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Title: Thermal tensor renormalization group simulations of square-lattice quantum spin models

Journal Article · · Physical Review B
 [1];  [2];  [1];  [3]; ORCiD logo [4];  [1]
  1. Beihang Univ., Beijing (China)
  2. Beihang Univ., Beijing (China); Ludwig-Maximilians-Univ., München (Germany)
  3. Ludwig-Maximilians-Univ., München (Germany)
  4. Brookhaven National Lab. (BNL), Upton, NY (United States); Ludwig-Maximilians-Univ., München (Germany)
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In this work, we benchmark the well-controlled and numerically accurate exponential thermal tensor renormalization group (XTRG) in the simulation of interacting spin models in two dimensions. Finite temperature introduces a finite thermal correlation length ξ, such that for system sizes L >> ξ finite-size calculations actually simulate the thermodynamic limit. Here in this paper, we focus on the square lattice Heisenberg antiferromagnet (SLH) and quantum Ising models (QIM) on open and cylindrical geometries up to width W = 10 . We explore various one-dimensional mapping paths in the matrix product operator (MPO) representation, whose performance is clearly shown to be geometry dependent. We benchmark against quantum Monte Carlo (QMC) data, yet also the series-expansion thermal tensor network results. Thermal properties including the internal energy, specific heat, and spin structure factors, etc. are computed with high precision, obtaining excellent agreement with QMC results. XTRG also allows us to reach remarkably low temperatures. For SLH, we obtain an energy per site u$$^*_g$$ ≃ -0.6694 (4) and a spontaneous magnetization m$$^*_S$$ ≃ 0.30 ( 1 ) already consistent with the ground-state properties, which is obtained from extrapolated low-T thermal data on W ≤ 8 cylinders and W ≤ 10 open strips, respectively. We extract an exponential divergence versus T of the structure factor S (M), as well as the correlation length ξ, at the ordering wave vector M = ($π,π$) , which represents the renormalized classical behavior and can be observed over a narrow but appreciable temperature window, by analyzing the finite-size data by XTRG simulations. For the QIM with a finite-temperature phase transition, we employ several thermal quantities, including the specific heat, Binder ratio, as well as the MPO entanglement to determine the critical temperature $$T_c$$.

Research Organization:
Brookhaven National Laboratory (BNL), Upton, NY (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Basic Energy Sciences (BES)
Grant/Contract Number:
SC0012704
OSTI ID:
1543401
Alternate ID(s):
OSTI ID: 1546376
Report Number(s):
BNL-211858-2019-JAAM; PRBMDO
Journal Information:
Physical Review B, Vol. 100, Issue 4; ISSN 2469-9950
Publisher:
American Physical Society (APS)Copyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 17 works
Citation information provided by
Web of Science

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Cited By (2)

Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase text January 2019
Tensor network simulation of the Kitaev-Heisenberg model at finite temperature text January 2019

Figures / Tables (16)

FIG. 1(p. 4)figure FIG. 1
FIG. 2(p. 5)figure FIG. 2
FIG. 3(p. 6)figure FIG. 3
FIG. 4(p. 7)figure FIG. 4
total shown in main panel (a), and finite size scaling of the extrapolated ata vs. $1/L^2$ shown in panel (b). In order to reduce the finite $D^*$ effects, we extrapolate the internal energy $$u$$ to $1/D^* = 0$, as seen in the inset of (a). In (b) we collect the low-temperature ($T≃0.01$) data extrapolated in (a) $$1/D^* \rightarrow 0$$ and analyze it here vs. $$1/L^2 \rightarrow 0$$. For "torus", we extrapolate the four largest system sizes (i.e., data in gray shaded area was excluded), to the thermodynamic limit, via a second-order polynomial fitting vs. $1/L^2$. The horizontal dashed line represents the ground state energy $$u_g≃-0.6694$$ from QMC. For comparison, (c-h) analyses the internal energy $$u$$ of SLH on YC6$$\times L$$ and YC8$$\times L$$ cylinders of lengths $L=8, 10, 12$. Exemplary extrapolations of $$u_{center}$$ vs. $$1/D^* \rightarrow 0$$ are shown in panels (c) and (f) for $T≃0.11$ and $0.06$, respectively, where $$u_{center}$$ is evaluated via a weighted average around the center as illustrated in the inset of (f) (see main text for more details). The results at $$1/D^* \rightarrow 0$$ are collected vs. $1/WL$ in panels (d, e, g, h) (green stars). There they are also compared to similarly extrapolated data for $$u_{tot}$$ (black squares), as well as to $$u_{subtr}$$ (blue horizontal line) obtained by subtracting the length $L=8$ from the $L'=12$ cylinder. With $$u_tot$$ also extrapolated to $$1/WL \rightarrow 0$$, we find good agreement across our data towards the thermodynamic limit." data-ostiid="1543401">
FIG. 5(p. 8)figure FIG. 5
FIG. 6(p. 9)figure FIG. 6
FIG. 7(p. 10)figure FIG. 7
FIG. 8(p. 10)figure FIG. 8
FIG. 9(p. 11)figure FIG. 9
FIG. 10(p. 12)figure FIG. 10
FIG. 11(p. 13)figure FIG. 11
FIG. 12(p. 13)figure FIG. 12
FIG. 13(p. 14)figure FIG. 13
FIG. A1(p. 15)figure FIG. A1
FIG. A2(p. 15)figure FIG. A2
FIG. A3(p. 16)figure FIG. A3

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