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arXiv:1912.13092 · Journal of High Energy Physics

Investigation of Complex ϕ4 Theory at Finite Density in Two Dimensions Using TRG

Daisuke Kadoh, Yoshinobu Kuramashi, Y. Nakamura, Ryo Sakai, Shinji Takeda, Yusuke Yoshimura

Published Feb 1, 2020• 31 citations
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Abstract

Abstract We study the two-dimensional complex ϕ4 theory at finite chemical potential using the tensor renormalization group. This model exhibits the Silver Blaze phenomenon in which bulk observables are independent of the chemical potential below the critical point. Since it is expected to be a direct outcome of an imaginary part of the action, an approach free from the sign problem is needed. We study this model systematically changing the chemical potential in order to check the applicability of the tensor renormalization group to the model in which scalar fields are discretized by the Gaussian quadrature. The Silver Blaze phenomenon is successfully confirmed on the extremely large volume V = 10242 and the results are also ensured by another tensor network representation with a character expansion.

DOI
10.1007/jhep02(2020)161

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We apply the tensor renormalization group method to the (1+11+11+1)-dimensional SU(2) principal chiral model at finite chemical potential with the use of the Gauss-Legendre quadrature to discretize the SU(2) Lie group. The internal energy at vanishing chemical potential $\ensuremath{\mu}=0$ shows good consistency with the prediction of the strong and weak coupling expansions. This indicates an effectiveness of the Gauss-Legendre quadrature for the partitioning of the SU(2) Lie group. In the finite density region with $\ensuremath{\mu}\ensuremath{\ne}0$ at the strong coupling we observe the Silver-Blaze phenomenon for the number density.
We present results of tensor-network simulations of the three-dimensional O(2)O(2)O(2) model at non-zero chemical potential and temperature, which were computed using the higher-order tensor-renormalization-group method (HOTRG). This necessitated enhancements to the HOTRG blocking procedure to reduce the truncation error in the case of anisotropic tensors. Moreover, the construction of the truncated vector spaces was adapted to strongly reduce the effect of systematic errors in the computation of observables using the finite-difference method. Our (improved) HOTRG results for the evolution of the number density with the chemical potential are in agreement with results obtained with the worm algorithm, and both the Silver Blaze phenomenon at zero temperature and the temperature dependence of the number density can be adequately reproduced.
Tensor-network study of the 3d O(2)O(2)O(2) model at non-zero chemical potential and temperature
We present results of tensor-network simulations of the three-dimensional O(2)O(2)O(2) model at non-zero chemical potential and temperature, which were computed using the higher-order tensor-renormalization-group method (HOTRG). This necessitated enhancements to the HOTRG blocking procedure to reduce the truncation error in the case of anisotropic tensors. Moreover, the construction of the truncated vector spaces was adapted to strongly reduce the effect of systematic errors in the computation of observables using the finite-difference method. Our (improved) HOTRG results for the evolution of the number density with the chemical potential are in agreement with results obtained with the worm algorithm, and both the Silver Blaze phenomenon at zero temperature and the temperature dependence of the number density can be adequately reproduced.
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We calculate thermodynamic potentials and their derivatives for the three-dimensional O(2)O(2)O(2) model using tensor-network methods to investigate the well-known second-order phase transition. We also consider the model at nonzero 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 nonzero magnetic field, temperature, and chemical potential are consistent with those obtained using other methods.