Quantum fidelity and quantum phase transitions in matrix product states

Abstract

Matrix product states, a key ingredient of numerical algorithms widely employed in the simulation of quantum spin chains, provide an intriguing tool for quantum phase transition engineering. At critical values of the control parameters on which their constituent matrices depend, singularities in the expectation values of certain observables can appear, in spite of the analyticity of the ground state energy. For this class of generalized quantum phase transitions we test the validity of the recently introduced fidelity approach, where the overlap modulus of ground states corresponding to slightly different parameters is considered. We discuss several examples, successfully identifying all the present transitions. We also study the finite size scaling of fidelity derivatives, pointing out its relevance in extracting critical exponents.

Quantum Critical Scaling of the Geometric Tensors

“…This sort of behavior has been observed in all the systems analyzed in Refs [5]- [9] and does amount to the critical fidelity drop at the QPTs. As we will show in the next section, in general the converse result i.e., QPT→ super-extensive growth of ds 2 (L) does not hold true: Q µν /L d can be finite in the thermodynamic limit even for gapless systems.…”

“…The intuition behind is extremely simple: at QPTs even the slightest move results in a major difference in some of the system's observables, in turn this latter has to show up in the degree of orthogonality i.e., fidelity, between the corresponding GSs. Systems of quasifree fermions have been analyzed [7], [8] as well as QPTs in matrix-product states [9]. Finite-temperature extensions have been also considered showing the robustness of the approach against mixing with low excited states [10].…”

Entanglement and quantum phase transitions in matrix-product spin-1 chains

“…How the entanglement varies when we approach a point of quantum phase transition? Answering these questions requires tools which have been developed only recently in the field of quantum information [1,2,3,4,5,6]. In this way a fruitful field of investigation at the borderline of condensed matter physics and quantum information has emerged.…”

Instabilities of the Dicke model with field interactions

“…It is known that quantum critical phenomena exhibits deep relations to the Berry phase (BP) [19,20,21,22] and the fidelity [23,24,25,26,27,28,29,30]. The BP has been extensively studied by the geometric time evolution of a quantum system, providing means to detect the quantum effects and critical behavior, such as quantum jumps and collapse [31,32,33].…”

“…The BP has been extensively studied by the geometric time evolution of a quantum system, providing means to detect the quantum effects and critical behavior, such as quantum jumps and collapse [31,32,33]. A recent proposal is to use the fidelity in identifying the quantum phase transition [24,25]. As a consequence of the dramatic changes in the structure of the ground states, the fidelity should drop at critical points.…”