Publications (122)456.53 Total impact
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ABSTRACT: The critical 2d classical Ising model on the square lattice has two topological conformal defects: the $\mathbb{Z}_2$ symmetry defect $D_{\epsilon}$ and the KramersWannier duality defect $D_{\sigma}$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function $Z_{D}$ of the critical Ising model in the presence of a topological conformal defect $D$ is expressed in terms of the scaling dimensions $\Delta_{\alpha}$ and conformal spins $s_{\alpha}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising CFT. This characteristic conformal data $\{\Delta_{\alpha}, s_{\alpha}\}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix $M_{D}$ for the partition function $Z_D$. In this paper we investigate the use of tensor network techniques to both represent and coarsegrain the partition functions $Z_{D_\epsilon}$ and $Z_{D_\sigma}$ of the critical Ising model with either a symmetry defect $D_{\epsilon}$ or a duality defect $D_{\sigma}$. We also explain how to coarsegrain the corresponding transfer matrices $M_{D_\epsilon}$ and $M_{D_\sigma}$, from which we can extract accurate numerical estimates of $\{\Delta_{\alpha}, s_{\alpha}\}_{D_{\epsilon}}$ and $\{\Delta_{\alpha}, s_{\alpha}\}_{D_{\sigma}}$. Two key new ingredients of our approach are (i) coarsegraining of the defect $D$, which applies to any (i.e. not just topological) conformal defect and yields a set of associated scaling dimensions $\Delta_{\alpha}$, and (ii) construction and coarsegraining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins $s_{\alpha}$.  [Show abstract] [Hide abstract]
ABSTRACT: In 1+1dimensional conformal field theory, the thermal state on a circle is related to a certain quotient of the vacuum on a line. We explain how to take this quotient in the MERA tensor network representation of the vacuum and confirm the validity of the construction in the critical Ising model. This result suggests that the tensors comprising MERA can be interpreted as performing local scale transformations, so that adding or removing them emulates conformal maps. In this sense, the optimized MERA recovers local conformal invariance, which is explicitly broken by the choice of lattice. Our discussion also informs the dialogue between tensor networks and holographic duality.  [Show abstract] [Hide abstract]
ABSTRACT: It is well known that the bulk physics of a topological phase constrains its possible edge physics through the bulkedge correspondence. Therefore, the different types of edge theories that a topological phase can host is a universal piece of data which can be used to characterize topological order. In this paper, we argue that beginning from only the fixed point wavefunction (FPW) of a nonchiral topological phase and by locally deforming it, all possible edge theories can be extracted from its entanglement Hamiltonian (EH). We illustrate our claim by deforming the FPW of the Wenplaquette model, the quantum double of $\mathbb{Z}_2$. We show that the possible EHs of the deformed FPWs reflect the known possible types of edge theories, which are generically gapped, but gapless if translationally symmetry is preserved. We stress that our results do not require an underlying Hamiltonian  thus, this lends support to the notion that a topological phase is indeed characterized by only a set of quantum states and can be studied through its FPWs.  [Show abstract] [Hide abstract]
ABSTRACT: Consider the partition function of a classical system in two spatial dimensions, or the Euclidean path integral of a quantum system in two spacetime dimensions, both on a lattice. We show that the tensor network renormalization (TNR) algorithm [\textit{G. Evenbly and G. Vidal, arXiv:1412.0732}] can be used to implement local scale transformations on these objects, namely a lattice version of conformal maps. Specifically, we explain how to implement the lattice equivalent of the logarithmic conformal map that transforms the Euclidean plane into a cylinder. As an application, and with the 2D critical Ising model as a concrete example, we use this map to build a lattice version of the scaling operators of the underlying conformal field theory, from which one can extract their scaling dimensions and operator product expansion coefficients. 
Article: LiebLiniger model with exponentially decaying interactions: A continuous matrix product state study
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ABSTRACT: The LiebLiniger model describes onedimensional bosons interacting through a repulsive contact potential. In this work, we introduce an extended version of this model by replacing the contact potential with a decaying exponential. Using the recently developed continuous matrix product states techniques, we explore the ground state phase diagram of this model by examining the superfluid and density correlation functions. At weak coupling superfluidity governs the ground state, in a similar way as in the LiebLiniger model. However, at strong coupling quasicrystal and superTonksGirardeau regimes are also found, which are not present in the original LiebLiniger case. Therefore the presence of the exponentiallydecaying potential leads to a superfluid/superTonksGirardeau/quasicrystal crossover, when tuning the coupling strength from weak to strong interactions. This corresponds to a Luttinger liquid parameter in the range $K \in (0, \infty)$; in contrast with the LiebLiniger model, where $K \in [1, \infty)$, and the screened longrange potential, where $K \in (0, 1]$.  [Show abstract] [Hide abstract]
ABSTRACT: Recent years have seen the discovery of a chiral spin liquid state  a bosonic analogue of a fractional Quantum Hall state first put forward by Kalmeyer and Laughlin in 1987  in several deformations of the Heisenberg model on the Kagome lattice. Here, we apply stateoftheart numerical techniques to one such model, where breaking of the timereversal symmetry drives the system into the chiral phase. Our methods allow us to obtain explicit matrixproduct state representations of the lowlying excitations of the chiral spin liquid state, including the topologically nontrivial semionic excitation. We characterize these excitations and study their energetics as the model is tuned towards a topological phase transition.  [Show abstract] [Hide abstract]
ABSTRACT: We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a manybody Hamiltonian $H$ by applying the recently proposed \textit{tensor network renormalization} (TNR) [G. Evenbly and G. Vidal, arXiv:1412.0732] to the Euclidean time evolution operator $e^{\beta H}$ for infinite $\beta$. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature $\beta$, produces a MERA representation of a thermal Gibbs state. Our construction endows TNR with a renormalization group flow in the space of wavefunctions and Hamiltonians (and not just in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems. 
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.  [Show abstract] [Hide abstract]
ABSTRACT: In a system with chiral topological order, there is a remarkable correspondence between the edge and entanglement spectra: the low energy spectrum of the system in the presence of a physical edge coincides with the lowest part of the entanglement spectrum (ES) across a virtual cut of the system, up to rescaling and shifting. In this paper, we explore whether the edgeES correspondence extends to nonchiral topological phases. Specifically, we consider the Wenplaquette model which has Z_2 topological order. The unperturbed model displays an exact correspondence: both the edge and entanglement spectra within each topological sector a (a = 1,...,4) are flat and equally degenerate. Here we show, through a detailed microscopic calculation, that in the presence of generic local perturbations: (i) the effective degrees of freedom for both the physical edge and the entanglement cut consist of a spin1/2 chain, with effective Hamiltonians H_edge^a and H_ent.^a respectively, both of which have a Z_2 symmetry enforced by the bulk topological order; (ii) there is in general no match between their low energy spectra, that is, there is no edgeES correspondence. However, if supplement the Z_2 topological order with a global symmetry (translational invariance along the edge/cut), i.e. by considering the Wenplaquette model as a symmetry enriched topological phase (SET), then there is a finite domain in Hamiltonian space in which both H_edge^a and H_ent.^a realize the critical Ising model, whose low energy effective theory is the c = 1/2 Ising CFT. This is achieved because the presence of the global symmetry implies that both Hamiltonians, in addition to being Z_2 symmetric, are KramersWannier selfdual. Thus, the bulk topological order and the global translational symmetry of the Wenplaquette model as a SET imply an edgeES correspondence at least in some finite domain in Hamiltonian space. 
Article: Fast convergence of imaginary time evolution tensor network algorithms by recycling the environment
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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. 
Article: Entanglement contour
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ABSTRACT: In the context of characterizing the structure of quantum entanglement in manybody systems, we introduce the entanglement contour, a tool to identify which realspace degrees of freedom contribute, and how much, to the entanglement of a region A with the rest of the system B. The entanglement contour provides a complementary, more re?fined approach to characterizing entanglement than just considering the entanglement entropy between A and B, with several concrete advantages. We illustrate this in the context of ground states and quantum quenches in fermionic quadratic systems. For instance, in a quantum critical system in $D = 1$ spatial dimensions, the entanglement contour allows us to determine the central charge of the underlying conformal field theory from just a single partition of the system into regions A and B, (using the entanglement entropy for the same task requires considering several partitions). In $D \geq 2$ dimensions, the entanglement contour can distinguish between gapped and gapless phases that obey a same boundary law for entanglement entropy. During a local or global quantum quench, the timedependent contour provides a detailed account of the dynamics of entanglement, including propagating entanglement waves, which offers a microscopic explanation of the behavior of the entanglement entropy as a function of time.  [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.  [Show abstract] [Hide abstract]
ABSTRACT: Entanglement negativity is a measure of mixedstate entanglement increasingly used to investigate and characterize emerging quantum manybody phenomena, including quantum criticality and topological order. We present two results for the entanglement negativity: a disentangling theorem, which allows the use of this entanglement measure as a means to detect whether a wavefunction of three subsystems $A$, $B$, and $C$ factorizes into a product state for parts $AB_1$ and $B_2C$; and a monogamy relation, which states that if $A$ is very entangled with $B$, then $A$ cannot be simultaneaously very entangled also with $C$.  [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.  [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.  [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. 
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. 
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.  [Show abstract] [Hide abstract]
ABSTRACT: Consider two quantum critical Hamiltonians H and ˜H on a ddimensional lattice that only differ in some region R. We study the relation between holographic representations, obtained through realspace renormalization, of their corresponding ground states ψ and ˜ψ. We observe that, even though ψ and ˜ψ disagree significantly both inside and outside region R, they still admit holographic descriptions that only differ inside the past causal cone C(R) of region R, where C(R) is obtained by coarsegraining region 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.  [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.
Publication Stats
12k  Citations  
456.53  Total Impact Points  
Top Journals
Institutions

20122015

Perimeter Institute for Theoretical Physics
Ватерлоо, Ontario, Canada


20062014

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


20022004

California Institute of Technology
 Institute for Quantum Information and Matter
Pasadena, CA, United States


20002002

University of Innsbruck
 Department of Theoretical Physics
Innsbruck, Tyrol, Austria


19981999

University of Barcelona
 Department of Structure and Constituents of Matter
Barcino, Catalonia, Spain
