Tensor renormalization group study of the three-dimensional O (2) model

Jacques Bloch,Raghav G Jha,Maximilian Meister,Robert Lohmayer

doi:10.1103/physrevd.104.094517

Jacques Bloch, Raghav G Jha + Show 2 more

Open Access

https://doi.org/10.1103/physrevd.104.094517

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Abstract

We calculate thermodynamic potentials and their derivatives for the three-dimensional $O(2)$ model using tensor-network methods to investigate the well-known second-order phase transition. We also consider the model at non-zero chemical potential to study the Silver Blaze phenomenon, which is related to the particle number density at zero temperature. Furthermore, the temperature dependence of the number density is explored using asymmetric lattices. Our results for both zero and non-zero magnetic field, temperature, and chemical potential are consistent with those obtained using other methods.

Highlights

Our understanding of quantum many-body systems has considerably improved in the past two decades mainly due to the refined understanding of the entangled ground state structure of systems with local Hamiltonians
We consider the model at nonzero chemical potential to study the Silver Blaze phenomenon, which is related to the particle number density at zero temperature
We have carried out the first tensor-network study of the three-dimensional classical Oð2Þ model at both zero and nonzero magnetic field, chemical potential, and temperature

Summary

INTRODUCTION

These ideas have been extended to two spatial dimensions (i.e., 2 þ 1-dimensional quantum systems) using the generalization of MPS known as projected entangled pair states (PEPS), but the success has been limited In addition to these methods for the continuous-time approach, an alternate method based on the idea of the tensor renormalization group (TRG) in discretized Euclidean space has been very successful. The three-dimensional Oð2Þ model has been extensively studied using bootstrap methods and Monte Carlo (MC) methods, and critical exponents have been determined directly in the conformal field theory (CFT) limit of the model. This can be understood as follows: In the CFT limit, the scaling dimension Δs of a charge-zero scalar was determined to be 1.51136(22) using bootstrap methods [23] From this one can compute ν 1⁄4 1=ð3 − ΔsÞ 1⁄4 0.67175ð10Þ and the critical exponent α 1⁄4 2 − dν 1⁄4 −0.01526ð30Þ. In the Appendix we briefly discuss the convergence of the tensor results as a function of the bond dimension

TENSOR-NETWORK FORMULATION

RESULTS

SUMMARY AND DISCUSSION