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arXiv:1610.02556 · Physical review. B./Physical review. B

Local quench, Majorana zero modes, and disturbance propagation in the Ising chain

Gianluca Francica, Tony J. G. Apollaro, Nicola Lo Gullo, Francesco Plastina

Published Dec 1, 2016• 18 citations
arXivHTMLPDF

Abstract

We study the generation and propagation of local perturbations in a quantum many-body spin system. In particular, we study the Ising model in transverse field in the presence of a local field defect at one edge. This system possesses a rich phase diagram with different regions characterized by the presence of one or two Majorana zero modes. We show that their localized character (i) enables a characterization of the Ising phase transition through a local-only measurement performed on the edge spin, and (ii) strongly affects the propagation of quasiparticles emitted after the sudden removal of the defect, so that the dynamics of the local magnetization show clear deviations from a ballistic behavior in the presence of the Majorana fermions.

DOI
10.1103/physrevb.94.245103

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We study the generation and propagation of local perturbations in a quantum many-body spin system. In particular, we study the Ising model in transverse field in the presence of a local field defect at one edge. This system possesses a rich phase diagram with different regions characterized by the presence of one or two Majorana zero modes. We show that their localized character (i) enables a characterization of the Ising phase transition through a local-only measurement performed on the edge spin, and (ii) strongly affects the propagation of quasiparticles emitted after the sudden removal of the defect, so that the dynamics of the local magnetization show clear deviations from a ballistic behavior in the presence of the Majorana fermions.
Quantum excitations in lattice systems always propagate at a finite maximum velocity. We probe this mechanism by considering a defect traveling at a constant velocity in the quantum Ising spin chain in transverse field. Independently of the microscopic details of the defect, we characterize the expectation value of local observables at large times and large distances from the impurity, where a local quasistationary state (LQSS) emerges. The LQSS is strongly affected by the defect velocity: for superluminal defects, it exhibits a growing region where translational invariance is spontaneously restored. We also analyze the behavior of the friction force exerted by the many-body system on the moving defect, which reflects the energy required by the LQSS formation. Exact results are provided in the two limits of extremely narrow and very smooth impurity. Possible extensions to more general free-fermion models and interacting systems are discussed.
Dynamical structure factor (S(k,ωωω)) is calculated for the one-dimensional (1D) transverse field Ising model, and its recent extension to include a three spin term, with quenched binary disorder. We study the low energy modes for lattices as large as 256 sites. We show that the intense zero energy modes appear whenever the binary disorder straddles two different winding numbers. We argue that these are Majorana modes, which reside on the boundaries of the rare regions. The size distribution of Majorana pairs has a fractal behavior at the critical points. With the longer ranged interactions a spin glass transition is observed as well.
We investigate Majorana modes in a quantum spin chain with bond-dependent exchange interactions by studying its dynamics. Specifically, we consider two-time correlations for the anisotropic Kitaev-Heisenberg (KH) Hamiltonian close to the so-called Kitaev critical point. Here the model coincides with a phase boundary of two uncoupled instances of Kitaev's model for ppp-wave superconductors, together supporting a degenerate ground state characterized by multiple Majorana modes. In this regime, the real-time dynamics of local spins reveal a set of strong zero modes, corresponding to a set of protruding frequencies in the two-time correlation function. We derive perturbative interactions that map the KH spin chain onto the topological regime of Kitaev's fermionic model, thus opening up a bulk gap while retaining almost degenerate modes in the mesoscopic regime, i.e., for finite system sizes. This showcases the emergence of Majorana modes in a chain of effective dimers. Here, the binding energy within each unit cell competes with the interdimer coupling to generate a finite-size energy gap, in analogy with local energy terms in the transverse-field Ising model. These modes give rise to long coherence times of local spins located at the system edges. By breaking the local symmetry in each dimer, one can also observe a second class of Majorana modes in terms of a beating frequency in the two-time correlations function of the edge spin. Furthermore, we develop a scenario for realizing these model predictions in ion-trap quantum simulators with collective addressing of the ions.
A one-dimensional Ising model in a transverse field can be mapped onto a system of spinless fermions with ppp-wave superconductivity. In the weak-coupling BCS regime, it exhibits a zero-energy Majorana mode at each end of the chain. Here, we consider a variation of the model, which represents a superconductor with longer-ranged kinetic energy and pairing amplitudes, as is likely to occur in more realistic systems. It possesses a richer zero-temperature phase diagram and has several quantum phase transitions. From an exact solution of the model, we find that these phases can be classified according to the number of Majorana zero modes of an open chain: zero, one, or two at each end. The model possesses a multicritical point where phases with zero, one, and two Majorana end modes meet. The number of Majorana modes at each end of the chain is identical to the topological winding number of the Anderson pseudospin vector that describes the BCS Hamiltonian. The topological classification of the phases requires a unitary time-reversal symmetry to be present. When this symmetry is broken, only the number of Majorana end modes modulo 2 can be used to distinguish two phases. In one of the regimes, the wave functions of the two phase-shifted Majorana zero modes decay exponentially in space but in an oscillatory manner. The wavelength of oscillation is identical to that in the asymptotic connected spin-spin correlation of the XYXYXY model in a transverse field, to which our model is dual.
Recently, single-site observables have been shown to be useful for probing critical slowing down in sudden quench dynamics [Da\ifmmode \breve{g}\else \u{g}\fi{} et al., Phys. Rev. B 107, L121113 (2023)]. Here, we demonstrate the potential of single-site magnetization as a probe of quantum phase transitions in integrable and nonintegrable transverse-field Ising chains (TFIC). We analytically prove the requirement of zero modes for the quasistationary regime to emerge at a probe site near the edge, and show how this regime gives rise to a nonanalytic behavior in the dynamical order profiles. Our t\mathit{t}t-DMRG calculations verify the results of the quench mean-field theory for near-integrable TFIC both with finite-size and finite-time scaling analyses. We find that both finite-size and finite-time analyses suggest a dynamical critical point for a strongly nonintegrable and locally connected TFIC. We finally demonstrate the presence of a quasistationary regime in the power-law interacting TFIC, and extract local dynamical order profiles for TFIC in the long-range Ising universality class with algebraic light cones.
The study of quasiparticle dynamics is central to understanding non-equilibrium phenomena in quantum many-body systems. Direct simulation of such dynamics on quantum hardware has been limited by circuit depth and noise constraints. In this work, we use a recently developed constant-depth circuit algorithm to examine the real-time evolution of site-resolved magnetization in a transverse-field Ising chain on noisy intermediate-scale quantum devices. By representing each spin as a pair of Majorana fermions, we identify two distinct dynamical regimes governed by the relative strength of spin interaction. Furthermore, we show how local impurities can serve as probes of Majorana modes, acting as dynamical barriers in the weak coupling regime. These results demonstrate that constant-depth quantum circuits provide a viable route for studying quasiparticle propagation and for probing Majorana signatures on currently available quantum processors.