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
Tensor Network States: Algorithms and Applications
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Contents
About the Workshop .................................................................................................... 2
Workshop Organization ............................................................................................... 3
Workshop Program ...................................................................................................... 4
Invited Speakers ........................................................................................................... 9
Abstract ...................................................................................................................... 10
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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]
Tensor Network States: Algorithms and Applications
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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)
Tensor Network States: Algorithms and Applications
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Tuesday Morning, Dec. 2, 2014, Room 236, Building M
Session Time Speaker/Affiliation/Title
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
Lunch, Wuke Hotel Restaurant
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
Ying-Jer Kao / Pochung Chen / Chung-Yu Lo
Uni10 tutorial (II): 1D MPS with symmetries
18:00- Banquet
Tensor Network States: Algorithms and Applications
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Wednesday, Dec. 3, 2014, Room 236, Building M
Session Time Speaker/Affiliation/Title
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
Garnet Chan, Princeton University, USA
Tensor network quantum Monte Carlo and entanglement based
quantum embeddings
Lunch, Wuke Hotel Restaurant
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
Ying-Jer Kao / Pochung Chen / Chung-Yu Lo
Uni10 tutorial (III): (2D iTEBD, TRG)
Tensor Network States: Algorithms and Applications
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Thursday, Dec. 4, 2014, Room 236, Building M
Session Time Speaker/Affiliation/Title
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
Lunch, Wuke Hotel Restaurant
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
Tensor Network States: Algorithms and Applications
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Friday, Dec. 5, 2014, Room 236, Building M
Session Time Speaker/Affiliation/Title
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
Lunch, Wuke Hotel Restaurant
Tensor Network States: Algorithms and Applications
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Invited Speakers
Gavin Brennen, Macquarie University, Australia
Garnet Chan, Princeton University, USA
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
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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).
Tensor Network States: Algorithms and Applications
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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
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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?
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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)]
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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
Tensor Network States: Algorithms and Applications
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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.
Tensor Network States: Algorithms and Applications
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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.