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arXiv:0912.2893 · Journal of Statistical Mechanics Theory and Experiment

Entanglement renormalization and boundary critical phenomena

Pietro Silvi, Vittorio Giovannetti, Pasquale Calabrese, Giuseppe E. Santoro, Rosario Fazio

Published Mar 4, 2010• 14 citations
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Abstract

The multiscale entanglement renormalization ansatz is applied to the study of boundary critical phenomena. We compute averages of local operators as a function of the distance from the boundary and the surface contribution to the ground state energy. Furthermore, assuming a uniform tensor structure, we show that the multiscale entanglement renormalization ansatz implies an exact relation between bulk and boundary critical exponents known to exist for boundary critical systems.

DOI
10.1088/1742-5468/2010/03/l03001

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The multiscale entanglement renormalization ansatz is applied to the study of boundary critical phenomena. We compute averages of local operators as a function of the distance from the boundary and the surface contribution to the ground state energy. Furthermore, assuming a uniform tensor structure, we show that the multiscale entanglement renormalization ansatz implies an exact relation between bulk and boundary critical exponents known to exist for boundary critical systems.
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We propose the use of entanglement renormalization techniques to study boundary critical phenomena on a lattice system. The multiscale entanglement renormalization ansatz (MERA), in its scale invariant version, offers a very compact approximation to quantum critical ground states. Here we show that, by adding a boundary to the MERA, an accurate approximation to the ground state of a semi-infinite critical chain with an open boundary is obtained, from which one can extract boundary scaling operators and their scaling dimensions. As in Wilson's renormalization-group formulation of the Kondo problem, our construction produces, as a side result, an effective chain displaying explicit separation of energy scales. We present benchmark results for the quantum Ising and quantum XX models with free and fixed boundary conditions.
Tensor network representations of many-body quantum systems can be described in terms of quantum channels. We focus on channels associated with the multiscale entanglement renormalization ansatz tensor network that has been recently introduced to efficiently describe critical systems. Our approach allows us to compute the multiscale entanglement renormalization ansatz correspondent to the thermodynamical limit of a critical system introducing a transfer matrix formalism, and to relate the system critical exponents to the convergence rates of the associated channels.
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We describe an algorithm to simulate time evolution using the multiscale entanglement renormalization ansatz and test it by studying a critical Ising chain with periodic boundary conditions and with up to $L\ensuremath{\approx}{10}^{6}$ quantum spins. The cost of a simulation, which scales as L log2(L)L\text{ }{\text{log}}_{2}(L)L log2​(L), is reduced to log2(L){\text{log}}_{2}(L)log2​(L) when the system is invariant under translations. By simulating an evolution in imaginary time, we compute the ground state of the system. The errors in the ground-state energy display no evident dependence on the system size. The algorithm can be extended to lattice systems in higher spatial dimensions.
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