Mean-field treatment of the long-range transverse field Ising model with fermionic Gaussian states

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“…For 𝛼 LR = 0.5, the deviation of 𝑐 ∞ from 1/2 is much larger than the error bar, implying the breakdown of the conformal symmetry of the SR Ising class. Our results are obtained from the second Rényi entropy under PBC, providing an interesting comparison with the previous results based on the von Neumann entropy under OBC [53,54,59]. All studies agree on a significant deviation from 1/2 for 𝛼 LR < 1.…”

“…A significant increase in the central charge from the Ising value 1/2 was observed at 𝛼 LR ≲ 1 in the TDVP [53] and density matrix renormalization group (DMRG) [54] calculations, based on which the breakdown of conformal invariance was proposed [54]. While here we focus on the critical ground state, a violation of the area law for the entanglement entropy was observed in the offcritical area [53,54,59], and it was shown that the area law of the noncritical ground state holds for 𝛼 LR > 2 [63].…”

“…The logarithmic system-size scaling of the entanglement entropy at a critical point in 1D is a useful universal property to measure the central charge of the CFT that characterizes the phase transition [71][72][73]. In the previous estimate of the central charge using the TDVP [53], DMRG [54], and generalized Hatree-Fock [59] methods, the von Neumann entanglement entropy was examined under open boundary conditions (OBC). Instead, we consider the second Rényi entropy for the measurement using the RBM wave function under PBC.…”

“…Because an exact solution is not available, constructing the picture of how its criticality deviates from the Ising class as 𝛼 LR decreases relies primarily on the collection of numerical observations. Despite various numerical studies characterizing the quantum phase transition in AF-LR-TFIC at equilibrium [53][54][55][56][57]59] and out of equilibrium [56,60], the picture remains incomplete in some parts, which requires more numerical evidence for clarification. Using the restricted Boltzmann machine (RBM) for the NQS ansatz [5], we consider the moments of staggered magnetization including the order parameter and the Binder ratio, the two-point correlation function, and the entanglement entropy to examine the present picture and find clearer signatures of the breakdown of the SR Ising class and the conformal invariance along the critical line.…”