Complex Langevin calculations in finite density QCD at large μ/T with the deformation technique

Abstract

It is well known that investigating QCD at finite density by standard Monte Carlo methods is extremely difficult due to the sign problem. Some years ago, the complex Langevin method with gauge cooling was shown to work at high temperature, i.e., in the deconfined phase. The same method was also applied to QCD in the so-called heavy dense limit in the whole temperature region. In this paper we attempt to apply this method to the large µ/T regime with moderate quark mass using four-flavor staggered fermions on a 4 3 × 8 lattice. While a straightforward application faces with the singular-drift problem, which spoils the validity of the method, we overcome this problem by the deformation technique proposed earlier. Explicit results for the quark number density and the chiral condensate obtained in this way for 3.2 ≤ µ/T ≤ 5.6 are compared with the results for the phase-quenched model obtained by the standard rational hybrid Monte Carlo calculation. This reveals a clear difference, which is qualitatively consistent with the Silver Blaze phenomenon.

Complex Langevin calculations in QCD at finite density

“…Let us also mention that the previous work [11] shows that the CLM works on a 4 3 × 8 lattice with the same β but only with the aid of the deformation technique [59], which is actually not needed for the lattice size in the present work. Thus we find that the situation becomes better for a larger lattice, which is also seen by comparing our results for 8 3 × 16 and 16 3 × 32 lattices in section 3.1.…”

“…[3] for a summary of the recent progress concerning this method and other methods for solving the sign problem. In particular, the CLM has been tested extensively in lattice QCD at finite density [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

“…See appendix A of ref. [11] for the details. The expectation value of an observable O(U ) is calculated as…”

Applying complex Langevin simulations to lattice QCD at finite density

“…Other methods have been suggested to try and obtain correct results from CLE simulations. One is to add terms to the action which cause the CLE to avoid the poles, with coefficients, which when taken to zero, yield the original action [43]. The question then is can these coefficients be taken small enough to allow them to be continued to zero, without them losing their effectiveness in avoiding the poles.…”

Dynamical Compactification of Extra Dimensions in the Euclidean IKKT Matrix Model via Spontaneous Symmetry Breaking

“…This method fails in many cases, and it was not until recently that with the help of new techniques [38,39,40,41,42] it was possible to meet the conditions for the equivalence of the stochastic process defined by the CLM and the original system, and obtain correct results for an extended range of parameters [43,44,45,21,46,47,48]. The CLM has been applied successfully to many systems in lattice quantum field theory [49,50,51,52,53,54,55,56,57,58,59,43,60,61,62,63,64,65,48,66,67,68,69,70], and matrix models [71,72,21,73,26,11,48,74,75,36,27].…”