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Register a new user$3d$ fermion-boson map with imaginary chemical potential
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We study the three-dimensional $U(N)$ Gross-Neveu and CP$^{N-1}$ models in the canonical formalism with fixed $U(1)$ charge. For large-$N$ this is closely related to coupling the models to abelian Chern-Simons in a monopole background. We show that the presence of the imaginary chemical potential for the $U(1)$ charge makes the phase structure of the models remarkably similar. We calculate their respective large-$N$ free energy densities and show that they are mapped into each other in a precise way. Intriguingly, the free energy map involves the Bloch-Wigner function and its generalizations introduced by Zagier. We expect that our results are connected to the recently discussed $3d$ bosonization.
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We study the phase structure of imaginary chemical potential. We calculate the Polyakov loop using clover-improved Wilson action and renormalization improved gauge action. We obtain a two-state signals indicating the first order phase transition for $beta = 1.9, mu_I = 0.2618, kappa=0.1388$ on $8^3times 4$ lattice volume We also present a result of the matrix reduction formula for the Wilson fermion.
We show that the three-dimensional map between fermions and bosons at finite temperature generalises for all odd dimensions $d>3$. We further argue that such a map has a nontrivial large $d$ limit. Evidence comes from studying the gap equations, the free energies and the partition functions of the $U(N)$ Gross-Neveu and CP$^{N-1}$ models for odd $dgeq 3$ in the presence of imaginary chemical potential. We find that the gap equations and the free energies can be written in terms of the Bloch-Wigner-Ramakrishnan $D_d(z)$ functions analysed by Zagier. Since $D_2(z)$ gives the volume of ideal tetrahedra in 3$d$ hyperbolic space our three-dimensional results are related to resent studies of complex Chern-Simons theories, while for $d>3$ they yield corresponding higher dimensional generalizations. As a spinoff, we observe that particular complex saddles of the partition functions correspond to the zeros and the extrema of the Clausen functions $Cl_d(theta)$ with odd and even index $d$ respectively. These saddles lie on the unit circle at positions remarkably well approximated by a sequence of rational multiples of $pi$.
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.
We extend a bottom up holographic model, which has been used in studying the color superconductivity in QCD, to the imaginary chemical potential ($mu_I$) region, and the phase diagram is studied on the $mu_I$-temperature (T) plane. The analysis is performed for the case of the probe approximation and for the background where the back reaction from the flavor fermions are taken into account. For both cases, we could find the expected Roberge-Weiss (RW) transitions. In the case of the back-reacted solution, a bound of the color number $N_c$ is found to produce the RW periodicity. It is given as $N_cgeq 1.2$. Furthermore, we could assure the validity of this extended model by comparing our result with the one of the lattice QCD near $mu_I=0$.
We investigate chemical-potential ($mu$) dependence of the static-quark free energies in both the real and imaginary $mu$ regions, using the clover-improved two-flavor Wilson fermion action and the renormalization-group improved Iwasaki gauge action. Static-quark potentials are evaluated from Polyakov-loop correlators in the deconfinement phase and the imaginary $mu=imu_{rm I}$ region and extrapolated to the real $mu$ region with analytic continuation. As the analytic continuation, the potential calculated at imaginary $mu=imu_{rm I}$ is expanded into a Taylor-expansion series of $imu_{rm I}/T$ up to 4th order and the pure imaginary variable $imu_{rm I}/T$ is replaced by the real one $mu_{rm R}/T$. At real $mu$, the 4th-order term weakens $mu$ dependence of the potential sizably. Also, the color-Debye screening mass is extracted from the color-singlet potential at imaginary $mu$, and the mass is extrapolated to real $mu$ by analytic continuation. The screening mass thus obtained has stronger $mu$ dependence than the prediction of the leading-order thermal perturbation theory at both real and imaginary $mu$.
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