Quantum dimensions from local operator excitations in the Ising model

TL;DR

The paper investigates how local operator excitations in the critical Ising chain drive the time evolution of entanglement, benchmarking lattice results against 2d CFT predictions. The main approach combines exact lattice simulations (via Jordan–Wigner mapped free fermions and linearized dynamics) with CFT expectations for Rényi entropies and the replica trick, focusing on primary operators and their descendants. A key finding is that the spin operator $\sigma$ reproduces the CFT prediction $\Delta S^{(n)}=\log\sqrt{2}$, while the energy operator $\varepsilon$ exhibits a residual lattice-induced mismatch, highlighting subtleties in lattice–continuum operator identification. The work further shows that entanglement and relative entropy dynamics can constrain operator mappings, extend to non-critical regimes and multiple excitations, and reveal universal features tied to the conformal data that govern the critical point.

Abstract

We compare the time evolution of entanglement measures after local operator excitation in the critical Ising model with predictions from conformal field theory. For the spin operator and its descendants we find that Renyi entropies of a block of spins increase by a constant that matches the logarithm of the quantum dimension of the conformal family. However, for the energy operator we find a small constant contribution that differs from the conformal field theory answer equal to zero. We argue that the mismatch is caused by the subtleties in the identification between the local operators in conformal field theory and their lattice counterpart. Our results indicate that evolution of entanglement measures in locally excited states not only constraints this identification, but also can be used to extract non-trivial data about the conformal field theory that governs the critical point. We generalize our analysis to the Ising model away from the critical point, states with multiple local excitations, as well as the evolution of the relative entropy after local operator excitation and discuss universal features that emerge from numerics.

Figures (9)

• Figure 1: Schematic illustration of the protocol. Local operator $\hat{\mathcal{O}}$ is inserted at a distance $l$ from the block $A$ of $L$ spins. We then calculate the change of entropy of the block $\Delta S_A$ resulting from the insertion as a function of time.
• Figure 2: Evolution of the cross-ratios on the cylinder for a single period of time for small but non-zero $\epsilon$.
• Figure 3: Evolution after excitation by $\sigma(n)$. Results for Ising Hamiltonian (solid lines) and linearized Ising Hamiltonian (dashed lines). Different system sizes $N=512$ (blue), $N=1024$ (red) and $N=2048$ (green). (a) Fixed block size $L = 64$ and distance $l = 64$. (b) Distance and block size as a fraction of the system size, $L=l=N/8$. (c) Excitation in the same distance from both ends of the block; Block size as a fraction of the system size $L = N/2+1$, $l=N/4$. See text for discussion.
• Figure 4: Evolution after excitation by $\varepsilon(n)$. Results for Ising Hamiltonian (solid lines) and linearized Ising Hamiltonian (dashed lines). System sizes $N=512$ (blue), $N=1024$ (red) and $N=2048$ (green). (a) Fixed block size $L = 64$ and distance from the block $l = 64$; (b) Block size and distance as a fraction of the system size, $L=l=N/8$. (c) Block next to the excitation with $l=1$ and the block size $L=128$. See text for discussion.
• Figure 5: Evolution after excitation by $\sigma(n+1)-\sigma(n)$. Results for Ising Hamiltonian (solid lines) and linearized Ising Hamiltonian (dashed lines). System sizes $N=512$ (blue), $N=1024$ (red) and $N=2048$ (green). (a) Fixed block size $L = 64$ and distance from the block $l = 64$; (b) Block size and distance as a fraction of the system size, $L=l=N/8$. (c) Block next to the excitation with $l=1$ and the block size $L=128$. See text for discussion.