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 . is a non-local free fermionic Hamiltonian on a ring with circumference , whose ground state is gapped and non-degenerate even at the critical point. The full spectrum of 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.
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
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. B270 186 [3] Wang X and Xiang T 1997 Phys. Rev. B56 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. B92 235105 [9] Pirvu B, Murg V, Cirac J I and Verstraete F 2010 New Journal of Physics12 025012 [10] Schultz T D, Mattis D C and Lieb E H 1964 Rev. Mod. Phys.36 856 [11] Pfeuty P 1970 Annals of Physics57 79 [12] Zuber J B and Itzykson C 1977 Phys. Rev. D15 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. B39 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 Experiment2004 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 Physics81 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. D104 114513
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
