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8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
http://slidepdf.com/reader/full/stefan-wessel-tensor-renormalization-group-beyond-the-ising-model 1/25
1
Tensor
Renormalization Group - Beyond the Ising Model
Stefan Wessel
Institut für Theoretische Physik IIIUniversität Stuttgart
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Collaborators Lars Bonnes
Hans Peter Büchler
L. Bonnes, H.P. Büchler, S.W., New Journal of Physics 12, 052027 (2010)
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Overview
TRG for Classical Systems Previous Applications
Three-body Interactions
Model and Phases
Results from TRG
Discussion
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Tensor NetworksClassical partition function
∑−
=}{
})({
i
i E e Z
σ
σ β
⋅⋅⋅= ∑=
D
k ji
kqr jnpilmijk T T T T 1,...,,
Cyclically symmetric tensor of rank z (=3)
and dimension D: Dk jiT ijk ,...,1,, , =
M. Levin, C. P. Nave, Phys. Rev. Lett. 99, 120601 (2007)
Z. Y. Xie
et al., Phys. Rev. Lett. 103, 160601 (2009)
H. H. Zhao et al ., Phys. Rev. B 81, 174411 (2010)
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Standard Example:
Ising Model on the Triangular Lattice
Bond variables: “domain wall” indicator
∑><
−= ji
jii SS J S E ,
})({
jiijb SS==><
σ σ
)(
2
1 z y x J z y x
xyz eT σ σ σ β
σ σ σ +++
= 2= D
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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TRG Scheme IIterative two-step coarse-graining procedure
1. 2.
1. 2.
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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TRG Scheme II1. Rewiring
Singular Value Decomposition
ν ν )(mimiv U S Λ=D2×D2
matrix
ν
ν
ν ν ν
ν
ν jl
D
mi jl
D
mi
D
k
klmijk jlmi SSV U T T M ∑∑∑===
=Λ==
22
1
)(
1
)(
1
))((
ν ν ν )( jl jl V S Λ=
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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TRG Scheme III2. Decimation
Keep only the largest Dmax singular values in S
Renormalized tensor
∑=
=max
1,,
' D
him
ihmihm SSST δ ν γ γδν
1. 2.
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Example:
Ising Model on the Triangular LatticeTracking the largest tensor elements during the
TRG flow allows to calculate free energy density
→ Obtain full
thermodynamics→ Critical
properties
max
M. Hinczewski, and A. N. Berker, Phys. Rev. E 77, 011104 (2008)
∑∞
=
−−=
0
)(
maxln32
i
iiT f β
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Another Example:
Shastry-Sutherland ModelSpin-1/2 Heisenberg model of coupled orthogonal
spin-dimers (→ Talk by S.R. Manmana, Thu.)Relevant for the 2D spin-gap material SrCu2
(BO3
)2
Ising limit was suggested to be relevant to the
magnetism in TmB4
E. Siemensmeyer et al., Phys. Rev. Lett. 101,177201 (2008)
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Magnetization ProcessResults for the Ising limit from TRG
1' == J J
M.-C. Chang, M.-F. Yang, Phys. Rev. B 79, 104411 (2009)
Agrees with MC results: Z.Y. Meng, and S.W., Phys Rev. B 78, 224416 (2008)
h
h
1/2-Plateau emerges in
the quantum model only
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Three-Body InteractionsHamiltonians in condensed-matter physics are effective
low-energy models - after integrating out higher-energydegrees of freedom
two-bodythree-body
In many systems two-body terms dominate
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Cold Polar Molecules
X1Σ(v = 0): Rigid rotor and electric dipole
E.g. LiCs: d = 6.3 Debye = 2.5 eaB
Two molecules: dipole-dipole interactions
rotation of
the molecule
electric
dipole
HP. Büchler, A. Micheli, P. Zoller, Nature Physics 3, 726 (2007)
Interaction engineering possible via
static electric and microwave fields
Li
Cs
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Effective Bose-Hubbard Model
B. Capogrossa-Sansone, S. W.,
H.P. Büchler, P. Zoller, G. Pupillo,
Phys. Rev. B 79, R020503 (2009)
K. Schmidt, J. Dorier, A. Läuchli,Phys. Rev. Lett. 101, 150405 (2008)
Prepare system inside an optical lattice
L. Bonnes, H.P. Bücher, S.W.,
N. J. of Phys. 12, 052027 (2010)
t
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Ground State Phase Diagram
V= 0
VBC
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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VBC Phase at Filling 9/16Results from QMC simulations:
local density and local kinetic energy
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Classical LimitClassical model with three-body interactions
Extensive ground state entropy
n=9/16
V= 0
034.032/)3ln(/ ≈= N S
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Ground State Phase Diagram from
Energy ConsiderationsCascade of density plateaus
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Tensor Elements for TRG
A B
1
2
3
4
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Two-Body-Interactions onlyTRG recovers the checkerboard solid at n=1/2
10
0
=
=
V
W
β
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Three-Body-Interactions onlyTRG finds the n=9/16 plateau!
10
0
=
=
W
V
β
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Numerical IssuesStable numerical iteration required a
customized LAPACK SVD routine with128 bit reals (floating point precision of ~10-34)
Still:No convergence for
chemical potentials
beyond W ≈μ
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Future Tasks Assess improvement from using the SRG
Study the thermal melting transition of the 9/16
VBC
Extract residual entropy of the VBC from the
TRG free energy current estimate from TRG: S/N = 0.037 > 0.034
at
Z. Y. Xie
et al., Phys. Rev. Lett. 103, 160601 (2009)
H. H. Zhao et al ., Phys. Rev. B
81, 174411 (2010)
10=W β
8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model
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Discussion TRG applied to a classical model with three-body
interactions and a complex phase diagram
Large numerical precision required for stable iterations of the TRG decimation procedure
Problem for the classical case: exponential range of tensor elements
Progressive loss of precisions during iterations
due to tensor contractions limits practical applicability
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Thank you for your attention!