Chiral topological spin liquids with projected entangled pair states

Topological chiral phases are ubiquitous in the physics of the Fractional Quantum Hall Effect. Non-chiral topological spin liquids are also well known. Here, using the framework of projected entangled pair states (PEPS), we construct a family of chiral spin liquids on the square lattice which are generalized spin-1/2 Resonating Valence Bond (RVB) states obtained from deformed local tensors with $d+i\, d$ symmetry. On a cylinder, we construct four topological sectors with even or odd number of spinons on the boundary and even or odd number of ($\mathbb{Z}_2$) fluxes penetrating the cylinder which, we argue, remain orthogonal in the limit of infinite perimeter. The analysis of the transfer matrix provides evidence of short-range (long-range) triplet (singlet) correlations as for the critical (non-chiral) RVB state. The Entanglement Spectrum exhibits chiral edge modes, which we confront to predictions of Conformal Field Theory, and the corresponding Entanglement Hamiltonian is shown to be long ranged.Comment: 7 pages, 6 figures, Final version (small changes

Field-induced superfluids and Bose liquids in Projected Entangled Pair States

In two-dimensional incompressible quantum spin liquids, a large enough magnetic field generically induces "doping" of polarized S=1 triplons or S=1/2 spinons. We review a number of cases such as spin-3/2 AKLT or spin-1/2 Resonating Valence Bond (RVB) liquids where the Projected Entangled Pair States (PEPS) framework provides very simple and comprehensive pictures. On the bipartite honeycomb lattice, simple PEPS can describe Bose condensed triplons (AKLT) or spinons (RVB) superfluids with transverse staggered (N\'eel) magnetic order. On the Kagome lattice, doping the RVB state with deconfined spinons or triplons (i.e. spinon bound pairs) yields uncondensed Bose liquids preserving U(1) spin-rotation symmetry. We find that spinon (triplon) doping destroys (preserves) the topological Z_2 symmetry of the underlying RVB state. We also find that spinon doping induces longer range interactions in the entanglement Hamiltonian, suggesting the emergence of (additive) log-corrections to the entanglement entropy.Comment: 7 pages, 7 figures Improved final version (as published

Ground-State Properties of Quantum Many-Body Systems: Entangled-Plaquette States and Variational Monte Carlo

We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a Variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any "sign" instability. We show results for various two dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatzs is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.Comment: 5 pages, 2 figures (see also arXiv:0907.4646

A generalization of the injectivity condition for Projected Entangled Pair States

We introduce a family of tensor network states that we term semi-injective Projected Entangled-Pair States (PEPS). They extend the class of injective PEPS and include other states, like the ground states of the AKLT and the CZX models in square lattices. We construct parent Hamiltonians for which semi-injective PEPS are unique ground states. We also determine the necessary and sufficient conditions for two tensors to generate the same family of such states in two spatial dimensions. Using this result, we show that the third cohomology labeling of Symmetry Protected Topological phases extends to semi-injective PEPS.Comment: 63 page

Approximating Gibbs states of local Hamiltonians efficiently with PEPS

We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with Projected Entangled Pair States (PEPS) as a function of the bond dimension ($D$), temperature ($\beta^{-1}$), and system size ($N$). First, we introduce a compression method in which the bond dimension scales as $D=e^{O(\log^2(N/\epsilon))}$ if $\beta<O(\log (N))$. Second, building on the work of Hastings [Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D=\left(N/\epsilon\right)^{O(\beta)}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with $D=N^{O(\log(N))}$ whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.Comment: 12 pages, 1 figur

Simulating two- and three-dimensional frustrated quantum systems with string-bond states

Simulating frustrated quantum magnets is among the most challenging tasks in computational physics. We apply String-Bond States, a recently introduced ansatz which combines Tensor Networks with Monte Carlo based methods, to the simulation of frustrated quantum systems in both two and three dimensions. We compare our results with existing results for unfrustrated and two-dimensional systems with open boundary conditions, and demonstrate that the method applies equally well to the simulation of frustrated systems with periodic boundaries in both two and three dimensions.Comment: 9 pages, 11 figures; v2: accepted version, Journal-Ref adde

Classification of Matrix-Product Unitaries with Symmetries

We prove that matrix-product unitaries (MPUs) with on-site unitary symmetries are completely classified by the (chiral) index and the cohomology class of the symmetry group $G$, provided that we can add trivial and symmetric ancillas with arbitrary on-site representations of $G$. If the representations in both system and ancillas are fixed to be the same, we can define symmetry-protected indices (SPIs) which quantify the imbalance in the transport associated to each group element and greatly refines the classification. These SPIs are stable against disorder and measurable in interferometric experiments. Our results lead to a systematic construction of two-dimensional Floquet symmetry-protected topological (SPT) phases beyond the standard classification, and thus shed new light on understanding nonequilibrium phases of quantum matter.Comment: 6+16 pages, 3+8 figure

RVB superconductors with fermionic projected entangled pair states

We construct a family of simple fermionic projected entangled pair states (fPEPS) on the square lattice with bond dimension $D=3$ which are exactly hole-doped resonating valence bond (RVB) wavefunctions with short-range singlet bonds. Under doping the insulating RVB spin liquid evolves immediately into a superconductor with mixed $d+is$ pairing symmetry whose pair amplitude grows as the square-root of the doping. The relative weight between $s$-wave and $d$-wave components can be controlled by a single variational parameter $c$. We optimize our ansatz w.r.t. $c$ for the frustrated $t-J_1-J_2$ model (including both nearest and next-nearest neighbor antiferromagnetic interactions $J_1$ and $J_2$, respectively) for $J_2\simeq J_1/2$ and obtain an energy very close to the infinite-PEPS state (using full update optimization and same bond dimension). The orbital symmetry of the optimized RVB superconductor has predominant d-wave character, although we argue a residual (complex s-wave) time reversal symmetry breaking component should always be present. Connections of the results to the physics of superconducting cuprates and pnictides are outlined.Comment: 6 pages, 4 figures and Supplemental Material (3 pages, 2 figures). Updated version including new iPEPS results using full update optimization scheme, showing excellent agreement with RVB wave functio

Information propagation for interacting particle systems

We show that excitations of interacting quantum particles in lattice models always propagate with a finite speed of sound. Our argument is simple yet general and shows that by focusing on the physically relevant observables one can generally expect a bounded speed of information propagation. The argument applies equally to quantum spins, bosons such as in the Bose-Hubbard model, fermions, anyons, and general mixtures thereof, on arbitrary lattices of any dimension. It also pertains to dissipative dynamics on the lattice, and generalizes to the continuum for quantum fields. Our result can be seen as a meaningful analogue of the Lieb-Robinson bound for strongly correlated models.Comment: 4 pages, 1 figure, minor change