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arXiv:0810.1414 · Physical Review B

Critical exponents with a multiscale entanglement renormalization Ansatz channel

Simone Montangero, Matteo Rizzi, Vittorio Giovannetti, Rosario Fazio

Published Sep 30, 2009• 28 citations
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Abstract

We show how to compute the critical exponents of one-dimensional quantum critical systems in the thermodynamic limit. The method is based on an iterative scheme applied to the multiscale entanglement renormalization Ansatz for the ground-state wave function. We test this scheme to compute the critical exponents of the Ising and $XXZ$ model for which we can compare the method with the exact values. The agreement is at worst within few percent of the exact results already for moderate dimensions of the tensor indices.

DOI
10.1103/physrevb.80.113103

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We show how to compute the critical exponents of one-dimensional quantum critical systems in the thermodynamic limit. The method is based on an iterative scheme applied to the multiscale entanglement renormalization Ansatz for the ground-state wave function. We test this scheme to compute the critical exponents of the Ising and XXZXXZXXZ model for which we can compare the method with the exact values. The agreement is at worst within few percent of the exact results already for moderate dimensions of the tensor indices.
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.