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We consider chiral symmetry breaking at nonzero chemical potential and discuss the relation with the spectrum of the Dirac operator. We solve the so called Silver Blaze Problem that the chiral condensate at zero temperature does not depend on the chemical potential while this is not the case for the Dirac spectrum and the weight of the partition function.
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The relation between the spectral density of the QCD Dirac operator at nonzero baryon chemical potential and the chiral condensate is investigated. We use the analytical result for the eigenvalue density in the microscopic regime which shows oscillations with a period that scales as 1/V and an amplitude that diverges exponentially with the volume $V=L^4$. We find that the discontinuity of the chiral condensate is due to the whole oscillating region rather than to an accumulation of eigenvalues at the origin. These results also extend beyond the microscopic regime to chemical potentials $mu sim 1/L$.
Lattice QCD at finite chemical potential is difficult due to the sign problem. We use stochastic quantization and complex Langevin dynamics to study this issue. First results for QCD in the hopping expansion are encouraging. U(1) and SU(3) one link models are used to gain further insight into why the method appears to be successful.
Inspired by recent discussions of inverse magnetic catalysis in the literature, we examine the effects of a uniform external magnetic field on the chiral phase transition in quenched ladder QED at nonzero chemical potential. In particular, we study the behaviour of the effective potential as the strength of the magnetic field is varied while the chemical potential is held constant. For a certain range of the magnetic field, the effective potential develops a local maximum. Inverse magnetic catalysis is observed at this maximum, whereas the usual magnetic catalysis is observed at the true minimum of the effective potential.
The chiral condensate in QCD at zero temperature does not depend on the quark chemical potential (up to one third the nucleon mass), whereas the spectral density of the Dirac operator shows a strong dependence on the chemical potential. The cancellations which make this possible also occur on the microscopic scale, where they can be investigated by means of a random matrix model. We show that they can be understood in terms of orthogonality properties of orthogonal polynomials. In the strong non-Hermiticity limit they are related to integrability properties of the spectral density. As a by-product we find exact analytical expressions for the partially quenched chiral condensate in the microscopic domain at nonzero chemical potential.
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 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.
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