Please wait a minute...
Chin. Phys. B, 2022, Vol. 31(11): 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
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:

Cite this article: 

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 2022 Chin. Phys. B 31 110205

[1] Tang W, Tu H H and Wang L 2020 Phys. Rev. Lett. 125 170604
[2] Cardy J L 1986 Nucl. Phys. B 270 186
[3] Wang X and Xiang T 1997 Phys. Rev. B 56 5061
[4] Verstraete F and Cirac J I 2010 Phys. Rev. Lett. 104 190405
[5] Draxler D, Haegeman J, Osborne T J, Stojevic V, Vanderstraeten L and Verstraete F 2013 Phys. Rev. Lett. 111 020402
[6] Vidal G, Latorre J I, Rico E and Kitaev A 2003 Phys. Rev. Lett. 90 227902
[7] Latorre J I, Rico E and Vidal G 2004 Quantum Informa-tion and Computation
[8] Rams M M, Zauner V, Bal M, Haegeman J and Verstraete F 2015 Phys. Rev. B 92 235105
[9] Pirvu B, Murg V, Cirac J I and Verstraete F 2010 New Journal of Physics 12 025012
[10] Schultz T D, Mattis D C and Lieb E H 1964 Rev. Mod. Phys. 36 856
[11] Pfeuty P 1970 Annals of Physics 57 79
[12] Zuber J B and Itzykson C 1977 Phys. Rev. D 15 2875
[13] Di Francesco P, Mathieu P and Sénéchal D 1997 Conformal field theory Graduate texts in contemporary physics (New York, NY: Springer)
[14] Boyanovsky D 1989 Phys. Rev. B 39 6744
[15] Bl?te H W J, Cardy J L and Nightingale M P 1986 Phys. Rev. Lett. 56 742
[16] Milsted A and Vidal G 2017 Phys. Rev. B 96 245105
[17] Zou Y, Milsted A and Vidal G 2020 Phys. Rev. Lett. 124 040604
[18] Haegeman J, Cirac J I, Osborne T J and Verstraete F 2013 Phys. Rev. B 88 085118
[19] Haldane F D M 1981 Phys. Rev. Lett. 47 1840
[20] Fisher M P A and Glazman L I 1996 Transport in a onedimensional luttinger liquid
[21] Calabrese P and Cardy J 2004 Journal of Statistical Mechanics: Theory and Experiment 2004 P06002
[22] McCoy B and Wu T T 1973 The Two-Dimensional Ising Model
[23] Its A, Jin B and Korepin V E 2006 arXiv: Quantum Physics
[24] Yang C N and Yang C P 1969 J. Math. Phys. 10 1115
[25] Koma T 1989 Progress of Theoretical Physics 81 783
[26] Destri C and de Vega H J 1992 Phys. Rev. Lett. 69 2313
[27] Gradshteyn I S, Ryzhik I M, Zwillinger D and Moll V 2014 Table of integrals, series, and products; 8th ed. (Amsterdam: Academic Press)
[28] Tang W, Xie X, Wang L and Tu H H 2021 Phys. Rev. D 104 114513
[1] Interface effect on superlattice quality and optical properties of InAs/GaSb type-II superlattices grown by molecular beam epitaxy
Zhaojun Liu(刘昭君), Lian-Qing Zhu(祝连庆), Xian-Tong Zheng(郑显通), Yuan Liu(柳渊), Li-Dan Lu(鹿利单), and Dong-Liang Zhang(张东亮). Chin. Phys. B, 2022, 31(12): 128503.
[2] Quantum walk search algorithm for multi-objective searching with iteration auto-controlling on hypercube
Yao-Yao Jiang(姜瑶瑶), Peng-Cheng Chu(初鹏程), Wen-Bin Zhang(张文彬), and Hong-Yang Ma(马鸿洋). Chin. Phys. B, 2022, 31(4): 040307.
[3] Theoretical study of (e, 2e) triple differential cross sections of pyrimidine and tetrahydrofurfuryl alcohol molecules using multi-center distorted-wave method
Yiao Wang(王亦傲), Zhenpeng Wang(王振鹏), Maomao Gong(宫毛毛), Chunkai Xu(徐春凯), and Xiangjun Chen(陈向军). Chin. Phys. B, 2022, 31(1): 010202.
[4] Mechanism of microweld formation and breakage during Cu-Cu wire bonding investigated by molecular dynamics simulation
Beikang Gu(顾倍康), Shengnan Shen(申胜男), and Hui Li(李辉). Chin. Phys. B, 2022, 31(1): 016101.
[5] Optimal control strategy for COVID-19 concerning both life and economy based on deep reinforcement learning
Wei Deng(邓为), Guoyuan Qi(齐国元), and Xinchen Yu(蔚昕晨). Chin. Phys. B, 2021, 30(12): 120203.
[6] Asymmetric coherent rainbows induced by liquid convection
Tingting Shi(施婷婷), Xuan Qian(钱轩), Tianjiao Sun(孙天娇), Li Cheng(程力), Runjiang Dou(窦润江), Liyuan Liu(刘力源), and Yang Ji(姬扬). Chin. Phys. B, 2021, 30(12): 124208.
[7] Real-space parallel density matrix renormalization group with adaptive boundaries
Fu-Zhou Chen(陈富州), Chen Cheng(程晨), and Hong-Gang Luo(罗洪刚). Chin. Phys. B, 2021, 30(8): 080202.
[8] Delta-Davidson method for interior eigenproblem in many-spin systems
Haoyu Guan(关浩宇) and Wenxian Zhang(张文献). Chin. Phys. B, 2021, 30(3): 030205.
[9] A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region
Tao Jiang(蒋涛), Rong-Rong Jiang(蒋戎戎), Jin-Jing Huang(黄金晶), Jiu Ding(丁玖), and Jin-Lian Ren(任金莲). Chin. Phys. B, 2021, 30(2): 020202.
[10] Modes decomposition in particle-in-cell software CEMPIC
Aiping Fang(方爱平)†, Shanshan Liang(梁闪闪), Yongdong Li(李永东), Hongguang Wang(王洪广), and Yue Wang(王玥). Chin. Phys. B, 2020, 29(10): 100205.
[11] Improved hybrid parallel strategy for density matrix renormalization group method
Fu-Zhou Chen(陈富州), Chen Cheng(程晨), Hong-Gang Luo(罗洪刚). Chin. Phys. B, 2020, 29(7): 070202.
[12] Investigation of active-region doping on InAs/GaSb long wave infrared detectors
Su-Ning Cui(崔素宁), Dong-Wei Jiang(蒋洞微), Ju Sun(孙矩), Qing-Xuan Jia(贾庆轩), Nong Li(李农), Xuan Zhang(张璇), Yong Li(李勇), Fa-Ran Chang(常发冉), Guo-Wei Wang(王国伟), Ying-Qiang Xu(徐应强), Zhi-Chuan Niu(牛智川). Chin. Phys. B, 2020, 29(4): 048502.
[13] Relaxation-rate formula for the entropic lattice Boltzmann model
Weifeng Zhao(赵伟峰), Wen-An Yong(雍稳安). Chin. Phys. B, 2019, 28(11): 114701.
[14] On-node lattices construction using partial Gauss-Hermite quadrature for the lattice Boltzmann method
Huanfeng Ye(叶欢锋), Zecheng Gan(干则成), Bo Kuang(匡波), Yanhua Yang(杨燕华). Chin. Phys. B, 2019, 28(5): 054702.
[15] Novel quantum secret image sharing scheme
Gao-Feng Luo(罗高峰), Ri-Gui Zhou(周日贵), Wen-Wen Hu(胡文文). Chin. Phys. B, 2019, 28(4): 040302.
No Suggested Reading articles found!