tensor network states: algorithms and...

Tensor Network States: Algorithms and Applications

Dec. 1-5, 2014

Beijing, China

Sponsored by

Key Laboratory of Condensed Matter Theory and Computation, Institute of Physics, Chinese

Academy of Sciences, China

Department of Physics and Astronomy, Shanghai Jiao Tong University, China


1

Contents

About the Workshop .................................................................................................... 2

Workshop Organization ............................................................................................... 3

Workshop Program ...................................................................................................... 4

Invited Speakers ........................................................................................................... 9

Abstract ...................................................................................................................... 10


2

About the Workshop

Tensor network states have in recent years emerged as a powerful theoretical tool to

study quantum collective phenomena. Some of the most popular tensor networks, such as the

matrix product state (MPS), the multi-scale entanglement renormalization ansatz (MERA),

and the projected entangled-pair states (PEPS), are currently used by many groups as the

basis for variational approaches to many-body systems. However, the tensor network

formalism goes well beyond numerical methods, and it is also used e.g., as a natural

framework to classify phases of quantum matter, or as a lattice realization of the holographic

principle of string theory.

The workshop "Tensor Network States: Algorithms and Applications" will be held in

the Institute of Physics, Chinese Academy of Sciences on December 1-5, 2014. In this

workshop we will bring together experts on tensor network algorithms and their wide

spectrum of applications, from statistical mechanics to condensed matter, from quantum

chemistry to nano-technology and high energy physics. Morning research talks will be

complemented with pedagogical lectures and tutorials in the afternoons.


3

Workshop Organization

Organizers:

Ying-Jer Kao National Taiwan University, Taiwan

Tomotoshi Nishino Kobe University, Japan

Guifre Vidal Perimeter Institute, Canada

Xiaoqun Wang Shanghai Jiao Tong University, China

Tao Xiang Institute of Physics, Chinese Academy of Sciences, China

Sponsors:

Key Laboratory of Condensed Matter Theory and Computation, Institute of Physics, Chinese

Academy of Sciences, China

Department of Physics and Astronomy, Shanghai Jiao Tong University, China

Contact:

Zhiyuan Xie, [email protected]

Qingmei Liu, Tel: 86-10-82649414, [email protected]

Jianwei Qi, Tel: 86-10-82649400, [email protected]


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Workshop Program

Monday, Dec. 1, 2014, Room 236, Building M

Session Time Speaker/Affiliation/Title

9:00-9:10 Welcome speech by Prof. Tao Xiang

Chair: Guifre Vidal

9:10-9:50

Bruce Normand, Renmin University Of China, China

New Physics from Tensor-Network Treatments of Potts and

Heisenberg Models

Session 9:50-10:30

Tomoyuki Morimae, Gunma University, Japan

I Measurement-based quantum computing on tensor-network state

10:30-11:10 Photo and Break

11:10-11:50

Yan Chen, Fudan University, China

Chiral and Time-reversal Invariant Spin Liquids in Anisotropic

Kagome Antiferromagnets

11:50-12:30

Ying-Jer Kao, National Taiwan University, Taiwan

The Universal Tensor Network Library

Lunch, Wuke Hotel Restaurant

Chair: Guangming Zhang

14:00-15:00

Garnet Chan, Princeton University, USA

Session Matrix product states(MPS)

II 15:00-15:45 Break

15:45-16:45

Ying-Jer Kao / Pochung Chen / Chung-Yu Lo

Uni10 tutorial (I): Basics, MPS (1D iTEBD)


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Tuesday Morning, Dec. 2, 2014, Room 236, Building M


Chair: Gang Su

9:00-9:40

Ian McCulloch, University of Queensland, Australia

Magnetic ordering in Triangular-Heisenberg J1-J2 cylinders.

9:40-10:20

Tomotoshi Nishino, Kobe University, Japan

Session Placket type local weight and tensor product state

III 10:20-10:40 Break

10:40-11:20

Chisa Hotta, Tokyo Univerity, Japan

Grand canonical analysis: A route to measuring bulk properties in an

applied field

11:20-12:00

Frank Pollmann, Max-Planck Institute, Germany

Entanglement and dynamics in many-body localized systems


Tuesday Afternoon, Dec. 2, 2014, Room 253, Building M

Chair: Garnet Chan

14:00-15:00

Guifre Vidal, Perimeter Institute, Canada

Session The multi-scale entanglement renormalization ansatz (MERA)

IV 15:00-15:45 Break

15:45-16:45


Uni10 tutorial (II): 1D MPS with symmetries

18:00- Banquet


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Wednesday, Dec. 3, 2014, Room 236, Building M


Chair: Ying-Jer Kao

9:00-9:40

Kouichi Okunishi, Niigata University, Japan

Symmetry-protected topological entanglement and negative sign

problem for SO(N) bilinear-biquadratic chains

9:40-10:20

Pochung Chen, National Tsinghua University, Taiwan

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

V 10:20-10:40 Break

10:40-11:20

Guangming Zhang, Tsinghua University, China

Critical entanglement spectrum of one-dimensional symmetry

protected topological phases

11:20-12:00


Tensor network quantum Monte Carlo and entanglement based

quantum embeddings


Session Chair: Roman Orus

VI

14:00-15:00

Zhiyuan Xie, Institute of Physics, Chinese Academy of Sciences,

China

Tensor Renormalization in classical statistical models and quantum

lattice models.

15:00-15:45 Break

15:45-16:45


Uni10 tutorial (III): (2D iTEBD, TRG)


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Thursday, Dec. 4, 2014, Room 236, Building M


Chair: Bruce Normand

9:00-9:40

Roman Orus, Johannes Gutenberg-University, Germany

Topological transitions and minimally entangled states from

multipartite entanglement with 2d PEPS

Session

9:40-10:20

Huanqiang Zhou, Chongqing University, China

VII Universal Order Parameters and Quantum Phase Transitions: A

Finite-Size Approach

10:20-10:40 Break

10:40-11:20

Gang Su, University of Chinese Academy of Sciences, China

Thermal tensor network renormalization group algorithms and

implications

11:20-12:00

Glen Evenbly, California Institute of Technology, USA

Tensor network renormalization


Chair: Tomotoshi Nishino

Session 14:00-15:00 Discussions on renormalization group method

VIII 15:00-15:45 Break

15:45-16:45 Poster Session


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Friday, Dec. 5, 2014, Room 236, Building M


Chair: Huanqiang Zhou

9:00-9:40

Gavin Brennen, Macquarie University, Australia

Simulating the physics of braiding anyons with matrix product

states

9:40-10:20

Naoki Nakatani, Hokkaido University, Japan

Session Tensor Network in Chemistry: Recent DMRG/TTNS Studies and

Perspectives for Catalysis Research

IX 10:20-10:40 Break

10:40-11:20

Yutaka Shikano, Institute for Molecular Science, Japan

Discrete-time quantum walk and Quantum dynamical simulation

11:20-12:00

Honggang Luo, Lanzhou University, China

Optimization and interaction of Hartree-Fock orbitals



9

Invited Speakers

Gavin Brennen, Macquarie University, Australia


Yan Chen, Fudan University, China

Pochung Chen, National Tsinghua University, Taiwan

Glen Evenbly, California Institute of Technology, USA

Chisa Hotta, Tokyo Univerity, Japan

Ying-Jer Kao, National Taiwan University, Taiwan

Honggang Luo, Lanzhou University, China

Ian McCulloch, University of Queensland, Australia

Tomoyuki Morimae, Gunma University, Japan

Naoki Nakatani, Hokkaido University, Japan

Tomotoshi Nishino, Kobe University, Japan

Bruce Normand, Renmin University, China

Kouichi Okunishi, Niigata University, Japan

Roman Orus, Johannes Gutenberg-University, Germany

Frank Pollmann, Max-Planck Institute, Germany

Yutaka Shikano, Institute for Molecular Science, Japan

Gang Su, University of Chinese Academy of Sciences, China

Guifre Vidal, Perimeter Institute, Canada

Zhiyuan Xie, Institute of Physics, Chinese Academy of Sciences, China

Guangming Zhang, Tsinghua University, China

Huanqiang Zhou, Chongqing University, China


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Abstract

Tensor network quantum Monte Carlo and entanglement based quantum embeddings

Garnet Chan

Princeton University, USA

In the first part of the talk, I will provide an update on our progress with using diffusion QMC

in conjunction with 1D and 2D tensor networks. In the second part, I will discuss a converged

zero-temperature phase diagram for the 2D Hubbard model produced using density matrix

embedding theory, and the relationship of this technique with tensor networks.


11

Chiral and Time-reversal Invariant Spin Liquids in Anisotropic Kagome

Antiferromagnets

Yan Chen

Fudan University, China

Kalmeyer-Laughlin (KL) chiral spin liquid (CSL) is a type of quantum spin liquid without

time-reversal symmetry, and it is considered as the parent state of exotic anyon superconductor. Such

an exotic state has been sought for more than twenty years; however, it remains unclear whether it

can exist in a realistic system where time-reversal symmetry is breaking spontaneously. By using the

density matrix renormalization group, we show that KL CSL exists in a frustrated anisotropic

kagome antiferromagnets, which has time-reversal symmetry breaking. We find that our model has

two topological degenerate ground states, which exhibit nonvanishing scalar chirality order and are

protected by finite excitation gap. Furthermore, we identify this state as KL CSL by the characteristic

edge conformal field theory from the entanglement spectrum and the quasiparticles braiding statistics

extracted from the modular matrix. Next we study spin-liquid phases of spin-1/2 XXZ kagome

antiferromagnets. We find that the emergence of the spin-liquid phase is independent of the

anisotropy of the XXZ interaction. In particular, the two extreme limits---the Ising and the XY---host

the same spin-liquid phases as the isotropic Heisenberg model. Both a time-reversal-invariant spin

liquid and a chiral spin liquid are obtained. We show that they evolve continuously into each other by

tuning the second- and the third-neighbor interactions.


12

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

Pochung Chen

National Tsinghua University, Taiwan

We study the quantum critical phase of an SU(2) symmetric spin-2 chain obtained from spin-2

bosons in a one-dimensional lattice. We obtain the scaling of the finite-size energies and

entanglement entropy by exact diagonalization and density-matrix renormalization group methods.

From the numerical results of the energy spectra, central charge, and scaling dimension, we

identify the 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 phase there is an emergent SU(3) symmetry in the thermodynamic limit.


13

Tensor network renormalization

Glen Evenbly

California Institute of Technology, USA

I will describe how to define a proper RG flow in the space of tensor networks, with

applications to the evaluation of classical partition functions, euclidean path integrals, and overlaps

of tensor network states.


14

Grand canonical analysis: A route to measuring bulk properties in an applied field

Chisa Hotta

Tokyo Univerity, Japan

The grand canonical numerical analysis is the technique we have developed to efficiently obtain

the physical quantities in an applied field in quantum many-body systems [1]. The observables are

the continuous and real functions of fields, mimicking their thermodynamic limit, even when a small

cluster is adopted. We first prepare an open system typically of length L~O (10), and systematically

scale down the Hamiltonian from center toward both ends. Then one could endow the role of small

particle bath to the edge sites with a negligibly small energy scale. The particles on the cluster are

self-organized to tune the particle number near the system center to their thermodynamic limit by

using these “particle baths”.

We briefly explain the overall mechanism [2] and show several examples of one- and

two-dimensional quantum spin systems where the bulk magnetization curve is obtained within the

accuracy of 10^{-3}-10^{-4}. We also refer to the successful evaluation of the singlet-triplet spin gap

of the spin 1/2 Kagome antiferromagnet using this method [3].

[1]. C. Hotta and N. Shibata, Phys. Rev. B 86, 041108(R) (2012)

[2]. C. Hotta, S. Nishimoto, and N. Shibata, Phys. Rev. B 87, 115128 (2013).

[3]. S. Nishimoto, N. Shibata and C. Hotta, Nature Comm. 4, 2287 (2013).


15

The Universal Tensor Network Library

Ying-Jer Kao

National Taiwan University, Taiwan

Tensors provide a natural and compact representation for multidimensional data, and algorithms

based on tensor networks have recently found their applications in quantum physics, quantum

information science, quantum chemistry, image/pattern recognition and data science. However,

programming tensor network algorithms is tedious and error prone due to the complexity in keeping

track of multiple tensor indices. There are also further complications in book keeping the indices as

many scientific applications require these tensors to obey certain symmetries. For a given tensor

network, storing the connectivity of the network and determining the optimal contraction sequence

are also crucial for further analysis. To address these issues, we develop a C++ framework geared

toward the application in tensor network algorithms called Uni10, the Universal Tensor Network

Library [1]. It provides basic symmetric tensor storage and operations with features such as Einstein

summation convention and easy-to-use interface, storage for graphical representations of networks,

and an engine to construct and analyze the contraction tree for a given network. A heuristic algorithm

is implemented to search for an optimal binary contraction order based on the computational and

memory constraints. I will also discuss our implementation of Uni10 on GPU.

[1]. http://www.uni10.org


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Measurement-based quantum computing on tensor-network state

Tomoyuki Morimae

Gunma University, Japan

In this talk, I will review the basics of measurement-based quantum computing on

tensor-network states and explain its recent developments including my two results,

measurement-based quantum computing on the string-net condensate and relation between

long-range correlation and the universality of measurement-based quantum computing on

matrix-product states.


17

Tensor Network in Chemistry: Recent DMRG/TTNS Studies and Perspectives for

Catalysis Research

Naoki Nakatani

Hokkaido University, Japan

Many-body wavefunction can be viewed as a rank-L tensor, where L is a number of sites. This

is mathematically decomposed into a product of lower-rank tensors to perform a tensor network.

Recently, it has received much attention in condensed matter physics, because it is known as a useful

tool to investigate strongly-correlated ground state.

In quantum chemistry, although the density matrix renormalization group (DMRG) is going to

be a useful tool to investigate strongly-correlated molecular systems such as transition metal

complexes and clusters, general tensor networks are not well investigated. In this talk, I would like to

present our recent DMRG/TTNS studies on molecular systems. Also I would like to discuss

perspectives of tensor network algorithms for quantum chemistry, from the viewpoint of Catalysis

Research.


18

A placket type local weight

Tomotoshi Nishino

Kobe University, Japan

Tensor Product State; Vertex type and IRF type: There are many ways of constructing a two

dimensional (variational) wave function as a product or tensor contraction of local factors. Here, we

consider so called a placket type local weight. How can one map such a state to the standard form of

the tensor product state?


19

New Physics from Tensor-Network Treatments of Potts and Heisenberg Models

Bruce Normand

Renmin University, China

Tensor-network formulations allow a new means of expressing and computing the partition

function of a classical lattice model and the wavefunction of a quantum system. The development of

increasingly sophisticated treatments of these tensor networks provides access to previously

unavailable physical insight. This two-part presentation highlights a number of such technical and

conceptual advances.

A wealth of classical statistical physics may be found in q-state Potts models, where advanced

projective techniques allow rapid and efficient calculation of, in principle, all thermodynamic

quantities for all q values on all lattices. These have been used to find new critical models

(``ordering'' at T = 0) and new phase transitions to states of partial site order on irregular lattices,

including for anomalously high q. By focusing on the tensor eigenvalue structure, a measure can be

constructed for determining the phase transition point with extreme accuracy, extending the regime

of applicability of tensor-based methods towards three spatial dimensions.

Among quantum lattice models, the spin-1/2 Heisenberg Hamiltonian in the kagome geometry

provides one of the most intriguing and enigmatic examples of a highly frustrated magnetic system

with a candidate spin-liquid ground state. The introduction of projected entangled simplex states

(PESS), which contain the multi-site entanglement of a frustrated lattice unit, leads to well-controlled

and rapidly convergent results. The PESS ground-state energy is the lowest variational result yet

obtained for this model and suggests that the true ground state of this system is in fact a Z2 (gapped)

spin liquid.


20

Symmetry-protected topological entanglement and negative sign problem for SO(N)

bilinear-biquadratic chains

Kouichi Okunishi1, Kenji Harada2

1Niigata University, Japan

2Kyoto University, Japan

We will discuss the relation between the symmetry-protected-topological entanglement and the

negative sign problem of quantum Monte Carlo (QMC) for the SO (N) bilinear-biquadratic (BLBQ)

chains. Using a generalized Jordan-Wigner transformation combined with the defining representation

of the SO (N) spin, we map the SO (N) BLBQ chains into the N-color bosonic particle models.

When the Jordan-Wigner transformation disentangles the symmetry-protected-topological

entanglement of the SO (N) BLBQ chains, this bubonic model becomes negative-sign free. For the

SO (3) case, Kennedy-Tasaki's transformation for the S=1 BLBQ chain also yields the same bosonic

model through dimer-R bases. We show some QMC results based on the world-line algorithm for the

N-color bosonic particle model.


21

Topological transitions and minimally entangled states from multipartite entanglement

with 2d PEPS

Roman Orus

Johannes Gutenberg-University, Germany

Topological order in a 2d quantum matter can be determined by the topological contribution to

the entanglement Renyi entropies. However, when close to a quantum phase transition, its calculation

becomes cumbersome. In this talk I will show how topological phase transitions in 2d systems can be

much better assessed by multipartite entanglement, as measured by the topological geometric

entanglement of blocks. Specifically, I will present an efficient tensor network algorithm based on

Projected Entangled Pair States (PEPS) to compute this quantity for a torus partitioned into cylinders,

and then use this method to find sharp evidence of topological phase transitions in 2d systems with a

string-tension perturbation. When compared to tensor network methods for Renyi entropies, this

approach produces almost perfect accuracies close to criticality and, on top, is orders of magnitude

faster. Moreover, I will show how the method also allows the identification of Minimally Entangled

States (MES), thus providing a very efficient and accurate way of extracting the topological

information of a 2d quantum lattice model from the multipartite entanglement structure of its ground

states.


22

Entanglement and dynamics in many-body localized systems

Frank Pollmann

Max-Planck Institute, Germany

Many-body localized (MBL) phases occur in isolated quantum systems when Anderson

localization persists in the presence of finite interactions. It turns out that the entanglement is a very

useful quantity to study these phases. First, we focus on the physics in the presence of strong disorder.

For this we study the time evolution of simple (unentangled) initial states for a system of interacting

spinless fermions in a one dimensional system. It is found that interactions induce a dramatic change

in the propagation of entanglement. Second, we use the bipartite entanglement of excited eigenstates

to pinpoint a phase transition from a localized to an extended phase in a random Ising chain with

short ranged interactions. A characterizing property of the MBL phase is that the area law also

applies to excited states. In one-dimensional systems, these states can be encoded efficiently using a

matrix-product state representation.


23

Discrete-time quantum walk and Quantum dynamical simulation

Yutaka Shikano

Institute for Molecular Science, Japan

The discrete time quantum walk defined as a quantum-mechanical analogue of the discrete time

random walk have recently been attracted from various and interdisciplinary fields. In this review,

the weak limit theorem, that is, the asymptotic behavior, of the one-dimensional discrete time

quantum walk is analytically shown. From the limit distribution of the discrete time quantum walk,

the discrete time quantum walk can be taken as the quantum dynamical simulator of some physical

systems.


24

Thermal tensor network renormalization group algorithms and implications

Gang Su

University of Chinese Academy of Sciences, China

It has been observed both experimentally and theoretically in recent years that a lot of quantum

correlated many-body systems can assume exotic quantum phases that yet need to be understood. As

the complexity of such systems leads usually to that the analytical or known numerical methods are

hard to obtain useful information, the development of novel numerical means is quite imperative in

tackling these intractable systems. In this talk, I will give a brief review on our recently developed

tensor network-based renormalization group methods (including LTRG, ODTNS and NCD) that can

be applied with high efficiency and accuracy to explore both ground state and thermodynamic

properties of low-dimensional quantum spin lattice systems. The implications of these methods in

several intriguing quantum spin systems will also be discussed.


25

The multi-scale entanglement renormalization ansatz (MERA)

Guifre Vidal

Perimeter Institute, Canada

In this pedagogical lecture, I will review the concept of entanglement renormalization, which

aims at producing a proper renormalization group flow for quantum systems on the lattice. I will also

describe the MERA, which is the tensor network state resulting from this coarse-graining

transformation. Then I will review the application of MERA to quantum critical systems.

Reading material:

[1]. Entanglement Renormalization: an introduction, G. Vidal, http://arxiv.org/abs/0912.1651,

[chapter 5 in "Understanding Quantum Phase Transitions", edited by Lincoln D. Carr

(Taylor & Francis, Boca Raton, 2010)]

[2]. Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz, G.

Evenbly, G. Vidal, http://arxiv.org/abs/1109.5334, [chapter 4 in "Strongly Correlated

Systems. Numerical Methods", edited by A. Avella and F. Mancini (Springer Series in

Solid-State Sciences, Vol. 176 2013)]


26

Tensor Renormalization in classical statistical models and quantum lattice models

Zhiyuan Xie

Institute of Physics, Chinese Academy of Sciences, China

Tensor renormalization method is a class of new methods which draws more and more

interests in the recent few years, and it flourishes the field of computational physics. In this talk, I

would like to talk about some algorithms developed in our team, including the Second

Renormalization Group (SRG) idea [1], Tensor Renormalization Group based on the Higher-order

Singular Value Decomposition (HOTRG) approach [2], and the mean-field entanglement approach in

the determination of the PESS ground state wavefunction [3].

Reference:

[1]. Phys.Rev.Lett.103, 160601 (2009), and arXiv:0809.0182

[2]. Phys. Rev. B 86, 045139 (2012), and arXiv:1204.1144

[3]. Phys. Rev. X 4, 011025 (2014), and arXiv:1307.5696


27

Critical entanglement spectrum of one-dimensional symmetry protected topological

phases

Guangming Zhang

Tsinghua University, China

Under an appropriate symmetric extensive bipartition in a one-dimension symmetry protected

topological (SPT) phase, a bulk critical entanglement spectrum can be obtained, resembling the

excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state.

Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry

breaking phase transitions. For the $S=1$ SPT (Haldane) phase with the

Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical entanglement spectrum

shows a delocalized version of the edge excitations in the SPT phase. From the wave function

corresponding to the lowest entanglement energy level, the central charge of the critical point can be

extracted and the critical theory can be identified as the same effective field theory as the spin-1/2

antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse square

long-range interaction.


28

Universal Order Parameters and Quantum Phase Transitions: A Finite-Size Approach

Huanqiang Zhou

Chongqing University, China

We propose a method to construct universal order parameters for quantum phase transitions in

many-body lattice systems. The method exploits the H-orthogonality of a few near-degenerate lowest

states of the Hamiltonian describing a given finite-size system, which makes it possible to perform

finite-size scaling and take full advantage of currently available numerical algorithms. An explicit

connection is established between the fidelity per site between two H-orthogonal states and the

energy gap between the ground state and low-lying excited states in the finite-size system. The

physical information encoded in this gap arising from finite-size fluctuations clarifies the origin of

the universal order parameter. We demonstrate the procedure for the one-dimensional quantum

formulation of the q-state Potts model, for q=2, 3, 4 and 5, as prototypical examples, using finite-size

data obtained from the density matrix renormalization group algorithm.

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