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arXiv:2601.18867

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

Katsumasa Nakayama, Kei Suzuki

cond-mat.stat-mechcond-mat.mes-hallcond-mat.str-elhep-thquant-phSubmitted Jan 26, 2026• Announced Jan 28, 2026• 0 citations
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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.

Comments
14 pages, 11 figures, 1 table

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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.
When studying the quantum Ising chain in a transverse field, we usually transform the spin operators to fermion operators by applying the Jordan-Wigner transformation,i.e.we take it for granted that the spin model and corresponding fermionic model are equivalent naively. But deliberate treatment could reveal that the equivalence accompanies some delicate conditions.Redundant degrees of freedom should be projected out carefully, especially for finite length of chain. In this paper, we compare the two models in detail so as to find the conditions of equivalence of them,including the interaction, the boundary conditions, and odevity of the number of lattice sites.We also exemplify in detail how to project out the redundant degrees of freedom for the fermionic representation.
The Jordan-Wigner transformation is known as a powerful tool in condensed matter theory, especially in the theory of low-dimensional quantum spin systems. The aim of this chapter is to review the application of the Jordan-Wigner fermionization technique for calculating dynamic quantities of low-dimensional quantum spin models. After a brief introduction of the Jordan-Wigner transformation for one-dimensional spin one-half systems and some of its extensions for higher dimensions and higher spin values we focus on the dynamic properties of several low-dimensional quantum spin models. We start from the famous s=1/2 XX chain. As a first step we recall well-known results for dynamics of the z-spin-component fluctuation operator and then turn to the dynamics of the dimer and trimer fluctuation operators. The dynamics of the trimer fluctuations involves both the two-fermion (one particle and one hole) and the four-fermion (two particles and two holes) excitations. We discuss some properties of the two-fermion and four-fermion excitation continua. The four-fermion dynamic quantities are of intermediate complexity between simple two-fermion (like the zz dynamic structure factor) and enormously complex multi-fermion (like the xx or xy dynamic structure factors) dynamic quantities. Further we discuss the effects of dimerization, anisotropy of XY interaction, and additional Dzyaloshinskii-Moriya interaction on various dynamic quantities. Finally we consider the dynamic transverse spin structure factor Szz(k,ω)S_{zz}({\bf{k}},ω)Szz​(k,ω) for the s=1/2 XX model on a spatially anisotropic square lattice which allows one to trace a one-to-two-dimensional crossover in dynamic quantities.
The Jordan-Wigner transformation is known as a powerful tool in condensed matter theory, especially in the theory of low-dimensional quantum spin systems. The aim of this chapter is to review the application of the Jordan-Wigner fermionization technique for calculating dynamic quantities of low-dimensional quantum spin models. After a brief introduction of the Jordan-Wigner transformation for one-dimensional spin one-half systems and some of its extensions for higher dimensions and higher spin values we focus on the dynamic properties of several lowdimensional quantum spin models. We start from the famous s = 1/2 XX chain. As a first step we recall well-known results for dynamics of the z-spin-component fluctuation operator and then turn to the dynamics of the dimer and trimer fluctuation operators. The dynamics of the trimer fluctuations involves both the two-fermion (one particle and one hole) and the four-fermion (two particles and two holes) excitations. We discuss some properties of the two-fermion and four-fermion excitation continua. The four-fermion dynamic quantities are of intermediate complexity between simple two-fermion (like the zz dynamic structure factor) and enormously complex multi-fermion (like the xx or xy dynamic structure factors) dynamic quantities. Further we discuss the effects of dimerization, anisotropy of XY interaction, and additional Dzyaloshinskii-Moriya interaction on various dynamic quantities. Finally we consider the dynamic transverse spin structure factor Szz(k, ω) for the s = 1/2 XX model on a spatially anisotropic square lattice which allows one to trace a one-to-two-dimensional crossover in dynamic quantities.
We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.
We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.
We revisit the Jordan-Wigner transformation, showing that-rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators-it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan-Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies