Publications (70)237.54 Total impact

Article: Tensor Network Renormalization
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ABSTRACT: We introduce a coarsegraining transformation for tensor networks that can be applied to the study of both classical statistical and quantum manybody systems, via contraction of the corresponding partition function or Euclidean path integral, respectively. The scheme is based upon the insertion of optimized unitary and isometric tensors into the tensor network and has, as its key feature, the ability to completely remove shortrange correlations at each coarsegraining step. As a result, it produces a renormalization group flow (in the space of tensors) that (i) has the correct structure of fixed points, and (ii) is computationally sustainable, even for systems at a critical point. We demonstrate the proposed approach in the context of the 2D classical Ising model both near and at the critical point.12/2014;  [Show abstract] [Hide abstract]
ABSTRACT: We propose an environment recycling scheme to speed up a class of tensor network algorithms that produce an approximation to the ground state of a local Hamiltonian by simulating an evolution in imaginary time. Specifically, we consider the timeevolving block decimation (TEBD) algorithm applied to infinite systems in 1D and 2D, where the ground state is encoded, respectively, in a matrix product state (MPS) and in a projected entangledpair state (PEPS). An important ingredient of the TEBD algorithm (and a main computational bottleneck, especially with PEPS in 2D) is the computation of the socalled environment, which is used to determine how to optimally truncate the bond indices of the tensor network so that their dimension is kept constant. In current algorithms, the environment is computed at each step of the imaginary time evolution, to account for the changes that the time evolution introduces in the manybody state represented by the tensor network. Our key insight is that close to convergence, most of the changes in the environment are due to a change in the choice of gauge in the bond indices of the tensor network, and not in the manybody state. Indeed, a consistent choice of gauge in the bond indices confirms that the environment is essentially the same over many time steps and can thus be reused, leading to very substantial computational savings. We demonstrate the resulting approach in 1D and 2D by computing the ground state of the quantum Ising model in a transverse magnetic field.11/2014;  [Show abstract] [Hide abstract]
ABSTRACT: A fundamental process in the implementation of any numerical tensor network algorithm is that of contracting a tensor network. In this process, a network made up of multiple tensors connected by summed indices is reduced to a single tensor or a number by evaluating the index sums. This article presents a MATLAB function ncon(), or "Network CONtractor", which accepts as its input a tensor network and a contraction sequence describing how this network may be reduced to a single tensor or number. As its output it returns that single tensor or number. The function ncon() may be obtained by downloading the source of this preprint.02/2014;  [Show abstract] [Hide abstract]
ABSTRACT: Topological phases in frustrated quantum spin systems have fascinated researchers for decades. One of the earliest proposals for such a phase was the chiral spin liquid put forward by Kalmeyer and Laughlin in 1987 as the bosonic analogue of the fractional quantum Hall effect. Elusive for many years, recent times have finally seen a number of models that realize this phase. However, these models are somewhat artificial and unlikely to be found in realistic materials. Here, we take an important step towards the goal of finding a chiral spin liquid in nature by examining a physically motivated model for a Mott insulator on the Kagome lattice with broken timereversal symmetry. We first provide a theoretical justification for the emergent chiral spin liquid phase in terms of a network model perspective. We then present an unambiguous numerical identification and characterization of the universal topological properties of the phase, including ground state degeneracy, edge physics, and anyonic bulk excitations, by using a variety of powerful numerical probes, including the entanglement spectrum and modular transformations.Nature communications. 01/2014; 5.  [Show abstract] [Hide abstract]
ABSTRACT: We propose algorithms, based on the multiscale entanglement renormalization ansatz, to obtain the ground state of quantum critical systems in the presence of boundaries, impurities, or interfaces. By exploiting the theory of minimal updates [G. Evenbly and G. Vidal, arXiv:1307.0831], the ground state is completely characterized in terms of a number of variational parameters that is independent of the system size, even though the presence of a boundary, an impurity, or an interface explicitly breaks the translation invariance of the host system. Similarly, computational costs do not scale with the system size, allowing the thermodynamic limit to be studied directly and thus avoiding finite size effects e.g. when extracting the universal properties of the critical system.Journal of Statistical Physics 12/2013; 157(45). · 1.40 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Matrix product states (MPS) have proven to be a very successful tool to study lattice systems with local degrees of freedom such as spins or bosons. Topologically ordered systems can support anyonic particles which are labeled by conserved topological charges and collectively carry nonlocal degrees of freedom. In this paper we extend the formalism of MPS to lattice systems of anyons. The anyonic MPS is constructed from tensors that explicitly conserve topological charge. We describe how to adapt the timeevolving block decimation (TEBD) algorithm to the anyonic MPS in order to simulate dynamics under a local and chargeconserving Hamiltonian. To demonstrate the effectiveness of anyonic TEBD algorithm, we used it to simulate (i) the ground state (using imaginary time evolution) of an infinite 1D critical system of (a) Ising anyons and (b) Fibonacci anyons both of which are well studied, and (ii) the real time dynamics of an anyonic Hubbardlike model of a single Ising anyon hopping on a ladder geometry with an anyonic flux threading each island of the ladder. Our results pertaining to (ii) give insight into the transport properties of anyons. The anyonic MPS formalism can be readily adapted to study systems with conserved symmetry charges, as this is equivalent to a specialization of the more general anyonic case.Physical Review B 11/2013; 89(7). · 3.66 Impact Factor 
Article: Scaling of entanglement entropy in the (branching) multiscale entanglement renormalization ansatz
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ABSTRACT: We investigate the scaling of entanglement entropy in both the multiscale entanglement renormalization ansatz (MERA) and in its generalization, the branching MERA. We provide analytical upper bounds for this scaling, which take the general form of a boundary law with various types of multiplicative corrections, including powerlaw corrections all the way to a bulk law. For several cases of interest, we also provide numerical results that indicate that these upper bounds are saturated to leading order. In particular we establish that, by a suitable choice of holographic tree, the branching MERA can reproduce the logarithmic multiplicative correction of the boundary law observed in Fermi liquids and spinBose metals in $D\geq 2$ dimensions.Physical Review B 10/2013; 89(23). · 3.66 Impact Factor 
Article: Global symmetries in tensor network states: symmetric tensors versus minimal bond dimension
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ABSTRACT: Tensor networks offer a variational formalism to efficiently represent wavefunctions of extended quantum manybody systems on a lattice. In a tensor network N, the dimension \chi of the bond indices that connect its tensors controls the number of variational parameters and associated computational costs. In the absence of any symmetry, the minimal bond dimension \chi^{min} required to represent a given manybody wavefunction \Psi> leads to the most compact, computationally efficient tensor network description of \Psi>. In the presence of a global, onsite symmetry, one can use a tensor network N_{sym} made of symmetric tensors. Symmetric tensors allow to exactly preserve the symmetry and to target specific quantum numbers, while their sparse structure leads to a compact description and lowers computational costs. In this paper we explore the tradeoff between using a tensor network N with minimal bond dimension \chi^{min} and a tensor network N_{sym} made of symmetric tensors, where the minimal bond dimension \chi^{min}_{sym} might be larger than \chi^{min}. We present two technical results. First, we show that in a tree tensor network, which is the most general tensor network without loops, the minimal bond dimension can always be achieved with symmetric tensors, so that \chi^{min}_{sym} = \chi^{min}. Second, we provide explicit examples of tensor networks with loops where replacing tensors with symmetric ones necessarily increases the bond dimension, so that \chi_{sym}^{min} > \chi^{min}. We further argue, however, that in some situations there are important conceptual reasons to prefer a tensor network representation with symmetric tensors (and possibly larger bond dimension) over one with minimal bond dimension.Physical Review B 07/2013; 88(11). · 3.66 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Consider two quantum critical Hamiltonians $H$ and $\tilde{H}$ on a $d$dimensional lattice that only differ in some region $\mathcal{R}$. We study the relation between holographic representations, obtained through realspace renormalization, of their corresponding ground states $\ket{\Psi}$ and $\ket{\tilde{\Psi}}$. We observe that, even though $\ket{\Psi}$ and $\ket{\tilde{\Psi}}$ disagree significantly both inside and outside region $\mathcal{R}$, they still admit holographic descriptions that only differ inside the past causal cone $\mathcal{C}(\mathcal{R})$ of region $\mathcal{R}$, where $\mathcal{C}(\mathcal{R})$ is obtained by coarsegraining region $\mathcal{R}$. We argue that this result follows from a notion of directed influence in the renormalization group flow that is closely connected to the success of Wilson's numerical renormalization group for impurity problems. At a practical level, directed influence allows us to exploit translation invariance when describing a homogeneous system with e.g. an impurity, in spite of the fact that the Hamiltonian is no longer invariant under translations.07/2013;  [Show abstract] [Hide abstract]
ABSTRACT: We use the entanglement negativity, a measure of entanglement for mixed states, to probe the structure of entanglement in the ground state of a topologically ordered system. Through analytical calculations of the negativity in the ground state(s) of the toric code model, we explicitly show that the entanglement of a region $A$ and its complement $B$ is the sum of two types of contributions. The first type of contributions consists of \textit{boundary entanglement}, which we see to be insensitive to tracing out the interior of $A$ and $B$. It therefore entangles only degrees of freedom in $A$ and $B$ that are close to their common boundary. As it is wellknown, each boundary contribution is proportional to the size of the relevant boundary separating $A$ and $B$ and it includes an additive, universal correction. The second contribution appears only when $A$ and $B$ are noncontractible regions (e.g. on a torus) and it consists of longrange entanglement, which we see to be destroyed when tracing out a noncontractible region in the interior of $A$ or $B$. Only the longrange contribution to the entanglement may depend on the specific ground state under consideration.Physical Review A 06/2013; 88(4). · 3.04 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Entanglement renormalization is a realspace renormalization group (RG) transformation for quantum manybody systems. It generates the multiscale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of manybody ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. In this paper we propose the use of symmetric tensors as a mechanism to build a symmetry protected RG flow, and discuss two important applications of this construction. First, we argue that symmetry protected entanglement renormalization produces the proper structure of RG fixedpoints, namely a fixedpoint for each symmetry protected phase. Second, in the context of holography, we show that by using symmetric tensors, a global symmetry at the boundary becomes a local symmetry in the bulk, thus explicitly realizing in the MERA a characteristic feature of the AdS/CFT correspondence.Physical review. B, Condensed matter 03/2013; 88(12). · 3.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Given a microscopic lattice Hamiltonian for a topologically ordered phase, we propose a numerical approach to characterize its emergent anyon model and, in a chiral phase, also its gapless edge theory. First, a tensor network representation of a complete, orthonormal set of ground states on a cylinder of infinite length and finite width is obtained through numerical optimization. Each of these ground states is argued to have a different anyonic flux threading through the cylinder. Then a quasiorthogonal basis on the torus is produced by chopping off and reconnecting the tensor network representation on the cylinder. From these two bases, and by using a number of previous results, most notably the recent proposal of Y. Zhang et al. [Phys. Rev. B 85, 235151 (2012)] to extract the modular U and S matrices, we obtain (i) a complete list of anyon types i, together with (ii) their quantum dimensions d_{i} and total quantum dimension D, (iii) their fusion rules N_{ij}^{k}, (iv) their mutual statistics, as encoded in the offdiagonal entries S_{ij} of S, (v) their selfstatistics or topological spins θ_{i}, (vi) the topological central charge c of the anyon model, and, in a chiral phase (vii) the low energy spectrum of each sector of the boundary conformal field theory. As a concrete application, we study the hardcore boson Haldane model by using the twodimensional density matrix renormalization group. A thorough characterization of its universal bulk and edge properties unambiguously shows that it realizes a ν=1/2 bosonic fractional quantum Hall state.Physical Review Letters 02/2013; 110(6):067208. · 7.73 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We describe a quantum circuit that produces a highly entangled state of N qubits from which one can efficiently compute expectation values of local observables. This construction yields a variational ansatz for quantum manybody states that can be regarded as a generalization of the multiscale entanglement renormalization ansatz (MERA), and to which we refer as the branching MERA. In a lattice system in D dimensions, the scaling of entanglement of a region of size L^D in the branching MERA is not subject to restrictions such as a boundary law L^{D1}, but can be proportional to the size of the region, as we demonstrate numerically for D=1,2 dimensions.10/2012;  [Show abstract] [Hide abstract]
ABSTRACT: The benefits of exploiting the presence of symmetries in tensor network algorithms have been extensively demonstrated in the context of matrix product states (MPSs). These include the ability to select a specific symmetry sector (e.g. with a given particle number or spin), to ensure the exact preservation of total charge, and to significantly reduce computational costs. Compared to the case of a generic tensor network, the practical implementation of symmetries in the MPS is simplified by the fact that tensors only have three indices (they are trivalent, just as the ClebschGordan coefficients of the symmetry group) and are organized as a onedimensional array of tensors, without closed loops. Instead, a more complex tensor network, one where tensors have a larger number of indices and/or a more elaborate network structure, requires a more general treatment. In two recent papers, namely (i) [Phys. Rev. A 82, 050301 (2010)] and (ii) [Phys. Rev. B 83, 115125 (2011)], we described how to incorporate a global internal symmetry into a generic tensor network algorithm based on decomposing and manipulating tensors that are invariant under the symmetry. In (i) we considered a generic symmetry group G that is compact, completely reducible and multiplicity free, acting as a global internal symmetry. Then in (ii) we described the practical implementation of Abelian group symmetries. In this paper we describe the implementation of nonAbelian group symmetries in great detail and for concreteness consider an SU(2) symmetry. Our formalism can be readily extended to more exotic symmetries associated with conservation of total fermionic or anyonic charge. As a practical demonstration, we describe the SU(2)invariant version of the multiscale entanglement renormalization ansatz and apply it to study the low energy spectrum of a quantum spin chain with a global SU(2) symmetry.Physical review. B, Condensed matter 08/2012; 86(19). · 3.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We propose the use of a dynamical window to investigate the realtime evolution of quantum manybody systems in a onedimensional lattice. In a recent paper [H. Phien et al, arxiv:????.????], we introduced infinite boundary conditions (IBC) in order to investigate realtime evolution of an infinite system under a local perturbation. This was accomplished by restricting the update of the tensors in the matrix product state to a finite window, with left and right boundaries held at fixed positions. Here we consider instead the use of a dynamical window, namely a window where the positions of left and right boundaries are allowed to change in time. In this way, all simulation efforts can be devoted to the spacetime region of interest, which leads to a remarkable reduction in computational costs. For illustrative purposes, we consider two applications in the context of the spin1 antiferromagnetic Heisenberg model in an infinite spin chain: one is an expanding window, with boundaries that are adjusted to capture the expansion in time of a local perturbation of the system; the other is a moving window of fixed size, where the position of the window follows the front of a propagating wave.Physical Review B 07/2012; 88(3). · 3.66 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We propose a formalism to study dynamical properties of a quantum manybody system in the thermodynamic limit by studying a finite system with infinite boundary conditions (IBC) where both finite size effects and boundary effects have been eliminated. For onedimensional systems, infinite boundary conditions are obtained by attaching two boundary sites to a finite system, where each of these two sites effectively represents a semiinfinite extension of the system. One can then use standard finitesize matrix product state techniques to study a region of the system while avoiding many of the complications normally associated with finitesize calculations such as boundary Friedel oscillations. We illustrate the technique with an example of time evolution of a local perturbation applied to an infinite (translationally invariant) ground state, and use this to calculate the spectral function of the S=1 Heisenberg spin chain. This approach is more efficient and more accurate than conventional simulations based on finitesize matrix product state and densitymatrix renormalizationgroup approaches.Physical review. B, Condensed matter 07/2012; 86(24). · 3.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We propose a real space renormalization group method to explicitly decouple into independent components a manybody system that, as in the phenomenon of spincharge separation, exhibits separation of degrees of freedom at low energies. Our approach produces a branching holographic description of such systems that opens the path to the efficient simulation of the most entangled phases of quantum matter, such as those whose ground state violates a boundary law for entanglement entropy. As in the coarsegraining transformation of [Phys. Rev. Lett. 99, 220405 (2007)], the key ingredient of this decoupling transformation is the concept of entanglement renormalization, or removal of shortrange entanglement. We demonstrate the feasibility of the approach, both analytically and numerically, by decoupling in real space the ground state of a critical quantum spin chain into two. Generalized notions of RG flow and of scale invariance are also put forward.05/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D) parameter plane, where N is the size of the spin chain and D is the dimension of the MPS matrices. In the first regime MPS can be used to perform finite size scaling (FSS). In the complementary regime the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large N limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS, as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendix, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.Physical review. B, Condensed matter 04/2012; 86(7). · 3.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving manybody systems. Here we put forward a variational Monte Carlo approach for the multiscale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with nonunitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just log(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two dimensional systems.Physical review. B, Condensed matter 01/2012; 85(16). · 3.77 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Tensor network states are powerful variational ans\"atze for manybody ground states of quantum lattice models. The use of Monte Carlo sampling techniques in tensor network approaches significantly reduces the cost of tensor contractions, potentially leading to a substantial increase in computational efficiency. Previous proposals are based on a Markov chain Monte Carlo scheme generated by locally updating configurations and, as such, must deal with equilibration and autocorrelation times, which result in a reduction of efficiency. Here we propose a perfect sampling scheme, with vanishing equilibration and autocorrelation times, for unitary tensor networks  namely tensor networks based on efficiently contractible, unitary quantum circuits, such as unitary versions of the matrix product state (MPS) and tree tensor network (TTN), and the multiscale entanglement renormalization ansatz (MERA). Configurations are directly sampled according to their probabilities in the wavefunction, without resorting to a Markov chain process. We also describe a partial sampling scheme that can result in a dramatic (basisdependent) reduction of sampling error.Physical review. B, Condensed matter 01/2012; 85(16). · 3.77 Impact Factor
Publication Stats
2k  Citations  
237.54  Total Impact Points  
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Institutions

2012–2014

Perimeter Institute for Theoretical Physics
Waterloo, Ontario, Canada 
University of Waterloo
Waterloo, Ontario, Canada


2006–2012

University of Queensland
 School of Mathematics and Physics
Brisbane, Queensland, Australia


2008

Chongqing University
Ch’ungch’ingshih, Chongqing Shi, China


2007

Boston University
 Department of Physics
Boston, MA, United States
