We demonstrate the existence of an anomaly-induced inhomogeneous phase in a class of vector-like gauge theories without sign problem, thus disproving the long-standing conjecture that the absence of sign problem precludes spontaneous breaking of translational invariance. The presence of the phase in the two-color modification of quantum chromodynamics can be tested by an independent nonperturbative evaluation of the neutral pion decay constant as a function of external magnetic field. Our results provide a benchmark for future lattice studies of inhomogeneous phases in dense quark matter.
Recently, it has been shown that the ground state of quantum chromodynamics (QCD) in sufficiently strong magnetic fields and at moderate baryon number chemical potential carries a crystalline condensate of neutral pions: the chiral soliton lattice (CSL) [1]. While the result was obtained in a model-independent manner using effective field theory techniques, its realization from first principles using lattice Monte Carlo simulation is hampered by the infamous sign problem. Here we show that CSL, or a similar inhomogeneous phase, also appears in the phase diagram of a class of vector-like gauge theories that do not suffer from the sign problem even in the presence of a baryon chemical potential and external magnetic field. We also show that the onset of nonuniform order manifests itself already in the adjacent homogeneous Bose-Einstein-condensation phase through a characteristic roton-like minimum in the dispersion relation of the lowest-lying quasiparticle mode. Last but not least, our work gives a class of explicit counterexamples to the long-standing conjecture that positivity of the determinant of the Dirac operator (that is, absence of the sign problem) in a vector-like gauge theory precludes spontaneous breaking of translational invariance, and thus implies the absence of inhomogeneous phases in the phase diagram of the theory.
We discuss the possible phases dual to the AdS hairy black holes in Horndeski theory. In the probe limit breaking the translational invariance, we study the conductivity and we find a non-trivial structure indicating a collective excitation of the charge carriers as a competing effect of momentum dissipation and the coupling of the scalar field to Einstein tensor. Going beyond the probe limit, we investigate the spontaneous breaking of translational invariance near the critical temperature and discuss the stability of the theory. We consider the backreaction of the charged scalar field to the metric and we construct numerically the hairy black hole solution. To determine the dual phases of a hairy black hole, we compute the conductivity. When the wave number of the scalar field is zero, the DC conductivity is divergent due to the conservation of translational invariance. For nonzero wave parameter with finite DC conductivity, we find two phases in the dual theory. For low temperatures and for positive couplings, as the temperature is lower, the DC conductivity increases therefore the dual theory is in metal phase, while if the coupling is negative we have the opposite behavior and it is dual to an insulating phase. We argue that this behavior of the coupling of the scalar field to Einstein tensor can be attributed to its role as an impurity parameter in the dual theory.
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