Abstract
We study the out-of-equilibrium probability distribution function of the local order parameter in the transverse field Ising quantum chain. Starting from a fully polarised state, the relaxation of the ferromagnetic order is analysed: we obtain a full analytical description of the late-time stationary distribution by means of a remarkable relation to the partition function of a 3-states classical model. Accordingly, depending on the phase whereto the post-quench Hamiltonian belongs, the probability distribution may locally retain memories of the initial long-range order. When quenching deep in the broken-symmetry phase, we show that the stationary order-parameter statistics is indeed related to that of the ground state. We highlight this connection by inspecting the ground-state equilibrium properties, where we propose an effective description based on the block-diagonal approximation of the nn-point spin correlation functions.
Highlights
Before proceeding with the direct computation of the probability distribution function, let us summarise some properties of the Probability Distribution Function (PDF) which are valid in the limit 1 whenever the state satisfies cluster decomposition and it is characterised by a finite correlation length
We have studied the full counting statistics of the local order parameter in the transverse field Ising quantum chain
At the equilibrium at zero temperature, we proposed a fairly accurate description for the corresponding generating function which is based on the diagonal approximation of the determinant representation
Summary
We mainly concentrate our analysis on the out-of-equilibrium and long-time stationary properties of the full counting statistics of the order parameter in the Ising quantum chain after a quench from the fully polarised initial state. Otherwise, when the state is quenched within the broken-symmetry phase, a simple Gaussian description is no more sufficient and the ground state of the post-quench Hamiltonian starts to play a crucial role in locally preserving the initial long-range order which eventually survives in the long-time limit. - In Sec. 6 we further propose an appealing interpretation of the stationary distribution in terms of the properties of the post-quench ground-state We argue that such relation is exact in a special scaling regime deep in the ferromagnetic phase. - in Sec. 7 we draw our conclusions; we relegate all supplementary calculations in the Appendices A-D
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