Nonzero chemical potential in the overlap Dirac operator and comparison to random matrix theory

Quantum chaos and QCD at finite chemical potential

We present a subset method that solves the sign problem for QCD at nonzero quark chemical potential in 0+1 dimensions. The subsets of gauge configurations are constructed using the center symmetry of the SU(3) group. These subsets completely solve the sign problem for up to five flavors. For a larger number of flavors the sign problem slowly reappears, and we propose an extension of the subsets that also solves the sign problem for these cases. The subset method allows for numerical simulations of the model at nonzero chemical potential. We also present some preliminary results on subsets for QCD in two, three, and four dimensions.

Topology and Staggered Fermion Action Improvement

The diffusion of titanium along with Ni, Co and Mn in pure synthetic forsterite has been studied as a function of temperature, oxygen fugacity, crystallographic orientation and chemical potentials in the four-component system MgO-SiO2-TiO2-TiO1.5. In over 100 different experimental conditions, no Ti diffusion profiles with the classical shape of the error function were observed. Instead, the profiles vary between 'stepped' at high fO(2) (probably owing to diffusion on octahedral sites and trapping in tetrahedral sites), to hockey-stick shapes at low fO(2) owing to concentration-dependent diffusion. The profile shapes vary systematically between these end-members according to oxygen fugacity. These profile shapes are also reproduced in experiments using natural San Carlos olivine. The change in profile shape is interpreted to be a function of valence-state change from Ti4+ to Ti3+, shown by a sigmoidal log fO(2)-diffusivity relationship. To distinguish between Ti4+ to Ti3+ in forsterite, a diffusion profile was also investigated using 'hydroxylation spectroscopy' whereby a crystal from a diffusion experiment was hydroxylated by annealing in an H2O-rich fluid at moderate pressure and temperature. The point defects associated with the added structural OH were then determined by infrared spectroscopy, providing evidence for the presence of Ti4+ to Ti3+ at intermediate oxygen fugacity. The transition from Ti4+ to Ti3+ occurs at considerably higher fO(2) in these diffusion experiments than in equilibrium experiments in similar systems. Titanium diffusion, qualified using a mobility parameter (M), is fastest along the c-axis, at high activity of silica, low oxygen fugacity and high temperature. The rate of Ti diffusion is broadly similar to those of Mn, Ni and Co, and closer to published rates ofMg self-diffusion in olivine than Si self-diffusion. The observed Ti4+ to Ti3+ diffusion occurs on the M-sites. From statistical examination of the large Mn, Ni and Co diffusivity dataset, we determine that there is a 10(2/3) dependence of diffusion on aSiO(2); the cations diffuse more rapidly at high aSiO(2). In addition, diffusivity has a 10(-1/8) dependence on fO(2), suggested to be the result of increased concentration of loosely bound Ti3+-vacancy pairs, which enhance the mobility of other cations. This behaviour is likely to be present in natural systems where trivalent (e. g. Fe3+, Al3+, Cr3+) and divalent cations diffuse together. Consideration should be given to diffusive interference by highly charged, fast-moving cations when extracting time scales from frozen diffusion profiles.

International audience

We discuss the spectrum of the QCD Dirac operator both at zero and at nonzero baryon chemical potential. We show that, in the ergodic domain of QCD, the Dirac spectrum can be obtained from the replica limit of a Toda lattice equation. At zero chemical potential this method explains the factorization of known results into compact and noncompact integrals, and at nonzero chemical potential it allows us to derive the previously unknown microscopic spectral density.

We consider the phase diagram of twisted mass Wilson fermions of two-flavor\nQCD in the parameter space of the quark mass, the isospin chemical potential,\nthe twist angle and the lattice spacing. This work extends earlier studies in\nthe continuum and those at zero chemical potential. We evaluate the phase\ndiagram as well as the spectrum of the (pseudo-)Goldstone bosons using the\nchiral Lagrangian for twisted mass Wilson fermions at non-zero isospin chemical\npotential. The phases are obtained from a mean field analysis. At zero twist\nangle we find that already an infinitesimal isospin chemical potential destroys\nthe Aoki phase. The reason is that in this phase we have massless Goldstone\nbosons with a non-zero isospin charge. At finite twist angle only two different\nphases are present, one phase which is continuously connected to the Bose\ncondensed phase at non-zero chemical potential and another phase which is\ncontinuously connected to the normal phase. For either zero or maximal twist\nthe phase diagram is more complicated as the saddle point equations allow for\nmore solutions.\n

The thermodynamics of massless ideal gas of overlap quarks has been investigated both analytically and numerically for both zero and nonzero baryon chemical potential. Any ${\ensuremath{\mu}}^{2}$ divergence is shown analytically to be absent for a class of actions with nonzero chemical potential. All such actions are shown to violate chiral invariance. While the parameter $M$ can be shown to be irrelevant in the continuum limit, as expected, it is shown numerically that the continuum limit can be reached with relatively coarser lattices for $1.5\ensuremath{\le}M\ensuremath{\le}1.6$. Numerical limitations of the existing method of introduction of chemical potential are demonstrated. Finally we also show that the energy density for the massive overlap fermions has the correct continuum limit.

We consider bosonic random matrix partition functions at nonzero chemical potential and compare the chiral condensate, the baryon number density and the baryon number susceptibility to the result of the corresponding fermionic partition function. We find that as long as results are finite, the phase transition of the fermionic theory persists in the bosonic theory. However, in case that the bosonic partition function diverges and has to be regularized, the phase transition of the fermionic theory does not occur in the bosonic theory, and the bosonic theory is always in the broken phase.

A Random Matrix Theory Approach to Denoise Single-Cell Data

At nonzero quark chemical potential lattice QCD simulations with dynamical quarks are hampered by the sign problem caused by the complex fermion determinant. In this talk we discuss the relation between QCD and random matrix theory and present a novel solution to the sign problem in dynamical random matrix simulations. The sign problem is solved by gathering matrices in subsets, whose sums of complex determinants are real and positive. Markov chains of relevant subsets were generated in Monte Carlo simulations and various observables, like the chiral condensate and quark number, were accurately computed. The subset formula, which is at the basis of the positivity proof for the subset weights, also provides a simple solution to the Silver Blaze puzzle.

The general issue of large ${N}_{c}$ QCD at nonzero chemical potential is considered with a focus on understanding the difference between large ${N}_{c}$ QCD with an isospin chemical potential and large ${N}_{c}$ QCD with a baryon chemical potential. A simple diagrammatic analysis analogous to 't Hooft's analysis at $\ensuremath{\mu}=0$ implies that the free energy with a given baryon chemical potential is equal to the free energy with an isospin chemical potential of the same value plus $1/{N}_{c}$ corrections. Phenomenologically, these two systems behave quite differently. A scenario to explain this difference in light of the diagrammatic analysis is explored. This scenario is based on a phase transition associated with pion condensation when the isospin chemical potential exceeds ${m}_{\ensuremath{\pi}}/2$; associated with this transition there is breakdown of the $1/{N}_{c}$ expansion---in the pion condensed phase there is a distinct $1/{N}_{c}$ expansion including a larger set of diagrams. While this scenario is natural, there are a number of theoretical issues which at least superficially challenge it. Most of these can be accommodated. However, the behavior of quenched QCD which raises a number of apparently analogous issues cannot be easily understood completely in terms of an analogous scenario. Thus, the overall issue remains open.

Fading is the time-dependent variations in signal strength measured at a receiver, due to temporally evolving multipath scattering and interference. In our previous work we introduced a statistical fading model for the time-reversal invariant case by combining the predictions of random matrix theory with the random coupling model that includes system-speci c properties such as the radiation impedance of the ports and short-orbit e ects. In the high-loss limit this random matrix theory model reduced to the most common fading models in the wireless communication eld. In this paper we discuss the theoretical model in more detail and extend it to the case of broken time-reversal invariance.

Baryons and the phase diagram for a large number of colors and flavors

Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) and c[over ](omega,x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of random matrix theory for Gaussian orthogonal ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Psi(x) using the inverse participation ratio and the amplitude distribution P(Psi(x)) .