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arXiv:2005.03145 · Physical review. B./Physical review. B

Confinement and bound states of bound states in a transverse-field two-leg Ising ladder

Flávia B. Ramos, Máté Lencsés, J. C. Xavier, Rodrigo G. Pereira

Published Jul 16, 2020• 21 citations
arXivHTMLPDF

Abstract

Weakly coupled Ising chains provide a condensed-matter realization of confinement. In these systems, kinks and antikinks bind into mesons due to an attractive interaction potential that increases linearly with the distance between the particles. While single mesons have been directly observed in experiments, the role of the multiparticle continuum and bound states of mesons in the excitation spectrum is far less clear. Using time-dependent density-matrix renormalization group methods, we study the dynamical structure factors of one- and two-spin operators in a transverse-field two-leg Ising ladder in the ferromagnetic phase. The propagation of time-dependent correlations and the two-spin excitation spectrum reveal the existence of interchain bound states, which are absent in the one-spin dynamical structure factor. We also identify two-meson bound states that appear at higher energies, above the thresholds of several two-meson continua.

DOI
10.1103/physrevb.102.014426

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Weakly coupled Ising chains provide a condensed-matter realization of confinement. In these systems, kinks and antikinks bind into mesons due to an attractive interaction potential that increases linearly with the distance between the particles. While single mesons have been directly observed in experiments, the role of the multiparticle continuum and bound states of mesons in the excitation spectrum is far less clear. Using time-dependent density-matrix renormalization group methods, we study the dynamical structure factors of one- and two-spin operators in a transverse-field two-leg Ising ladder in the ferromagnetic phase. The propagation of time-dependent correlations and the two-spin excitation spectrum reveal the existence of interchain bound states, which are absent in the one-spin dynamical structure factor. We also identify two-meson bound states that appear at higher energies, above the thresholds of several two-meson continua.
The Heisenberg–Ising spin ladder is one of the few short-range models showing confinement of elementary excitations without the need of an external field, neither transverse nor longitudinal. This feature makes the model suitable for an experimental realization with ultracold atoms. In this paper, we combine analytic and numerical techniques to precisely characterize its spectrum in the regime of Hamiltonian parameters showing confinement. We find two kinds of particles, which we dub intrachain and interchain mesons, that correspond to bound states of kinks within the same chain or between different ones, respectively. The ultimate physical reasons leading to the existence of two families of mesons is a residual double degeneracy of the ground state: the two types of mesons interpolate either between the same vacuum (intrachain) or between the two different ones (interchain). While the intrachain mesons can also be qualitatively assessed through an effective mean field description and were previously known, the interchain ones are new and they represent general features of spin ladders with confinement.
Abstract The Heisenberg–Ising spin ladder is one of the few short-range models showing confinement of elementary excitations without the need of an external field, neither transverse nor longitudinal. This feature makes the model suitable for an experimental realization with ultracold atoms. In this paper, we combine analytic and numerical techniques to precisely characterize its spectrum in the regime of Hamiltonian parameters showing confinement. We find two kinds of particles, which we dub intrachain and interchain mesons, that correspond to bound states of kinks within the same chain or between different ones, respectively. The ultimate physical reasons leading to the existence of two families of mesons is a residual double degeneracy of the ground state: the two types of mesons interpolate either between the same vacuum (intrachain) or between the two different ones (interchain). While the intrachain mesons can also be qualitatively assessed through an effective mean field description and were previously known, the interchain ones are new and they represent general features of spin ladders with confinement.
A bstract When two transverse-field Ising chains (TFICs) with magnetic order are coupled, the original free excitations become confined, giving rise to meson-like bound states. In this work, we study such bound states systematically. The mesons are characterized by their fermion number parity and chain-exchanging properties, which lead to distinct sets of mesonic states. The meson masses are determined by solving the Bethe-Salpter equation. An interesting observation is the additional degeneracy in the chain-exchanging odd sectors. Beyond the two particle approximation, we exploit the truncated free fermionic space approach to calculate the spectrum numerically. Corrections to the meson masses are obtained, and the degeneracy is further confirmed. The characterization and degeneracy can be connected to the situation when each chain is tuned to be quantum critical, where the system is described by the Isingh2 {\textrm{Ising}}_h^2 Isingh2​ Ising h 2 integrable model, a sine-Gordon theory with ℤ 2 orbifold. Here we establish a clear correspondence between the particles in the bosonized form and their fermionic counterparts. Near this point, the stability of these particles is analyzed using the form factor perturbation scheme, where four particles are always present. Additionally, we calculate the evolution of the dominant dynamical structure factor for local spin operators, providing further insight into the low-energy excitations and their role in the system’s behavior. The two-particle confinement framework as well as the parity classifications may inspire the study for other coupled bi-partite systems.
The infinite Heisenberg XXZ spin-12\frac{1}{2}21​ chain in the gapped antiferromagnetic regime has two degenerate vacua and kink topological excitations (which are also called spinons) interpolating between these vacua as elementary excitations. Application of an arbitrary weak staggered longitudinal magnetic field hhh induces a long-range attractive potential between two adjacent spinons leading to their confinement into ``meson'' bound states. Exploiting the integrability of the XXZ model in the deconfined phase h=0h=0h=0, we perform perturbative calculations of the energy spectra of the two-spinon bound states in the weak confinement regime at $h\ensuremath{\rightarrow}+0$, using the strength of the staggered magnetic field hhh as a small parameter. Both transverse and longitudinal dynamical structure factors of the local spin operators are calculated as well in the two-spinon approximation in the weak confinement regime to the leading order in hhh.
We study the quasiparticle excitation and quench dynamics of the one-dimensional transverse-field Ising model with power-law (1/r^{α}) interactions. We find that long-range interactions give rise to a confining potential, which couples pairs of domain walls (kinks) into bound quasiparticles, analogous to mesonic states in high-energy physics. We show that these quasiparticles have signatures in the dynamics of order parameters following a global quench, and the Fourier spectrum of these order parameters can be exploited as a direct probe of the masses of the confined quasiparticles. We introduce a two-kink model to qualitatively explain the phenomenon of long-range-interaction-induced confinement and to quantitatively predict the masses of the bound quasiparticles. Furthermore, we illustrate that these quasiparticle states can lead to slow thermalization of one-point observables for certain initial states. Our work is readily applicable to current trapped-ion experiments.
We investigate the interplay between confinement and Krylov complexity in the long-range transverse-field Ising chain, where power-law interactions ∼1/rα\sim 1/r^α∼1/rα generate an effective confining potential between domain walls without an explicit symmetry-breaking field. Using the time-averaged Krylov state complexity after a quench as a diagnostic, we find that the complexity is strongly suppressed and only weakly system size-dependent at small ααα and small transverse field hxh_xhx​, where confinement is strongest, and crosses over to rapid growth at larger ααα, where it increases with system size. Within the confined regime, the oscillation frequencies of the complexity agree with the meson masses of an effective two-kink model, with the dominant frequency growing linearly with the lightest meson mass and the time-averaged complexity decreasing approximately as its inverse square. We further study operator Krylov complexity at finite temperature using the Wightman inner product. At low temperature, it exhibits bounded oscillations that directly encode the meson spectrum, whereas this structure is progressively washed out as the temperature increases. These results identify Krylov complexity as a spectroscopic probe of interaction-induced confinement and reproduce, in a microscopic lattice model, the hallmarks of Krylov complexity identified in confining field theories.