arXiv subjects

Shinichiro Akiyama

Publications and source records attributed to Shinichiro Akiyama.

31 records · Page 2Linked to original sources

arXiv2023

Critical endpoint of (3+1)-dimensional finite density $Z_3$ gauge-Higgs model with tensor renormalization group

The critical endpoint of the (3+1)-dimensional $Z_3$ gauge-Higgs model at finite density is determined by the tensor renormalization group method. This work is an extension of the previous one on the $Z_2$ model. The vital difference between them is that the $Z_3$ model suffers from the sign problem, while the $Z_2$ model does not. We show that the tensor renormalization group method allows us to locate the critical endpoint for the $Z_3$ gauge-Higgs model at finite density, regardless of the sign problem.

hep-lat↗

arXiv2023

Matrix product decomposition for two- and three-flavor Wilson fermions: Benchmark results in the lattice Gross-Neveu model at finite density

We formulate the path integral of two- and three-flavor Wilson fermion in two dimensions as a multilayer Grassmann tensor network by the matrix product decomposition. Thanks to this new description, the memory cost scaling is reduced from $\mathrm{O}(\mathrm{e}^{N_{f}})$ for the conventional construction to $\mathrm{O}(N_{f})$. Based on this representation, we develop a coarse-graining algorithm where spatially or temporally adjacent Grassmann tensors are converted into a canonical form along a virtual direction before we carry out the spacetime coarse-graining. Benchmarking with the lattice Gross-Neveu model at finite density, we see that the Silver Blaze phenomenon in the pressure and number density is captured with relatively small bond dimensions.

hep-lat↗

arXiv2022

Bond-weighting method for the Grassmann tensor renormalization group

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 in the fermionic system and provide numerical evidence that the optimal choice of the hyperparameter is not affected by whether the system is bosonic or fermionic. In addition, by monitoring the singular value spectrum, we find that the scale-invariant structure of the renormalized Grassmann tensor is successfully kept by the bond-weighting technique.

hep-lat↗

arXiv2022

Tensor renormalization group study of (3+1)-dimensional $Z_2$ gauge-Higgs model at finite density

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 the known estimates by the mean-field approximation and the Monte Carlo studies. This is the first application of the tensor renormalization group method to a four-dimensional lattice gauge theory and a key stepping stone toward the future investigation of the phase structure of the finite density QCD.

hep-lat↗

arXiv2022

Metal-insulator transition in (2+1)-dimensional Hubbard model with tensor renormalization group

We investigate the doping-driven metal-insulator transition of the (2+1)-dimensional Hubbard model in the path-integral formalism with the tensor renormalization group method. We calculate the electron density $\langle n\rangle$ as a function of the chemical potential $μ$ choosing three values of the Coulomb potential with $U=80$, $8$, and $2$ as representative cases of the strong, intermediate, and weak couplings. We have determined the critical chemical potential at each $U$, where the Hubbard model undergoes the metal-insulator transition from the half-filling plateau with $\langle n\rangle=1$ to the metallic state with $\langle n\rangle > 1$. Our results indicate that the model exhibits the metal-insulator transition over the vast region of the finite coupling $U$.

cond-mat.str-el↗

arXiv2021

Quantum Field Theories with Tensor Renormalization Group

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 problem in practical calculations and applicable to the four-dimensional models.

hep-lat↗

arXiv2021

More about the Grassmann tensor renormalization group

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 and staggered fermions and find that it properly works for them. We also find that Wilson fermions show better performance than staggered fermions in the tensor renormalization group approach, unlike the Monte Carlo method.

hep-lat↗

arXiv2021

Phase transition of four-dimensional lattice $ϕ^4$ theory with tensor renormalization group

We investigate the phase transition of the four-dimensional single-component $ϕ^4$ theory on the lattice using the tensor renormalization group method. We have examined the hopping parameter dependence of the bond energy and the vacuum condensation of the scalar field $\langleϕ\rangle$ at a finite quartic coupling $λ$ on large volumes up to $V=1024^4$ in order to detect the spontaneous breaking of the $\mathbb{Z}_2$ symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional $ϕ^4$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.

hep-lat↗

arXiv2021

Tensor renormalization group approach to (1+1)-dimensional Hubbard model

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.

hep-lat↗

arXiv2020

Restoration of chiral symmetry in cold and dense Nambu--Jona-Lasinio model with tensor renormalization group

We analyze the chiral phase transition of the Nambu--Jona-Lasinio model in the cold and dense region on the lattice developing the Grassmann version of the anisotropic tensor renormalization group algorithm. The model is formulated with the Kogut--Susskind fermion action. We use the chiral condensate as an order parameter to investigate the restoration of the chiral symmetry. The first-order chiral phase transition is clearly observed in the dense region at vanishing temperature with $μ/T\sim O(10^3)$ on a large volume of $V=1024^4$. We also present the results for the equation of state.

hep-lat↗

arXiv2020

Tensor renormalization group approach to four-dimensional complex $ϕ^4$ theory at finite density

Tensor network is an attractive approach to field theory with negative sign problem. The complex $ϕ^4$ theory at finite density is a test bed for numerical algorithms to verify their effectiveness. The model shows a characteristic feature called the Silver Blaze phenomenon associated with the sign problem in the large volume limit at low temperature. We analyze the four-dimensional model employing the anisotropic tensor renormalization group algorithm. We find a clear signal of the Silver Blaze phenomenon on a large volume of $V=1024^4$, which implies that the tensor network approach is effective even for four-dimensional field theory beyond two dimensions.

hep-lat↗

arXiv2019

Phase transition of four-dimensional Ising model with tensor network scheme

We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to $1024^4$, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.

hep-lat↗

arXiv2019

Phase transition of four-dimensional Ising model with higher-order tensor renormalization group

We apply the higher-order tensor renormalization group (HOTRG) to the four-dimensional ferromagnetic Ising model, which has been attracting interests in the context of the triviality of the scalar $ϕ^4_{d=4}$ theory. We investigate the phase transition of this model with HOTRG enlarging the lattice size up to $1024^4$ with parallel computation. The results for the internal energy and the magnetization are consistent with the weak first-order phase transition.

hep-lat↗