Holographic Aspects of Dynamical Mean-Field Theory

TL;DR

This work reframes dynamical mean-field theory (DMFT) for electrons on a Bethe lattice with semicircular density of states as a holographic renormalization group, deriving a recursion for branch Green's functions that flows to a DMFT self-consistent fixed point. An emergent AdS$_2$-like geometry arises from smearing lattice nodes, and the boundary correlation functions acquire scaling dimensions governed by the fixed-point Green's function, connecting the interior bulk physics to edge observables. Numerical DMFT results on the Bethe-lattice Hubbard model reveal a first-order Mott transition with a metallic regime ($ riangle_D o 1$) and an insulating regime ($ riangle_D$ larger), showing that boundary scaling dimensions encode deep interior physics. The framework suggests a novel AdS/DMFT perspective and points to extensions to magnetically ordered states and looped/hyperbolic lattices, with implications for understanding quantum fluctuations in holographic formulations of strongly correlated matter.

Abstract

Dynamical mean-field theory (DMFT) is one of the most standard theoretical frameworks for addressing strongly correlated electron systems. Meanwhile, the concept of holography, developed in the field of quantum gravity, provides an intrinsic relationship between quantum many-body systems and space-time geometry. In this study, we demonstrate that these two theories are closely related to each other by shedding light on holographic aspects of DMFT, particularly for electrons with a semicircle density of states. We formulate a holographic renormalization group for the branch Green's function from the outer edge to the interior of the Bethe lattice network, and then find that its fixed point can be interpreted as a self-consistent solution of Green's function in DMFT. By introducing an effective two-dimensional anti-de Sitter space, 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. 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.

Figures (5)

• Figure 1: The Bethe lattice with $q=3$ and $N=4$ with the imaginary time axis, where the index $n$ for dotted circles represents the generation from the center node and $\tau$ denotes the imaginary time. Note that the antiperiodic boundary condition is assumed in the $\tau$ direction, where $\beta$ denotes an inverse temperature. The distance between $\alpha$ and $\alpha'$ in the circumference direction is defined by counting the number of nodes along the outer edge, while the red line represents the shortest network path connecting the edge spins.
• Figure 2: The network connectivity for a parent node [($n-1$)th generation]
• Figure 3: Imaginary part of $G^*(i\omega)$ for $U=0.0$, 1.0, 2.0, and 3.0. Solid curve represents the free electron curve $\mathrm{Im} G^*(i\omega) = 2(\omega \mp \sqrt{\omega^2 +1})$.
• Figure 4: The scaling dimension $\Delta_D$ for $p=100$ at $\beta=100$. The lower branch indicates the metallic solution, while the upper branch corresponds to the insulating one. The vertical dotted lines represent $U_{c1}\simeq 2.36$ and $U_{c2}\simeq 2.55$.
• Figure 5: The scaling dimension spectrum with $p=100$ at $\beta=100$ for (a) $U=1.0$ and (b) 3.0. $\Delta_D^{(l,l')}$ is plotted against $l_s\equiv l+l' -2 l^*$ of the horizontal axis. The origin of the vertical axis is also sifted by $\Delta_D$. In this figure we plot the contributions only form $l, l' \ge 0$, for simplicity.