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Register a new userPerformance of Complex Langevin Simulation in 0+1 dimensional massive Thirring model at finite density
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Statistical sampling with the complex Langevin (CL) equation is applied to (0+1)-dimensional Thirring model, and its uniform-field variant, at finite fermion chemical potential $mu$. The CL simulation reproduces a crossover behavior which is similar to but actually deviating from the exact solution in the transition region, where we confirm that the CL simulation becomes susceptible to the drift singularities, i.e., zeros of the fermion determinant. In order to simulate the transition region with the CL method correctly, we examine two approaches, a reweighting method and a model deformation, in both of which a single thimble with an attractive fixed point practically covers the integration domain and the CL sampling avoids the determinant zeros. It turns out that these methods can reproduce the correct crossover behavior of the original model with using reference ensembles in the complexified space. However, they need evaluation of the reweighting factor, which scales with the system size exponentially. We discuss feasibility of applying these methods to the Thirring model and to more realistic theories.
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We demonstrate that the complex Langevin method (CLM) enables calculations in QCD at finite density in a parameter regime in which conventional methods, such as the density of states method and the Taylor expansion method, are not applicable due to the severe sign problem. Here we use the plaquette gauge action with $beta = 5.7$ and four-flavor staggered fermions with degenerate quark mass $m a = 0.01$ and nonzero quark chemical potential $mu$. We confirm that a sufficient condition for correct convergence is satisfied for $mu /T = 5.2 - 7.2$ on a $8^3 times 16$ lattice and $mu /T = 1.6 - 9.6$ on a $16^3 times 32$ lattice. In particular, the expectation value of the quark number is found to have a plateau with respect to $mu$ with the height of 24 for both lattices. This plateau can be understood from the Fermi distribution of quarks, and its height coincides with the degrees of freedom of a single quark with zero momentum, which is 3 (color) $times$ 4 (flavor) $times$ 2 (spin) $=24$. Our results may be viewed as the first step towards the formation of the Fermi sphere, which plays a crucial role in color superconductivity conjectured from effective theories.
We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multi thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multiple thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.
In this paper we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass is inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling cooling solves the convergence problems as was shown before in the literature.
We investigate the sign problem in 0+1 dimensional QCD at finite chemical potential by using the path optimization method. The SU(3) link variable is complexified to the SL(3,$mathbb{C}$) link variable, and the integral path is represented by a feedforward neural network. The integral path is then optimized to weaken the sign problem. The average phase factor is enhanced to be greater than 0.99 on the optimized path. Results with and without diagonalized gauge fixing are compared and proven to be consistent. This is the first step of applying the path optimization method to gauge theories.
We study a random matrix model for QCD at finite density via complex Langevin dynamics. This model has a phase transition to a phase with nonzero baryon density. We study the convergence of the algorithm as a function of the quark mass and the chemical potential and focus on two main observables: the baryon density and the chiral condensate. For simulations close to the chiral limit, the algorithm has wrong convergence properties when the quark mass is in the spectral domain of the Dirac operator. A possible solution of this problem is discussed.
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