Tensor renormalization group study of the 3d $O(2)$ model
Jacques Bloch, Raghav G. Jha, Robert Lohmayer, Maximilian Meister
备注：
v1: 9 pages with 8 figures, v2: Minor corrections with revised Fig. 7a closer to the critical value, version matches the one accepted for publication in Phys. Rev. D
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.
The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilibrium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and backward tensor modes are projected on the same lower dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor tr…
We investigate the critical endpoints of the (3+1)-dimensional $Z_2$ gauge-Higgs model at finite density together with the (2+1)-dimensional one at zero density as a benchmark using the tensor renormalization group method. We focus on the phase transition between the Higgs phase and the confinement phase at finite chemical potential along the critical end line. In the (2+1)-dimensional model, the resulting endpoint is consistent with a recent numerical estimate by the Monte Carlo simulation. In the (3+1)-dimensional case, however, the location of the critical endpoint shows disagreement with t…
We present a tensor-network approach for two-dimensional strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a Grassmann higher-order tensor renormalization group method, specifically tailored for this model. During the coarsening procedure, the blocking of adjacent Grassmann tensors is performed analyti…
We study the $q$-state Potts models on a cubic lattice in the thermodynamic limit using tensor renormalization group transformations with the triad approximation. By computing the thermodynamic potentials, we locate the first-order phase transitions for $10 < q \le 20$ which has not been explored using any method. We also examine the efficiency of the triad approximation method in obtaining the fixed-point tensor and comment on how this can be improved.
We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)$d$) complex $ϕ^4$ theory at finite density, (1+1)$d$ pure U(1) lattice gauge theory with a $θ$ term, (3+1)$d$ Nambu--Jona-Lasinio model at finite density and (1+1)$d$ and (2+1)$d$ Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign …
Renormalization group flows of Hamiltonians using tensor networks
利用张量网络构造了二维经典配分函数的哈密顿量的重整化群流。类似于张量网络重整化（[G. Evenbly和G. Vidal，Phys. Rev. Lett. 115，180405（2015）]，[S.杨志- C. Gu和X。- G Wen，Phys. Rev. Lett. 118，110504（2017）]），我们获得了临界状态下的近似不动点张量网络。然而，我们的形式主义保持积极的张量在每一步，因此产生的解释，在哈密尔顿流。我们强调，张量网络方法和Kadanoff的自旋阻塞方法之间的关键区别可以理解为在每一个抽取步骤中局部基的变化，这是克服互信息面积定律的关键属性。我们推导出代数关系的不动点张量，计算临界指数，并基准我们的方法上的伊辛模型和六顶点模型。
We propose a method to represent the path integral over gauge fields as a tensor network. We introduce a trial action with variational parameters and generate gauge field configurations with the weight defined by the trial action. We construct initial tensors with indices labelling these gauge field configurations. We perform the tensor renormalization group with the initial tensors and optimize the variational parameters. As a first step to the TRG study of non-Abelian gauge theory in more than two dimensions, we apply this method to three-dimensional pure SU(2) gauge theory. Our result for t…
Recently, the tensor network description with bond weights on its edges has been proposed as a novel improvement for the tensor renormalization group algorithm. The bond weight is controlled by a single hyperparameter, whose optimal value is estimated in the original work via the numerical computation of the two-dimensional critical Ising model. We develop this bond-weighted tensor renormalization group algorithm to make it applicable to the fermionic system, benchmarking with the two-dimensional massless Wilson fermion. We show that the accuracy with the fixed bond dimension is improved also …
We investigate the metal-insulator transition of the (1+1)-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. The critical chemical potential $μ_{\rm c}$ and the critical exponent $ν$ are determined from the $μ$ dependence of the electron density in the thermodynamic limit. Our results for $μ_{\rm c}$ and $ν$ show consistency with an exact solution based on the Bethe ansatz. Our encouraging results indicate the applicability of the tensor renormalization group method to the analysis of higher-dimensional Hubbard models.
We derive a general formula of the tensor network representation for $d$-dimensional lattice fermions with ultra-local interactions, including Wilson fermions, staggered fermions, and domain-wall fermions. The Grassmann tensor is concretely defined with auxiliary Grassmann variables that play a role in bond degrees of freedom. Compared to previous works, our formula does not refer to the details of lattice fermions and is derived by using the singular value decomposition for the given Dirac matrix without any ad-hoc treatment for each fermion. We numerically test our formula for free Wilson an…
We present preliminary results of the non-perturbative renormalization group (RG) running of the flavor non-singlet tensor operator. We employ the $χ$SF scheme for $N_f=3$ QCD using ensembles generated by the ALPHA collaboration for the computation of the quark mass running. The $χ$SF property of automatic $O(a)$ improvement prevents the $O(a)$ mixing of the correlation functions.
We propose a second renormalization group (SRG) in the triad representation of tensor networks. The SRG method improves two parts of the triad tensor renormalization group, which are the decomposition of intermediate tensors and the preparation of isometries, taking the influence of environment tensors into account. Every fundamental tensor including environment tensor is given as a rank-3 tensor, and the computational cost of the proposed algorithm scales with ${\cal O}(χ^5)$ employing the randomized SVD where $χ$ is the bond dimension of tensors. We test this method in the classical Ising mo…
To assist the matching of lattice field theory results to the high energy continuum limit we evaluate the Green's function where the tensor quark bilinear operator is inserted at zero momentum in a quark $2$-point function for an arbitrary covariant gauge. This is carried out in both the MSbar and RI' schemes to four loops. The tensor current anomalous dimension is also calculated to four loops in both schemes for an arbitrary colour group.
In this paper, we propose a parallel computing method for the Higher Order Tensor Renormalization Group (HOTRG) applied to a $d$-dimensional $( d \geq 2 )$ simple lattice model. Sequential computation of the HOTRG requires $O ( χ^{4 d - 1} )$ computational cost, where $χ$ is bond dimension, in a step to contract indices of tensors. When we simply distribute elements of a local tensor to each process in parallel computing of the HOTRG, frequent communication between processes occurs. The simplest way to avoid such communication is to hold all the tensor elements in each process, however, it req…
