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

Abstract

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.

On the structure of the emergent 3D expanding space in the Lorentzian type IIB matrix model

“…Monte Carlo simulation of the interpolating model for θ = 0 is difficult since the complex weight e β(cos θ+i sin θ)S b causes the sign problem. As a promising approach to overcome this problem, we may use the complex Langevin method [38,39], which has attracted much attention recently [40][41][42][43][44][45][46]. It was successful also in investigating the SSB of rotational symmetry in the 6d Euclidean type IIB matrix model [7].…”

Complex Langevin analysis of the space-time structure in the Lorentzian type IIB matrix model

“…Unfortunately, this is not straightforward since the complex Langevin method may suffer from the singular-drift problem due to the near-zero eigenvalues of the Dirac operator. The deformation technique [12] used successfully in studying the Euclidean version [13] is worth trying, though. We consider that the dominance of classical solutions at late times [26,27] supported by our results is important because it enables us to understand possible late-time behaviors of this model by solving classical equations of motion.…”

“…If we treat the phase of the complex weight by reweighting, huge cancellation among configurations with different phases occurs, which makes the calculation impractical. Recently the complex Langevin method (CLM) [4,5] has been attracting much attention as a promising approach to this problem [6][7][8][9][10][11][12]. In particular, it has been applied successfully to the Euclidean version of the 6D type IIB matrix model [13], and the spontaneous breaking of the rotational SO(6) symmetry to SO(3) suggested by the Gaussian expansion method [14] has been confirmed.…”

Combining the complex Langevin method and the generalized Lefschetz-thimble method

“…While the gauge cooling technique [27] (See refs. [10,28] for its justification) has enlarged this range of applicability to the extent that finite density QCD either with heavy quarks [29][30][31] or in the deconfined phase [32,33] can now be investigated, it is not yet clear whether one can investigate it even in the confined phase with light quarks [34][35][36][37][38][39][40][41].…”