Jordan-Wigner mapping between quantum-spin and fermionic Casimir effects

TL;DR

This work addresses how finite-size corrections to ground-state energies in one-dimensional spin chains can be interpreted as fermionic Casimir effects through the Jordan-Wigner transformation. It establishes a consistent definition of Casimir energy across spin and fermion representations, builds a comprehensive dictionary of Casimir types (massless, massive, damped, oscillatory, remnant), and analyzes the transverse-field Ising and XY chains under open and periodic boundaries. The key contributions include analytic expressions for Casimir energies at critical points, identification of universal 1/N scaling tied to conformal data, and the demonstration of surface-energy contributions and ground-state parity switching as distinct Casimir phenomena. The findings highlight that spin-chain platforms can realize and probe fermionic Casimir effects experimentally, offering a concrete route to observe lattice fermion Casimir physics in controllable quantum simulators and materials.

Abstract

The Jordan-Wigner transformation connects spin operators in one-dimensional spin systems and fermionic operators. In this work, we elucidate the relationship between the finite-size corrections in the spin representation and the fermionic Casimir effect in the corresponding fermion representation. In particular, we focus on the ground-state energy of one-dimensional transverse-field Ising and XY models, and show that all finite-size corrections can be interpreted as lattice fermionic Casimir effects. We further find several types of Casimir phenomena, such as the conventional Casimir energy from massless fields, damping behavior from massive fields, vanishing behavior from flat or nonrelativistic bands, and oscillating behavior from the finite-density effect. Our findings establish a dictionary between finite-size corrections in spin chains and fermionic Casimir effects, and provide experimentally relevant platforms for the fermionic Casimir phenomena.

Figures (11)

• Figure 1: Dispersion relations of fermions in the TFIM under some transverse fields $h$, based on Eq. (\ref{['eq:TFIM_lambda_bulk']}). The dispersion relations at $h>0$ are shifted by $\pi$, compared with those for $h<0$.
• Figure 2: Casimir energy in the transverse-field Ising (TFI) chain under the open boundary condition.
• Figure 3: Casimir energy in the transverse-field Ising chain under the periodic boundary condition.
• Figure 4: Phase diagram of Casimir effect in the one-dimensional transverse-field Ising model under the open or periodic boundary condition (OBC or PBC).
• Figure 5: Dispersion relations of fermions in TFXY models under some transverse fields $h$, based on Eq. (\ref{['eq:TFXY_lambda_bulk']}). (i) $\gamma=0$ (the TFXX model). (ii) $\gamma=0.5$. (iii) $\gamma=\sqrt{2}$. The dispersion relations at $h>0$ are shifted by $\pi$, compared with those for $h<0$.