Exact holographic mapping and emergent space-time geometry

TL;DR

The paper introduces exact holographic mapping (EHM), a unitary boundary-to-bulk construction that yields an emergent space-time geometry from boundary correlations. Through a detailed 1+1D lattice Dirac fermion example, it shows how the bulk can reproduce AdS-like geometry for critical states, BTZ-like horizons at finite temperature, and IR-term terminating geometries for finite mass; it also demonstrates a wormhole geometry arising from entangled chains and its dynamical evolution under a quantum quench. Beyond free fermions, the work discusses causal structure, links to AdS/CFT concepts, and the potential for self-consistent or interacting bulk descriptions, outlining significant open questions. The framework provides a concrete route to geometrizing quantum many-body states and exploring holographic ideas in condensed-mmatter contexts.

Abstract

In this paper, we propose an {\it exact holographic mapping} which is a unitary mapping from the Hilbert space of a lattice system in flat space (boundary) to that of another lattice system in one higher dimension (bulk). By defining the distance in the bulk system from two-point correlation functions, we obtain an emergent bulk space-time geometry that is determined by the boundary state and the mapping. As a specific example, we study the exact holographic mapping for $(1+1)$-dimensional lattice Dirac fermions and explore the emergent bulk geometry corresponding to different boundary states including massless and massive states at zero temperature, and the massless system at finite temperature. We also study two entangled one-dimensional chains and show that the corresponding bulk geometry consists of two asymptotic regions connected by a worm-hole. The quantum quench of the coupled chains is mapped to dynamics of the worm-hole. In the end we discuss the general procedure of applying this approach to interacting systems, and other open questions.

Figures (7)

• Figure 1: (a) Network representing a MERA state (without disentanglers). (b) Network of the exact holographic mapping. Each node in the network stands for a unitary transformation $U$ which maps the states of two input sites $\left|s_1,s_2\right\rangle$ to one bulk state (red dot) labeled by $\left|\alpha\right\rangle$ and one auxiliary state $\left|t\right\rangle$. More details of the definition are given in Sec. \ref{['sec:def']}. (c) A simplified representation of the EHM network in (b), in which all the unitary transformations in the same layer are combined together to one unitary mapping (grey triangle). The boundary theory (yellow square) is mapped to auxiliary degrees of freedom (blue arrow) and bulk states (red filled circle) in each step. After $N$ steps, the boundary theory is mapped to the bulk theory consisting of all red dots.
• Figure 2: Distance between two points with equal time and different spatial location ((a) and (b)) and that between two points at the same spatial location with different time ((c) and (d)) for the critical system $T=m=0$. (a) and (b) shows the distance between two points separated along the horizontal direction and vertical direction, respectively. The $x$ axis is the geometrical distance between the two points in the AdS space. The AdS radius $R$ and correlation length $\xi$ is obtained from fitting in (a). (c) and (d) shows the distance between two points at the same spatial location and time difference of $\tau$. The fitting yields values of $R$ and $\xi$ independently from the spatial correlation functions. In all panels, the circles are the numerical results and the lines are the fitting with AdS space geodesic distance. All numerical calculations in Sec. \ref{['sec:freefermion']} and \ref{['sec:finiteTm']} are done for a chain with $2^17$ sites.
• Figure 3: (a) Spatial distance $d_{12}$ between two points $(j_1,n)$ and $(j_2,n)$ for different $n$. Distance increases with the increase of $n$. The inset is the colorplot of the ratio of mutual information $I_{12}(T)$ of the finite temperature system and $I_{12}(0)$ that of the critical system, as a function of horizontal and vertical coordinates. (b) Temporal distance $d_n(\tau)$ between two points $(j,n,0)$ and $(j,n,\tau)$ for different $n$. Distance decreases with the increase of $n$. The dashed lines are the fitting with the analytic formula (\ref{['geodesicT']}). (c) The radius $\rho$ as a function of $n$, obtained from the fitting (red line with circles). The blue dashed line labels $r=b$ and the black dotted line shows the zero temperature value $\rho=2^{N-n}/2\pi$. (d) Entropy per site as a function of $n$ for finite temperature (red circles) and zero temperature (blue line). The black dotted line shows the maximal entropy value $2\log 2$. All calculations are done for $T=0.005$.
• Figure 4: (a) Spatial distance $d_{12}$ between two points $(j_1,n)$ and $(j_2,n)$ for different $n$. Distance increases with the increase of $n$. The inset is the colorplot of the ratio of mutual information $I_{12}(m)$ of the massive system and $I_{12}(0)$ that of the critical system, as a function of horizontal and vertical coordinates. (b) Temporal distance $d_n(\tau)$ between two points $(j,n,0)$ and $(j,n,\tau)$ for different $n$. Distance decreases with the increase of $n$. (c) Long time behavior of $d_n(\tau)$ as a function of $n$ for a $\tau=1580\gg 1/m$. (d) Entropy per site as a function of $n$, for finite mass (red line with circles) and massless system (blue dashed line). The black dotted line show the maximal entropy value $2\log 2$. All calculations are done for $m=0.005, T=0$.
• Figure 5: (a) Schematic picture of the wormhole geometry. The green line illustrates a geodesic path between the two end points. (b) Distance $d_{n}^{12}$ between two sites with coordinate $(j,n)$ in the two layers. The main figure and the inset shows the distance in linear scale and log scale, respectively. All the results in this section are done for $2^16$ sites and $\lambda=0.05$.