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arXiv:2204.06354 · Physical review. E

Complexity in the Lipkin-Meshkov-Glick model

Kunal Pal, Kuntal Pal, Tapobrata Sarkar

Published Apr 27, 2023• 21 citations
arXivHTMLPDF

Abstract

We study complexity in a spin system with infinite-range interaction, via the paradigmatic Lipkin-Meshkov-Glick (LMG) model, in the thermodynamic limit. Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived, which helps us to establish several distinguishing features compared to complexity in other known spin models. In a time-independent LMG model, close to phase transition, the NC diverges logarithmically, much like the entanglement entropy. Remarkably, however, in a time-dependent scenario, this divergence is replaced by a finite discontinuity, as we show by using the Lewis-Riesenfeld theory of time-dependent invariant operators. The FSC of a variant of the LMG model shows different behavior compared to quasifree spin models. Namely, it diverges logarithmically when the target (or reference) state is near the separatrix. Numerical analysis indicates that this is due to the fact that geodesics starting with arbitrary boundary conditions are "attracted" toward the separatrix and that near this line, a finite change in the affine parameter of the geodesic produces an infinitesimal change of the geodesic length. The same divergence is shared by the NC of this model as well.

DOI
10.1103/physreve.107.044130

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We study complexity in a spin system with infinite-range interaction, via the paradigmatic Lipkin-Meshkov-Glick (LMG) model, in the thermodynamic limit. Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived, which helps us to establish several distinguishing features compared to complexity in other known spin models. In a time-independent LMG model, close to phase transition, the NC diverges logarithmically, much like the entanglement entropy. Remarkably, however, in a time-dependent scenario, this divergence is replaced by a finite discontinuity, as we show by using the Lewis-Riesenfeld theory of time-dependent invariant operators. The FSC of a variant of the LMG model shows different behavior compared to quasifree spin models. Namely, it diverges logarithmically when the target (or reference) state is near the separatrix. Numerical analysis indicates that this is due to the fact that geodesics starting with arbitrary boundary conditions are "attracted" toward the separatrix and that near this line, a finite change in the affine parameter of the geodesic produces an infinitesimal change of the geodesic length. The same divergence is shared by the NC of this model as well.
Abstract We use the spread complexity (SC) of a time-evolved state after a sudden quantum quench in the Lipkin–Meshkov–Glick (LMG) model prepared in the ground state as a probe of the quantum phase transition when the system is quenched toward the critical point. By studying the growth of the effective number of elements of the Krylov basis that contributes to the SC more than a preassigned cutoff, we show how the two phases of the LMG model can be distinguished. We also explore the time evolution of spread entropy after both non-critical and critical quenches. We show that the sum contributing to the spread entropy converges slowly in the symmetric phase of the LMG model compared to that in the broken phase, and for a critical quench, the spread entropy diverges logarithmically at late times.
Abstract We study information theoretic quantities in models with three and four spin interactions. These models show distinctive characteristics compared to their nearest neighbour (NN) counterparts. Here, we quantify these in terms of the Nielsen complexity (NC) in static and quench scenarios, the Fubini–Study complexity (FSC), and the entanglement entropy (EE). The models that we study have a rich phase structure, and we show how the difference in the nature of phase transitions in these, compared to ones with NN interactions, result in different behaviour of information theoretic quantities, from ones known in the literature. For example, the derivative of the NC does not diverge but shows a discontinuity near continuous phase transitions, and the FSC may be regular and continuous across such transitions. We also study multiple quench scenarios in these models and contrast these with quenches in the transverse XY model. The EE shows a novel discontinuity both at first and second order quantum phase transitions.
We study the dynamics of the Lipkin-Meshkov-Glick (LMG) model with a finite number of spins. In the thermodynamic limit, the ground state of the LMG model with an isotropic Hamiltonian in the broken phase breaks to a mean-field ground state with a certain direction. However, when the spin number NNN is finite, the exact ground state is always unique and is not given by a classical mean-field ground state. Here, we prove that when NNN is large but finite, through a tiny external perturbation, a localized state which is close to a mean-field ground state can be prepared, which mimics spontaneous symmetry breaking. Also, we find the localized in-plane spin polarization oscillates with two different frequencies $\ensuremath{\sim}O(1/N)$, and the lifetime of the localized state is long enough to exhibit this oscillation. We numerically test the analytical results and find that they agree very well with each other. Finally, we link the phenomena to quantum time crystals and time quasicrystals.
The spectrum of the Lipkin-Meshkov-Glick model is exactly derived in the thermodynamic limit by means of a spin-coherent-state formalism. In the first step, a classical analysis allows one to distinguish between four distinct regions in the parameter space according to the nature of the singularities arising in the classical energy surface; these correspond to spectral critical points. The eigenfunctions are then analyzed more precisely in terms of the associated roots of the Majorana polynomial, leading to exact expressions for the density of states in the thermodynamic limit. Finite-size effects are also analyzed, leading in particular to logarithmic corrections near the singularities occurring in the spectrum. Finally, we also compute expectation values of the spin operators in a semiclassical analysis in order to illustrate some subtle effects occurring in one region of the parameter space.
The fidelity metric has recently been proposed as a useful and elegant approach to identify and characterize both quantum and classical phase transitions. We study this metric on the manifold of thermal states for the Lipkin–Meshkov–Glick (LMG) model. For the isotropic LMG model, we find that the metric reduces to a Fisher–Rao metric, reflecting an underlying classical probability distribution. Furthermore, this metric can be expressed in terms of derivatives of the free energy, indicating a relation to Ruppeiner geometry. This allows us to obtain exact expressions for the (suitably rescaled) metric in the thermodynamic limit. The phase transition of the isotropic LMG model is signalled by a degeneracy of this (improper) metric in the paramagnetic phase. Due to the integrability of the isotropic LMG model, ground-state level crossings occur, leading to an ill-defined fidelity metric at zero temperature.
We study the adiabatic dynamics of the Lipkin-Meshkov-Glick (LMG) model close to its quantum critical point by linearly switching the transverse field from an initial large value to zero. We concentrate our attention on the residual energy after the quench in order to characterize the level of diabaticity of the evolution. As a function of the characteristic time of the quench $\ensuremath{\tau}$ we identify three different regimes. For fast quenches the residual energy Eres{E}_{\text{res}}Eres​ is almost independent on $\ensuremath{\tau}$. For slower quenches a second intermediate region appears in which a powerlike decay emerges with ${E}_{\text{res}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}3/2}$. By further slowing the quench rate, we find a third large-$\ensuremath{\tau}$ regime characterized by a different power law, ${E}_{\text{res}}\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}2}$. All these findings can be accounted for by means of an effective Landau-Zener approximation for the finite-size LMG model. We complete our description of the adiabatic dynamics of the LMG model through the analysis of the entanglement entropy of the evolved state.