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Path-integral Monte Carlo method for the local $Z_{2}$ Berry phase

Yuichi Motoyama¹ and Synge Todo²

¹Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
²Institute for Solid State Physics, University of Tokyo, 7-1-26-R501 Port Island South, Kobe 650-0047, Japan

Phys. Rev. E 87, 021301(R) – Published 6 February, 2013

DOI: https://doi.org/10.1103/PhysRevE.87.021301

Abstract

We present a loop cluster algorithm Monte Carlo method for calculating the local $Z_{2}$ Berry phase of the quantum spin models. The Berry connection, which is given as the inner product of two ground states with different local twist angles, is expressed as a Monte Carlo average on the worldlines with fixed spin configurations at the imaginary-time boundaries. The “complex weight problem” caused by the local twist is solved by adopting the meron cluster algorithm. We present the results of simulation on the antiferromagnetic Heisenberg model on an out-of-phase bond-alternating ladder to demonstrate that our method successfully detects the change in the valence bond pattern at the quantum phase transition point. We also propose that the gauge-fixed local Berry connection can be an effective tool to estimate precisely the quantum critical point.

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