Chiral entanglement in massive quantum field theories in 1+1 dimensions

TL;DR

The authors quantify entanglement between left- and right-moving sectors during massive RG flows in 1+1D diagonal CFTs on a cylinder. They implement Cardy’s variational Ansatz using smeared conformal boundary states to approximate the ground state and validate this approach with extensive Truncated Conformal Space Approach (TCSA) computations. A key result is the universal O(1) term γ in the chiral entanglement entropy, which, together with the linear in L slope 𝔅 = π c/(24 τ^*), characterizes the IR ground state and can distinguish degenerate vacua. Applications to Ising and tricritical Ising field theories demonstrate accurate agreement between analytics and numerics, supporting the universality and utility of the chiral entanglement framework for RG flows in 1+1 dimensions.

Abstract

We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy's Ansatz. The chiral entanglement entropy contains a universal $O(1)$ term $γ$ for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.

Figures (12)

• Figure 1: Conformal partition function $\mathcal{Z}_{ab}$ on an annulus. The system is quantised on a cylinder of length $L$, the smearing parameter $\tau$ has the dimension of the inverse of the mass gap (correlation length). The conformal Hamiltonian $H_{\text{CFT}}$ generates translation along the annulus, between two boundary states $a$ and $b$.
• Figure 2: Coefficients of the ground state eigenvector as a function of the conformal energy (eigenvalue of $L_0+\bar{L}_0-c/12$) in Ising field theory perturbed by the spin field $\sigma$ ($t=0,\ h>0$) at the dimensionless volume $mL=80$. Discrete dots are TCSA data at level $20$ with $28624$ states, while the lines show the prediction of \ref{['overgs']} with $\tau_*$ given in Table \ref{['tab:taus']} (with an overall sign difference which is due to the choice of the numerics). Different colours correspond to different modules: blue--$1$, red--$\sigma$ and black--$\varepsilon$; the point lying on (or close to) the horizontal axis correspond to non-diagonal contributions. Inset: zooming on the conformal energy region $\approx14-15$ shows the deviations from the diagonal form of the Ansatz discussed in the text.
• Figure 3: Coefficients of the ground state vector as a function of the conformal energy (eigenvalue of $L_0+\bar{L}_0-c/12$) of the CFT state in the thermal perturbation of the Ising fixed point at $mL=10$. Dots are from TCSA at level $20$ with $28624$ states, the lines are predicted by using \ref{['overgs']} the smearing from Cardy's Ansatz (see Table \ref{['tab:taus']}) and $c_{a^*h}$ corresponding to $|\text{NS}\rangle$ and $|\text{R}\rangle$. It is clear the overlaps are not in full quantitative agreement. The reason is that the conformal weight of the perturbation is high: the energy levels are logarithmically divergent, and the cut-off effects are relatively high. However, it turns out that the cut-off extrapolated chiral entanglement entropy calculated from the overlaps agrees very well with the theoretical prediction (cf. Section \ref{['sec:TCSA']}).
• Figure 4: Coefficients of the ground state vector as a function of the conformal energy (eigenvalue of $L_0+\bar{L}_0-c/12$) of the CFT state in the $t<0$ perturbation of the tricritical Ising fixed point at $mL=30$. Dots are from TCSA at level $14$ with $22559$ states in the NS sector and $18751$ in the R sector. The lines are predicted by using \ref{['overgs']} the smearing from Cardy's Ansatz (see Table \ref{['tab:taus']}) and $c_{a^*h}$ corresponding to $|\text{tNS}\rangle$ and $|\text{tR}\rangle$.
• Figure 5: Coefficients of the ground state vector as a function of the conformal energy (eigenvalue of $L_0+\bar{L}_0-c/12$) of the CFT state in the $h'>0$ perturbation of the tricritical Ising fixed point at $mL=5$. Dots are from TCSA at level $14$ with $13373$ states in the sector $\{1,\sigma',\varepsilon"\}$ and $27937$ in $\{\sigma,\varepsilon,\varepsilon'\}$. The lines are predicted by using \ref{['overgs']} the smearing from Cardy's Ansatz (see Table \ref{['tab:taus']}) and $c_{a^*h}$ corresponding to $|\text{GS}_+\rangle$ and $|\text{GS}_-\rangle$.