Operator content of real-space entanglement spectra at conformal critical points

Abstract

We provide numerical evidence that the low-lying part of the entanglement spectrum of a real-space block (i.e. a single interval) of a one-dimensional quantum many body system at a conformal critical point corresponds to the energy spectrum of a boundary conformal field theory (CFT). This correspondence allows to uncover a subset of the operator content of a conformal field theory by inspection of the entanglement spectrum of a single wave function, thus providing important information on a CFT beyond its central charge. As a practical application we show that for many systems described by a compactified boson CFT, one can infer the compactification radius (governing e.g. the power law decay of correlation functions) of the theory in a simple way based on the entanglement spectrum.

Figures (5)

• Figure 1: (Color online). Setup of the bipartitions and boundary conditions of the one dimensional lattice models investigated here. The orange shaded region denotes the size and position of block A in the case of (a) open boundary conditions (OBC) and (b) periodic boundary conditions (PBC).
• Figure 2: (Color online) (a) / (c) Entanglement spectrum $\xi$ for a $L=256$ Bose-Hubbard chain at $U/J=2$ with OBC / PBC and a block size of $L_A=128$. (b) / (d): Structure of the ES in each $\delta N_A$ sector for OBC / PBC, obtained after subtracting the lowest $\xi$ value in each $\delta N_A$ sector, and setting the difference between the first and second $\xi$ value in the $\delta N_A=0$ sector to one. The levels with an assigned value are indicated by the dashed red line, while the relative position of all other levels highlights the emergent CFT structure of the ES. The approximate degeneracy at energy level $l$ is compatible with $p(l)$, the number of integer partitions of $l$. (e) Schematic representation of the boundary CFT energy spectrum of a compactified boson with free boundary conditions CFTYellowBookAlcaraz1987Cazalilla2004. Each orange shaded circle denotes a primary field with scaling dimension $\eta (\delta N_A)^2$, while the equally spaced white circles on top of each primary field complete the Virasoro towers with a degeneracy count of $p(l)$. The schematic spectrum has been plotted for $\eta=1/2$. The energy differences denoted $\Delta \xi(0,0)$ and $\Delta \xi(0,\pm1)$ in (a) and (c) enter the formula \ref{['eq:eta_estimator']}$\eta= {\Delta \xi(0,\pm1)} / {\Delta \xi(0,0)}$.
• Figure 3: (Color online) Scaling dimension $\eta$ of the Bose-Hubbard chain at filling $n=N/L=1$ as a function of the interaction $U/J$. The critical value $\eta_c=1/4$ and location of the transition from the superfluid to the Mott insulator $(U/J)_c$Kuehner1998GiamarchiBook are indicated with horizontal and vertical lines. The red dashed curve shows $\eta$ from the particle number fluctuations analysis in Ref. Rachel2012. The particle number cutoff per site is set to $N_\mathrm{max}=4$ in all the results.
• Figure 4: (Color online) (a) Scaling dimension $\eta_x$ of the $S=1/2$ XXZ model in zero magnetic field \ref{['eq:Ham_XXZ']} obtained from Eq. \ref{['eq:eta_estimator']} as a function of $\Delta=J_z/J_{xy}$ for different system sizes and boundary conditions. The dashed line denotes the exact Bethe ansatz result for $\eta_x$. (b) Scaling dimension $\eta_x$ of the $S=1/2$ Heisenberg model (XXZ model at $\Delta=1$) as a function of the magnetization per site. The dashed line denotes the numerically exact Bethe ansatz result for $\eta_x$ from Ref. Essler2004. (c) The same quantity $\eta_x$ of the $S=1$ Heisenberg chain as a function of the magnetization per site. The blue circles denote the DMRG data of Ref. Fath2003, where the exponent $\eta_x$ was determined based on fits of the decay of the correlation functions.
• Figure 5: (Color online) Left panel: (a) / (c) shifted and rescaled ES for the transverse field Ising model at the critical point for OBC / PBC. The levels which have assigned values are encircled by the dashed red line. (b) catalog of the conformal towers of the $\mathcal{M}_3$$c=1/2$ CFT. The filled symbols denote the two towers realized in the numerical ES. Right panel: (d) / (f) shifted and rescaled ES for the quantum three-state Potts model at the critical point for OBC / PBC. The levels which have assigned values are encircled by the dashed red line. (e) catalog of the conformal towers of the $\mathcal{M}_5$$c=4/5$ CFT. The filled symbols denote the three distinct towers realized in the numerical ES.