Chin. Phys. B, 2022, Vol. 31(11): 110205    DOI: 10.1088/1674-1056/ac8a8f
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# 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
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.
Keywords:  continuous matrix product operator      transverse field Ising model      state-operator-correspondence
Received:  13 June 2022      Revised:  16 August 2022      Accepted manuscript online:  18 August 2022
 PACS: 02.70.-c (Computational techniques; simulations) 05.10.Cc (Renormalization group methods) 05.70.Jk (Critical point phenomena) 03.65.-w (Quantum mechanics)
Fund: 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).
Corresponding Authors:  Lei Wang     E-mail:  wanglei@iphy.ac.cn