Scaling of geometric phase and fidelity susceptibility across the critical points and their relations

Abstract

It was shown via numerical simulations that geometric phase (GP) and fidelity susceptibility (FS) in some quantum models exhibit universal scaling laws across phase transition points. Here we propose a singular function expansion method to determine their exact form across the critical points as well as their corresponding constants. For the models such as anisotropic XY model where the energy gap is closed and reopened at the special points (k0 = 0, π), scaling laws can be found as a function of system length N and parameter deviation λ − λc (where λc is the critical parameter). Intimate relations for the coefficients in GP and FS have also been determined. However in the extended models where the gap is not closed and reopened at these special points, the scaling as a function of system length N breaks down. We also show that the second order derivative of GP also exhibits some intriguing scaling laws across the critical points. These exact results can greatly enrich our understanding of GP and FS in the characterization of quantum phase transitions. [14][15][16]. GP has become a central concept in amounts of investigations in recent decades as an important tool to study the geometric feature of Hamiltonians [17][18][19]; especially, it can even be used to characterize topological phase transitions [20][21][22], which are beyond the accessibility of Landau theory of phase transition. This phase can also be directly measured in experiments [3,[23][24][25][26]. Across the critical points the derivative of GP exhibits universal scaling laws [27][28][29].Fidelity susceptibility (FS) based on the overlap of ground state functions is another way beyond the Landau paradigm to characterize quantum phase transitions [30][31][32][33][34][35][36][37][38][39][40][41][42][43]. The FS is not defined along a closed trajectory in parameter space, thus it is not directly related to the global geometric feature of the ground state. However since the structure of wave functions in two different phases are different, we see that FS also exhibits some scaling laws across critical points; see reviews in Refs. [29,32].In previous literatures all these scaling laws are exploited by numerical simulations, thus our understanding of these laws are limited although they have been widely investigated [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. Here in this work these scaling laws are obtained exactly using a singular function expansion method, in which all coefficients are also determined exactly. We show that these two measurements are essentially determined by the same physics across the critical points, thus their coefficients also have some intimate relations. The coefficients of the divergent terms only reflect how and in which way the energy gap is closed and reopened during phase transition, thus do not carry information about the topological properties of ground state wave functions. We also find that the constant term in FS is accompanied by a discontinuous jump across the critical points, thus does not...