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# Correlation Properties of anisotropic XY Model with a Sudden Quench 

Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 

Hongli Guo, Zhao Liu, Heng Fan, and Shu Chen 

(Dated: October 31, 2018) 

Starting from a general Hamiltonian which may undergo a quantum phase transition (QPT) with the change of a controllable parameter, we obtain a general conclusion that in a sudden quench system, when the final Hamiltonian is fixed, the behavior of the time-averaged expectation of any observable has close relationship with the gapless excitation of the initial Hamiltonian. To clarify our conclusion, we investigate the two-spin correlation of a XY chain in a transverse field under a sudden quench at zero temperature. The critical property of the derivative of quench two-spin correlation and the long-range correlation of the quench system are analyzed. 

Quantum phase transition (QPT), which is driven by quantum fluctuation and occurs at zero temperature, is a very important research area recently [1]. Besides the static properties of QPT, the non-equilibrium dynamics induced by the quench of the parameter in the Hamilto- nian through a critical point has always been an attrac- tive topic in condensed matter physics [2–18]. 

Generally speaking, a dynamical evolution can be in- duced by either a sudden quench or a slow quench. Tak- ing the spin chain in the magnetic field as an example, the typical paradigm of sudden quench is as follows. Ini- tially ( t < 0) the magnetic field h ( t < 0) = a . Then at time t = 0 the magnetic field is changed to b suddenly, namely h ( t ≥ 0) = b and the system begins to evolve from the initial state. After a long enough time evolution, the time-averaged expectation value of an observable A reaches a steady value, which we define as quench A . Quench A is 〈 A ( a, b ) 〉 = lim$_{t}$$_{→∞}$ 1 t ∫ t $_{0}$〈 A ( a, b, τ ) 〉 dτ [13], with 〈 A ( a, b, τ ) 〉 the expectation value of observable A at time t = τ with initial magnetic field a and final mag- netic field b . One can fix b ( a ) to investigate the relation between quench A and a ( b ). 

The magnetization M and its derivative with respect to magnetic field ∂$_{h}$M of transverse Ising model are an- alyzed in Ref. [14]. It is found that ∂$_{h}$M has similar behavior for static system and sudden quench system. Here interesting questions arise: why does this similarity happen and whether other quantities still exhibit such similarity? 

In this work, we start from a general Hamiltonian H ( λ ) containing a controllable parameter λ to search for the answers. We obtain a general conclusion that in a sud- den quench system, when the final Hamiltonian is fixed, the behavior of the time-averaged expectation of any ob- servable has close relationship with the gapless excitation of the initial Hamiltonian. There is a similarity between the critical phenomena of sudden quench and static sys- tem. Then we numerically investigate the sudden quench properties of two-spin correlation of an anisotropic XY chain in a transverse field [19]. The results are consistent with our conclusion. We also research the long-range 

PACS numbers: 64.60.Ht, 75.10.Pq, 64.60.Cn 

## I. INTRODUCTION 

correlation of the system in both directions parallel and perpendicular to the magnetic field. There is no quali- tative difference between static and sudden quench case in the direction parallel to the magnetic field, but a re- markable difference exists in the direction perpendicular to the magnetic field, which is consistent with the result in Ref.[3]. At last, we briefly discuss the case in which initial Hamiltonian is fixed. 

Here we consider a general Hamiltonian H ( λ ) = H$_{0}$ + λH$_{I}$ which contains a controllable parameter λ . We sup- pose the ground state of H ( λ ) is non-degenerate and with the varying of λ the system can undergo a QPT hap- pening at λ = λ$_{c}$ , where the energy gap of H ( λ ) van- ishes in the thermodynamics limit (TL). This is the case for many condensed matter systems, especially for spin chains. For example H$_{I}$ = $^{∑}$$_{i}$σ z i and λ is the magnetic field for transverse Ising model. 

Now we try to derive the singular behavior of the time- averaged expectation value of any observable A when QPT happens, both for static case and sudden quench case. At time t < 0, λ = a and the eigenvectors and eigenvalues of H ( a ) are denoted as | ψ$_{n}$ ( a ) 〉 and E$_{n}$ ( a ) respectively. At zero temperature the system is in the ground state of this initial Hamiltonian H ( a ), namely | Ψ( t < 0) 〉 = | ψ$_{0}$ ( a ) 〉 . Therefore for static case, which means λ = a for t ≥ 0, the the time-averaged expec- tation value is just 〈 A ( a ) 〉 = 〈 ψ$_{0}$ ( a ) | A | ψ$_{0}$ ( a ) 〉 . By the perturbation theory, we know 

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which leads to 

<img src='content_image/395113.jpg'>

## II. GENERAL MODEL 

