Finite-size scaling in the transverse Ising model on a square lattice

Abstract

Energy eigenvalues and order parameters are calculated by exact diagonalization for the transverse Ising model on square lattices of up to 6x6 sites. Finite-size scaling is used to estimate the critical parameters of the model, confirming universality with the three-dimensional classical Ising model. Critical amplitudes are also estimated for both the energy gap and the ground-state energy.

Fidelity susceptibility in the two-dimensional transverse-field Ising andXXZmodels

“…Comparing this value to the critical point 1/h c ≃ 0.328 obtained in previous works [20,21], our result here is consistent with them up to two digits. Moreover, we can also see that the averaged fidelity susceptibility peaks sharper for a larger N and is in fact scales approximately with N 0.51 as shown in the inset of Fig.…”

“…For a finite system, the ground state of the Ising model is non-degenerate in each subspace, thus the perturbation expansion as introduced in the previous section is valid as long as the lattice is finite. In the thermodynamic limit, the model exhibits a quantum phase transition at 1/h c ≃ 0.328 [20,21]. For h ≫ h c , the transverse field dominates and the ground state is a paramagnetic phase, with spins almost fully polarized in the z-direction.…”

“…Periodic boundary conditions are assumed. This model was originally introduced by de Gennes to describe potassiumdihydrogen-phosphate type ferroelectrics [16] and has been studied extensively via various approaches, like realspace renormalization group [18], density-matrix renormalization [19], numerical diagonalization [20,21], and entanglement [22]. Obviously, the Hamiltonian of the model commutes with the parity operator P = i σ z i .…”

Entanglement renormalization and gauge symmetry

“…44. We also review a deformation of the toric code model obtained by adding a magnetic field on thex direction, as analyzed 55 is denoted by a vertical line. The blue (top) curve is a lower bound to the entanglement entropy of the deformed toric code model without W exact .…”

Critical behavior of one-particle spectral weights in the transverse Ising model

“…10 The square lattice is bipartite, and hence one finds that the Hamiltonian is symmetric under a spin rotation by about the z axis on the B sublattice, followed by a coupling inversion → −. Correspondingly, there are symmetrical critical points at couplings = ± c , where c has been estimated from series expansions 11 at 0.328 51͑8͒, and from a finitesize scaling analysis 12 at 0.328 41͑2͒. We have computed series for the one-particle spectral weight S 1p xx ͑k͒ and the quasiparticle energy E͑k͒ to order 14 ͑further terms can be supplied on request͒.…”