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Register a new userTowards extremely dense matter on the lattice
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QCD is expected to have a rich phase structure. It is empirically known to be difficult to access low temperature and nonzero chemical potential $mu$ regions in lattice QCD simulations. We address this issue in a lattice QCD with the use of a dimensional reduction formula of the fermion determinant. We investigate spectral properties of a reduced matrix of the reduction formula. Lattice simulations with different lattice sizes show that the eigenvalues of the reduced matrix follow a scaling law for the temporal size $N_t$. The properties of the fermion determinant are examined using the reduction formula. We find that as a consequence of the $N_t$ scaling law, the fermion determinant becomes insensitive to $mu$ as $T$ decreases, and $mu$-independent at T=0 for $mu<m_pi/2$. The $N_t$ scaling law provides two types of the low temperature limit of the fermion determinant: (i) for low density and (ii) for high-density. The fermion determinant becomes real and the theory is free from the sign problem in both cases. In case of (ii), QCD approaches to a theory, where quarks interact only in spatial directions, and gluons interact via the ordinary Yang-Mills action. The partition function becomes exactly $Z_3$ invariant even in the presence of dynamical quarks because of the absence of the temporal interaction of quarks. The reduction formula is also applied to the canonical formalism and Lee-Yang zero theorem. We find characteristic temperature dependences of the canonical distribution and of Lee-Yang zero trajectory. Using an assumption on the canonical partition function, we discuss physical meaning of those temperature dependences and show that the change of the canonical distribution and Lee-Yang zero trajectory are related to the existence/absence of $mu$-induced phase transitions.
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A three-dimensional effective lattice theory of Polyakov loops is derived from QCD by expansions in the fundamental character of the gauge action, u, and the hopping parameter, kappa, whose action is correct to kappa^n u^m with n+m=4. At finite baryon density, the effective theory has a sign problem which meets all criteria to be simulated by complex Langevin as well as by Monte Carlo on small volumes. The theory is valid for the thermodynamics of heavy quarks, where its predictions agree with simulations of full QCD at zero and imaginary chemical potential. In its region of convergence, it is moreover amenable to perturbative calculations in the small effective couplings. In this work we study the challenging cold and dense regime. We find unambiguous evidence for the nuclear liquid gas transition once the baryon chemical potential approaches the baryon mass, and calculate the nuclear equation of state. In particular, we find a negative binding energy per nucleon causing the condensation, whose absolute value decreases exponentially as mesons get heavier. For decreasing meson mass, we observe a first order liquid gas transition with an endpoint at some finite temperature, as well as gap between the onset of isospin and baryon condensation.
The properties of matter at finite baryon densities play an important role for the astrophysics of compact stars as well as for heavy ion collisions or the description of nuclear matter. Because of the sign problem of the quark determinant, lattice QCD cannot be simulated by standard Monte Carlo at finite baryon densities. I review alternative attempts to treat dense QCD with an effective lattice theory derived by analytic strong coupling and hopping expansions, which close to the continuum is valid for heavy quarks only, but shows all qualitative features of nuclear physics emerging from QCD. In particular, the nuclear liquid gas transition and an equation of state for baryons can be calculated directly from QCD. A second effective theory based on strong coupling methods permits studies of the phase diagram in the chiral limit on coarse lattices.
Lattice QCD with heavy quarks reduces to a three-dimensional effective theory of Polyakov loops, which is amenable to series expansion methods. We analyse the effective theory in the cold and dense regime for a general number of colours, $N_c$. In particular, we investigate the transition from a hadron gas to baryon condensation. For any finite lattice spacing, we find the transition to become stronger, i.e. ultimately first-order, as $N_c$ is made large. Moreover, in the baryon condensed regime, we find the pressure to scale as $psim N_c$ through three orders in the hopping expansion. Such a phase differs from a hadron gas with $psim N_c^0$, or a quark gluon plasma, $psim N_c^2$, and was termed quarkyonic in the literature, since it shows both baryon-like and quark-like aspects. A lattice filling with baryon number shows a rapid and smooth transition from condensing baryons to a crystal of saturated quark matter, due to the Pauli principle, and is consistent with this picture. For continuum physics, the continuum limit needs to be taken before the large $N_c$ limit, which is not yet possible in practice. However, in the controlled range of lattice spacings and $N_c$-values, our results are stable when the limits are approached in this order. We discuss possible implications for physical QCD.
After combined character and hopping expansions and integration over the spatial gauge links, lattice QCD reduces to a three-dimensional $SU(3)$ Polyakov loop model with complicated interactions. A simple truncation of the effective theory is valid for heavy quarks on reasonably fine lattices and can be solved by linked cluster expansion in its effective couplings. This was used ealier to demonstrate the onset transition to baryon matter in the cold and dense regime. Repeating these studies for general $N_c$, one finds that for large $N_c$ the onset transition becomes first-order, and the pressure scales as $psim N_c$ through three consecutive orders in the hoppoing expansion. These features are consistent with the formal definition of quarkyonic matter given in the literature. We discuss the implications for $N_c=3$ and physical QCD.
In this paper we carry out a low-temperature scan of the phase diagram of dense two-color QCD with $N_f=2$ quarks. The study is conducted using lattice simulation with rooted staggered quarks. At small chemical potential we observe the hadronic phase, where the theory is in a confining state, chiral symmetry is broken, the baryon density is zero and there is no diquark condensate. At the critical point $mu = m_{pi}/2$ we observe the expected second order transition to Bose-Einstein condensation of scalar diquarks. In this phase the system is still in confinement in conjunction with non-zero baryon density, but the chiral symmetry is restored in the chiral limit. We have also found that in the first two phases the system is well described by chiral perturbation theory. For larger values of the chemical potential the system turns into another phase, where the relevant degrees of freedom are fermions residing inside the Fermi sphere, and the diquark condensation takes place on the Fermi surface. In this phase the system is still in confinement, chiral symmetry is restored and the system is very similar to the quarkyonic state predicted by SU($N_c$) theory at large $N_c$.
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