Quantum dimensions from local operator excitations in the Ising model

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 Rényi 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.

Subsystem distance after a local operator quench

“…This fact has been already observed in Ref. [29] for the entanglement and Rényi entropies (as well as for the relative entropies): even for values of of the order of few hundreds, deviations from the CFT have been observed (see also [59][60][61][62][63][64][65] for similar issues in other local quenches). These discrepancies have a double origin: on the one hand, they are intrinsically connected with the finite lattice spacing, on the other, they come from the non-linearity of the dispersion relation (3.10).…”

“…In this paper, we consider the subsystem trace distance in latter category of quenches: at an initial time t = 0, a local operator is inserted somewhere in the ground state of the theory, and then the system is let evolve without further disturbance. The time evolution of entanglement and Rényi entropies, as well as of local operators, has been intensively studied in the literature [10][11][12][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The main goal of this paper is to investigate the distance between the RDMs at different times after a local operator quench.…”

“…Specifically, we consider the critical Ising spin chain (whose continuum limit is the 2D free massless fermion theory, which is a 2D CFT with central charge c = 1/2), and the XX chain (2D free massless compact boson theory with central charge c = 1). The entanglement and Rényi entropies after some local operator quenches in the Ising spin chain have been already investigated in [29]. In this paper, we will calculate numerically the entanglement entropy, the Rényi entropies, the trace distance, and the Schatten distances in the critical Ising spin chain, as well as in the XX spin chain.…”

Entanglement Entropy after Double Excitation as an Interaction Measure

“…This statement has already shown in [10,18,22] in another way. For the same reason, this sum rule also appears in free massless scalar field theories [7].…”

“…This is because the entanglement entropy in RCFTs behaves as if correlations were carried by free quasiparticles [21]. Actually this rule has already shown in [10,18,22]. However, there is no reason that the sum rule can be also applied to other CFT, in particular, holographic CFTs because the quasiparticle picture breaks down in holographic CFTs.…”