Singular-drift Problem Research Articles

Complex Langevin analysis of the spontaneous breaking of 10D rotational symmetry in the Euclidean IKKT matrix model

The IKKT matrix model is a promising candidate for a nonperturbative formulation of superstring theory. In this model, spacetime is conjectured to emerge dynamically from the microscopic matrix degrees of freedom in the large-N limit. Indeed in the Lorentzian version, Monte Carlo studies suggested the emergence of (3+1)-dimensional expanding spacetime. Here we study the Euclidean version instead, and investigate an alternative scenario for dynamical compactification of extra dimensions via the spontaneous symmetry breaking (SSB) of 10D rotational symmetry. We perform numerical simulations based on the complex Langevin method (CLM) in order to avoid a severe sign problem. Furthermore, in order to avoid the singular-drift problem in the CLM, we deform the model and determine the SSB pattern as we vary the deformation parameter. From these results, we conclude that the original model has an SO(3) symmetric vacuum, which is consistent with previous results obtained by the Gaussian expansion method (GEM). We also apply the GEM to the deformed matrix model and find consistency with the results obtained by the CLM.

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

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 $\mu/T$ regime with moderate quark mass using four-flavor staggered fermions on a $4^3\times 8$ 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 $3.2\leq \mu/T\leq 5.6$ 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.

Comparative studies of the deformation techniques for the singular-drift problem in the complex Langevin method

In application of the complex Langevin method to QCD at high density and low temperature, the singular-drift problem occurs due to the appearance of near-zero eigenvalues of the Dirac operator. In order to avoid this problem, we proposed to de-form the Dirac operator in such a way that the near-zero eigenvalues do not appear and to extrapolate the deformation parameter to zero from the available data points. Here we test three different types of deformation in a simple large-N matrix model, which under-goes an SSB due to the phase of the fermion determinant, and compare them to see the consistency with one another.

Complex Langevin simulation of QCD at finite density and low temperature using the deformation technique

We study QCD at finite density and low temperature by using the complex Langevin method. We employ the gauge cooling to control the unitarity norm and intro-duce a deformation parameter in the Dirac operator to avoid the singular-drift problem. The reliability of the obtained results are judged by the probability distribution of the magnitude of the drift term. By making extrapolations with respect to the deformation parameter using only the reliable results, we obtain results for the original system. We perform simulations on a 43 × 8 lattice and show that our method works well even in the region where the reweighing method fails due to the severe sign problem. As a result we observe a delayed onset of the baryon number density as compared with the phase-quenched model, which is a clear sign of the Silver Blaze phenomenon.

The complex Langevin analysis of spontaneous symmetry breaking induced by complex fermion determinant

In many interesting physical systems, the determinant which appears from integrating out fermions becomes complex, and its phase plays a crucial role in the determination of the vacuum. An example of this is QCD at low temperature and high density, where various exotic fermion condensates are conjectured to form. Another example is the Euclidean version of the type IIB matrix model for 10d superstring theory, where spontaneous breaking of the SO(10) rotational symmetry down to SO(4) is expected to occur. When one applies the complex Langevin method to these systems, one encounters the singular-drift problem associated with the appearance of nearly zero eigenvalues of the Dirac operator. Here we propose to avoid this problem by deforming the action with a fermion bilinear term. The results for the original system are obtained by extrapolations with respect to the deformation parameter. We demonstrate the power of this approach by applying it to a simple matrix model, in which spontaneous symmetry breaking from SO(4) to SO(2) is expected to occur due to the phase of the complex fermion determinant. Unlike previous work based on a reweighting-type method, we are able to determine the true vacuum by calculating the order parameters, which agree with the prediction by the Gaussian expansion method.

Gauge cooling for the singular-drift problem in the complex Langevin method — a test in Random Matrix Theory for finite density QCD

Recently, the complex Langevin method has been applied successfully to finite density QCD either in the deconfinement phase or in the heavy dense limit with the aid of a new technique called the gauge cooling. In the confinement phase with light quarks, however, convergence to wrong limits occurs due to the singularity in the drift term caused by small eigenvalues of the Dirac operator including the mass term. We propose that this singular-drift problem should also be overcome by the gauge cooling with different criteria for choosing the complexified gauge transformation. The idea is tested in chiral Random Matrix Theory for finite density QCD, where exact results are reproduced at zero temperature with light quarks. It is shown that the gauge cooling indeed changes drastically the eigenvalue distribution of the Dirac operator measured during the Langevin process. Despite its non-holomorphic nature, this eigenvalue distribution has a universal diverging behavior at the origin in the chiral limit due to a generalized Banks-Casher relation as we confirm explicitly.