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arXiv:1310.0205

Path-integral Monte Carlo for the Gauge-fixed Berry Connection and the Local Z_2 Berry Phase

Yuichi Motoyama, Synge Todo

Published Oct 1, 2013• 0 citations
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Abstract

We present a general formula of the gauge-fixed Berry connection which can be evaluated by path- integral Monte Carlo method. We also propose that the gauge-fixed local Berry connection can be an- other effective tool to estimate precisely the quantum critical point. For a demonstration, we calculate the gauge-fixed Berry connection and the local Z2 Berry phase of the antiferromagnetic Heisenberg model on a staggered bond-alternating ladder, and estimated quantum critical point is consistent with other methods.

DOI
10.48550/arxiv.1310.0205

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The local ZN{Z}_{N}ZN​ quantized Berry phase for the SU(N)\mathrm{SU}(N)SU(N) antiferromagnetic Heisenberg spin model is formulated. This quantity, which is a generalization of the local Z2{Z}_{2}Z2​ Berry phase for SU(2) symmetry, has a direct correspondence to the number of singlet pairs spanning a particular bond, and is effective as a tool to characterize and classify the various symmetry-protected topological phases of one-dimensional SU(N)\mathrm{SU}(N)SU(N) spin systems. We extend the path-integral quantum Monte Carlo method for the Z2{Z}_{2}Z2​ Berry phase in order to calculate the ZN{Z}_{N}ZN​ Berry phase numerically. We demonstrate our method by calculating the Z4{Z}_{4}Z4​ Berry phase for the bond-alternating SU(4)\mathrm{SU}(4)SU(4) antiferromagnetic Heisenberg chain, which is represented by the Young diagram of four columns.
We study quantum wires and point contacts with a fixed-node path integral Monte Carlo technique. The fixed-node technique uses a variational principle to map fermionic problems into effective bosonic problems, which are then evaluated with standard quantum Monte Carlo techniques. While fixed-node is an approximation, it has the useful properties of being variational and being able to recover the exact answer when the exact nodes or phases of the density matrix are known. In these finite-temperature simulations we use the free particle density matrix as a fixed-node constraint to efficiently simulate hundreds of interacting electrons. We have calculated charge densities, pair correlation functions, and the currentcurrent Matsubura Green's functions for quantum wires and quantum point contacts.
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