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### Efficient simulation of low-temperature physics in one-dimensional gapless systems

##### Yuya Kusuki, Kotaro Tamaoka, Zixia Wei, and Yasushi Yoneta

##### Phys. Rev. B **110**, L041122 – Published 29 July 2024

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#### Abstract

We discuss the computational efficiency of the finite-temperature simulation with minimally entangled typical thermal states (METTS). To argue that METTS can be efficiently represented as matrix product states, we present an analytic upper bound for the average entanglement Rényi entropy of METTS for a Rényi index $0<q\le 1$. In particular, for one-dimensional (1D) gapless systems described by conformal field theories, the upper bound scales as $O(c{N}^{0}log\beta )$ where $c$ is the central charge and $N$ is the system size. Furthermore, we numerically find that the average Rényi entropy exhibits a universal behavior characterized by the central charge and is roughly given by half of the analytic upper bound. Based on these results, we show that METTS can provide a speedup compared to employing the purification method to analyze thermal equilibrium states at low temperatures in 1D gapless systems.

- Received 23 February 2024
- Revised 28 May 2024
- Accepted 24 June 2024

DOI:https://doi.org/10.1103/PhysRevB.110.L041122

©2024 American Physical Society

#### Physics Subject Headings (PhySH)

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Entanglement in field theoryQuantum entanglementQuantum statistical mechanics

- Techniques

Conformal field theoryDensity matrix renormalization groupTensor network methods

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsQuantum Information, Science & TechnologyParticles & Fields

#### Authors & Affiliations

Yuya Kusuki^{1,2}, Kotaro Tamaoka^{3}, Zixia Wei^{4,2,5,*}, and Yasushi Yoneta^{6,7}

^{1}Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA^{2}Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan^{3}Department of Physics, College of Humanities and Sciences, Nihon University, Tokyo 156-8550, Japan^{4}Center for the Fundamental Laws of Nature and Society of Fellows, Harvard University, Cambridge, Massachusetts 02138, USA^{5}Yukawa Institute for Theoretical Physics, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan^{6}Center for Quantum Computing, RIKEN, Wako, Saitama 351-0198, Japan^{7}Department of Basic Science, The University of Tokyo, Meguro, Tokyo 153-8902, Japan

^{*}Contact author: zixiawei@fas.harvard.edu

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##### Issue

Vol. 110, Iss. 4 — 15 July 2024

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Article part of CHORUS

Accepted manuscript will be available starting29 July 2025.#### Images

###### Figure 1

Average entanglement entropy of METTS between the left part (sites $n\le {N}_{A}$) and the right part (sites $n>{N}_{A}$) as a function of the size ${N}_{A}$ of the subsystem $A$ for the critical transverse-field Ising chain with $\nu =z$ and $\beta =4$. Imaginary-time evolution is carried out using second-order Trotter decomposition with a time step $\delta \tau =0.04$. The data are averaged over $10\phantom{\rule{0ex}{0ex}}000$ samples.

###### Figure 2

Growth of the average entanglement Rényi entropy of METTS for the half chain $A=\{1,2,...,N/2\}$ (a), (b)in the critical transverse-field Ising model of length $N=640$ and (c)in the critical Heisenberg model with the next-nearest-neighbor interaction of length $N=1280$. The insets show the standard deviation of the entanglement Rényi entropy. Imaginary-time evolution is carried out using second-order Trotter decomposition with a time step $\delta \tau =0.04$ for the Ising model and $\delta \tau =0.01$ for the Heisenberg model. The data are averaged over $10\phantom{\rule{0ex}{0ex}}000$ samples for the Ising model and 5000 samples for the Heisenberg model.

###### Figure 3

Minimum bond dimension $D$ necessary to approximate the TFD state and METTS with a truncation error of less than ${10}^{-10}$ for the critical transverse-field Ising chain with $N=640$ sites. The data points for METTS are the average of $D$ of $10\phantom{\rule{0ex}{0ex}}000$ samples. The error bars indicate the standard deviation. We set the quantization axis $\nu $ to $z$. Imaginary-time evolution is carried out using second-order Trotter decomposition with a time step $\delta \tau =0.04$.

###### Figure 4

Typical trajectory of the cumulative moving average of the energy per site $u=E/N$ in the Markov chain of the METTS simulation for the critical transverse-field Ising chain with $N=640$ sites at $\beta =128$. We set the quantization axis $\nu $ to $z$ for the odd-numbered steps and $x$ for the even-numbered steps to reduce correlations between successive samples [22]. To avoid initial transients, we discard the first five samples when calculating the sample averages. The solid line shows the result of the TFD approach, considered exact, and the shaded region shows a range of 0.1 times the standard deviation ${\delta u}_{\beta}^{\left(\mathrm{can}\right)}$ of $u$ in ${\rho}_{\beta}^{\left(\mathrm{can}\right)}$ from the exact value. The inset shows the total computation time in the METTS simulation as a function of the number of steps in the Markov chain. The solid line shows the computation time in the TFD simulation.