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arXiv:1501.00661

Identification of Degeneracy and Criticality of Two-Dimensional Statistical and Quantum Systems by the Boundary States of Tensor Networks

Shi-Ju Ran, Cheng Peng, Wei Li, Gang Su

Published Jan 4, 2015• 0 citations
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Abstract

We propose a systematic scheme to reach the properties of two-dimensional (2D) statistical and quantum systems by studying the effective (1+1)-dimensional theory that is constructed from the tensor network representation. On on hand, we discover that the degeneracy of the 2D system can be determined by the purity of the boundary thermal state, which is the density operator of the effective theory at zero (effective) temperature. On the other hand, we find that the gapped (or critical) 2D system leads to a gapped (or critical) effective (1+1)-dimensional theory whose criticality can be accessed by the entanglement entropy $S$ of its ground state dubbed as boundary pure state. We also uncover that for the critical systems, $S$ obeys the same logarithmic law as that found in the critical 1D quantum chains, which reads $S = (κc/6)\log_2 D + const.$, with $c$ the central charge and $κ$ a constant related to the scaling property of the correlation length $ξ$ as $ξ\sim D^κ$. Such a scaling law presents an efficient way to characterize the critical universality class of the original 2D systems. An important implication of our work is that many well-established theories for 1D quantum chains become available for studying 2D systems with the help of the proposed lower dimensional correspondence.

DOI
10.48550/arxiv.1501.00661

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We propose a systematic scheme to reach the properties of two-dimensional (2D) statistical and quantum systems by studying the effective (1+1)-dimensional theory that is constructed from the tensor network representation. On on hand, we discover that the degeneracy of the 2D system can be determined by the purity of the boundary thermal state, which is the density operator of the effective theory at zero (effective) temperature. On the other hand, we find that the gapped (or critical) 2D system leads to a gapped (or critical) effective (1+1)-dimensional theory whose criticality can be accessed by the entanglement entropy SSS of its ground state dubbed as boundary pure state. We also uncover that for the critical systems, SSS obeys the same logarithmic law as that found in the critical 1D quantum chains, which reads S=(κc/6)log⁡2D+const.S = (κc/6)\log_2 D + const.S=(κc/6)log2​D+const., with ccc the central charge and κκκ a constant related to the scaling property of the correlation length ξξξ as ξ∼Dκξ\sim D^κξ∼Dκ. Such a scaling law presents an efficient way to characterize the critical universality class of the original 2D systems. An important implication of our work is that many well-established theories for 1D quantum chains become available for studying 2D systems with the help of the proposed lower dimensional correspondence.
Determination and characterization of criticality in two-dimensional (2D) quantum many-body systems is one of the most important challenges and problems in modern physics. Based on the scaling theory in one-dimensional (1D) systems and tensor networks, we propose an efficient scheme to access the criticality of 2D quantum states. We demonstrate that the boundary state of a critical infinite projected entangled pair state can be described by 1D conformal field theory, which provides a solid indicator that allows one to identify and characterize the criticality of 2D states. Our scheme is verified on the resonating valence bond (RVB) states on kagome and square lattices, where the boundary state of the honeycomb RVB is found to be described by a c=1c=1c=1 conformal field theory. We apply our scheme also to the ground state of the spin-1/2 XXZ model on honeycomb lattice, illustrating the difficulties of standard variational tensor network approaches to study the critical ground states. Our scheme is of high versatility and flexibility, and can be further applied to investigate the quantum criticality in many other phenomena, such as finite-temperature and topological phase transitions.
We describe an algorithm for studying the entanglement entropy and spectrum of two-dimensional (2D) systems, as a coupled array of NNN one-dimensional chains in their continuum limit. Using the algorithm to study the quantum Ising model in 2D (both in its disordered phase and near criticality), we confirm the existence of an area law for the entanglement entropy and show that near criticality there is an additive piece scaling as cefflog(N)/6{c}_{\mathrm{eff}}\phantom{\rule{0.16em}{0ex}}\mathrm{log}(N)/6ceff​log(N)/6 with ${c}_{\mathrm{eff}}\ensuremath{\approx}1$. Studying the entanglement spectrum, we show that entanglement gap scaling can be used to detect the critical point of the 2D model. When short-range (area law) entanglement dominates we find (numerically and perturbatively) that this spectrum reflects the energy spectrum of a single quantum Ising chain.
(Please refer to arXiv:1810.08050, which has completely different aims but contains all the main contents of this paper) In this work, we propose to access the information of criticality and excitations of one-dimensional quantum systems by a matrix product state (MPS) defined in the (imaginary) time direction. This state, dubbed as time MPS (tMPS), is a boundary state of tensor network (TN) that represents the ground-state simulation after Trotter-Suzuki decomposition. We show that the tMPS exhibits the structure of the continuous MPS originally proposed for the field theories. The information of excitations, e.g., dynamic correlation length and energy gap, can be accurately calculated from the tMPS. The non-universal renormalization of velocity of the excitations is given by a ratio between the correlations of the ground state and the tMPS. When the system is at the quantum critical point, the tMPS is found to show the logarithmic scaling law, where the scaling coefficient gives the central charge that characterizes the criticality of the theory. Our work also implies that the spectra of the transfer matrices defined by the ground state and tMPS provide information of the low-lying masses of the theory in the continuous limit. This could help to understand the role of the finiteness of the bond dimension as an infra red regulator. Furthermore, we show from the perspective of methodology that, the tMPS emerges from a generalized TN ab-initio optimization principle scheme, which unifies the infinite density matrix renormalization group and the infinite time-evolving block decimation algorithms.
Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground-state degeneracy. In this paper, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigenequation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our general theory by numerically solving 2D fully frustrated XY spin models on the kagome, square, and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets.
In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages
Thermal quantum critical systems, with partition functions expressed as conformal tensor networks, are revealed to exhibit universal entropy corrections on nonorientable manifolds. Through high-precision tensor network simulations of several quantum chains, we identify the universal entropy SK=lnk{S}_{\mathcal{K}}=lnkSK​=lnk on the Klein bottle, where kkk relates to quantum dimensions of the primary fields in conformal field theory (CFT). Different from the celebrated Affleck-Ludwig boundary entropy lnglnglng (g(g(g reflects noninteger ground-state degeneracy), SK{S}_{\mathcal{K}}SK​ has nonono boundary dependence or surface energy terms accompanying it, and can be very conveniently extracted from thermal data. On the M\"obius-strip manifold, we uncover an entropy SM=12(lng+lnk){S}_{\mathcal{M}}=\frac{1}{2}(lng+lnk)SM​=21​(lng+lnk) in CFT, where 12lng\frac{1}{2}lng21​lng is associated with the only open edge of the M\"obius strip and 12lnk\frac{1}{2}lnk21​lnk is associated with the nonorientable topology. As a useful application, we employ the universal entropy to accurately pinpoint the quantum phase transitions, even for those without local order parameters.