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arXiv:1902.06649 · Journal of Statistical Mechanics Theory and Experiment

Phase transition in complex-time Loschmidt echo of short and long range spin chain

Leonardo Santilli, Miguel Tierz

Published Jun 1, 2020• 9 citations
arXivHTMLPDF

Abstract

Abstract We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for an XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as e − α l − j / l − j p + 1 , with p integer or natural number and α ⩾ 0, we show that there are third order phase transitions in a scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz determinants with a pure Fisher–Hartwig singularity, to obtain exactly the Loschmidt echo for complex times and discuss the associated Stokes phenomena. We also study the case of a finite chain for one of the generalized XX models.

DOI
10.1088/1742-5468/ab837b

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Abstract We explain and exploit the random matrix formulation of the Loschmidt echo for the XX spin chain, valid for multiple domain wall initial states and also for an XX spin chain generalized with additional interactions to more neighbours. For models with interactions decaying as e − α l − j / l − j p + 1 , with p integer or natural number and α ⩾ 0, we show that there are third order phase transitions in a scaling limit of the complex-time Loschmidt echo amplitudes. For the long-range version of the chain, we use an exact result for Toeplitz determinants with a pure Fisher–Hartwig singularity, to obtain exactly the Loschmidt echo for complex times and discuss the associated Stokes phenomena. We also study the case of a finite chain for one of the generalized XX models.
We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transition persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.
A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that nonanalyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices, we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples---both in the integrable and the nonintegrable cases---where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.
Abstract We have considered the spin-1/2 XXZ chain model which is exactly solvable with the Bethe anzats method. It is shown adding a single defect away from edges is sufficient to see a chaos signature in the model. In this work, we have studied the time behavior of the Loschmidt-echo (LE) by considering domain-wall and Neel states as two product initial states and also a Bell state as an entangled initial state. We have used the numerical full diagonalization method to have full spectrum of the model. It is shown that very short times behavior of the LE is independent from defect and initial state, and the decay is quadratic. Moreover, results show that in the chaotic regime and for intermediate times, the exponential decay behavior of the LE shows some changes by removing defect from the chain in the domain-wall and Bell states, while no changes is seen in the Neel state. To get some intuition of these changes, we have addressed the relation between the dynamics of the LE and the local density of states features.
We study the the ground state fidelity and the ground state Loschmidt echo of a three-site interacting XX chain in presence of a staggered field which exhibits special types of quantum phase transitions due to change in the topology of the Fermi surface, apart from quantum phase transitions from gapped to gapless phases. We find that, on one hand, the fidelity is able to detect only the boundaries separating the gapped from the gapless phase; it is completely insensitive to the phase transition from the two Fermi points region to the four Fermi points region lying within this gapless phase. On the other hand, the Loschmidt echo shows a dip only at a special point in the entire phase diagram and hence fails to detect any quantum phase transition associated with the present model. We provide appropriate arguments in support of this anomalous behavior.
Abstract. We study the time evolution of an integrable many-particle system, described by the q -boson Hamiltonian in the limit of strong bosonic interactions q → ∞ . It is shown that, for a particular class of pure initial states, the analytical calculation of certain observables simplifies considerably. Namely, we provide exact formulas for the calculation of the Loschmidt-echo and the emptiness formation probability, where the computational time scales polynomially with the particle number. Moreover, we construct a non-local mapping of the q -boson model to the XX spin chain, and show how this can be utilized to obtain the time evolution of various local bosonic observables for translationally invariant initial states. The results obtained via the bosonic and fermionic picture show perfect agreement. In the infinite volume and large time limits, we rigorously verify the prediction of the generalized Gibbs ensemble for homogeneous initial Fock states.
We address the computation of the Loschmidt echo in interacting integrable spin chains after a quantum quench. We focus on the massless regime of the XXZ spin-1/2 chain and present exact results for the dynamical free energy (Loschmidt echo per site) for a special class of integrable initial states. For the first time we are able to observe and describe points of non-analyticities using exact methods, by following the Loschmidt echo up to large real times. The dynamical free energy is computed as the leading eigenvalue of an appropriate Quantum Transfer Matrix, and the non-analyticities arise from the level crossings of this matrix. Our exact results are expressed in terms of “excited-state” thermodynamic Bethe ansatz equations, whose solutions involve non-trivial Riemann surfaces. By evaluating our formulas, we provide explicit numerical results for the quench from the Néel state, and we determine the first few non-analytic points.