8/3/2019 Stefan Wessel- Tensor Renormalization Group - Beyond the Ising Model

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Tensor 

Renormalization Group -  Beyond the Ising Model

Stefan Wessel

Institut für Theoretische Physik IIIUniversität Stuttgart

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Collaborators Lars Bonnes

Hans Peter Büchler 

L. Bonnes, H.P. Büchler, S.W., New Journal of Physics 12, 052027 (2010)

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Overview

TRG for Classical Systems Previous Applications

Three-body Interactions

Model and Phases

Results from TRG

Discussion

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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)

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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

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TRG Scheme IIterative two-step coarse-graining procedure

1. 2.

1. 2.

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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=

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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.

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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 

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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)

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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

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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 

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Ground State Phase Diagram

V= 0

VBC

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VBC Phase at Filling 9/16Results from QMC simulations:

local density and local kinetic energy

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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

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Ground State Phase Diagram from

Energy ConsiderationsCascade of density plateaus

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Tensor Elements for TRG

 A B

1

2

3

4

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Two-Body-Interactions onlyTRG recovers the checkerboard solid at n=1/2

10

0

=

=

V 

W 

 β 

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Three-Body-Interactions onlyTRG finds the n=9/16 plateau!

10

0

=

=

W 

V 

 β 

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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 

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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!