时间矩阵乘积态理论及其应用

doi:10.7523/j.issn.2095-6134.2019.06.004

中国科学院大学学报 ›› 2019, Vol. 36 ›› Issue (6): 745-751.DOI: 10.7523/j.issn.2095-6134.2019.06.004

时间矩阵乘积态理论及其应用

彭程¹, 冉仕举²

1. 中国科学院大学物理科学学院, 北京 100049;
2. 西班牙光子科学研究所, 卡斯特利德费尔斯 08860

收稿日期:2018-05-16 修回日期:2018-05-28 发布日期:2019-11-15
通讯作者: 彭程
基金资助:
国家自然科学基金（11574309）资助

Time matrix product state: theory and applications

PENG Cheng¹, RAN Shiju²

1. School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
2. ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels 08860, Spain

Received:2018-05-16 Revised:2018-05-28 Published:2019-11-15

摘要/Abstract

摘要： 提出一种有效地刻画二维或高维量子临界系统的时间矩阵乘积态理论。利用数值重整化群，建立实空间矩阵乘积态与时间矩阵乘积态在描述高维量子多体系统的基态纠缠熵与关联长度两方面的等价性。在蜂窝状六角格子上的自旋1/2各向异性海森堡反铁磁模型中观察到两种不同类型的时间矩阵乘积态纠缠熵标度行为，还在kagome格子上的自旋1/2各向同性海森堡反铁磁体中观察到时间矩阵乘积态纠缠熵的对数型发散行为。这意味着高维量子系统的临界性有可能通过建立在一维时间矩阵乘积态基础上的（1+1）维共形场论来描述。

关键词: 量子临界, 纠缠熵, 关联长度, 标度律, 时间矩阵乘积态

Abstract: In this work, we propose an efficient approach to identify the criticality of finite-size quantum systems in higher-dimensions. Starting from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and an one-dimensional (1D) quantum state defined in the imaginary time direction in terms of the so-called time matrix product state (tMPS). We show that the criticality of the targeted model can be identified by the tMPS. We benchmark our proposal with the results obtained from the spin-1/2 Heisenberg antiferromagnet on honeycomb lattice. We also demonstrate critical scaling behavior of the tMPS on the spin-1/2 kagome Heisenberg antiferromagnet. The present study indicates that the 1D conformal field theory in the imaginary time provides a useful tool to characterize the criticality of higher-dimensional quantum systems.

Key words: criticality, entanglement entropy, correlation length, scaling law, time matrix product state

中图分类号:

O469

彭程, 冉仕举. 时间矩阵乘积态理论及其应用[J]. 中国科学院大学学报, 2019, 36(6): 745-751.

PENG Cheng, RAN Shiju. Time matrix product state: theory and applications[J]. , 2019, 36(6): 745-751.

参考文献

[1] Holzhey C, Larsen F, Wilczek F. Geometric and renormalized entropy in conformal field theory[J]. Nuclear Physics B, 1994, 424(3):443-467.
[2] Osterloh A, Amico L, Falci G, et al. Scaling of entanglement close to a quantum phase transition[J]. Nature, 2002, 416(6 881):608-610.
[3] Osborne T J, Nielsen M A. Entanglement in a simple quantum phase transition[J]. Physical Review A, 2002, 66(3):032 110.
[4] Vidal G, Latorre J I, Rico E, et al. Entanglement in quantum critical phenomena[J]. Physical Review Letters, 2003, 90(22):227 902.
[5] Calabrese P, Cardy J. Entanglement entropy and quantum field theory[J]. Journal of Statistical Mechanics:Theory and Experiment, 2004, 2004(6):P06 002.
[6] Calabrese P, Cardy J. Entanglement entropy and quantum field theory:a non-technical introduction[J]. International Journal of Quantum Information, 2006, 4(3):429-438.
[7] Tagliacozzo L, Oliveira T, Iblisdir S, et al. Scaling of entanglement support for matrix product states[J]. Physical Review B, 2008, 78(2):024 410.
[8] Andersson M, Boman M, Östlund S. Density-matrix renormalization group for a gapless system of free fermions[J]. Physical Review B, 1999, 59(16):10 493-10 503.
[9] Calabrese P, Lefevre A. Entanglement spectrum in one-dimensional systems[J]. Physical Review A, 2008, 78(3):032 329.
[10] Pollmann F, Mukerjee S, Turner A M, et al. Theory of finite-entanglement scaling at one-dimensional quantum critical points[J]. Physical Review Letters, 2009, 102(25):255 701.
[11] Stojevic V, Haegeman J, McCulloch I P, et al. Conformal data from finite entanglement scaling[J]. Physical Review B, 2015, 91(3):035 120.
[12] Ran S J, Peng C, Li W, et al. Criticality in two-dimensional quantum systems:tensor network approach[J]. Physical Review B, 2017, 95(15):155 114.
[13] Verstraete F, Cirac J I. Continuous matrix product states for quantum fields[J]. Physical Review Letters, 2011, 104(19):190 405.
[14] Trotter H F. On the product of semi-groups of operators[J]. Proceedings of the American Mathematical Society, 1959, 10(4):545-551.
[15] Suzuki M, Inoue M. The ST-transformation approach to analytic solutions of quantum systems. I:General formulations and basic limit theorems[J]. Progress of Theoretical Physics, 1987, 78(4):787-799.
[16] Inoue M, Suzuki M. The ST-transformation approach to analytic solutions of quantum systems. II:Transfer-matrix and Pfaffian methods[J]. Progress of theoretical physics, 1988, 79(3):645-664.
[17] Li W, Gong S S, Zhao Y, et al. Quantum phase transition, O(3) universality class, and phase diagram of the spin-1/2 Heisenberg antiferromagnet on a distorted honeycomb lattice:a tensor renormalization-group study[J]. Physical Review B, 2010, 81(18):184 427.
[18] Imai T, Nytko E A, BartLetters B M, et al. ⁶³Cu,³⁵Cl, and ¹H NMR in the S=1/2 kagome lattice ZnCu₃(OH)₆Cl₂[J]. Physical Review Letters, 2008, 100(7):077 203.
[19] Helton J S, Matan K, Shores M, et al. Spin dynamics of the spin-1/2 kagome lattice antiferromagnet ZnCu₃(OH)₆Cl₂[J]. Physical Review Letters, 2007, 98(10):107 204.
[20] Olariu A, Mendels P, Bert F, et al. ¹⁷O NMR study of the intrinsic magnetic susceptibility and spin dynamics of the quantum kagome antiferromagnet ZnCu₃(OH)₆Cl₂[J]. Physical Review Letters, 2008, 100(8):087 202.
[21] Wulferding D, Lemmens P, Scheib P, et al. Interplay of thermal and quantum spin fluctuations in the kagome lattice compound herbertsmithite[J]. Physical Review B, 2010, 82(14):144 412.
[22] Han T H, Helton J S, Chu S Y, et al. Fractionalized excitations in the spin-liquid state of a kagome-lattice antiferromagnet[J]. Nature, 2012, 492(7 429):406-410.
[23] Ryu S, Motrunich O I, Alicea J, et al. Algebraic vortex liquid theory of a quantum antiferromagnet on the kagome lattice[J]. Physical Review B, 2007, 75(18):184 406.
[24] Ran Y, Hermele M, Lee P A, et al. Projected-wave-function study of the spin-1/2 Heisenberg model on the kagome lattice[J]. Physical Review Letters, 2007, 98(11):117 205.
[25] Hermele M, Ran Y, Lee P A, et al. Properties of an algebraic spin liquid on the kagome lattice[J]. Physical Review B, 2008, 77(22):224 413.
[26] Sindzingre P, Lhuillier C. Low-energy excitations of the kagome antiferromagnet and the spin-gap issue[J]. EPL (Europhysics Letters), 2009, 88(2):270 09.
[27] Iqbal Y, Becca F, Sorella S, et al. Gapless spin-liquid phase in the kagome spin-1/2 Heisenberg antiferromagnet[J]. Physical Review B, 2013, 87(6):060 405.
[28] Iqbal Y, Poilblanc D, Becca F. Vanishing spin gap in a competing spin-liquid phase in the kagome Heisenberg antiferromagnet[J]. Physical Review B, 2014, 89(2):020 407.
[29] Liao H J, Xie, Z Y, Chen J et al. Gapless spin-liquid ground state in the S=1/2 kagome antiferromagnet[J]. Physical Review Letters, 2017, 118(13):137 202.
[30] He Y C, Zaletel M P, Oshikawa M, et al. Signatures of Dirac cones in a DMRG study of the kagome Heisenberg model[J]. Physical Review X, 2017, 7(3):031 020.

时间矩阵乘积态理论及其应用

Time matrix product state: theory and applications

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