functions
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5.2 The 1/r2 (inverse square) t-J model in one dimension
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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)
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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)
Ground-state: Gutzwiller wavefunctionY. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67, 1338 (1991)
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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)
Ground-state: Gutzwiller wavefunctionY. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67, 1338 (1991)
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
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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)
Ground-state: Gutzwiller wavefunctionY. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67, 1338 (1991)
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’
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Spectral function for electron addition
M. Arikawa, Y. Saiga, and Y. Kuramoto, Phys. Rev. Lett. 86, 3096 (2001)
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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
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Analytic expressions
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Analytic expressions
Spectral function for electron addition (0 ≤ k < 2π)
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Analytic expressions
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 )
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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
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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)
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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
.
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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
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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)
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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
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Antiholons at J = 0.5 t vs. doping
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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
ρ = 0.9 J/t = 2ρ = 0.9 J/t = 2ρ = 0.9 J/t = 2ρ = 0.9 J/t = 2ρ = 0.9 J/t = 2
ω/tω/tω/tω/tω/t
πkF 2kF
EF
0
-6
-4
-2
0
2
4
PSfrag
replacements
ρ = 0.8 J/t = 2ρ = 0.8 J/t = 2ρ = 0.8 J/t = 2ρ = 0.8 J/t = 2ρ = 0.8 J/t = 2
ω/tω/tω/tω/tω/t
πkF 2kF
EF
0
-6
-4
-2
0
2
4
PSfrag
replacements
ρ = 0.7142 J/t = 2ρ = 0.7142 J/t = 2ρ = 0.7142 J/t = 2ρ = 0.7142 J/t = 2ρ = 0.7142 J/t = 2
ω/tω/tω/tω/tω/t
πkF 2kFE
F
0
-6
-4
-2
0
2
4
PSfrag
replacements
ρ = 0.6 J/t = 2ρ = 0.6 J/t = 2ρ = 0.6 J/t = 2ρ = 0.6 J/t = 2ρ = 0.6 J/t = 2
ω/tω/tω/tω/tω/t
πkF 2kF
EF
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Antiholons are generic to the nearest neighbor t-J model at finite J
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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
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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?
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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?
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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 help to understand D = 2?
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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
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5.4 Outlook of on-going and future work
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5.4 Outlook of on-going and future work
• Spin and density correlation and spectral functions.−→ spin-gap and phase separation
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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
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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
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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
• Extension to finite temperature
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New representation and the minus-sign problem
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New representation and the minus-sign problem
Minus sign problem in one dimension: non-existent for N↑+N↓ = 4m+ 2,with m integer
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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
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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
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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
|↓> is a barrier for holes
![Page 45: Monte Carlo simulations of quantum systems with global updatesscs.physics.unisa.it/TCVIII/Download/Muramatsu/Lecture5.pdf · 5.1 The t-J model in one dimension Phase diagram M. Ogata,](https://reader031.fdocuments.us/reader031/viewer/2022011922/6044441b54b65554ff7f2d3c/html5/thumbnails/45.jpg)
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
|↓> is a barrier for holes
expect also less severe minus signproblem for J small, low doping, andlarge systems in two dimensions
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Summary of Lecture 1
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Summary of Lecture 1
• Checkerboard decomposition −→ reduction to a 2-sites problem
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Summary of Lecture 1
• Checkerboard decomposition −→ reduction to a 2-sites problem
• World-lines −→ useful in 1-D or with bosons in higher dimensions
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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.
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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.
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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.
• 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)
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Summary of Lecture 2
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Summary of Lecture 2
• Loop-algorithm −→ particular form of cluster algorithms.Statistical mechanics of graphsC. Fortuin and P. Kasteleyn, Physica 57, 536 (1972)
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Summary of Lecture 2
• 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
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Summary of Lecture 2
• 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)
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Summary of Lecture 2
• 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)
• Minus-sign problem −→ eliminated in special cases with the Meronmethod
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Summary of Lecture 3
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Summary of Lecture 3
• t-J model −→ strongly correlated fermions
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Summary of Lecture 3
• t-J model −→ strongly correlated fermions
• Canonical transformation −→ pseudospins + spinless holes
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Summary of Lecture 3
• t-J model −→ strongly correlated fermions
• Canonical transformation −→ pseudospins + spinless holes
• Single hole dynamics and loop-algorithm −→ exact treatment of single holedynamics
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Summary of Lecture 3
• t-J model −→ strongly correlated fermions
• Canonical transformation −→ pseudospins + spinless holes
• 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
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Summary of Lecture 4
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Summary of Lecture 4
• Hybrid-loop algorithm −→ new algorithm for doped antiferromagnets
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Summary of Lecture 4
• Hybrid-loop algorithm −→ new algorithm for doped antiferromagnets
• Loop-algorithm −→ pseudospins
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Summary of Lecture 4
• Hybrid-loop algorithm −→ new algorithm for doped antiferromagnets
• Loop-algorithm −→ pseudospins
• Determinantal algorithm −→ holes
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Summary of Lecture 4
• Hybrid-loop algorithm −→ new algorithm for doped antiferromagnets
• Loop-algorithm −→ pseudospins
• Determinantal algorithm −→ holes
• Stabilization −→ like in simulations of the Hubbard model
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Summary of Lecture 4
• Hybrid-loop algorithm −→ new algorithm for doped antiferromagnets
• Loop-algorithm −→ pseudospins
• Determinantal algorithm −→ holes
• Stabilization −→ like in simulations of the Hubbard model
• Static and dynamical correlation functions −→ spectral functions
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Summary of Lecture 5
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Summary of Lecture 5
• t-J model in 1-D −→ rich phase diagram reminiscent of high Tcsuperconductors
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Summary of Lecture 5
• t-J model in 1-D −→ rich phase diagram reminiscent of high Tcsuperconductors
• 1/r2 t-J model −→ free excitations with fractional statistics
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Summary of Lecture 5
• 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
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Summary of Lecture 5
• 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
• Antiholon −→ new generic excitation of the nearest neighbort-J model
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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)