Random matrix theory of unquenched two-colour QCD with nonzero chemical potential

Abstract

We solve a random two-matrix model with two real asymmetric matrices whose primary purpose is to describe certain aspects of quantum chromodynamics with two colours and dynamical fermions at nonzero quark chemical potential µ. In this symmetry class the determinant of the Dirac operator is real but not necessarily positive. Despite this sign problem the unquenched matrix model remains completely solvable and provides detailed predictions for the Dirac operator spectrum in two different physical scenarios/limits: (i) the ε-regime of chiral perturbation theory at small µ, where µ 2 multiplied by the volume remains fixed in the infinite-volume limit and (ii) the high-density regime where a BCS gap is formed and µ is unscaled. We give explicit examples for the complex, real, and imaginary eigenvalue densities including N f = 2 non-degenerate flavours. Whilst the limit of two degenerate masses has no sign problem and can be tested with standard lattice techniques, we analyse the severity of the sign problem for non-degenerate masses as a function of the mass split and of µ.On the mathematical side our new results include an analytical formula for the spectral density of real Wishart eigenvalues in the limit (i) of weak non-Hermiticity, thus completing the previous solution of the corresponding quenched model of two real asymmetric Wishart matrices.

Dirac spectrum in complex Langevin simulations of QCD

“…It is essential in the microscopic domain of QCD [20,21,32], for Cooper pairing at large chemical potential [33], in one-dimensional QED at any chemical potential [4,34], in two-color QCD with unmatched quark masses [35], and for odd-flavored QCD at zero chemical potential and a negative quark mass [36], All cases are characterized by a fermonic sign problem and the results obtained here also apply in these cases: if complex Langevin is applied the resulting Dirac spectrum must be drastically different from the one obtained with real fields.…”

Exact results for two-color QCD at low and high density

“…In the meantime, the generalization to the unquenched case has been worked out [14]. Since the analytical results are rather cumbersome we will not present them here.…”

“…It is evident from the plots that the sign problem (i) increases with increasingμ, (ii) decreases with increasingm, and (iii) decreases with increasing ν (in agreement with [15]). A quantitative analysis [14] reveals that in the thermodynamic limit the average sign makes a first-order transition from 1 to 0 atμ = m/2, which in physical units corresponds to a critical chemical potential µ phys = m π /2.…”

Banks-Casher-type relations for complex Dirac spectra

“…In this case (3.2) with d rep = 2 was already proposed in [6]. The corresponding RMT was constructed in [13], and the analytical RMT result for ρ s (ξ ) was computed in [14]. From this result one finds…”