We propose a new mean-field approach to analyze many-body systems of fermions in the gauge/gravity duality. We introduce a non-vanishing classical fermionic field in the gravity dual, which we call the holographic mean field. The holographic mean field reflects the dynamics of the fermions in the bulk, and generates the expectation values of the bi-linear operators of the boundary fermions. This enables us to analyze finite-density systems of baryons in the confinement phase. Our method provides a new bulk condition which relates the chemical potential to the charge density in the GKP-Witten prescription. Many-body physics of strongly-correlated fermions is one of the central subjects in modern physics. Stronglycorrelated fermions appear in various places, such as the systems of cold atoms, high-Tc superconductors, neutron stars and the quark-gluon plasma. However, the theoretical description of many-body systems of stronglycorrelated fermions is still a challenge owing to their non-perturbative natures. One of the successful computational frameworks for strongly-interacting fermions is the lattice gauge theory, but the Monte Carlo simulation at finite baryon density suffers from the sign problem. Recently, a great deal of attention has been paid to the AdS/CFT correspondence [1–3] or the holographic method because of its ability for non-perturbative computations and the absence of the sign problem. However, the current framework of the AdS/CFT correspondence still has room for improvement in dealing with the systems at finite baryon density, as we will explain later. It can be summarized as follows: 1) dynamical baryons at finite density have not been satisfactorily incorporated, and 2) the application of the GKP-Witten prescription [2, 3] to relate the baryon chemical potential to the baryon density has an ambiguity in the grand canonical ensemble. In the present paper, we shall propose a new approach, a holographic mean-field theory for fermions, to resolve these two problems. The self-consistent incorporation of dynamical baryons in our mean-field approach provides an eigenvalue equation which eliminates the ambiguity. We shall revisit the details of these problems below. The quarks are realized as fundamental strings attached to the flavor brane in the gravity dual [4]. Each end point of the strings carries a unit charge with respect to a U(1) gauge field on the flavor brane, and the quark density can be evaluated from the electric field in the radial direction at the boundary through the Gausslaw constraint. The quark chemical potential is thus the time component of the U(1) vector potential, A0, at the boundary, which is conjugate to the electric field [5, 6]. This fits the conventional GKP-Witten prescription: The boundary value of A0 gives the source (the chemical potential); the expectation value of the conjugate operator (the density) is given by the derivative of A0 at the boundary with respect to the radial direction. These quantities are boundary conditions for the Gauss-law constraint which may be chosen arbitrarily. However, another physical condition in the bulk, i.e., the bulk condition, fixes the ambiguity in the relationship between them whenever the GKP-Witten prescription works. To the best of the authors’ knowledge, only the example where the bulk condition is yet to be known is the finite density systems of baryons.

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