Time matrix product state: theory and applications

doi:10.7523/j.issn.2095-6134.2019.06.004

Abstract

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

CLC Number:

O469

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

References

[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.