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arXiv:cond-mat/0111203 · AcPPB

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

Oleg Derzhko, Johannes Richter, Taras Verkholyak

Published Nov 11, 2001• 0 citations
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Abstract

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.
DOI
10.48550/arxiv.cond-mat/0111203

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The Jordan–Wigner transformation is applied to the spatially anisotropic spin-1/2 Heisenberg model on a square lattice with the nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions. The transformed Hamiltonian describes the interacting spinless fermions that hop between neighbor sites in a gauge field. Using the mean-field-type approximation to both the direct interaction between fermions and the phase factors, which represent the gauge field, the problem is reduced to that concerning a free Fermi gas. Two types of antiferromagnetic ordering (the N´eel and collinear ones) are considered. By calculating the ground-state energies, the phase transitions induced by the interaction frustration were analyzed.
The two-dimensional Jordan-Wigner transformation is used to investigate the zero-temperature spin-Peierls transition for an anisotropic two-dimensional XY\mathrm{XY}XY model in the adiabatic limit. The phase diagram between the dimerized (D) state and uniform (U) state is shown in the parameter space of dimensionless interchain coupling $h{(=J}_{\ensuremath{\perp}}/J)$ and spin-lattice coupling \ensuremath{\eta}. It is found that the spin-lattice coupling $\ensuremath{\eta}$ must exceed some critical value ${\ensuremath{\eta}}_{c}$ in order to reach the D phase for any finite h. The dependence of ${\ensuremath{\eta}}_{c}$ on h is given by $\ensuremath{-}1/\mathrm{ln}h$ for $\stackrel{\ensuremath{\rightarrow}}{h}0$ and the transition between the U and D phases is of first order for at least $h>{10}^{\ensuremath{-}3}.$
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 present a numerical self-consistent variational approach based on the Jordan-Wigner transformation for two-dimensional spin systems. We apply it to the study of the well-known quantum (S=1/2)(S=1/2)(S=1/2) antiferromagnetic XXZ\mathrm{XXZ}XXZ system as a function of the easy-axis anisotropy $\ensuremath{\Delta}$ on a periodic square lattice. For the SU(2)\mathrm{SU}(2)SU(2) case the method converges to a N\'eel ordered ground state irrespective of the input density profile used and in accordance with other studies. This shows the potential utility of the proposed method to investigate more complicated situations such as frustrated or disordered systems.