Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

doi:10.1088/1674-1056/ac8a8f

中国物理B ›› 2022, Vol. 31 ›› Issue (11): 110205-110205.doi: 10.1088/1674-1056/ac8a8f

Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

Yueshui Zhang(张越水)^1,2 and Lei Wang(王磊)^3,4,†

1 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
2 University of Chinese Academy of Sciences, Beijing 100049, China;
3 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
4 Songshan Lake Materials Laboratory, Dongguan 523808, China

收稿日期:2022-06-13 修回日期:2022-08-16 接受日期:2022-08-18 出版日期:2022-10-17 发布日期:2022-10-19
通讯作者: Lei Wang E-mail:wanglei@iphy.ac.cn
基金资助:
This project is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB30000000) and the National Natural Science Foundation of China (Grant Nos. 11774398 and T2121001).

Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

Yueshui Zhang(张越水)^1,2 and Lei Wang(王磊)^3,4,†

1 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
2 University of Chinese Academy of Sciences, Beijing 100049, China;
3 Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China;
4 Songshan Lake Materials Laboratory, Dongguan 523808, China

Received:2022-06-13 Revised:2022-08-16 Accepted:2022-08-18 Online:2022-10-17 Published:2022-10-19
Contact: Lei Wang E-mail:wanglei@iphy.ac.cn
Supported by:
This project is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB30000000) and the National Natural Science Foundation of China (Grant Nos. 11774398 and T2121001).

摘要/Abstract

摘要： We study the structure of the continuous matrix product operator (cMPO)^[1] for the transverse field Ising model (TFIM). We prove TFIM's cMPO is solvable and has the form $T=\rm{e}^{-\frac{1}{2}\hat{H}_{\rm F}}$. $\hat{H}_{\rm F}$ is a non-local free fermionic Hamiltonian on a ring with circumference $\beta$, whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of $\hat{H}_{\rm F}$ is determined analytically. At the critical point, our results verify the state-operator-correspondence^[2] in the conformal field theory (CFT). We also design a numerical algorithm based on Bloch state ansatz to calculate the low-lying excited states of general (Hermitian) cMPO. Our numerical calculations coincide with the analytic results of TFIM. In the end, we give a short discussion about the entanglement entropy of cMPO's ground state.

关键词: continuous matrix product operator, transverse field Ising model, state-operator-correspondence

Abstract: We study the structure of the continuous matrix product operator (cMPO)^[1] for the transverse field Ising model (TFIM). We prove TFIM's cMPO is solvable and has the form $T=\rm{e}^{-\frac{1}{2}\hat{H}_{\rm F}}$. $\hat{H}_{\rm F}$ is a non-local free fermionic Hamiltonian on a ring with circumference $\beta$, whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of $\hat{H}_{\rm F}$ is determined analytically. At the critical point, our results verify the state-operator-correspondence^[2] in the conformal field theory (CFT). We also design a numerical algorithm based on Bloch state ansatz to calculate the low-lying excited states of general (Hermitian) cMPO. Our numerical calculations coincide with the analytic results of TFIM. In the end, we give a short discussion about the entanglement entropy of cMPO's ground state.

Key words: continuous matrix product operator, transverse field Ising model, state-operator-correspondence

中图分类号: (Computational techniques; simulations)

02.70.-c

05.10.Cc (Renormalization group methods) 05.70.Jk (Critical point phenomena) 03.65.-w (Quantum mechanics)

Yueshui Zhang(张越水) and Lei Wang(王磊). Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study[J]. 中国物理B, 2022, 31(11): 110205-110205.

Yueshui Zhang(张越水) and Lei Wang(王磊). Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study[J]. Chin. Phys. B, 2022, 31(11): 110205-110205.

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Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

Structure of continuous matrix product operator for transverse field Ising model: An analytic and numerical study

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