Mesons in a quantum Ising ladder (2502.15463v2)

Abstract: 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 Ising$_h^2$ integrable model, a sine-Gordon theory with $\mathbb{Z}_2$ 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 paper demonstrates that coupling two transverse-field Ising chains results in meson-like bound states through confinement phenomena.
• The research employs analytical Bethe-Salpeter equations alongside numerical TFFSA to precisely calculate meson masses and renormalized tension factors.
• These findings extend confinement concepts from high-energy physics to condensed matter, offering promising avenues for experimental quantum simulation.

Analysis of "Mesons in a Quantum Ising Ladder" (2502.15463)

Introduction

The paper investigates the unique phenomena emerging from a quantum Ising ladder, specifically when two transverse-field Ising chains (TFIC) with magnetic order are coupled. This coupling results in the confinement of previously free excitations, leading to the formation of meson-like bound states. The paper systematically explores these meson formations, characterized by fermion number parity and chain-exchanging properties. Key to this paper is the derivation and solution of the Bethe-Salpeter equation to determine the meson masses and analyze the system's excitation spectrum.

Theoretical Background

The concept of confinement, familiar from quantum chromodynamics (QCD), is here applied to a condensed matter context, where similar mechanisms appear in low-dimensional systems. Historically, soliton confinement in integrable quantum field theories (IQFT), such as the Ising model with perturbations, has drawn parallels to particle interactions in QCD due to integrability breaking. The paper employs this analogy to elucidate the meson-like interactions and spectra in the coupled Ising systems.

Specifically, when the TFICs are weakly coupled, the resulting model captures the confinement of two fermions, producing distinct sets of mesonic states. These states can be described by “intrachain” and “interchain” mesons, referring to domain walls within the same chain or between different chains, respectively.

Methodology

The research uses an array of analytical and numerical techniques. Initially, a linear potential approximation provides a qualitative picture of confinement akin to that of Ising chains, using a framework where the energy increases linearly with the separation of domain walls. However, for a more precise characterization of meson masses, the authors solve a set of Bethe-Salpeter (BS) equations under the infinite momentum frame, capturing the relativistic two-particle bound state dynamics.

Additionally, the paper employs the truncated free fermionic space approach (TFFSA), allowing for an exploration beyond the two-particle subspace and enabling accurate numerical calculations of the meson spectra across different sectors characterized by fermion number parity and chain-exchanging symmetry.

Analysis and Results

Figure 1: Illustration on formation of intrachain meson (a-c) and interchain meson (d-f) (J<0). Black dashed lines represent domain walls in each chain (intrachain domain walls) and red ones denote domain walls in between (interchain domain walls).

Meson masses were obtained analytically for weak coupling and numerically for general coupling, revealing a degeneracy between different symmetry sectors and a richer mesonic structure due to the additional degrees of freedom in the ladder system. The TFFSA confirmed these findings and provided a renormalized tension factor, essential for accurately describing the meson spectrum near criticality.

Implications

The implications of these findings span both theoretical and experimental domains. Theoretically, the results extend concepts of confinement beyond QCD, demonstrating how mesonic structures can arise from simple lattice models when combined with chain interactions. Practically, the Ising ladder model serves as a candidate for experimental realization in quantum simulators, such as cold atom or Rydberg atom setups, potentially allowing direct observation of confinement phenomena.

Future Directions

Future research could focus on deeper investigation into the non-equilibrium dynamics of such confined systems. In addition, exploring the quantum critical points and examining their robust signatures in the spectrum could provide further insights into coupled quantum systems and their analogies to high-energy physics phenomena. Finally, experimental validation in quantum simulation platforms would be pivotal to corroborating the theoretical predictions outlined.

Conclusion

The paper advances our understanding of confinement in low-dimensional systems by demonstrating the formation of meson-like states in a coupled TFIC ladder. The combined use of analytical approximations and sophisticated numerical techniques highlights the diversity and complexity of excitations that can emerge in such seemingly simple systems. These insights not only enrich the theoretical landscape of statistical mechanics but also pave the way for future experimental explorations of quantum spin systems.