Ion-Olimpiu Stamatescu

Complex Langevin method: When can it be trusted?

We analyze to what extent the complex Langevin method, which is in principle capable of solving the so-called sign problems, can be considered as reliable. We give a formal derivation of the correctness and then point out various mathematical loopholes. The detailed study of some simple examples leads to practical suggestions about the application of the method.

Complex Langevin: etiology and diagnostics of its main problem

The complex Langevin method is a leading candidate for solving the so-called sign problem occurring in various physical situations. Its most vexing problem is that sometimes it produces ‘convergence to the wrong limit’. In this paper we carefully revisit the formal justification of the method, identifying points at which it may fail and derive a necessary and sufficient criterion for correctness. This criterion is, however, not practical, since its application requires checking an infinite tower of identities. We propose instead a practical test involving only a check of the first few of those identities; this raises the question of the ‘sensitivity’ of the test. This sensitivity as well as the general insights into the possible reasons of failure (the etiology) are then tested in two toy models where the correct answer is known. At least in those models the test works perfectly.

Linearly rising quark potential and non-strange baryon spectrum

Abstract We investigate the consequences of a linearly rising effective quark-quark two-particle potential within the baryons. We restricted ourselves to a discussion of the nucleonic states. The Schrodinger equation is solved by an approximation technique which makes use of the well-known fact that the three-body problem can be solved for an oscillator potential. The relativistic corrections abstracted from gauge theories are also considered. Some features of the spectrum emerge in a natural way. The degeneracy of the excited oscillator levels is removed and the [70, 1−]1 is pushed up in energy into the region of the second excited band of states. The relativistic corrections are very important and, for usual values of the parameters, the motion of the quarks is relativistic. While the spin-spin splittings come out roughly correct, the spin-orbit splittings become much too large.

Gauge cooling in complex Langevin for lattice QCD with heavy quarks

Abstract We employ a new method, “gauge cooling”, to stabilize complex Langevin simulations of QCD with heavy quarks. The results are checked against results obtained with reweighting; we find agreement within the estimated errors, except for strong gauge coupling in the confinement region. The method allows us to go to previously unaccessible high densities.

Stochastic quantization at finite chemical potential

A nonperturbative lattice study of QCD at finite chemical potential is complicated due to the complex fermion determinant and the sign problem. Here we apply the method of stochastic quantization and complex Langevin dynamics to this problem. We present results for U(1) and SU(3) one link models and QCD at finite chemical potential using the hopping expansion. The phase of the determinant is studied in detail. Even in the region where the sign problem is severe, we find excellent agreement between the Langevin results and exact expressions, if available. We give a partial understanding of this in terms of classical flow diagrams and eigenvalues of the Fokker-Planck equation.

THE SU(3) DECONFINEMENT PHASE TRANSITION IN THE PRESENCE OF QUARKS

Abstract The fate of the deconfinement phase transition is studied as a function of the quark mass on a 8 3 × 2 lattice. The first order phase transition present in the pure SU(3) lattice gauge theory weakens rapidly as the quark mass is decreased and no such transition is observed below a critical mass value. This critical mass appears to be large, of the order of GeV.

Topology of the SU (2) vacuum: a lattice study using improved cooling

We study the topological structure of the SU(2) vacuum at zero temperature: topological susceptibility, size, shape and distance distributions of the instantons. We use a cooling algorithm based on an improved action with scale invariant instanton solutions. This algorithm needs no monitoring or calibration, has an inherent cut off for dislocations and leaves unchanged instantons at physical scales. The physical relevance of our results is checked by studying the scaling and finite volume dependence. We obtain a susceptibility of (200(15) MeV)4. The instanton size distribution is peaked around 0.43 fm, and the distance distribution indicates a homogeneous, random spatial structure.

Renormalization group flow of SU(3) lattice gauge theory : Numerical studies in a two coupling space

We investigate the renormalization group (RG) flow of SU(3) lattice gauge theory in a two coupling space with couplings β11 and β12 corresponding to 1×1 and 1×2 loops, respectively. Extensive numerical calculations of the RG flow are made in the fourth quadrant of this coupling space, i.e., β11>0 and β12<0 . Swendsens factor two blocking and the Schwinger–Dyson method are used to find an effective action for the blocked gauge field. The resulting renormalization group flow runs quickly towards an attractive stream which has an approximate line shape. This is a numerical evidence of a renormalized trajectory which locates close to the two coupling space. A model flow equation which incorporates a marginal coupling (asymptotic scaling term), an irrelevant coupling and a non-perturbative attraction towards the strong coupling limit reproduces qualitatively the observed features. We further examine the scaling properties of an action which is closer to the attractive stream than the currently used improved actions. It is found that this action shows excellent restoration of rotational symmetry even for coarse lattices with a∼0.3 fm.

QCD ON ANISOTROPIC LATTICES

Most lattice simulations of QCD are performed on isotropic lattices, i.e. lattices with identical lattice spacings in the space and time directions. This is certainly the most natural approach for many problems. However, in particular the analysis of temperature effects on isotropic euclidean lattices is limited in several respects. The temperature is related to the time-like extent of the lattice through 1/T= N~a, where N, denotes the number of lattice sites in this direction and a is the lattice spacing. Thus, the highest temperature one can reach on an isotropic lattice is limited by Tm~ x = 1/a. It has recently been observed that this is a severe limitation in the analysis of the SU(2) Higgs model where the symmetry-restoring phase transition seems to take place at temperatures which are of the order of a-1 [1,2]. Clearly, having only one site in the time direction will also lead to large finite-size effects. Moreover, the introduction of an anisotropy, i.e. different lattice spacings in space and time directions, may help in this case. Another problem of considerable interest is the calculation of spectral functions for QCD at finite temperature. This is relevant for the calculation of transport

Controlling complex Langevin dynamics at finite density

At nonzero chemical potential the numerical sign problem in lattice field theory limits the use of standard algorithms based on importance sampling. Complex Langevin dynamics provides a possible solution, but it has to be applied with care. In this review, we first summarise our current understanding of the approach, combining analytical and numerical insight. In the second part we study SL(C, ℂ) gauge cooling, which was introduced recently as a tool to control complex Langevin dynamics in nonabelian gauge theories. We present new results in Polyakov chain models and in QCD with heavy quarks and compare various adaptive cooling implementations.