Session Q26: Focus Session: Computational Frontiers in Quantum Spin Systems I

Computing Entanglement Entropy in Quantum Monte Carlo

The scaling of entanglement entropy in quantum many-body wavefunctions is expected to be a fruitful resource for studying quantum phases and phase transitions in condensed matter. However, until the recent development of estimators for Renyi entropy in quantum Monte Carlo (QMC), we have been in the dark about the behaviour of entanglement in all but the simplest two-dimensional models. In this talk, I will outline the measurement techniques that allow access to the Renyi entropies in several different QMC methodologies. I will then discuss recent simulation results demonstrating the richness of entanglement scaling in 2D, including: the prevalence of the ``area law''; topological entanglement entropy in a gapped spin liquid; anomalous subleading logarithmic terms due to Goldstone modes; universal scaling at critical points; and examples of emergent conformal-like scaling in several gapless wavefunctions. Finally, I will explore the idea that ``long range entanglement'' may complement the notion of ``long range order'' for quantum phases and phase transitions which lack a conventional order parameter description.   

Scaling of entanglement entropy in the 2D Heisenberg ground state

We use a Loop-Ratio Valence Bond quantum Monte Carlo algorithm to study the scaling of the bipartite Renyi entanglement entropy in the 2D Heisenberg ground state. We uncover the surprising result that finite-size scaling supports a logarithmic correction to the entropic area law even with the absence of corners in the entangled region. In addition, examining the scaling within a single system, we observe an aspect-ratio dependent scaling term resembling the ``conformal distance'' term that appears in one-dimensional systems with conformal symmetry.   

Entanglement scaling of the 2D RVB wavefunction

The resonating valence bond (RVB) state on a two-dimensional lattice is a superposition of all permutations of singlet spin pairs. This wavefunction was first proposed by Anderson as a simple spin liquid ground state, showing no long range order at T=0. Using a loop-algorithm Monte Carlo method that samples all nearest-neighbor singlet pairs, we examine the entanglement entropy of the nearest neighbor SU(2) RVB wavefunction on the square lattice. In addition to the area law, we show that the entanglement entropy splits into two branches, due to the different topological sectors of the RVB wavefunction. These branches individually scale with a logarithmic dependence on the size of the entangled region, the functional form of which appears to be similar to the conformal distance observed in scaling at conformal critical points in 1D. We comment on the implication for the search for topological order, and on generalizations of this wavefunction, including models involving SU(N) spins.   

Entanglement entropy at the quantum critical point of the 2D transverse field Ising model

Entanglement entropy is a quantity that is desirable to examine at quantum critical points in condensed matter systems, because it is expected that sub-leading scaling terms should contain universal coefficients. In dimensions higher than one, these universal coefficients (that are sub-leading to the area law) may possibly be used to identify the universality class of the quantum critical point, much like the central charge in 1D systems. The recent development of zero temperature projector methods for the transverse field Ising model in combination with replica methods for stochastic series expansion quantum Monte Carlo (QMC) allows us to examine this idea, using measurements of Renyi entanglement entropies. We compare zero- and finite- temperature QMC results with series expansion, and discuss the scaling of the Renyi entropies at the 2D critical point in the transverse field Ising model.   

Finite entanglement scaling at novel phase transitions in the Bose-Hubbard model with pairing terms

With the introduction of pair hopping, the 1+1D Bose-Hubbard model contains string like defects and half vortices in addition to the familiar vortices that drive the Kosterlitz - Thouless (KT) transition. Recent work [1] proposed the existence of a novel phase transition directly from the insulating to superfluid phase which is partly of an Ising, rather than KT type, contrary to expectations based on symmetry. To characterize the transition we demonstrate an approach to the study of 1+1D critical phenomena using infinite matrix product state algorithms (iMPS), in which critical fluctuations are cut off not by a finite system size, but by the finite entanglement of the iMPS ansatz. Starting from the ``finite entanglement'' scaling of the correlation length [2], we show that scaling and correlation functions also admit a universal ``finite entanglement'' collapse, avoiding boundary effects and validating an elegant alternative to finite size scaling methods for critical phases. \\[4pt] [1] Shi, Lamacraft and Fendley, arxiv:1108.5744v1\\[0pt] [2] Pollmann, Mukerjee, Turner and Moore, Phys. Rev. Lett. 102, 255701 (2009)   

Renormalization of tensor-network states

We have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently proposed for studying the tensor-network states/models in two dimensions. A second renormalization scheme is introduced to take into account the environment contribution in the calculation of the partition function of classical tensor network models or the expectation values of quantum tensor network states. It improves significantly the accuracy of the coarse grained tensor renormalization group method. In the study of the quantum tensor-network states, we point out that the renormalization effect of the environment can be efficiently and accurately described by the bond vector. This, combined with the imaginary time evolution of the wave function, provides an accurate projection method to determine the tensor-network wave function. It reduces significantly the truncation error and enables a tensor-network state with a large bond dimension, which is difficult to be accessed by other methods, to be accurately determined.   

Cluster update for tensor network states

We propose a novel recursive way of updating the tensors in projected entangled pair states by evolving the tensor in imaginary time evolution on clusters of different sizes. This generalizes the so-called simple update method of Jiang et al. [Phys. Rev. Lett. 101, 090603 (2008)] and the updating schemes in the single layer picture of Pizorn et al. [Phys. Rev. A 83, 052321 (2011)]. A finite-size scaling of the observables as a function of the cluster size provides a remarkable improvement in the accuracy as compared to the simple update scheme. We benchmark our results on the hand of the spin 1/2 staggered dimerized antiferromagnetic model on the square lattice, and accurate results for the magnetization and the critical exponents are determined. Reference L. Wang and F. Verstraete, arXiv:1110.4362.   

The spin 1/2 J1-J2 Antiferromagnetic Heisenberg model on Square Lattice

We studied the spin 1/2 J1-J2 Heisenberg antiferromagnets on square lattice using the recently proposed cluster update for tensor network states [arXiv:1110.4362]. Ground state wavefunction with tensor network bond dimension upto $D=9$ was obtained. Through a finite size scaling analysis, we observe a second order phase transition from an antiferromagnetic ordered state to a paramagnetic phase with plaquette valence bond order at a coupling constant ratio J2/J1~ 0.44. The ground state energies in the thermodynamic limit are in the order of 10$^{-3}$J1 per site difference from the state of art exact diagonalization study [Eur. Phys. J. B 73, 117 (2010)].   

The spin-$1/2$ $J_1-J_2$ Heisenberg antiferromagnet on a square lattice:a plaquette renormalized tensor network study

We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E, \textbf{83}, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg ${J}_{1}$-${J}_{2}$ model on an $L\times L$ square lattice with $L$=8,16 and 32. By treating tensor elements as variational parameters, we obtain the ground states for different $J_2/J_1$ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations and plaquette order parameters. In addition to the well-known N\'{e}el-order and collinear-order at low and high ${J}_2/{J}_1$, we observe a plaquette-like order at ${J}_2/J_1\approx 0.5$. A continuous transition between the N\'{e}el order and the plaquette-like order near $J_2^{c_1}\approx 0.40 J_1$ is observed. The collinear order emerges at ${J}_2^{c_2} \approx 0.62J_1$ through a first-order phase transition.   

MERA Study of Spatially Anisotropic Triangular Antiferromagnets

We report variational calculations for the ground states in the spatially anisotropic triangular antiferromagnets. The variational wave function is based on the tensor network with an entanglement renormalization [1]. The entanglement renormalization improves the ability of describing a quantum state. We construct a three-dimensional MERA tensor network for the triangular lattice models. The model in this study has two groups of the antiferromagnetic Heisenberg couplings on a triangular lattice: one on links along a lattice axis and the other on other links. $J_1$ and $J_2$ denote the coefficient of their couplings, respectively. We calculate the ground states of finite lattices ($N=114, 2166$) and an infinite lattice. We confirm a magnetic phase in the region of $0.7 < J_2/J_1 \le 1$. The magnetic structure is incommensurate, and the wave vector is not consistent with that of a classical model except for $J_1=J_2$.\\[4pt] [1] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007).   

General framework of non-Abelian SU(N) symmetries for matrix product states

We present a generic numerical framework for the treatment of an arbitrary set of non-abelian quantum symmetries within the matrix product states (MPS) approach and generalization thereof. The framework is based on the simple observation that Clebsch Gordan coefficient spaces can be split off as tensor products for all objects relevant within MPS [1]. As such it is applicable, for example, to the numerical as well as the density matrix renormalization group (NRG or DMRG, respectively). The framework is applied to a generalized SU(3) channel-symmetric Anderson impurity model within the NRG. This model of a fully screened spin $S=\frac{3}{2}$ Anderson model has been suggested recently as the effective microscopic Kondo model for Fe impurities in gold or silver [2]. Results are presented on the explicit treatment of U(1)$\otimes$SU(2)$\otimes$SU(3) for charge, spin, and channel, respectively. This is compared to the alternative description in terms of SU(2)$^{\otimes4}$ symmetries for total spin and particle-hole symmetry in every channel. \\[1ex] [1] Singh et al, PRA {\bf 82}, 050301 (2010) \\{} [2] Costi et al, PRL {\bf 102}, 056802 (2009).   

Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition

We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements. In contrast to variational approaches and most implementations of DMRG, Lanczos rotations towards the ground state do not involve incremental minimizations, (e.g. sweeping procedures) which may get stuck in false local minima. The lattice of size N is partitioned into two subclusters. At each iteration the rotating Lanczos vector is compressed into two sets of $n_{{\rm svd}}$ small subcluster vectors using singular value decomposition. For low entanglement entropy $S_{ee}$, (satisfied by short range Hamiltonians), the truncation error is bounded by $\exp(-n_{{\rm svd}}^{1/S_{ee}})$. Convergence is tested for the Heisenberg model on Kagom\'e clusters of 24, 30 and 36 sites, with no lattice symmetries exploited, using less than 15GB of dynamical memory. Generalization of the Lanczos-SVD algorithm to multiple partitioning is discussed, and comparisons to other techniques are given. Reference: arXiv:1105.0007   

An efficient basis for the modeling of doped and undoped S=1/2 antiferromagnet

We formulate an efficient numerical basis to model both doped and undoped S=$\frac{1}{2}$ Heisenberg antiferromagnet (AFM) with two-dimensional periodic boundary condition (2DPBC). Using a linear combination of Slater determinants with total-spin symmetries, a variational approach is developed to systematically and combinatorially decreases the Hilbert space of the problems, allowing the application of exact diagonalization to record-breaking system sizes. We can now model explicitly the wavefunction of an undoped 64-spin AFM square lattice with 2DPBC. For the doped scenarios, we solve a half-filled lattice with 32 coppers and 64 oxygens with one or two electrons removed. This allows, for the first time, a direct comparison of 32-unit-cell exact diagonalization between multi-band model and the t-J model, quantifying several oxygen-specific properties relevant to the lightly doped cuprate structures.