Scaling Behavior of the Ground-State Fidelity in the Lipkin–meshkov–glick Model

Abstract

In this paper, we study the scaling behavior of the ground-state logarithmic fidelity in the Lipkin-Meshkov-Glick (LMG) model. We find that the logarithmic fidelity shows a different scaling behavior in different phases of the model. It is an extensive quantity in the model's symmetry-broken phase while it behaves intensively in the polarized phase. Moreover, we also find that the logarithmic fidelity shows a singular behavior around the vicinity of the critical point and the critical exponent of the correlation length is also obtained numerically.

Decay of quantum Loschmidt echo and fidelity in the broken phase of the Lipkin-Meshkov-Glick model

“…In the symmetric phase, our result reduces to those given in Refs. [31,36]; in the broken phase, our result predicts an N dependence of the fidelity, which was observed numerically previously [31].…”

“…This implies that, for large N, the logarithm of the fidelity is linear in the system's size N, a phenomenon observed numerically in Ref. [31]. Furthermore, the fidelity has different scaling behaviors on the two sides of the critical point hc = 1.…”

“…This leads to the study of two other quantities at QPTs, one being the fidelity, defined as the overlap of the ground states corresponding to two neighboring values of X (say, X and X'), and the other being the quantum Loschmidt echo (LE), which gives a measure of the deviation of quantum motions under X and X' for the same initial state. At QPTs, dramatic decay has been observed both in the fidelity (with respect to X) [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and in the LE (with respect to both X and time) [32][33][34][35][36][37][38][39].…”

“…Numerical simulations show that this model has the following property: The decay rate of the fidelity is linear in the particle number N in the broken phase, in contrast to the finite decay rate in the symmetric phase in the thermodynamic limit [26,31,42], This looks peculiar, since, in most of the models that have been studied, the fidelity (also the LE) shows the same type of decaying behavior on the two sides of the critical point. In fact, as shown in Ref.…”

Genuine multipartite correlations distribution in the criticality of the Lipkin-Meshkov-Glick model

“…Even though we are using the anisotropy parameter γ = 0.5 for all of our calculations, the ground state is still an approximation of the GHZ-like state, so the spins are correlated. Some previous works in quantum information theory used entanglement [4][5][6][22][23][24][25][26][27] and others correlations [28][29][30][31][32][33][34] as order parameter to detect the second order phase transition of this model. Furthermore, there are also some previous works in quantum information that make use of FSS [4, 5, 27, 29-31, 33, 34] for the calculus of exponents in the LMG model.…”