Isometric Tensor Network States in Two Dimensions

Tensor network states (TNS) are a promising but numerically challenging tool for simulating two-dimensional (2D) quantum many-body problems. We introduce an isometric restriction of the TNS ansatz that allows for highly efficient contraction of the network. We consider two concrete applications using this ansatz. First, we show that a matrix-product state representation of a 2D quantum state can be iteratively transformed into an isometric 2D TNS. Second, we introduce a 2D version of the time-evolving block decimation algorithm (TEBD$^2$) for approximating the ground state of a Hamiltonian as an isometric TNS, which we demonstrate for the 2D transverse field Ising model.Comment: 5 pages, 4 figure

Are you sure you are using the correct model? Model Selection and Averaging of Impulse Responses

Impulse responses can be estimated to analyze the effects of a shock to a variable over time. Typically, (vector) autoregressive models are estimated and the impulse responses implied by the coefficients calculated. In general, however, there is no knowledge of the correct autoregressive order. In fact, when models are seen as approximations to the data generating process (DGP), all models are imperfect and there is no a priori difference in their validity. Hence, a lag length should be chosen by a sensible method, for instance an information criterion. In Monte Carlo simulations, this paper studies what characteristics influence the optimal autoregressive order when all models are only approximations to the DGP. It finds that the precise coefficients in the DGP, the sample size, and the impulse response horizon to be estimated all influence the mean squared error-minimizing lag length. Furthermore, it evaluates the performance of model selection and averaging methods for estimating impulse responses. Across the characteristics found to be relevant, averaging outperforms model selection, and in particular Mallows' Model Averaging and a smoothed Hannan-Quinn Information Criterion perform best. Finally, the study is extended to vector autoregressive models. In addition to the characteristics relevant in the univariate case, the optimal lag length also depends on which (cross) impulse response is to be estimated. Many issues remain for vector autoregressive models, however, and more work is necessary

Characterization and stability of a fermionic \nu=1/3 fractional Chern insulator

Using the infinite density matrix renormalization group method on an infinite cylinder geometry, we characterize the $1/3$ fractional Chern insulator state in the Haldane honeycomb lattice model at $\nu=1/3$ filling of the lowest band and check its stability. We investigate the chiral and topological properties of this state through (i) its Hall conductivity, (ii) the topological entanglement entropy, (iii) the $U(1)$ charge spectral flow of the many body entanglement spectrum, and (iv) the charge of the anyons. In contrast to numerical methods restricted to small finite sizes, the infinite cylinder geometry allows us to access and characterize directly the metal to fractional Chern insulator transition. We find indications it is first order and no evidence of other competing phases. Since our approach does not rely on any band or subspace projection, we are able to prove the stability of the fractional state in the presence of interactions exceeding the band gap, as has been suggested in the literature. As a by-product we discuss the signatures of Chern insulators within this technique.Comment: published versio

Ergodicity-breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians

We show that the combination of charge and dipole conservation---characteristic of fracton systems---leads to an extensive fragmentation of the Hilbert space, which in turn can lead to a breakdown of thermalization. As a concrete example, we investigate the out-of-equilibrium dynamics of one-dimensional spin-1 models that conserve charge (total $S^z$) and its associated dipole moment. First, we consider a minimal model including only three-site terms and find that the infinite temperature auto-correlation saturates to a finite value---showcasing non-thermal behavior. The absence of thermalization is identified as a consequence of the strong fragmentation of the Hilbert space into exponentially many invariant subspaces in the local $S^z$ basis, arising from the interplay of dipole conservation and local interactions. Second, we extend the model by including four-site terms and find that this perturbation leads to a weak fragmentation: the system still has exponentially many invariant subspaces, but they are no longer sufficient to avoid thermalization for typical initial states. More generally, for any finite range of interactions, the system still exhibits non-thermal eigenstates appearing throughout the entire spectrum. We compare our results to charge and dipole moment conserving random unitary circuit models for which we reach identical conclusions.Comment: close to published version: 10 pages + Appendices. Updated discussions and conten

Signatures of Dirac cones in a DMRG study of the Kagome Heisenberg model

The antiferromagnetic spin-$1/2$ Heisenberg model on a kagome lattice is one of the most paradigmatic models in the context of spin liquids, yet the precise nature of its ground state is not understood. We use large scale density matrix normalization group simulations (DMRG) on infinitely long cylinders and find indications for the formation of a gapless Dirac spin liquid. First, we use adiabatic flux insertion to demonstrate that the spin gap is much smaller than estimated from previous DMRG simulation. Second, we find that the momentum dependent excitation spectrum, as extracted from the DMRG transfer matrix, exhibits Dirac cones that match those of a $\pi$-flux free fermion model (the parton mean-field ansatz of a $U(1)$ Dirac spin liquid)Comment: 15 pages, 16 figure

Infinite density matrix renormalization group for multicomponent quantum Hall systems

While the simplest quantum Hall plateaus, such as the $\nu = 1/3$ state in GaAs, can be conveniently analyzed by assuming only a single active Landau level participates, for many phases the spin, valley, bilayer, subband, or higher Landau level indices play an important role. These `multi-component' problems are difficult to study using exact diagonalization because each component increases the difficulty exponentially. An important example is the plateau at $\nu = 5/2$, where scattering into higher Landau levels chooses between the competing non-Abelian Pfaffian and anti-Pfaffian states. We address the methodological issues required to apply the infinite density matrix renormalization group to quantum Hall systems with multiple components and long-range Coulomb interactions, greatly extending accessible system sizes. As an initial application we study the problem of Landau level mixing in the $\nu = 5/2$ state. Within the approach to Landau level mixing used here, we find that at the Coulomb point the anti-Pfaffian is preferred over the Pfaffian state over a range of Landau level mixing up to the experimentally relevant values.Comment: 12 pages, 9 figures. v2 added more data for different amounts of Landau level mixing at 5/2 fillin

Isometric tensor network representations of two-dimensional thermal states

Tensor networks provide a useful tool to describe low-dimensional complex many-body systems. Finding efficient algorithms to use these methods for finite temperature simulations in two dimensions is a continuing challenge. Here, we use the class of recently introduced isometric tensor network states (isoTNS), which can also be directly realized with unitary gates on a quantum computer. We utilize a purification ansatz to efficiently represent thermal states of the transverse field Ising model. By performing an imaginary time evolution starting from infinite temperature, we find that this approach offers a new way with low computational complexity to represent states at finite temperatures.Comment: 9 pages, 7 figure