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arXiv:2509.19704 · Physical review. B./Physical review. B

Holographic aspects of dynamical mean-field theory

Kouichi Okunishi, Akihisa Koga

Published Jun 10, 2026• 0 citations
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Abstract

Dynamical mean-field theory (DMFT) is one of the most standard theoretical frameworks for addressing strongly correlated electron systems. In this study, we explore a holographic renormalization-group (RG)-like structure inherent in DMFT, which we refer to as "AdS/DMFT", by focusing on the background Bethe-lattice network behind DMFT for electrons with a semicircle density of states. We formulate an RG transformation for the branch Green's function from the outer edge to the interior of the background Bethe-lattice network, and then find that its fixed point can be interpreted as a self-consistent solution of Green's function in DMFT. Moreover, we clarify that the scaling dimensions for the branch Green's function and the boundary correlation functions of electrons at the outer edge of the Bethe-lattice network are characterized by the fixed-point Green's function, analogous to the behavior of a scalar field in an effective two-dimensional anti-de Sitter space. We also perform DMFT computations for the Bethe-lattice Hubbard model, which illustrate that the scaling dimensions capture the Mott transition in the deep interior.

DOI
10.1103/z1c9-55ff

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Dynamical mean-field theory (DMFT) is one of the most standard theoretical frameworks for addressing strongly correlated electron systems. In this study, we explore a holographic renormalization-group (RG)-like structure inherent in DMFT, which we refer to as "AdS/DMFT", by focusing on the background Bethe-lattice network behind DMFT for electrons with a semicircle density of states. We formulate an RG transformation for the branch Green's function from the outer edge to the interior of the background Bethe-lattice network, and then find that its fixed point can be interpreted as a self-consistent solution of Green's function in DMFT. Moreover, we clarify that the scaling dimensions for the branch Green's function and the boundary correlation functions of electrons at the outer edge of the Bethe-lattice network are characterized by the fixed-point Green's function, analogous to the behavior of a scalar field in an effective two-dimensional anti-de Sitter space. We also perform DMFT computations for the Bethe-lattice Hubbard model, which illustrate that the scaling dimensions capture the Mott transition in the deep interior.
Abstract The Bethe lattice Ising model—a classical model of statistical mechanics for the phase transition—provides a novel and intuitive understanding of the prototypical relationship between tensor networks and the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. After analytically formulating a holographic renormalization group for the Bethe lattice model, we demonstrate the underlying mechanism and the exact scaling dimensions for the power-law decay of boundary-spin correlations by introducing the relation between the lattice network and an effective Poincaré metric on a unit disk. We compare the Bethe lattice model in the high-temperature region with a scalar field in AdS2, and then discuss its more direct connection to the p-adic AdS/CFT. In addition, we find that the phase transition in the interior induces a crossover behavior of boundary-spin correlations, depending on the depth of the corresponding correlation path.
The dynamical mean-field theory (DMFT), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [K. Byczuk and D. Vollhardt, Phys. Rev. B 77, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green's function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-type regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\ensuremath{\mu}/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane and on the $T/U\text{\ensuremath{-}}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}/U$ plane are produced ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.
We apply a recently introduced hybridization flow functional renormalization group scheme for Anderson-like impurity models as an impurity solver in a dynamical mean-field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard model on the Bethe lattice in infinite dimensions we calculate the local two-particle vertex for the metallic and the insulating phase. Then we go to a two-site cluster DMFT scheme for the two-dimensional Hubbard model that includes short-range antiferromagnetic fluctuations and obtain the local and nonlocal two-particle vertex functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.
An approximate analytical scheme of the dynamical mean-field theory (DMFT), that is used for electron systems with Hubbard correlations and is exact in the limit of the infinite dimensionality of a space, is developed.The effective single-site problem arising in the framework of this method is formulated in terms of the auxiliary Fermi-field.The irreducible Green's function technique with the projecting on the Hubbard basis of Fermi-operators is used for its solution.A system of DMFT equations is obtained in the approximation which is a generalization of the Hubbard-III approximation and combines it with a self-consistent renormalization of the local electron levels.It is shown that the proposed approach includes as simple specific cases a number of known approximations (Hubbard-III, AA, MAA, . . . ) based on the assumption of the single-site structure of the electron selfenergy.
We study the Hubbard model at half band-filling on a Bethe lattice with infinite coordination number in the paramagnetic insulating phase at zero temperature. We use the dynamical mean-field theory (DMFT) mapping to a single-impurity Anderson model with a bath whose properties have to be determined self-consistently. For a controlled and systematic implementation of the self-consistency scheme we use the fixed-energy (FE) approach to the DMFT. In FE-DMFT the onset and the width of the Hubbard bands are adjusted self-consistently but the energies of the bath levels are kept fixed relatively to both band edges during the calculation of self-consistent hybridization strengths between impurity and bath sites. Using the dynamical density-matrix renormalization group method (DDMRG) we calculate the density of states with a resolution ranging from 3% of the bare bandwidth W=4tW=4tW=4t at high energies to 0.5% in the vicinity of the gap. The DDMRG resolution and accuracy for the density of states and the gap is superior to those obtained with other numerical methods in previous DMFT investigations. We find that the Mott gap closes at a critical coupling Uc/t=4.45±0.05U_{\rm c}/t=4.45 \pm 0.05Uc​/t=4.45±0.05. At U=4.5tU=4.5tU=4.5t, we observe prominent shoulders near the onset of the Hubbard bands. They are the remainders of the quasi-particle resonance in the metallic phase which appears to split when the gap opens at UcU_{\rm c}Uc​.
Since the first investigation of the Hubbard model in the limit of infinite dimensions by Metzner and Vollhardt, dynamical mean-field theory (DMFT) has become a very powerful tool for the investigation of lattice models of correlated electrons. In DMFT the lattice model is mapped on an effective quantum impurity model in a bath which has to be determined self-consistently. This approach led to significant progress in our understanding of typical correlation problems such as the Mott transition; furthermore, the combination of DMFT with ab-initio methods now allows for a realistic treatment of correlated materials. The focus of these lecture notes is on the relation between quantum impurity physics and the physics of lattice models within DMFT. Issues such as the observability of impurity quantum phase transitions in the corresponding lattice models are discussed in detail.