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We present results of tensor-network simulations of the three-dimensional$O(2)$ model at non-zero chemical potential and temperature, which werecomputed using the higher-order tensor-renormalization-group method (HOTRG).This necessitated enhancements to the HOTRG blocking procedure to reduce thetruncation error in the case of anisotropic tensors. Moreover, the constructionof the truncated vector spaces was adapted to strongly reduce the effect ofsystematic errors in the computation of observables using the finite-differencemethod. Our (improved) HOTRG results for the evolution of the number densitywith the chemical potential are in agreement with results obtained with theworm algorithm, and both the Silver Blaze phenomenon at zero temperature andthe temperature dependence of the number density can be adequately reproduced.

Tensor-network study of the 3d $O(2)$ model at non-zero chemical potential and temperature