Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0812.4750

JORDAN-WIGNER FERMIONIZATION AND THE THEORY OF LOW-DIMENSIONAL QUANTUM SPIN MODELS.: DYNAMIC PROPERTIES

O. DERZHKO

Published Jun 1, 2008• 3 citations
arXivHTMLPDF

Abstract

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 $S_{zz}({\bf{k}},ω)$ 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.

DOI
10.1142/9789812709455_0002

Related papers

JORDAN-WIGNER FERMIONIZATION AND THE THEORY OF LOW-DIMENSIONAL QUANTUM SPIN MODELS. DYNAMIC PROPERTIES

Oleg Derzhko

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.

•Jan 1, 2008•2 citations

Dynamics of dimer and z spin component fluctuations in spin-1/2 XY chain

Oleg Derzhko, Taras Krokhmalskii, Petro A. Hlushak

One-dimensional quantum spin-1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors).This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions.A number of dynamic quantities (e.g.related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.We consider the spin-1/2 XY chain in a transverse ( z) magnetic field with the Hamiltonianand calculate the dynamic structure factorsis the dimer operator.The results for the dynamic transverse structure factor S zz (κ, ω) and for the dynamic dimer structure factor S DD (κ, ω) are known, whereas the analysis of the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been reported so far.We compare different two-fermion dynamic quantities contrasting their generic and specific features.

Condensed Matter Physics•Jan 1, 2005•0 citations

Jordan-Wigner Fermions and Dynamic Probes of Quantum Spin Chains

Oleg Derzhko, Taras Verkholyak

With the help of the Jordan–Wigner transformation the spin-12 XY chains can be reformulated in terms of noninteracting spinless fermions and, as a result, many statistical-mechanics calculations can be performed rigorously, i.e. without making any simplifying approximations. We are interested in dynamic properties of such chains (time-dependent spin correlation functions, dynamic structure factors, dynamic susceptibilities) which are of great importance for interpretation of experimental data. We have worked out a number of dynamic quantities explicitly as well as have performed a general analysis of the two-fermion continua which are relevant for different dynamic quantities.

Presented at•Jan 1, 2003•0 citations

Jordan–Wigner fermionization for spin-½ systems in two dimensions: A brief review

Oleg Derzhko

We review the papers on the Jordan{Wigner transformation in two dimensions to comment on a possibility of examining the statistical mechanics properties of two{dimensional spin{ 1 2 models.We discuss in some detail the two{dimensional spin{ 1 2 isotropic XY and Heisenberg models.

arXiv:cond-mat/0101188•Journal of Physical Studies•Jan 1, 2001•25 citations

Dimer and trimer fluctuations in the s=12 transverse XX chain

Oleg Derzhko, Taras Krokhmalskii, Joachim Stolze +1 more

Exact results for the dynamic dimer and trimer structure factors of the one-dimensional $s=\frac{1}{2}\phantom{\rule{0.3em}{0ex}}XX$ model in a transverse magnetic field $(\ensuremath{\Vert}z)$ are presented and discussed in relation to known exact results for the dynamic spin structure factors. In the framework of the Jordan-Wigner representation, the accessible spectrum of the dimer fluctuation operator is limited to two-fermion excitations, whereas that of the trimer fluctuation operator involves two- and four-fermion excitations. The spectral boundaries, soft modes, and singularity structure of the four-fermion excitation continuum as probed by the dynamic trimer structure factor are examined and compared to corresponding properties of the two-fermion excitation continuum, as probed by the dynamic dimer and transverse spin structure factors.

arXiv:cond-mat/0507203•Physical Review B•Mar 31, 2005•7 citations

Jordan-Wigner fermions and the spin-1/2 anisotropic XY model on a square lattice

Oleg Derzhko, Johannes Richter, Taras Verkholyak

Using the two-dimensional Jordan-Wigner fermionization we calculate the thermodynamic quantities of the (spatially anisotropic) square-lattice spin-1/2 anisotropic XY (XZ) model. We compare the results of different approaches for the ground-state and thermodynamic properties of the model.

arXiv:cond-mat/0111203•AcPPB•Nov 11, 2001•0 citations
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.
One-dimensional quantum spin-1/2 XY models admit the rigorous analysis not only of their static properties (i.e. the thermodynamic quantities and the equal-time spin correlation functions) but also of their dynamic properties (i.e. the different-time spin correlation functions, the dynamic susceptibilities, the dynamic structure factors).This becomes possible after exploiting the Jordan-Wigner transformation which reduces the spin model to a model of spinless noninteracting fermions.A number of dynamic quantities (e.g.related to transverse spin operator or dimer operator fluctuations) are entirely determined by two-fermion excitations and can be examined in much detail.We consider the spin-1/2 XY chain in a transverse ( z) magnetic field with the Hamiltonianand calculate the dynamic structure factorsis the dimer operator.The results for the dynamic transverse structure factor S zz (κ, ω) and for the dynamic dimer structure factor S DD (κ, ω) are known, whereas the analysis of the dynamic structure factor S zD (κ, ω) = (S Dz (κ, ω)) has not been reported so far.We compare different two-fermion dynamic quantities contrasting their generic and specific features.
With the help of the Jordan–Wigner transformation the spin-12 XY chains can be reformulated in terms of noninteracting spinless fermions and, as a result, many statistical-mechanics calculations can be performed rigorously, i.e. without making any simplifying approximations. We are interested in dynamic properties of such chains (time-dependent spin correlation functions, dynamic structure factors, dynamic susceptibilities) which are of great importance for interpretation of experimental data. We have worked out a number of dynamic quantities explicitly as well as have performed a general analysis of the two-fermion continua which are relevant for different dynamic quantities.
Exact results for the dynamic dimer and trimer structure factors of the one-dimensional s=12XXs=\frac{1}{2}\phantom{\rule{0.3em}{0ex}}XXs=21​XX model in a transverse magnetic field $(\ensuremath{\Vert}z)$ are presented and discussed in relation to known exact results for the dynamic spin structure factors. In the framework of the Jordan-Wigner representation, the accessible spectrum of the dimer fluctuation operator is limited to two-fermion excitations, whereas that of the trimer fluctuation operator involves two- and four-fermion excitations. The spectral boundaries, soft modes, and singularity structure of the four-fermion excitation continuum as probed by the dynamic trimer structure factor are examined and compared to corresponding properties of the two-fermion excitation continuum, as probed by the dynamic dimer and transverse spin structure factors.