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arXiv:1609.06852 · Physical review. B./Physical review. B

Criticality in two-dimensional quantum systems: Tensor network approach

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

Published Apr 10, 2017• 12 citations
arXivHTMLPDF

Abstract

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 $c=1$ 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.

DOI
10.1103/physrevb.95.155114

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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.
Characterizing criticality in quantum many-body systems of dimension ≥2\ge 2≥2 is one of the most important challenges of the contemporary physics. In principle, there is no generally valid theoretical method that could solve this problem. In this work, we propose an efficient approach to identify the criticality of quantum systems in higher dimensions. Departing from the analysis of the numerical renormalization group flows, we build a general equivalence between the higher-dimensional ground state and a 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 faithfully identified by the tMPS, using the mature scaling schemes of correlation length and entanglement entropy in 1D quantum theories. We benchmark our proposal with the results obtained for the Heisenberg anti-ferromagnet on honeycomb lattice. We demonstrate critical scaling relation of the tMPS for the gapless case, and a trivial scaling for the gapped case with spatial anisotropy. The critical scaling behaviors are insensitive to the system size, suggesting the criticality can be identified in small systems. Our tMPS scheme for critical scaling shows clearly that the spin-1/2 kagomé Heisenberg antiferromagnet has a gapless ground state. More generally, the present study indicates that the 1D conformal field theories in imaginary time provide a very useful tool to characterize the criticality of higher dimensional quantum systems.
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.
We introduce a tensor network method for approximating thermal equilibrium states of quantum many-body systems at low temperatures. Whereas the usual approach starts from infinite temperature and evolves the state in imaginary time (toward lower temperature), our ansatz is constructed from the zero-temperature limit, the ground state, which can be found with a standard tensor network approach. Motivated by properties of the ground state for conformal field theories, our ansatz is especially well suited near criticality. Moreover, it allows an efficient computation of thermodynamic quantities and entanglement properties. We demonstrate the performance of our approach with a tree tensor network ansatz, although it can be extended to other tensor networks, and present results illustrating its effectiveness in capturing the finite-temperature properties in one- and two-dimensional scenarios. In particular, in the critical one-dimensional case we show how the ansatz reproduces the finite temperature scaling of entanglement in a conformal field theory.
Progress in describing thermodynamic phase transitions in quantum systems is obtained by noticing that the Gibbs operator ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ for a two-dimensional (2D) lattice system with a Hamiltonian HHH can be represented by a three-dimensional tensor network, the third dimension being the imaginary time (inverse temperature) $\ensuremath{\beta}$. Coarse graining the network along $\ensuremath{\beta}$ results in a 2D projected entangled-pair operator (PEPO) with a finite bond dimension DDD. The coarse graining is performed by a tree tensor network of isometries. The isometries are optimized variationally, taking into account full tensor environment, to maximize the accuracy of the PEPO. The algorithm is applied to the isotropic quantum compass model on an infinite square lattice near a symmetry-breaking phase transition at finite temperature. From the linear susceptibility in the symmetric phase and the order parameter in the symmetry-broken phase, the critical temperature is estimated at Tc=0.0606(4)J{\mathcal{T}}_{c}=0.0606(4)JTc​=0.0606(4)J, where JJJ is the isotropic coupling constant between S=12S=\frac{1}{2}S=21​ pseudospins.
The Tensor Network (TN) algorithm for strongly related electronic quantum lattice systems has become the uppermost algorithm for studying strongly related electronic quantum lattice systems in numerical computation. Focusing on the TN algorithm based on 2-D spin systems, this paper discusses the theoretical principles of the Projected Entangled Pair States (PEPS) algorithm and its implementation process. It also explores the infinite PEPS (iPEPS) algorithm for simulating the spin-1/2 infinite 2-D square lattice quantum XYX model. The critical phenomena of the two-dimensional XYX quantum system are analyzed from the perspectives of ground-state fidelity and order parameters, and the quantum critical point is identified. It is anticipated that with the further development of TN algorithms, the single-site fidelity theory of quantum phase transitions (QPTs) can be applied to various quantum many-body systems in condensed matter physics, thereby deepening our understanding of the extremely rich states of matter in the quantum world. Research indicates that quantum information theory provides a universal method for the study of QPTs, which can identify several ground-state fidelities and order parameters through the ground-state wave function of the system, several ground state fidelity and local order parameters can be identified to fully describe some quantum phase transition points, greatly enriching the traditional Landau theory of symmetry breaking QPTs.
Emergent phenomena of interacting quantum many-body systems arguably pose some of the greatest challenges to modern theoretical physics. Many of these systems are characterized by drastic changes in their properties as they are driven into a regime where quantum effects and interactions become relevant. Examples are the fractional quantum Hall effect and high-temperature superconductors, to name just two. New theoretical tools are necessary to study such systems, as even simple model Hamiltonians are not fully understood after decades of intense research. Numerical simulations are emerging as an indispensable tool alongside more traditional analytic approaches and have seen immense progress in recent years. In this thesis, we study tensor network states, a new class of numerical methods that have been developed in recent years drawing on insights from quantum information theory. Combining renormalization group methods with knowledge about the entanglement structure of ground states of quantum systems, tensor network states aim to efficiently describe ground states of strongly correlated quantum systems. Center stage is taken by the projected entangled-pair states. This ansatz for two-dimensional systems is discussed in detail and its capabilities are assessed by comparison to established methods and by exploring a frustrated system. We then develop a general formalism to exploit Abelian symmetries in a tensor network algorithm and demonstrate its application to projected entangled-pair states. A large part of this thesis is devoted to the discussion of a class of supersymmetric models for interacting lattice fermions. After introducing the concept of supersymmetry in quantum mechanics and an extensive overview of conformal field theory and its relation to numerical simulations, we study the critical theory of the supersymmetric model on the square ladder. We use two tensor network state algorithms, the density-matrix renormalization group and multi-scale entanglement renormalization along with exact diagonalization to explore this challenging system. In the final two chapters, we discuss two applications of tensor network states, namely simulations of the SU(3) Heisenberg model in two dimensions using two different tensor network state algorithms with the goal of illuminating the nature of its ground state, and the indistinguishability, a measure to detect phase transitions and classify wave functions which is particularly suitable for calculations based on tensor network states.