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

Abstract

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 ∼ O(103) on a large volume of V = 10244. We also present the results for the equation of state.

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

“…Figure 6 tells us that the first-order transitions and the critical endpoint are actually on the self-dual line as expected. Moreover, the location of the critical endpoint (β c , η c ) = (0.70051 (7), 0.12575(3)) is consistent with the previous result β c ≈ 0.701 [34]. These do assure the validity of the current TRG-based determination of the critical endpoint, which is characterized as a point with vanishing ∆ L .…”

“…At the initial stage of the study of particle physics with the TRG method, we have focused on developing an efficient method to treat the scalar, gauge, and fermion fields and verifying the following advantages of the TRG method employing the lower-dimensional models: (i) no sign problem [3,[9][10][11][12][13][14], (ii) logarithmic computational cost on the system size, (iii) direct manipulation of the Grassmann variables [3,4,16], (iv) evaluation of the partition function or the path-integral itself. Recently, the authors and their collaborators have successfully applied the TRG method to analyze the phase transitions of the (3+1)d complex φ 4 theory at finite density [15], the (3+1)d real φ 4 theory [17], and the (3+1)d Nambu−Jona-Lasinio (NJL) model at high density and very low temperature [7]. From these previous studies, it is shown that the TRG method efficiently works to investigate the (3+1)d scalar field theories with some field regularization technique and the method allows us to directly evaluate the path integral of the (3+1)d lattice fermions.…”

Gauge invariant input to neural network for path optimization method

“…We evaluate it by the exponential moving average (EMA) in the previous optimization so that θ 0 = θ pre EMA . Using this cost function (14), the neural network finds the best path that enhances e iθ(t) as much as possible.…”

“…It is also a non-Monte Carlo method. Although the computational cost is extremely high, it has been vigorously tested even in 4dimensional theoretical models [12][13][14][15]. Recent improved algorithms considerably reduce the cost [16,17].…”

Tensor renormalization group and the volume independence in 2D U(N) and SU(N) gauge theories

“…The method was originally applied to the 2D Ising model [1], and it was extended to other spin models [2][3][4][5] and theories with continuous variables such as scalar field theories [6,7]. As yet another notable feature of the TRG, it allows direct implementation of fermionic degrees of freedom as Grassmann variables [8][9][10][11][12][13][14][15][16][17][18] unlike in Monte Carlo methods, which inevitably require some sort of "bosonization" leading to huge increase in the computational JHEP12(2021)011 cost. While extension to higher dimensional theories is not as straightforward as in Monte Carlo methods, there are various proposals for efficient schemes to construct a coarse-grained tensor network [19,20], which have been successfully applied to simple four-dimensional theories [21][22][23].…”