We introduce the multiscale entanglement renormalization ansatz, a class of quantum many-body states on a D-dimensional lattice that can be efficiently simulated with a classical computer, in that ...the expectation value of local observables can be computed exactly and efficiently. The multiscale entanglement renormalization ansatz is equivalent to a quantum circuit of logarithmic depth that has a very characteristic causal structure. It is also the ansatz underlying entanglement renormalization, a novel coarse-graining scheme for many-body quantum systems on a lattice.
We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it ...into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block.
Invariance under translation is exploited to efficiently simulate one-dimensional quantum lattice systems in the limit of an infinite lattice. Both the computation of the ground state and the ...simulation of time evolution are considered.
We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a ...quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.
We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed tensor network ...renormalization G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015) to the Euclidean time evolution operator e(-βH) for infinite β. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β, produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.
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 off-diagonal entries S(ij) of S, (v) their self-statistics 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 hard-core boson Haldane model by using the two-dimensional 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.
Cephalopods (nautiluses, cuttlefishes, squids and octopuses) exhibit direct development and display two major developmental modes: planktonic and benthic. Planktonic hatchlings are small and go ...through some degree of morphological changes during the planktonic phase, which can last from days to months, with ocean currents enhancing their dispersal capacity. Benthic hatchlings are usually large, miniature-like adults and have comparatively reduced dispersal potential. We examined the relationship between early developmental mode, hatchling size and species latitudinal distribution range of 110 species hatched in the laboratory, which represent 13% of the total number of live cephalopod species described to date. Results showed that species with planktonic hatchlings reach broader distributional ranges in comparison with species with benthic hatchlings. In addition, squids and octopods follow an inverse relationship between hatchling size and species latitudinal distribution. In both groups, species with smaller hatchlings have broader latitudinal distribution ranges. Thus, squid and octopod species with larger hatchlings have latitudinal distributions of comparatively minor extension. This pattern also emerges when all species are grouped by genus (n = 41), but was not detected for cuttlefishes, a group composed mainly of species with large and benthic hatchlings. However, when hatchling size was compared to adult size, it was observed that the smaller the hatchlings, the broader the latitudinal distributional range of the species for cuttlefishes, squids and octopuses. This was also valid for all cephalopod species with benthic hatchlings pooled together. Hatchling size and associated developmental mode and dispersal potential seem to be main influential factors in determining the distributional range of cephalopods.
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 many-body states that can be regarded as a generalization of the multiscale entanglement renormalization ansatz (MERA), which we refer to 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(D-1), but can be proportional to the size of the region, as we demonstrate numerically.
Tensor Network States and Geometry Evenbly, G.; Vidal, G.
Journal of statistical physics,
11/2011, Volume:
145, Issue:
4
Journal Article
Peer reviewed
Open access
Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in
D
spatial dimensions. Different types of tensor network states can be seen to generate different ...geometries. Matrix product states (MPS) in
D
=1 dimensions, as well as projected entangled pair states (PEPS) in
D
>1 dimensions, reproduce the
D
-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (
D
+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for
D
=1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law—that is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in
D
>1 dimensions.
The generalization of the multiscale entanglement renormalization ansatz (MERA) to continuous systems, or cMERA Haegeman et al., Phys. Rev. Lett. 110, 100402 ...(2013)PRLTAO0031-900710.1103/PhysRevLett.110.100402, is expected to become a powerful variational ansatz for the ground state of strongly interacting quantum field theories. In this Letter, we investigate, in the simpler context of Gaussian cMERA for free theories, the extent to which the cMERA state |Ψ^{Λ}⟩ with finite UV cutoff Λ can capture the spacetime symmetries of the ground state |Ψ⟩. For a free boson conformal field theory (CFT) in 1+1 dimensions, as a concrete example, we build a quasilocal unitary transformation V that maps |Ψ⟩ into |Ψ^{Λ}⟩ and show two main results. (i) Any spacetime symmetry of the ground state |Ψ⟩ is also mapped by V into a spacetime symmetry of the cMERA |Ψ^{Λ}⟩. However, while in the CFT, the stress-energy tensor T_{μν}(x) (in terms of which all the spacetime symmetry generators are expressed) is local, and the corresponding cMERA stress-energy tensor T_{μν}^{Λ}(x)=VT_{μν}(x)V^{†} is quasilocal. (ii) From the cMERA, we can extract quasilocal scaling operators O_{α}^{Λ}(x) characterized by the exact same scaling dimensions Δ_{α}, conformal spins s_{α}, operator product expansion coefficients C_{αβγ}, and central charge c as the original CFT. Finally, we argue that these results should also apply to interacting theories.
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