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TNQMP2016International Workshop on

Tensor Networks andQuantum Many-Body Problems

June 27 - July 15, 2016The Institute for Solid State PhysicsThe University of TokyoKashiwa, Japan

TNQMP2016 10th Annual ISSP International Workshop on Tensor Networks and Quantum Many-Body Problems

Date:

June 27 – July 15, 2016

Venue:

The Institute for Solid State Physics (ISSP), The University of Tokyo, Kashiwa, Japan

Scope:

The main objective of this workshop is to achieve a better understanding of

tensor-network states and tensor-network methods in quantum many-body problems.

Related physical systems and numerical methods will be also discussed.

Advisors:

Frank Verstraete (Vienna) and Guifre Vidal (Perimeter)

Program Committee:

Naoki Kawashima (chair, ISSP), Kenji Harada (co-chair, Kyoto),

Shoji Hashimoto (KEK), Masaki Oshikawa (ISSP), Synge Todo (Tokyo),

and Takeshi Yanai (IMS)

Local Organizing Committee:

Takahiro Misawa (ISSP), Satoshi Morita (ISSP), Yuichi Motoyama (ISSP),

Tsuyoshi Okubo (ISSP), Jun Yamazaki (ISSP), and Kazuyoshi Yoshimi (ISSP)

Sponsors:

The Institute for Solid State Physics (ISSP)

Elements Strategy Initiative Center for Magnetic Materials (ESICMM)

Post ‘K’ Computer Priority Issue SEVEN (PostK-7)

Professional development Consortium for Computational Materials Scientists (PCoMS)

Leading Research Network on Topological Phenomena in Novel Quantum Matter

(TOPNET)

Seminar Rooms Symposium: A632 Lectures: A615 Poster: Lounge on the 6th floor List of Lecturers 1st week June 28 (Tue.) June 29 (Wed.) June 30 (Thu.) July 1 (Fri.)

10:00 – 11:30 N. Kawashima M.-C. Bañuls M. Oshikawa T. Takayanagi

15:00 – 16:30 G. Evenbly G. Evenbly R. Orús R. Orús

2nd week July 5 (Tue.) July 6 (Wed.) July 7 (Thu.) July 8 (Fri.)

10:00 – 11:30 Ö. Legeza F. Verstraete F. Verstraete T. Nishino

15:00 – 16:30 F. Pollmann F. Pollmann N. Schuch N. Schuch

3rd week July 12 (Tue.) July 13 (Wed.) July 14 (Thu.) July 15 (Fri.)

10:00 – 11:30 P. Corboz P. Corboz L. Tagliacozzo T. Tohyama

15:00 – 16:30 T. Xiang T. Xiang Y.-J. Kao N. Kawashima

(13:00 – 14:30)

Symposium 1 (June 27) 10:00 – 10:15 Naoki Kawashima (ISSP)

Opening

10:15 – 10:45 Glen Evenbly (UC Irvine)

Entanglement Renormalization and Wavelets

10:45 – 11:15 Zhi-Yuan Xie (RUC, Beijing)

Some Progress on Tensor Renormalization Group

11:15 – 11:45 Mari Carmen Bañuls (MPQ)

Finding almost conserved local quantities in non-integrable

quantum systems

11:45 – 13:30 Lunch

13:30 – 14:00 Naoki Nakatani (Hokkaido)

Matrix Product Multi-Linear Algebra Library

14:00 – 14:30 Satoshi Morita (ISSP)

Development of parallel library for tensor network methods

14:30 – 15:00 Matthew T. Fishman (Caltech)

Compression of Correlation Matrices and an Efficient Method for

Forming Matrix Product States of Fermionic Gaussian States

15:00 – 15:30 Coffee break + Photo

15:30 – 16:00 Ian P. McCulloch (Queensland)

Topological order and space group symmetry fractionalization in

the frustrated J1-J2 Heisenberg model

16:00 – 16:30 Kenji Harada (Kyoto)

Branching and tensor network

16:30 – 17:00 Hiroshi Ueda (AICS, RIKEN)

Real-space parallel infinite-size density matrix renormalization

group

17:00 – 17:30 Román Orús (Mainz)

Kitaev honeycomb tensor networks: exact unitary circuits and

applications

Symposium 2 (July 4) 10:00 – 10:30 Haijun Liao (CAS, Beijing)

Heisenberg antiferromagnet on the Husimi lattice

10:30 – 11:00 Tsuyoshi Okubo (ISSP)

Magnetization process of kagome lattice Heisenberg

antiferromagnets: 1/3 plateau state and effects of

Dzyaloshinskii-Moriya interaction

11:00 – 11:30 Hyunyong Lee (SKKU)

Featureless Quantum Insulator on the Honeycomb Lattice and

Square lattice

11:30 – 12:00 Frank Verstraete (Viena)

TBA

12:00 – 13:00 Lunch

13:00 – 13:30 Tadashi Takayanagi (Kyoto)

Boundary States in CFTs and Continuous MERA

13:30 – 14:00 Shuo Yang (Perimeter)

Tensor networks with loop optimization

14:00 – 14:30 Tomotoshi Nishino (Kobe)

Mean-Field Behavior in Uniform Tensor Product States

14:30 – 16:00 Poster session + Photo

16:00 – 16:30 Pochung Chen (NTHU)

Quantum Critical Spin-2 Chain with Emergent SU(3) Symmetry

16:30 – 17:00 Soichiro Mohri (ISSP)

Order parameters of bond-type symmetry protected topological

phases in two dimensions

17:00 – 17:30 Örs Legeza (Wigner RCP)

Tensor product methods and entanglement optimization for

models with long range interactions

Symposium 3 (July 11) 10:00 – 10:15 Masashi Takigawa (ISSP)

From the Director of ISSP

10:15 – 10:45 Ying-Jer Kao (NTU)

Steady States of Infinite-Size Dissipative Quantum Chains via

Imaginary Time Evolution

10:45 – 11:15 Tao Xiang (CAS, Beijing)

Majorana Positivity and the Fermion Sign Problem of Quantum

Monte Carlo Simulations

11:15 – 11:45 Norbert Schuch (MPQ)

Chiral Projected Entangled Pair States

11:45 – 13:00 Lunch

13:00 – 13:30 Luca Tagliacozzo (U. Strathclyde)

Finite bond dimension effects in tensor network states

13:30 – 14:00 Takeshi Yanai (IMS)

Molecular electronic structure theory based on ab initio density

matrix renormalization group

14:00 – 14:30 Takami Tohyama (TUS)

Density-matrix renormalization group study of Kitaev-Heisenberg

models on honeycomb and triangular lattices

14:30 – 15:00 Coffee break + Photo

15:00 – 15:30 Frank Pollmann (MPIPSK)

Many-body localization: Entanglement and efficient numerical

simulations

15:30 – 16:00 Philippe Corboz (Amsterdam)

Recent advances in simulating the 2D Hubbard and t-J models with

iPEPS

16:00 – 16:30 Hui-Hai Zhao (ISSP)

Variational Monte Carlo Study of fermionic models with tensor

networks

16:30 – 16:45 Naoki Kawashima (ISSP)

Closing

Poster Session (July 4, 14:30 – 16:00)

Andrew S. Darmawan (Sherbrooke)

Tensor-network simulations of topological codes under realistic noise

Lars-Hendrik Frahm (Humburg)

Coherent Dynamics of Magnetic Atoms in Spin-Polarized Environments

Ryo Igarashi (UTokyo)

Randomization algorithm in MPS/PEPS

Chung-Yu Lo (NTHU)

Long range correlations of one-dimensional impurity systems: DMRG study with

infinite boundary condition

Valentin Stauber (Vienna)

Symmetry Breaking and the Geometry of Reduced Density Matrices

Takafumi Suzuki (Hyogo)

Dynamical and thermal properties of Kitaev-Heisenberg magnets

Masahiko G. Yamada (ISSP)

Design of Ru-based honeycomb metal-organic frameworks and JKΓ model

Youhei Yamaji (UTokyo)

Open-source Numerical Diagonalization Application HΦ

Yilin Zhao (McMaster)

A tree tensor method for the simultaneous determination of multiple eigenstates

Syllabuses of Workshop Lectures

Lectures on 1st week June 28 10:00 – 11:30

Naoki Kawashima (ISSP)

Introduction

In this first lecture of the workshop, I give a brief overview of the coming lectures and

events. I also give some discussions on why we should focus on the tensor network, in

particular, among many new promising numerical techniques.

June 28 15:00 – 16:30

Glen Evenbly (UC Irvine)

1st lecture: Introduction to the MERA

I will offer an introduction to entanglement renormalization, a coarse-graining

transformation for quantum systems on the lattice, and demonstrate how this leads to a

class of tensor network state known as the MERA. Properties of the MERA, including

the scaling of correlators and entanglement entropy, will be discussed.

June 29 10:00 – 11:30

Mari-Carmen Bañuls (MPQ)

Using TNS for Lattice Gauge Theories

Lattice Gauge Theories, in their Hamiltonian version, offer a challenging scenario for

Tensor Network (TN) techniques. While the dimensions and sizes of the systems

amenable to TNS studies are still far from those achievable by 4-dimensional systems,

TN can be readily used for problems which more standard techniques cannot easily

tackle, such as the presence of a chemical potential, or out-of-equilibrium dynamics.

We have explored the performance of Matrix Product States (MPS) in the case of the

Schwinger model, as a widely used testbench for lattice techniques, and also in a

non-Abelian scenario. Using finite-size, open boundary MPS, we are able to determine

low energy states of the model away from any perturbative regime. The precision

achieved by the method allows for accurate finite size and continuum limit

extrapolations of the ground state energy, but also of the mass gaps and temperature

dependent quantities, thus showing the feasibility of these techniques for gauge theory

problems.

June 29 15:00 – 16:30

Glen Evenbly (UC Irvine)

2nd lecture: Introduction to tensor network renormalization

I will offer an overview of tensor renormalization algorithms for partition functions (of

classical many-body systems) and path integrals (of quantum many-body systems), and

then introduce the newly developed tensor network renormalization (TNR) algorithm,

which is based upon incorporation of disentangling into the coarse-graining step.

June 30 10:00 – 11:30

Masaki Oshikawa (ISSP)

Matrix-Product States and Symmetry-Protected Phases

I will discuss Matrix-Product States (MPS), which is a powerful formulation to describe

gapped phases in one dimension. In particular, I will discuss how global symmetries

are represented in terms of MPS. This leads to characterization of distinct quantum

phases in the presence of symmetries, namely Symmetry-Protected Topological (or

trivial) phases.

June 30 15:00 – 16:30

Román Orús (Mainz)

1st lecture: From qubits to entanglement, and then to Matrix Product States

In this first lecture I will explain basic properties of entanglement in quantum

mechanics. After reviewing how this behaves in some quantum many-body systems, I

will motivate the picture of tensor networks. I will then explain Matrix Product States,

which is a particular tensor network useful to describe 1d quantum lattice systems.

July 1 10:00 – 11:30

Tadashi Takayanagi (Kyoto)

Continuous MERA and Holography

After we introduce the idea of holography (or AdS/CFT correspondence), we would like

to give an overview of recent interpretations of holography in terms of tensor networks.

In particular, we would like to emphasize that continuous MERA offers us an excellent

candidate of such holographic tensor networks for CFTs.

July 1 15:00 – 16:30

Román Orús (Mainz)

2nd lecture: Simulating 2d systems with PEPS

In these lectures I will explain the basic properties of PEPS and how they can be used to

simulate 2d quantum lattice systems. I will show how to implement algorithms for

finite systems, infinite systems, fermions, as well as different strategies to implement

the simulations in the thermodynamic limit such as corner transfer matrices and fast

full update.

Lectures on 2nd week

July 5 10:00 – 11:30

Örs Legeza (Wigner RCP)

Tensor product methods and entanglement optimization for models with long range

interactions

Hierarchical Tucker tensor (HT) format and Tensor Trains (TT) have been introduced

recently for low rank tensor product approximation. TT representation, also known as

Matrix Product States (MPS), and HT representation, apparent in tensor network

states (TNS), have been used in quantum physics for several years. Hierarchical tensor

decompositions are based on subspace approximation by extending Tucker

decomposition into a multilevel framework. Therefore, they inherit the favorable

properties of Tucker tensors, i.e., they offer a stable and robust approximation, but still

enabling low order scaling with respect to the dimensions. For many high dimensional

problems, hard to treat so far, this approach may offer a novel strategy to circumvent

the curse of dimensionality.

In this contribution, we overview tensor network states techniques that can be used for

the treatment of high-dimensional optimization tasks used in many-body quantum

physics with long range interactions and ab initio quantum chemistry. Among the

various optimization tasks, we will discuss those which are connected to a controlled

manipulation of entanglement, which is in fact the key ingredient of such methods.

July 5 15:00 – 16:30

July 6 15:00 – 16:30

Frank Pollmann (MPIPKS)

Detecting topological orders from Matrix-Product State based simulations

In this lecture, I will discuss how to numerically obtain several quantities that

characterize different classes of topological order starting from a microscopic

Hamiltonian. First, I will consider symmetry protected topological phases and show

that these can be classified using matrix-product state based methods. Second, I will

demonstrate how characteristic properties of topological excitations can be extracted

for systems with intrinsic topological order.

July 6 10:00 – 11:30

July 7 10:00 – 11:30

Frank Verstraete (Vienna)

TBA

July 7 15:00 – 16:30

July 8 15:00 – 16:30

Norbert Schuch (MPQ)

Topological order in Projected Entangled Pair States

In my lecture, I will discuss how topological order manifests itself in Projected

Entangled Pair States. In particular, I will explain conditions on the local entanglement

structure necessary for topological order and its relation to the physical symmetries of

the system, and the way in which these conditions can give rise to a range of different

topological phases in the system.

July 8 10:00 – 11:30

Tomotoshi Nishino (Kobe)

Tensor Product States applied to Statistical Lattice Models

We consider variational estimation of the free energy of statistical lattice model, such as

the Ising model, by means of the matrix product state (MPS) and its two-dimensional

generalization, the tensor product state (TPS). It should be noted that most of the

important concepts were established in 1968 by Baxter.

Lectures on 3rd week July 12 10:00 – 11:30

July 13 10:00 – 11:30

Philippe Corboz (Amsterdam)

Introduction to iPEPS

In these two lectures I provide a practical introduction to simulations with iPEPS

(infinite projected entangled-pair states or infinite tensor product states). Topics

covered include discussion of the ansatz, contraction methods (with special focus on

the corner transfer-matrix method), optimization methods (imaginary time evolution

using simple- and full update, and variational optimization), optimization strategies,

systematic study of competing states, and generalization to fermionic systems.

July 12 15:00 – 16:30

July 13 15:00 – 16:30

Tao Xiang (CAS, Beijing)

Renormalization of Tensor Network Models

In this talk, I will give an extensive introduction to the tensor renormalization group

and discuss its application in the two or three dimensional quantum lattice models.

Emphasis will be given to the coarse graining tensor renormalization using the

higher-order singular value decomposition (HOTRG) and the projected entangled

simplex representation for the ground state wave function of quantum lattice models.

The HOTRG provides an accurate but low computational cost technique for studying

both classical and quantum lattice models in two or three dimensions. I will

demonstrate this method using the Ising model on the cubic lattices. The projected

entangled simplex state (PESS), on the other hand, provides a good representation of

tensor network states for highly frustrated quantum lattice models. The PESS extends

the pair correlation in the projected entangled pair states (PEPS) to a simplex. For the

spin-1/2 Heisenberg model on the Kagome lattice, we obtain an accurate result for the

ground state energy, which agrees with other numerical calculations and sets a new

upper bound for the ground state energy.

July 14 10:00 – 11:30

Luca Tagliacozzo (U. Strathclyde)

Constructing lattice gauge theories with Tensor Networks

In this lecture I will formalize the concept of a lattice gauge theory from scratch by

using tensor networks. In this way students with background on many body quantum

systems and tensor networks will be able to understand what a lattice gauge theory is.

July 14 15:00 – 16:30

Ying-Jer Kao (NTU)

Programming tensor network algorithms

In this lecture, we will give some concrete examples and programming tips on how to

program the tensor network algorithms using an open source tensor network library

uni10 (http://uni10.org). In particular we will give specific instructions on how to

implement infinite time-evolution bond-decimation (iTEBD) algorithm in 1D and 2D,

and the tensor renormalization group (TRG).

July 14 10:00 – 11:30

Takami Tohyama (TUS)

Dynamical properties of strongly correlated electron systems studied by

density-matrix renormalization group

We are developing dynamical density-matrix renormalization group (DDMRG) for

studying dynamical properties of strongly correlated electron systems not only in one

dimension but also in two dimensions. Recent results obtained by DDMRG on

K-computer will be discussed.

July 15 13:00 – 14:30

Naoki Kawashima (ISSP)

Closing Remarks

Abstracts of Symposium Talks

Entanglement Renormalization and Wavelets

Glen Evenbly

Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575

USA

I shall establish a precise connection between discrete wavelet transforms (WTs) and entan-

glement renormalization (ER) in the context of free particle systems. Specifically, I employ

Daubechies wavelets to build approximations to the ground state of the critical Ising model, then

demonstrate that these states correspond to instances of the multi-scale entanglement renormal-

ization ansatz (MERA), producing the first known analytic MERA for critical systems. I will also

discuss how the wavelet/MERA connection could also lead to useful advances in the design of

wavelet transforms and in wavelet applications.

Some Progress on Tensor Renormalization Group

Z. Y. Xie

Department of Physics, Renmin University of China, Beijing 100872, China

In this talk, I will talk about some recent progress on coarse-graining tensor renormalization group method in our group, which including the higher order tensor renormalization group (abbreviated as HOTRG) method[1], and the recent applications of this method in the classical Ising model on the cubic lattice[2], the classical XY model[3], and the finite size Kitaev model[4]. If it is possible, I will also talk about our latest work on tensor renormalization based on the variational calculation.

[1] Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang, Phys. Rev. B 86, 045139

(2012) [2] S. Wang, Z. Y. Xie, J. Chen, B. Normand, and T. Xiang, Chin. Phys. Lett. 31, 070503

(2014) [3] J. F. Yu, Z. Y. Xie, Y. Meurice, Yuzhi Liu, A. Denbleyker, H. Y. Zou, M. P. Qin, J. Chen,

and T. Xiang, Phys. Rev. E 89, 013308 (2014) [4] H. H. Zhao, Z. Y. Xie, T. Xiang, and M. Imada, Phys. Rev. B 93, 125115 (2016)

Finding almost conserved local quantities in non-integrablequantum systems

H. Kim1, M. C. Bañuls2, J. I. Cirac2, D. Huse3, M. Hastings4

1Rutgers University, NJ, USA

2Max Planck Intitut für Quantenoptik, Germany

3Princeton University, NJ, USA

4Microsoft Research, USA

The question of thermalization of closed quantum systems refers to whether local expectationvalues attain stationary values, independent of the details of the initial state, and it is of fundamen-tal interest. Generically, non-integrable systems, in which the only conserved local quantity is theenergy density, are expected to reach thermalization. But experimental and numerical results havehinted at the existence of very slow time scales in systems without local conserved quantities.

Using tensor network techniques and exact diagonalization, we have shown how very largetime scales can appear in the dynamics of one dimensional spin chains. Our method finds opera-tors evolving slower than the slowest energy mode in a non-integrable system.

Our method opens the door to exploring the long-time physics in other scenarios, by system-atically exploring the existence of almost conserved local quantities. In particular the problem ofmany body localization suggests several setups where this new technique can be of use.

[1] H. Kim, M. C. Bañuls, J. Cirac, M. Hastings and D. Huse, Slowest local operators in quan-

tum spin chains, Phys. Rev. E92 (2015), 012128, [arXiv:1410.4186].

Matrix Product Multi-Linear Algebra Library

N. Nakatani

Institute for Catalysis, Hokkaido University, Japan

Density matrix renormalization group (DMRG) is now widely used as a powerful tool to theoretically investigate many-body systems. A wavefunction derived from the DMRG calculation is of a product of site-independent matrices, so called matrix product state (MPS), which enables to describe a many-body wavefunction in a very compact formalism. Similarly, an operator can be written in terms of matrix product operator (MPO), which gives a very compact formalism of many-body operators.

In the traditional quantum mechanics, a wavefunction and an operator are written down in a vector and a matrix, respectively. And most of all numerical algorithms had been implemented in terms of matrix-vector linear algebra. Here, let MPS and MPO be alternative representations of a wavefunction and an operator, respectively, and formulate a new numerical algorithm in terms of MPS-MPO multi-linear algebra. This is of great use to implement a many-body wavefunction theory in the formulation as it is, i.e. no working equations and no further approximations are required, since matrix product representations involve the schematic approximation procedure themselves, and the accuracy of computation can easily be controlled by a single parameter, M (or D), the dimension of each matrix.

In working implementations of quantum chemistry calculations, BLAS/LAPACK library is widely utilized to carry out matrix-vector linear algebra efficiently. Here, I would provide a BLAS-like library program to carry out MPS by MPS, MPO by MPS, and MPO by MPO multiplications and compressions of MPS and MPO, in a simple way. These library functions has implemented in terms of the DMRG sweep algorithm to efficiently compute them.

I will show some demonstrative applications using the MPS-MPO multi-linear algebra. One of the simplest applications is the imaginary time-evolution to compute the ground state. In quantum chemistry, it is very difficult to explicitly write a propagator or even employ the Trotter-type decomposition because of the complicated structure of molecular Hamiltonians. Therefore, the Taylor expansion for the up to the first-order has been often considered to express a propagator. In the MPO representation, on the other hand, it is easy to include the higher-order terms of Taylor series to express the propagator without much increase of computational scaling. Figure 1 showed that including the higher-order terms well improves the convergence of the imaginary time-evolution simulation.

Figure 1: Imaginary time-evolution tocompute the ground state of C20H22 molecule.

Development of parallel library for tensor network methods

S. Morita

Institute for Solid State Physics, The University of Tokyo, Japan

Tensor network method is one of numerical methods for quantum and classical many-body systems. Uni10 [1] and ITensor [2] are well-known libraries for tensor network algorithms. To achieve highly accurate simulation for interesting problems, one needs to increase bond dimensions of tensors. However, the computational cost and the amount of memory rapidly increase with bond dimension. Since these libraries do not support parallel computation, parallelized library for tensor network methods is desired.

In the first part of my talk, I will introduce our developing library for parallel tensor calculations [3]. Common operations appearing tensor network methods are implemented using C++ language with hybrid parallelization (MPI + OpenMP). In our library, a tensor is stored as a matrix on distributed memory and main operations in tensor network methods, “contraction” and “decomposition”, are done by calling routines in ScaLAPACK [4]. We adopted application programming interface similar to NumPy, which is the fundamental Python package with multidimensional array. One can easily translate a Python test code into a parallel C++ code with our library.

In the second part, I will show performance of randomized algorithm for tensor decomposition. In the tensor network methods, singular value decomposition (SVD) and QR decomposition typically appear and they would need the heaviest computational cost. In the most case, however, the exact decomposition is not necessary. For example, one 4-rank tensor with bond dimension 𝜒 is decomposed into two 3-rank tensors connected by a χ-dim bond, in which only 𝑂(𝜒) singular values are necessary. For such a problem, we developed parallelized codes of the randomized algorithm for the low-rank approximation, which was recently proposed [5]. Its computational cost is 𝑂(𝜒5) for the previous example in contrast to the 𝑂(𝜒6) cost of the exact decomposition. We applied the randomized algorithm to the tensor renormalization group (TRG), and succeeded in reducing its computational cost from 𝑂(𝜒6) to 𝑂(𝜒5) by avoiding a creation of 4-rank tensors [1] https://github.com/yingjerkao/uni10/ [2] http://itensor.org/ [3] https://github.com/smorita/mptensor/ [4] http://www.netlib.org/scalapack/ [5] N. Halko, P. G. Martinsonn, J. A. Tropp, SIAM Review 53, 217 (2011).

Compression of Correlation Matrices and an E�cientMethod for Forming Matrix Product States of Fermionic

Gaussian States

Matthew T. Fishman1, Steven R. White2

1Institute for Quantum Information and Matter, California Institute of Technology,

Pasadena, CA 91125, USA

2Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA

Here we present an e�cient and numerically stable procedure for compressing a correlationmatrix into a set of local unitary single-particle gates, which leads to a very e�cient way offorming the matrix product state (MPS) approximation of a pure fermionic Gaussian state, such asthe ground state of a quadratic Hamiltonian. The procedure involves successively diagonalizingsubblocks of the correlation matrix to isolate local states which are purely occupied or unoccupied.A small number of nearest neighbor unitary gates isolates each local state. The MPS of this stateis formed by applying the many-body version of these gates to a product state. We treat the simplecase of compressing the correlation matrix of spinless free fermions with definite particle numberin detail, though the procedure is easily extended to fermions with spin and more general BCSstates (utilizing the formalism of Majorana modes). We also present a DMRG-like algorithm toobtain the compressed correlation matrix directly from a hopping Hamiltonian. In addition, wediscuss a slight variation of the procedure which leads to a simple construction of the multiscaleentanglement renormalization ansatz (MERA) of a fermionic Gaussian state, and present a simplepicture of orthogonal wavelet transforms in terms of the gate structure we present in this paper. Asa simple demonstration we analyze the Su-Schrie�er-Heeger model (free fermions on a 1D latticewith staggered hopping amplitudes).

Topological order and space group symmetryfractionalization in the frustrated J1-J2 Heisenberg model

I. P. McCulloch1, S. Saadatmand1

1Centre for Engineered Quantum Systems, School of Mathematics and Physics, The

University of Queensland, Brisbane, Australia

We report on the results of a large-scale numerical study of the spin-1/2 Heisenberg model onthe triangular lattice, with nearest- and next-nearest neighbor interactions. Using SU (2)-invariantiDMRG for infinite cylinders up to width 12, we obtain 4 candidate groundstates, corresponding toeven/odd spinon sectors, each with linear and projective representations of the cylinder geometry.The only topological order that is consistent with these symmetries is the Z2⇥ Z2 Toric Code. Themomentum-resolved entanglement spectrum reveals the structure of the low-lying excitations,which has some unusual features, suggesting Dirac nodes indicating gapless excitations in thethermodynamic limit. We find no evidence for chiral symmetry breaking, contrary to some recentworks.

Branching and tensor network

Kenji HaradaGraduate School of Informatics, Kyoto University, Japan

I will propose a “branching” operation of a tensor index for tensor network algorithms[1].To define a branching operation explicitly, we will introduce a branching operator in a tensornetwork. We can determine it by a local optimization problem. The concept of a branchingoperation has many applications for tensor network algorithms. Using a branching operator, wecan transform a tensor to a tensor network defined by multiple tensors. For example, a many-bodydecomposition from a single tensor to a four-body tensor network. Apart from complexity, using amany-body decomposition technique, we can derive a projected entangled pair state from a generalwave-function, and we can solve a perfect disentanglement problem of a local tensor network.

[1] Kenji Harada, “Branching and tensor network”, 20pBT-5, the JPS Annual (71-th) MarchMeeting, 2016, Japan.

Real-space parallel infinite-size density matrixrenormalization group

Hiroshi UedaRIKEN Advanced Institute for Computational Science (AICS), Japan

We develop a procedure of parallel infinite-size density matrix renormalization group (iDMRG)applicable for a quantum system represented by a matrix product operator with long-periodstructures. In the procedure, we combine the Hida’s iDMRG for one-dimensional random spinsystems [1] with the McCulloch’s wave function prediction [2]. This procedure can be regarded asan infinite-size version of the real-space parallel DMRG proposed by Stoudenmire and White [3].One of merits of our parallel iDMRG is that we can implement it easily by inserting slight MPIcommunications to multiple iDMRG calculations. In addition, our parallel iDMRG does notcontain any kind of finite-size sweeping in contrast to the traditional procedure of iDMRG fora system represented by a matrix product operator with long-period structures. Therefore, ouriDMRG is suitable for analyzing properties of multi-leg ladder/cylinder systems with the infinitelength.

We discuss details of our parallel iDMRG, and show benchmark calculations for physicalquantities of a ground state of the spin-1/2 Heisenberg model on the YC8 kagome cylinder withthe infinite length. The Hamiltonian of this system can be represented by a matrix product operatorwith a 12-site period. The YC8 cylinders with finite lengths have already been investigated by usingthe finite-size DMRG [4] and non-Abelian DMRG [5, 6]. Applying the appropriate subtractionmethods [7] and extrapolations to the limit of infinitesimal truncation error in DMRG [8] to rawdata of our parallel iDMRG, we find that our algorithm can evaluate proper energy per site, bondstrength on nearest neighbors, spin-spin correlation functions and its correlation length despiteusing relatively small number of kept states in iDMRG, namely up to 2800 states. Also we showour algorithm can be parallelized e�ciently for both shared-memory and distributed memorysystems [9].

[1] K. Hida, J. Phys. Soc. Jpn. 65, 895 (1996).[2] I. P. McCulloch, arXiv:0804.2509.[3] E. M. Stoudenmire and S. R. White, Phys. Rev. B 87, 155137 (2013).[4] S. Yan, D. A. Huse, and S. R. White, Science 332, 1173 (2011).[5] S. Depenbrock, I. P. McCulloch, and U. Schollwök, Phys. Rev. Lett. 109, 067201 (2012).[6] F. Kolley, S. Depenbrock, I. P. McCulloch, U. Schollwöck, and V. Alba, Phys. Rev. B 91,

104418 (2015).[7] E. Stoudenmire and S. R. White, Annu. Rev. Condens. Matter Phys. 3, 111 (2012).[8] S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004 (2007).[9] HU, submitted to Phys. Rev. B.

Kitaev honeycomb tensor networks: exact unitary circuitsand applications

P. Schmoll and R. Orús1

1Institute of Physics, Johannes Gutenberg University, 55099 Mainz, Germany

The Kitaev honeycomb model is a paradigm of exactly-solvable models, showing non-trivialphysical properties such as topological quantum order, abelian and non-abelian anyons, and chi-rality. Its solution is one of the most beautiful examples of the interplay of di↵erent mathematicaltechniques in condensed matter physics. In this talk, I will show how to derive a tensor network(TN) description of the eigenstates of this spin-1/2 model in the thermodynamic limit, and in par-ticular for its ground state. In our setting, eigenstates are naturally encoded by an exact 3d TNstructure made of fermionic unitary operators, corresponding to the unitary quantum circuit build-ing up the many-body quantum state. In the derivation I will review how the di↵erent ÒsolutioningredientsÓ of the Kitaev honeycomb model can be accounted for in the TN language, namely:Jordan-Wigner transformation, braidings of Majorana modes, fermionic Fourier transformation,and Bogoliubov transformation. The TN built in this way allows for a clear understanding ofseveral properties of the model. In particular, we show how the fidelity diagram is straightforwardboth at zero temperature and at finite temperature in the vortex-free sector. Finally, I will alsodiscuss the pros and cons of contracting of our 3d TN down to a 2d Projected Entangled Pair State(PEPS) with finite bond dimension.

[1] P. Schmoll and R. Orús, in preparation.

Heisenberg antiferromagnet on the Husimi lattice

H. J. Liao1, Z. Y. Xie1, 2, J. Chen1, X. J. Han1, H. D. Xie1, B. Normand2, and T. Xiang1, 3

1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics, Renmin University of China, Beijing 100872, China 3Collaborative Innovation Center of Quantum Matter, Beijing 100190, China

We perform a systematic study of the antiferromagnetic Heisenberg model on the Husimi

lattice[1] using numerical tensor-network methods based on Projected Entangled Simplex States (PESS). The nature of the ground state varies strongly with the spin quantum number, S. For S = 1/2, it is an algebraic (gapless) quantum spin liquid. For S = 1, it is a gapped, non-magnetic state with spontaneous breaking of triangle symmetry (a trimerized simplex-solid state). For S = 2, it is a simplex-solid state with a spin gap and no symmetry-breaking; both integer-spin simplex-solid states are characterized by specific degeneracies in the entanglement spectrum. For S = 3/2, and indeed for all spin values S ≥ 5/2, the ground states have 120-degree antiferromagnetic order. In a finite magnetic field, we find that, irrespective of the value of S, there is always a plateau in the magnetization at m = 1/3. [1] H. J. Liao, Z. Y. Xie, J. Chen, X. J. Han, H. D. Xie, B. Normand, and T. Xiang, Phys. Rev.

B 93, 075154 (2016).

Magnetization process of kagome lattice Heisenberg

antiferromagnets: 1/3 plateau state and effects of

Dzyaloshinskii-Moriya interaction

T. Okubo, N. Kawashima

Institute for Solid State Physics, The University of Tokyo, Japan

The kagome lattice Heisenberg antiferromagnet is a typical example of the two-dimensional frustrated spin systems. Due to strong quantum fluctuations, the ground state of the S=1/2 quantum spin kagome lattice Heisenberg model is expected to be a spin-liquid state without any magnetic long-range orders. Under magnetic fields, it has been proposed that several magnetization plateaus at 1/9, 1/3, 5/9, and 7/9 of the saturation magnetization appear based on a density matrix renormalization group (DMRG) calculation [1]. Recently, a tensor network calculation has also shown the existence of these magnetization plateaus [2]. On the other hand, based on the exact diagonalization (ED), Nakano and Sakai proposed that the expected 1/3 plateau has peculiar critical exponents compared to other two-dimensional systems such as the triangular lattice Heisenberg antiferromagnets [3,4].

In this talk, I will present the ground state properties of S=1/2 kagome lattice Heisenberg antiferromagnet under external magnetic fields using an infinite Projected Entangled Pair State (iPEPS) tensor network method. In this iPEPS method, we represent the ground state wave-function as the two-dimensional network of tensors. By optimizing each tensor so as to minimize the total energy, we obtained wave-functions close to the ground state under magnetic fields. The magnetization curve obtained by iPEPS contains clear 1/9, 1/3, 5/9 and 7/9 plateaus that are consistent with the previous calculations. The 1/3-plateau state obtained by the simple update optimization was semi classical up-up-down state, which was different from a resonating state observed in DMRG [1]. We will discuss about the nature of the 1/3 plateau state by introducing a cluster update optimization. We also investigate effects of the Dzyaloshinskii-Moriya (DM) interaction, which exists in real kagome lattice compounds. Our calculation shows that the plateau width becomes smaller when we increase the amplitude of the DM interaction and the plateaus disappear for Dz/J > 0.1. [1] S. Nishimoto, N. Shibata, and C. Hotta, Nat. Commun. 4, 2287 (2013). [2] T. Picot, M. Ziegler, R. Orus, and D. Poilblanc, Phys. Rev. B 93, 060407(R) (2016). [3] H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 79, 053707 (2010). [4] H. Nakano and T. Sakai, J. Phys. Soc. Jpn. 83, 104710 (2014).

Featureless Quantum Insulator on the Honeycomb Latticeand Square lattice

Hyunyong Lee1, Jung Hoon Han1,1

Department of Physics, Sungkyunkwan University, Suwon 16419, Korea

We show how to construct fully symmetric, gapped states without topological order on ahoneycomb lattice for S = 1/2 spins using the language of projected entangled pair states(PEPS).An explicit example is given for the virtual bond dimension D = 4. Four distinct classes di�ering bylattice quantum numbers are found by applying the systematic classification scheme introduced byRef. [1]. Lack of topological degeneracy or other conventional forms of symmetry breaking, andthe existence of energy gap in the proposed wave functions, are checked by numerical calculationsof the entanglement entropy and various correlation functions. Our work provides the first explicitrealization of a featureless quantum insulator for spin-1/2 particles on a honeycomb lattice.

Secondly, we classify QSL wave functions of spin-1 preserving all the lattice symmetries,spin rotation, and time reversal on the square lattice. Several explicit constructions of such wavefunctions are given for bond dimensions ranging from two to four, along with thorough numericalanalyses to identify their physical characters. We found four distinct phases, i.e. the plaquette-ordered phase, gapped and gapless RVB phases and critical phase where both spin and dimercorrelations decay algebraically. The phase boundary has been detected e�ciently by the fidelityanalysis. Details of the phase transition and numerical results will be given.

[1] S. Jiang and Y. Ran, Phys. Rev. B 92, 104414 (2015).

Boundary States in CFTs and Continuous MERA

T. Takayanagi1

1 Yukawa Institute for Theoretical Physics, Kyoto University, Japan

The holographic principle or gauge/gravity correspondence in string theory tells us that a gravitational theory in a (d+1) dimensional spacetime is equivalent to a non-gravitational theory (i.e. a quantum theory of matter) in a d dimensional spacetime. One of the most famous examples of holography is an equivalence between gravity in anti de-Sitter (AdS) space and conformal field theory (CFT), so called AdS/CFT correspondence. Recently, it has been pointed out that the basic mechanism of AdS/CFT can be regarded as a special class of tensor networks which describe quantum critical points (i.e. CFTs), such as MERA and its generalizations. This is mainly because in holography, the structure of quantum entanglement is directly interpreted as the geometry of gravitational spacetime. In this talk we would like to focus on continuous version of MERA so called cMERA and apply this idea to CFTs. First we point out that the infrared state in a cMERA for a CFT, which is expected to have vanishing quantum entanglement, can be constructed from so called boundary states (or Cardy states) in CFTs [1]. This in principle gives us a general formulation of cMERA for any CFTs. Then we will explain how various geometrical properties of anti de-Sitter space such as killing symmetry and metric can be derived from purely conformal field theoretic data [2,3]. [1] M. Miyaji, S. Ryu, T. Takayanagi, X. Wen, JHEP 1505(2015)152. [2] M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi, K. Watanabe, Phys.Rev.Lett.115 (2015) 17, 171602.

[3] T.Takayanagi, work in progress.

Tensor networks with loop optimization

Shuo Yang

1

1

Perimeter Institute for Theoretical Physics, Waterloo, N2L 2Y5, Canada

I will present our recent work about Loop optimization for tensor network renormalization [1]

and its new generalizations and applications [2]. In Ref. [1], we introduce a tensor renormalization

group scheme for coarse-graining a two-dimensional tensor network, which can be successfully

applied to both classical and quantum systems on and o↵ criticality. The key idea of our scheme

is to deform a 2D tensor network into small loops and then optimize tensors on each loop. In

this way we remove short-range entanglement at each iteration step, and significantly improve the

accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical

Ising model and a frustrated 2D quantum model. In Ref. [2], we show new generalizations and

applications of Ref. [1].

[1] S. Yang, Z.-C. Gu and X.-G. Wen, arXiv:1512.04938.

[2] S. Yang et al., in preparation.

Mean-Field Behavior in Uniform Tensor Product States

Tomotoshi Nishino1, Andrej Gendiar2

1Department of Physics, Graduate School of Science, Kobe University, Japan 2Institute of Physics, Slovak Academy of Sciences, SLOVAKIA

In variational studies by means of matrix product states (MPS), the variational (free-)

energy F can be minimized only through the tuning of matrix elements. Since the MPS is finitely correlated when the matrix dimension m is finite, F is an analytic function of finite number of parameters when the MPS is uniform. As a result, variational MPS always show mean-field behavior just around the phase transition point. [1-4] We briefly review this phenomenon in the case of Kramers-Wannier approximation. [1] The mean-field region is normally negligible in DMRG since sufficiently large degrees of freedom can be kept for the matrix indices. Note that on the hyperbolic space the phase transition is always mean-field like. [5] Similar behavior is observed also in the uniform 2D tensor product state (TPS), the 2-dimensional generalization of uniform MPS. [6,7] A TPS can be critical, but normally an optimized TPS after numerical minimization of (free-) energy is off-critical, as long as finite degrees of freedom is allowed for auxiliary space. To detect this “fictitious” mean-field behavior is a checkpoint for numerical optimization of uniform TPS; the corner-transfer matrix renormalization group [2, 6-9] makes it possible to evaluate the (free-) energy F rapidly.

(Terminologies: TPS = PEPS, uniform TPS = iPEPS)

[1] H.A. Kramers, G.H. Wannier, “Statistics of the Two-dimensional Ferromagnet, Part II”, Phys. Rev. 60, 263 (1941). [2] R.J. Baxter, “Dimers on a Rectangular Lattice”, J. Math. Phys. 9, 650 (1968). [3] S.K. Tsang, “Square Lattice Variational Approximations applied to the Ising Model”, J. Stat. Phys. 20, 95 (1979). [4] C. Liu, L. Wang, A.W. Sandvik, Yu-Cheng Su, Ying-Jer Kao, “Symmetry Breaking and Criticality in Tensor-Product States”, Phys. Rev. B 82, 060410(R)(2010) [5] T. Iharagi, A. Gendiar, H. Ueda, T. Nishino, “Phase Transition of the Ising Model on a Hyperbolic Lattice”, J. Phys. Soc. Jpn. 79, 104001, (2010). [6] T. Nishino, K. Okunishi, Y. Hieida, N. Maeshima, Y. Akutsu, “Self-consistent Tensor Product Variational approximation for 3D classical models”, Nucl. Phys. B 3, 504 (2000). [7] T. Nishino, Y. Hieida, K. Okunishi, N. Maeshima, Y. Akutsu, A. Gendiar, “Two-dimensional Tensor Product Variational Formulation”, Prog. Theor. Phys. 3, 409 (2001). [8] R. Orus, G. Vidal, “Simulation of Two-dimensional Quantum Systems on an Infinite Lattice Revisited: CTM for Tensor Contraction”, Phys. Rev. B 80, 094403 (2009). [9] P. Corboz, “Variational Optimization with Infinite Projected Entangled-Pair States”, arXiv:1605.03006.

Quantum Critical Spin-2 Chain with Emergent SU(3)Symmetry

Pochung Chen1, Zhi-Long Xue1, I. P. McCulloch2, Ming-Chiang Chung3, Chao-ChunHuang4, and S.-K. Yip4

1Department of Physics, National Tsing Hua University, Taiwan2Centre for Engineered Quantum Systems, The University of Queensland, Australia

3Department of Physics, National Chung Hsing University, Taiwan4Institute of Physics and Institute of Atomic and Molecular Sciences, Academia Sinica,

Taiwan

We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained fromspin-2 bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies andentanglement entropy by exact diagonalization and density-matrix renormalization group methods.From the numerical results of the energy spectra, central charge, and scaling dimension we identifythe conformal field theory describing the whole critical phase to be the SU(3)1 Wess-Zumino-Witten model. We find that, while the Hamiltonian is only SU(2) invariant, in this critical phasethere is an emergent SU(3) symmetry in the thermodynamic limit.

[1] Pochung Chen, Zhi-Long Xue, I. P. McCulloch, Ming-Chiang Chung, Chao-Chun Huang,and S.-K. Yip, Phys. Rev. Lett. 114, 145301 (2015).

[2] Pochung Chen, Zhi-Long Xue, I. P. McCulloch, Ming-Chiang Chung, and S.-K. Yip, Phys.Rev. A 85, 011601 (2012).

Order parameters of bond-type symmetry protected

topological phases in two dimensions

Soichiro Mohri and Masaki Oshikawa

Institute for Solid State Physics, The University of Tokyo, Japan

One-dimensional symmetry protected topological (SPT) phases can be classified by equivalence classes of projective presentations [1]. And the projective representations can be calculated using matrix product state (MPS) representations [2]. Therefore, one-dimensional SPT phases can be detected by numerical methods based on MPS representations such as density matrix renormalization group and time-evolving block decimation. For example, SPT phases protected by the spin rotation symmetry can be detected by the following order parameter.

Here, U is the projective representation of the symmetry, � is the dominant eigenvalue of the transfer matrix and � is the bond dimension.

We extend the order parameter of SPT phases to two-dimensional system. Representing the states in terms of projected entangled pair states (PEPS) allows us to calculate projective representations (Fig. 1). Therefore, two-dimensional bond-type SPT phases such as AKLT states can be detected by PEPS formalism. To verify this scheme, we performed numerical calculation of the anisotropic Heisenberg model on the square lattice (Fig. 2).

[1] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Phys. Rev. B 81, 064439 (2010); X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011). [2] F. Pollmann and A. M. Turner, Phys. Rev. B 86, 125441 (2012).

Figure 2: The order parameters of the anisotropic Heisenebrg model on the square lattice.

Figure 1: The method to calculate a projective representation of a PEPS. The wavy and solid lines indicate the physical and virtual degrees of freedom, respectively. By contracting the virtual degrees of freedom, the problem reduce to the case of MPS.

UU-1

u

= ei

Tensor product methods and entanglement optimization formodels with long range interactions

Ö. Legeza1, L. Veis1, V. Murg2, F. Verstraete2, C. Krumnow3, J. Eisert3,G. Ehlers4, and R. M. Noack4

1“Lendület” Research Group, Wigner Research Centre for Physics, Budapest, Hungary2Fakultät für Physik, Universität Wien, Vienna, Austria

3Dahlem Center for Complex Quantum Systems, Freie Universität, Berlin, Germany4Philipps-Universität Marburg, Marburg, Germany

Tensor network states and specifically matrix-product states have proven to be a powerful toolfor simulating ground states of strongly correlated spin and fermionic models. In this contribu-tion, we focus on tensor network states techniques that can be used for the treatment of high-dimensional optimization tasks in strongly correlated quantum many-body systems with longrange interactions [1, 2, 3]. We will present our recent developments on fermionic orbital op-timization and tree-tensor network states and discuss properties of various strongly correlatedsystems in light of the entanglement which gives insight into the fundamental nature of the cor-relations in their ground states. Examples will be shown for extended periodic systems, the two-dimensional Hubbard model at weak coupling, ⇡-conjugated polymers and graphene nanoribbons.

[1] S. Szalay, M. Pfe↵er, V. Murg, G. Barcza, F. Verstraete, R. Schneider, and Ö. Legeza, Int. J.Quant. Chem., 115, 1342 (2015).

[2] C. Krumnow, L. Veis, Ö. Legeza, J. Eisert, arXiv:1504.00042 (2015).

[3] G. Ehlers, J. Sólyom, Ö. Legeza, R. M. Noack, Phys. Rev. B 92, 235116 (2015).

Steady States of Infinite-Size Dissipative Quantum Chains

via Imaginary Time Evolution

Adil A. Gangat

1

, Te I

1

, and Ying-Jer Kao

1,2

1Department of Physics, National Taiwan University, Taipei 10607, Taiwan2National Center of Theoretical Sciences, National Tsing-Hua University, Hsinchu ,

30043, Taiwan

We show how to use imaginary time evolution of matrix product density operators with the in-

finite time-evolving block decimation algorithm to determine the nonequilibrium steady states of

one-dimensional dissipative quantum lattices in the thermodynamic limit. We provide a demon-

stration with the transverse field quantum Ising chain. The approach is also amenable to higher

dimensions.

Majorana Positivity and the Fermion Sign Problem of

Quantum Monte Carlo Simulations Tao Xiang

Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

The sign problem is a major obstacle in quantum Monte Carlo simulations for many-body fermion systems. We examine this problem with a new perspective based on the Majorana reflection positivity and Majorana Kramers positivity. Two sufficient conditions are proven for the absence of the fermion sign problem. Our proof provides a unified description for all the interacting lattice fermion models previously known to be free of the sign problem based on the auxiliary field quantum Monte Carlo method. It also allows us to identify a number of new sign-problem-free interacting fermion models including, but not limited to, lattice fermion models with repulsive interactions but without particle-hole symmetry and interacting topological insulators with spin-flip terms.

[1] Z. C. Wei, C. J. Wu, Y. Li, S. Zhang, T. Xiang, arXiv:1601.01994.

Chiral Projected Entangled Pair States

N. SchuchMax-Planck-Institute of Quantum Optics, Garching, Germany

Systems with chiral topological order are of particular interest in condensed matter physicssince they exhibit exotic features such as protected quantized edge modes, and also since theycan be realized experimentally in the fractional quantum Hall e�ect. For a long time, it has beenan open question whether Projected Entangled Pair States (PEPS) are capable of capturing thephysics of chiral models, which only recently has been answered in the a�rmative [1, 2]. In mytalk, I will explain the way in which PEPS can be used to describe systems with chiral topologicalorder [1, 3] and discuss recent results on chiral PEPS, including the way in which the chiral natureof fermionic PEPS can be understood from the local symmetries in the PEPS description [4], anda novel construction for chiral PEPS based on spin degrees of freedom which yields models witha chiral edge described by a SU(2)1 conformal field theory, such as in the ⌫ = 1/2 fractionalquantum Hall state [5, 6].

[1] T. Wahl, H.-H. Tu, N. Schuch, and J. Cirac, Phys. Rev. Lett. 111, 236805 (2013),arXiv:1308.0316.

[2] J. Dubail and N. Read, Phys. Rev. B 92, 205307 (2015), arXiv:1307.7726.

[3] S. Yang, T. B. Wahl, H.-H. Tu, N. Schuch, and J. I. Cirac, Phys. Rev. Lett. 114, 106803 (2015),arXiv:1411.6618.

[4] T. B. Wahl, S. T. Haßler, H.-H. Tu, J. I. Cirac, and N. Schuch, Phys. Rev. B 90, 115133 (2014),arXiv:1405.0447.

[5] D. Poilblanc, J. I. Cirac, and N. Schuch, Phys. Rev. B 91, 224431 (2015), arXiv:1504.05236.

[6] D. Poilblanc, N. Schuch, and I. A�eck, (2016), arXiv:1602.05969.

Finite bond dimension e↵ects in tensor network states

L. Tagliacozzo

1

, A. Coser

2

,

1

The University of Strathclyde Glasgow

2

Universidad Complutense de Mardird

We take a fresh look on the e↵ects that a finite bond dimension induces on tensor network

states in the vicinity of quantum critical points. In 1D the finite bond dimension of a matrix

product state close to a critical point induces a finite correlation lenght in the state it describes.

We discuss about generalizations of this phenomenon to higher dimensions.

Molecular electronic structure theory based on ab initio

density matrix renormalization group

T. Yanai1

1Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki, Japan

The density matrix renormalization group (DMRG) method has been emerging as a new

powerful tool to investigate multireference electronic structures for ab initio quantum chemistry calculations. The strength of the DMRG method arises in the applications where mean-field methods, such as density functional theory, are no longer reliable because of strong (or multireference) electron correlation effects. In my talk, we will describe recent methodological advances in quantum chemical DMRG and associated dynamical correlation methods [1-6], and demonstrate their applicability for chemical applications including π-conjugated organic systems [7,8] and multi-nuclear transition metal complexes [9,10]. [1] D. Ghosh, J. Hachmann, T. Yanai and G. K-L. Chan, J. Chem. Phys. 128, 144117 (2008) [2] Y. Kurashige and T. Yanai, J. Chem. Phys. 130, 234114 (2009). [3] T. Yanai, Y. Kurashige, E. Neuscamman, G. K-L. Chan, J. Chem. Phys. 132, 024105 (2010) [4] Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011). [5] M. Saitow, Y. Kurashige, and T. Yanai, J. Chem. Phys. 139, 044118 (2013). [6] Y. Kurashige, J. Chalupský, T. N. Lan, and T. Yanai, J. Chem. Phys. 141, 174111 (2014). [7] W. Mizukami, Y. Kurashige, and T. Yanai, J. Chem. Phys. 133, 091101 (2010). [8] W. Mizukami, Y. Kurashige, and T. Yanai, J. Chem. Theo. Comp. 9, 401-407 (2012). [9] Y. Kurashige, G. K-L. Chan, and T. Yanai, Nature Chem. 5, 660-666 (2013). [10] J. Chalupský, T. A. Rokob, Y. Kurashige, T. Yanai, E. I. Solomon, L. Rulíšek, and M. Srnec, J. Am. Chem. Soc, 136, 15977 (2014).

Density-matrix renormalization group study of Kitaev-

Heisenberg models on honeycomb and triangular lattices

Takami Tohyama

Department of Applied Physic, Tokyo University of Science, Japan

The Kitaev-Heisenberg (KH) honeycomb lattice model has recently been proposed to describe magnetic properties in A2IrO3 (A=Na, Li). The model includes an isotropic Heisenberg term and strongly anisotropic Kitaev terms. However, it has turned out that the KH model cannot explain a zigzag-type antiferromagnetic order observed in Na2IrO3. This discrepancy has inspired further studies about more suitable effective spin models for Na2IrO3. For example, anisotropic interactions due to trigonal distortions have been introduced to the KH model [1]. Not only honeycomb lattice but also triangular lattice has attracted attention. From experimental side, Ba3IrTi2O9 has been suggested as a possible candidate of a spin-liquid material with the triangular-lattice structure [2]. From theoretical side, because the KH model on the triangular lattice has both geometrical frustration and Kitaev-type frustration, the quantum effect on the model is expected to be highly non-trivial.

Motivated by these facts, we examine KH models on honeycomb and triangular lattices by using two-dimensional density-matrix renormalization group method (2D-DMRG).

In honeycomb lattice, we use an extended KH model including additional anisotropic interactions [3]. We make a phase diagram of the extended KH model around the Kitaev spin-liquid phase from the ground-state energy and spin-spin correlation functions. We also investigate entanglement entropy (EE) and entanglement spectrum (ES). We find that the lowest level of ES at magnetically ordered states is nondegenerate. This is in contrast to the Kitaev spin-liquid state, where all of ES form pairs. As a result, the nature of Schmidt gap changes at the phase boundary between the Kitaev spin liquid and other magnetically ordered phases. In triangular lattice, we use the KH model [4]. The phase diagram obtained by 2D-DMRG is found to be consistent with classical one. As is the case for spin correlation, EE discontinuously changes at all the phase boundaries. The Schmidt gap is closing at the boundaries. This is in contrast with the extended KH model on honeycomb lattice.

These works were done in collaboration with Kazuya Shinjo, Shigetoshi Sota, Seiji Yunoki, and Keisuke Totsuka. [1] Y. Yamaji et al., Phys. Rev. Lett. 113, 107201 (2014). [2] T. Dey et al., Phys. Rev. B 86, 140405(R) (2012). [3] K. Shinjo, S. Sota, and T. Tohyama, Phys. Rev. B 91, 054401 (2015). [4] K. Shinjo, S. Sota, S. Yunoki, K. Totsuka, and T. Tohyama, arXiv:1512.02334.

Many-body localization: Entanglement and e�cient

numerical simulations

F. Pollmann

1

1

Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany

Many-body localization (MBL) occurs in isolated quantum systems when Anderson localiza-

tion persists in the presence of finite interactions. To understand this phenomenon, the development

of new e�cient numerical methods to find highly excited many-body eigenstates is essential. In

this talk, we will discuss two approaches to simulate MBL systems: First, we introduce a variant of

the density-matrix renormalization group (DMRG) method that obtains individual highly excited

eigenstates of MBL systems to machine precision accuracy at moderate to large disorder [1, 2] (see

Fig. 1). This DMRG-X method explicitly takes advantage of the local spatial structure and the low

entanglement which is characteristic for MBL eigenstates. Second, we generalize the algorithm

to obtain Floquet eigenstates and study the fate of MBL in periodically driven systems.

�30

�15

0

15

30

Ene

rgy

Spe

ctru

m

Exact DMRG-X0.0

0.5

1.0

0.79

0.84

0.02

0.06

200 800n

0

0.4

�n ExactDMRG-X

Figure 1: Comparison between eigenen-

ergies obtained for a disordered XXZ

chain using exact diagonalization (blue)

and DMRG-X (red) for a system of size

L = 12.

[1] V. Khemani, F. Pollmann, S. L. Sondhi,

arXiv:1509.00483.

[2] F. Pollmann, V. Khemani, J. I. Cirac, S. L. Sondhi,

arXiv:1506.07179.

Recent advances in simulating the 2D Hubbard andt-J models with iPEPS

P. Corboz1

1Institute for Theoretical Physics, University of Amsterdam, The Netherlands

In this talk I report on recent progress in simulating the 2D Hubbard and t-J models with in-finite projected entangled-pair states (iPEPS) - a tensor network ansatz for 2D wave functions inthe thermodynamic limit. Our simulation results reveal an extremely close competition between auniform d-wave superconducting state and di↵erent types of stripe states, where iPEPS yields bet-ter variational energies than other state-of-the-art variational methods for large 2D systems [1]. Akey factor to determine the true ground state among several competing states is to have an accurateestimate of the energies in the infinite bond dimension D limit. However, a simple extrapolationin 1/D often does not provide an accurate result. We show how to improve these estimates byan extrapolation based on a truncation error which is computed within the iPEPS imaginary timeevolution algorithm. Finally, I present a variational optimization scheme for iPEPS which yieldsa higher accuracy than previous ground state algorithms.

[1] P. Corboz, Phys. Rev. B 93, 45116 (2016).

Variational Monte Carlo Study of fermionic models with

tensor networks Hui-Hai Zhao1,2

1 Department of Applied Physics, The University of Tokyo, Japan 2 Institute for Solid State Physics, The University of Tokyo, Japan

The conventional matrix product states (MPS) and projected entangled pair states (PEPS) employ the real space product state as reference wave function. By using variational Monte Carlo to optimize the tensor elements, various kinds of reference wave functions can be used[1, 2], such as free fermion, BCS or AFM states, which can be chosen according to the phase of the ground state. Moreover, we can apply quantum number projections, such as spin, momentum and lattice symmetry projections, to recover the symmetry of the wave function to further improve the accuracy. As a result, more accurate ground state wave function can be obtained without increasing tensor bond dimension. In this study, we combine many-variable variational Monte Carlo[3] with tensor networks to study fermionic models. The variational wave function is composed of pairing wave function with quantum number projections, Gutzwiller, Jastrow, doublon-holon correlation factors and tensor networks. In order to provide more flexible representation, we optimize all the variational parameters simultaneous in the wave function. [1] Chung-Pin Chou, Frank Pollmann, and Ting-Kuo Lee Phys. Rev. B 86, 041105(R)

(2012). [2] Olga Sikora, Hsueh-Wen Chang, Chung-Pin Chou, Frank Pollmann, and Ying-Jer Kao

Phys. Rev. B 91, 165113 (2015). [3] Daisuke Tahara and Masatoshi Imada, J. Phys. Soc. Jpn. 77, 114701 (2008).

Abstracts of Posters

Tensor-network simulations of topological codes underrealistic noise

A. S. Darmawan1, D. Poulin1,1Départmenet de Physique, Université de Sherbrooke, Québec, Canada

The surface code is a promising candidate for quantum error correction in many architectures,requiring only nearest-neighbour interactions on a two-dimensional square lattice. Our under-standing of the performance of the surface code is mostly based on numerical simulations whichhave found a high threshold relative to other error correction schemes. These simulations usuallyassume Pauli noise which has the advantage that it allows e�cient simulation within the stabilizerformalism. However most realistic noise processes (e.g. amplitude damping and systematic overrotation), are non-Pauli. In this work we present an improved simulation scheme for the surfacecode under local non-Pauli noise. Our algorithm is based on the tensor network description of thesurface code as a projected entangled pair state (PEPS). While this description of the surface codehas been used extensively in the field of condensed matter physics, it has not yet been expoitedto study the code’s error-correction properties. Syndrome sampling, computation of the processmatrix and logical error rates can all be performed by contracting an appropriate tensor network.

Our algorithm is exact, allowing us to probe the performance of the code when the logicalnoise rate is low (well below threshold). While exponential, the algorithm is much more e�cientthan brute-force simulation. Under systematic rotation, we have simulated codes up to size 11⇥11with 121 physical qubits. Under amplitude damping, codes of up to size 9 ⇥ 17 with 153 physicalqubits were simulated. These system sizes are su�ciently large to allow noise thresholds to beestimated.

Coherent Dynamics of Magnetic Atoms in Spin-PolarizedEnvironments

Lars-Hendrik Frahm1, Christoph Hübner1, Benjamin Baxevanis1,2, and DanielaPfannkuche1

1I. Institut für Theoretische Physik, Universität Hamburg, Germany2Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

We investigate transport e�ects of a magnetic atom that is exchange connected to two electronreservoirs. Using a description in the framework of open quantum systems, the atom evolvesnon-unitary dynamics, which we tackle by finding the Liouville operator of lowest order in atom-reservoir coupling. An e�ective crystal field arises from the substrate the atom is living on, whichgives the spin of the atom an easy axis for alignment. Further, a spin-polarized electron reservoirbreaks the rotational symmetry around the spin quantization axis. A proper description of thedynamics of the quantum spin requires to consider all elements of the reduced density operator,where its knowledge allows to calculate magnetization dynamics and transport properties on anequal footing. We discuss the electron transport through the atomic system by especially focusingon the non-linear influence of the spin torque e�ect. Extension of these investigations to the utmostinteresting regime of strong coupling to the reservoirs seems feasible, given the new concepts oftensor networks for open quantum systems currently under development.

Randomization algorithms in MPS/PEPS

R. Igarashi1

1Information Technology Center, The University of Tokyo, Japan

Matrix Product State (MPS) and Projected Entangled Pair States (PEPS)[1,2] algorithms become one of the standard method for solving low dimensional quantum lattice models such as Heisenberg models and/or Hubbard model since MPS/PEPS are free from negative sign problem which is inevitable in quantum Monte Carlo simulation when system have lattice frustration and/or fermion degree of freedom. Moreover, Not only the ground state energy and wavefunction but also real time evolution of the system can be calculated by MPS/PEPS. This feature enables us to compare dynamic structure factor directly between MPS simulation and inelastic neutron scattering experiments The computational complexity of MPS/PEPS algorithms are polynomial. One of the heaviest part in MPS/PEPS is rank-revealing QR decomposition or truncated SVD. Therefore, reducing calculation cost in rank-revealing QR or truncated SVD is very demanding. Recently, randomization techniques is introduced to calculate them[3]. I will show how these algorithms works and how much time we can reduce using this technique. [1]. U. Schollwöck, Ann. Phys. 326 (2011), 96-192. [2]. R. Orus, Ann. Phys. 349 (2014), 117-158. [3]. N. Halko, P. G. Martinsson, and J. A. Tropp, SIAM Rev. 53(2) (2011), 217-288.

Long range correlations of one-dimensional impuritysystems: DMRG study with infinite boundary condition

Chung-Yu Lo1, Shuai Yin1, Ying-Jer Kao2, and Pochung Chen1

1Department Physics, National Tsing Hua University, Taiwan2Department of Physics, National Taiwan University, Taiwan

We study the power-law correlations of Luttinger-liquid wire with a single impurity usingdensity-matrix renormalization group (DMRG). Although it is generally believed that finite-sizeDMRG algorithms fall short of capturing scale-invariant properties, we propose that, by apply-ing infinite boundary conditions (IBC) [1], DMRG calculations can be e↵ective to determine therenormalization group (RG) fixed points and obtain the long range correlations of semi-infiniteand impurity systems. We model the single impurity locating at the center of a one-dimensionalIBC quantum lattice, which can also be viewed as two semi-infinite wires connected througha weak link. We compute the correlation functions between the two semi-wires at di↵erentRG fixed point regions. Our results agree with the theoretical predictions from RG analysis[2]. Comparison to other numerical studies, including multiscale entanglement renormalizationansatz (MERA) [3] and finite-size DMRG with conformal mapping [4], is also discussed.

[1] Ho N. Phien, Guifre Vidal, and Ian P. McCulloch, Phys. Rev. B 86, 245107 (2012).

[2] C. L. Kane and Matthew P. A. Fisher, Phys. Rev. Lett. 68, 1220 (1992).

[3] Ya-Lin Lo, Yun-Da Hsieh, Chang-Yu Hou, Pochung Chen, and Ying-Jer Kao, Phys. Rev. B90, 235124 (2014).

[4] Armin Rahmani, Chang-Yu Hou, Adrian Feiguin, Claudio Chamon, and Ian A✏eck, Phys.Rev. Lett. 105, 226803 (2010).

Symmetry Breaking and the Geometry of Reduced DensityMatrices

V. Stauber1, D. Draxler1, L. Vanderstraeten2, J. Haegeman2, and F. Verstraete1,2

1Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090

Wien, Austria

2Faculty of Physics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

The concept of symmetry breaking and the emergence of corresponding local order parame-ters constitute the pillars of modern day many body physics. We demonstrate that the existenceof symmetry breaking is a consequence of the geometric structure of the convex set of reduceddensity matrices of all possible many body wavefunctions. The surfaces of these convex bodiesexhibit non-analyticities, which signal the emergence of symmetry breaking and of an associatedorder parameter. We illustrate this with a few paradigmatic examples of many body systems ex-hibiting symmetry breaking: the quantum Ising model, the classical Ising and Potts model in twodimensions at finite temperature and the ideal Bose gas in three dimensions at finite temperature.This state based viewpoint on phase transitions provides a unique novel tool for studying exoticmany body phenomena.

[1] V. Stauber, D. Draxler, L. Vanderstraeten, J. Haegeman, and F. Verstraete, arXiv:1412.7642

Dynamical and thermal properties of

Kitaev-Heisenberg magnets

T. Suzuki1, Y. Yamaji2 , T. Yamada1, and S. Suga1

1Graduate School of Engineering, University of Hyogo, Japan 2Quantum-Phase Electronics Center, University of Tokyo, Japan

One of the focusing topics in condensed matter physics is realization of the Kitaev’s spin

liquid state [1]. In Na2IrO3, Jeff=1/2 magnetic moments are carried by Ir4+ ions. These magnetic moments constitute a honeycomb lattice plane and the strong anisotropic interactions, namely the Kitaev interactions, exist due to the edge-shared structure of IrO6 octahedra [2,3]. Thus, the effective model of this compound can be descried by the Kitaev-Heisenberg (KH) model on the honeycomb lattice whose ground state includes the Kitaev’s spin-liquid state. Experimentally, it has been observed that Na2IrO3 exhibits a magnetic phase transition at TN ~ 15 [K] and a zigzag order is stabilized in the lower temperature region [4]. While the coupling parameters have been estimated in Refs. [4-8] to explain the origin of zigzag order and to discuss the closeness of the Kitaev spin liquid phase in a parameter space, there is a large discrepancy among these previous studies.

From the above background, we calculate dynamical structure factors (DSFs) for proposed models in Refs. [4-8] by using a numerical diagonalization method. We compare numerical results with the inelastic neutron scattering (INS) measurements for powder samples [4], and clarify which model properly reproduces the INS spectra of Na2IrO3. To clarify the characteristics of the low-lying excitations, we also perform linearized spin-wave analysis and compare the results with the DSFs. We find that the spin-wave excitations fail to explain the low-lying excitations of the DSFs, if the system is located in the vicinity of the Kitaev’s spin liquid phase [9]. Furthermore, in such region, the temperature dependence of the specific heat shows a two-peaks structure, which can promise the emergence of Majorana fermions [10]. [1] A. Kitaev, Ann. Phys. 321, 2 (2006). [2] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205(2009). [3] J. Chaloupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 105, 027204 (2010). [4] S. K. Choi, et al., Phys. Rev. Lett. 108, 127204 (2012). [5] J. Chaoupka, G. Jackeli, and G. Khaliullin, Phys. Rev. Lett. 110, 097204 (2013). [6] Y. Singh, et al., Phys. Rev. Lett. 108, 127203 (2012). [7] Y. Yamaji, et al., Phys. Rev. Lett. 113, 107201 (2014). [8] Y. Sizyuk, et al., Phys. Rev. B 90, 155126 (2014). [9] T. Suzuki, et al., Phys. Rev. B 92 184411 (2015). [10] Y. Yamaji, et al., arXiv:1601.05512. (accepted in Phys. Rev. B)

Design of Ru-based honeycomb metal-organic frameworksand JK� model

M. G. Yamada1, H. Fujita1, and M. Oshikawa1

1Institute for Solid State Physics, The University of Tokyo, Japan

We propose Ru-based honeycomb metal-organic frameworks (MOFs) as new candidates forthe realization of the extended Kitaev-Heisenberg model a.k.a. the JK� model [1]. Thanks tothe strong suppression of a direct exchange interaction, it is more likely to realize a spin liquidground state in MOFs than in the other inorganic candidates, such as iridates and RuCl3. In orderto achieve the Kitaev-dominant regime of the JK� model, we here propose two types of ideal or-ganic ligands, oxalate-based ligands and tetraaminopyrazine-based ligands in these MOFs, usingthe same mechanism as that proposed by Jackeli and Khaliullin [2]. Then, we discuss controlparameters to obtain a Kitaev-dominant MOF and show that the almost degenerate nature of thehighest occupied molecular orbitals of the proposed ligands is important.

[1] J. G. Rau, E. K.-H. Lee and H.-Y. Kee, Phys. Rev. Lett. 112, 077204 (2014).

[2] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).

Open-source Numerical Diagonalization Application HΦ

Y. Yamaji1, T. Misawa2, K. Yoshimi2, M. Kawamura2, Synge Todo3,

and N. Kawashima2

1Quantum-Phase Electronics Center, The University of Tokyo, Japan2Institute for Solid State Physics, The University of Tokyo, Japan

3Department of Physics, The University of Tokyo, Japan

HΦ is an open-source numerical solver for finite-size quantum lattice models. The Lanc-

zos method[1] for calculations of the ground state and few excited states properties, and

finite temperature calculations based on thermal pure quantum states[2] are implemented

in HΦ, with an easy-to-use and flexible user interface. By using HΦ, you can analyze

a wide range of quantum lattice hamiltonians including simple Hubbard and Heisenberg

models, multi-band extensions of the Hubbard model, exchange couplings that break SU(2)

symmetry of quantum spins such as Dzyaloshinskii-Moriya and Kitaev interactions, and

Kondo lattice models describing itinerant electrons coupled with quantum spins. In the

poster presentation, as examples of HΦ simulations, we will show the magnetization curve

of a S = 1/2 kagome-lattice Heisenberg antiferromagnet, and temperature dependence of

specific heat and spin structure factors of the ab initio spin Hamiltonian of Na2IrO3[3].

[1] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).

[2] S. Sugiura and A. Shimizu, Phys. Rev. Lett. 108, 240401 (2012).

[3] Y. Yamaji, Y. Nomura, M. Kurita, R. Arita, and M. Imada, Phys. Rev. Lett. 113,

107201 (2014).

A tree tensor method for the simultaneous determination ofmultiple eigenstates

Y. Zhao1, and P. W. Ayers1

1Department of Chemistry and Chemical Biology, McMaster University, Canada

When studying phenomena like photochemistry and molecular magnetism, it is important to beable to characterize multiple electronic states, ideally all simultaneously and at the same level oftheory. An additional complication arises for transition-metal complexes and rare-earth materials,which tend to be strongly correlated, so that traditional single-reference methods are inappro-priate. Here we propose a way to simultaneously describe multiple excited states of strongly-correlated systems with a tree-tensor structure.

A full-CI wavefunction can be written as a linear combination of Slater determinants of spin-orbitals �ic (qc ).

| (1, . . . ,N )i =B1X

i1

· · ·BNX

iN

Fi1, ...iN |�i1 (q1)�i2 (q2) . . . �iN (qN )i (1)

Our idea is that instead of using single-electron basis functions (orbitals) we will design many-eletron functions �iw (qw ) , that depend on several orbitals. Specifically, we use a tree-tensorbasis, where the basis function corresponding to a node in the tree is sum of (antisymmetrized)products of the basis functions of its children in the previous layer, ` � 1.

� (`;J )iw

(q (`;J )w ) =

B(1,`�1;J, )X

i1

· · ·B(d (w,`;J ),`�1,J,w)X

id (w,`;J )=1F (w,`;J )i1, . . .id (w,`;J )

|� (`�1;J,w)i1

(q (`�1;Jw)1 ) . . . � (`�1;J,w)

iN(q (`�1;J,w)

N)i (2)

The resulting hierarchical basis lets us write the wavefunction in a canonical polyadic (CP)format. We then perform configuration interaction calculations in the many-electron basis to solvemultiple excited states simultaneously with a common basis. We believe that this approach cansignificant enhance the accuracy and e�ciency of this procedure, compared to more traditionalapproaches.

Figure 1: Pictorial representation of the tree-tensor basis, where qv refers to the primitive orbital.

[1] P. S. Thomas and T. C. Third, J. Phys. Chem. A 119, 13074-13091 (2015).

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