A holographic correspondence from tensor network states

Nathan A. McMahon, Sukhwinder Singh, Gavin K. Brennen

Javier Molina-Vilaplana

Xiao-Liang Qi

Random tensor network states are toy models for holographic duality, which have entanglement properties determined by graph geometry. In this paper, we propose a generalization of the random tensor network states which describe an ensemble of states preserving a given global symmetry. We show that Renyi entropy for this family of states can be described by a quantum extremal surface formula, with corrections to the area law term determined by a bulk gauge theory wavefunction. This provides a toy model of the correspondence between boundary global symmetry and bulk gauge symmetry in holographic duality. We discuss the boundary physical consequences of the bulk deconfined and confined phases.

Rafael N. Alexander, Amr A. Ahmadain, Zhao Zhang +1 more

Sukhwinder Singh

In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study quantum many-body systems at low energies. In this paper, we describe a particular way in which any given tensor network can be viewed as a representation of two different quantum many-body states. The two quantum many-body states are said to correspond to each other by means of the tensor network. We apply this ``tensor network state correspondence''---a correspondence between quantum many-body states mediated by tensor networks as we describe---to the multi-scale entanglement renormalization ansatz (MERA) representation of ground states of one dimensional (1D) quantum many-body systems. Since the MERA is a 2D hyperbolic tensor network (the extra dimension is identified as the length scale of the 1D system), the two quantum many-body states obtained from the MERA, via tensor network state correspondence, are seen to live in the bulk and on the boundary of a discrete hyperbolic geometry. The bulk state so obtained from a MERA exhibits interesting features, some of which caricature known features of the holographic correspondence of String theory. We show how (i) the bulk state admits a description in terms of ``holographic screens'', (ii) the conformal field theory data associated with a critical ground state can be obtained from the corresponding bulk state, in particular, how pointlike boundary operators are identified with extended bulk operators. (iii) We also present numerical results to illustrate that bulk states, dual to ground states of several critical spin chains, have exponentially decaying correlations, and that the bulk correlation length generally decreases with increase in central charge for these spin chains.

Grzegorz Czelusta, Jakub Mielczarek A B

Tensor networks provide a powerful tool for studying many-body quantum systems, particularly making quantum simulations more efficient. In this article, we construct a tensor network representation of the spin network states, which correspond to S U ( 2 ) gauge-invariant discrete field theories. Importantly, the spin network states play a central role in the loop quantum gravity (LQG) approach to the Planck scale physics. Our focus is on the Ising-type spin networks, which provide a basic model of quantum space in LQG. It is shown that the tensor network approach improves the previously introduced methods of constructing quantum circuits for the Ising spin networks by reducing the number of qubits involved. It is also shown that the tensor network approach is convenient for implementing holographic states involving the bulk-boundary conjecture, which contributes to establishing a link between LQG and holographic duality.