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Ferroelectrics

1D S = 1/2 anisotropic xy model in transverse field with dzyaloshinskii-moriya interaction

Oleg Derzhko, A. P. Moina

Published Mar 1, 1994• 14 citations
Publisher

Abstract

Using Jordan-Wigner fermionization we examined the statistical mechanics of 1D s=1/2 XY model in transverse field with Dzyaloshinskii-Moriya interaction. In particular we calculated exactly the transverse dynamical susceptibility. The obtained results change dramatically in the presence of Dzyaloshinskii-Moriya interaction that can be of considerable interest in connection with the corresponding experimental studies.

DOI
10.1080/00150199408016541

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In this paper, we study the quantum coherence of one-dimensional transverse XY model with Dzyaloshinskii-Moriya interaction, which is given by the following Hamiltonian:HXY=∑i=1N((1+γ/2) σixσi+1x+(1-γ/2) σiyσi+1y-hσiz) ∑i=1ND(σixσi+1y-σiyσi+1x).(8)Here, 0 ≤ γ ≤ 1 is the anisotropic parameter, h is the magnitude of the transverse magnetic field, D is the strength of Dzyaloshinskii-Moriya (DM) interaction along the z direction. The limiting cases such as γ=0 and 1 reduce to the isotropic XX model and the Ising model, respectively. We use the Jordan-Winger transform to map explicitly spin operators into spinless fermion operators, and then adopt the discrete Fourier transform and the Bogoliubov transform to solve the Hamiltonian Eq.(8) analytically. When the DM interactions appear, the excitation spectrum becomes asymmetric in the momentum space and is not always positive, and thus a gapless chiral phase is induced. Based on the exact solutions, three phases are identified by varying the parameters:antiferromagnetic phase, paramagnetic phase, and gapless chiral phase. The antiferromagnetic phase is characterized by the dominant x-component nearest correlation function, while the paramagnetic phase can be characterized by the z component of spin correlation function. The two-site correlation functions Grxy and Gryx (r is the distance between two sites) are nonvanishing in the gapless chiral phase, and they act as good order parameters to identify this phase. The critical lines correspond to h=1, γ=2D, and h=√4D2 -γ2 + 1 for γ>0. When γ=0, there is no antiferromagnetic phase. We also find that the correlation functions undergo a rapid change across the quantum critical points, which can be pinpointed by the first-order derivative. In addition, Grxy decreases oscillatingly with the increase of distance r. The correlation function Grxy for γ=0 oscillates more dramatically than for γ=1. The upper boundary of the envelope is approximated as Grxy~r-1/2, and the lower boundary is approximately Grxy~r-3/2, so the long-range order is absent in the gapless chiral phase. Besides, we study various quantum coherence measures to quantify the quantum correlations of Eq.(8). One finds that the relative entropy CRE and the Jensen-Shannon entropy CJS are able to capture the quantum phase transitions, and quantum critical points are readily discriminated by their first derivative. We conclude that both quantum coherence measures can well signify the second-order quantum phase transitions. Moreover, we also point out a few differences in deriving the correlation functions and the associated density matrix in systems with broken reflection symmetry.

Acta Physica Sinica•Jan 1, 2018•5 citations
Using Jordan-Wigner fermionization we examined the statistical mechanics of 1D s=1/2 XY model in transverse field with Dzyaloshinskii-Moriya interaction. In particular we calculated exactly the transverse dynamical susceptibility. The obtained results change dramatically in the presence of Dzyaloshinskii-Moriya interaction that can be of considerable interest in connection with the corresponding experimental studies.
We consider the spin-12\frac{1}{2}21​ anisotropic XYXYXY chain in a transverse (z)(z)(z) field with the Dzyaloshinskii-Moriya interaction directed along the zzz axis in spin space to examine the effect of the Dzyaloshinskii-Moriya interaction on the zzzzzz, xxxxxx, and yyyyyy dynamic structure factors. Using the Jordan-Wigner fermionization approach we analytically calculate the dynamic transverse spin structure factor. It is governed by a two-fermion excitation continuum. We analyze the effect of the Dzyaloshinskii-Moriya interaction on the two-fermion excitation continuum. Other dynamic structure factors which are governed by many-fermion excitations are calculated numerically. We discuss how the Dzyaloshinskii-Moriya interaction manifests itself in the dynamic properties of the quantum spin chain at various fields and temperatures.
Within the real space renormalization group we obtain the phase portrait of the anisotropic quantum XYXYXY model on square lattice in presence of Dzyaloshinskii-Moriya (DM) interaction. The model is characterized by two parameters, $\ensuremath{\lambda}$ corresponding to XYXYXY anisotropy, and DDD corresponding to the strength of DM interaction. The flow portrait of the model is governed by two global Ising-Kitaev attractors at $(\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}1,D=0)$ and a repeller line, $\ensuremath{\lambda}=0$. This line itself is characterized by an attractor at D=0D=0D=0. Renormalization flow of concurrence suggests that the $\ensuremath{\lambda}=0$ line corresponds to a topological phase transition. The gap starts at zero on this repeller line corresponding to superfluid phase of underlying bosons; and flows towards a finite value at the Ising-Kitaev points. At these two fixed points the spin fields become purely classical, and hence the resulting Ising degeneracy can be interpreted as topological degeneracy of dual degrees of freedom. The state of affairs at the Ising-Kitaev fixed point is consistent with the picture of a ppp-wave pairing of strength $\ensuremath{\lambda}$ of Jordan-Wigner fermions coupled with Chern-Simons gauge fields.
The anisotropic XY-model in a transverse field (s=1/2) on the one-dimensional alternating superlattice (closed chain) is considered. The solution of the model is obtained by introducing a generalized Jordan-Wigner transformation which maps the system onto a non-interacting fermion gas. The exact excitation spectrum is determined by reducing the problem to a diagonalization of a block matrix, and it is shown that it is numerically identical to the one obtained by using the approximate transfer matrix method . The induced magnetization and the susceptibility χzzχ^{zz}χzz are determined as a function of the transverse field, and it is shown that, at T=0, the susceptibility presents multiple singularities. It is also shown, as expected, that this critical behaviour driven by transverse field belongs to the same universality class of the model on the alternating chain.
In this paper, we study the quantum coherence of one-dimensional transverse XY model with Dzyaloshinskii-Moriya interaction, which is given by the following Hamiltonian:HXY=∑i=1N((1+γ/2) σixσi+1x+(1-γ/2) σiyσi+1y-hσiz) ∑i=1ND(σixσi+1y-σiyσi+1x).(8)Here, 0 ≤ γ ≤ 1 is the anisotropic parameter, h is the magnitude of the transverse magnetic field, D is the strength of Dzyaloshinskii-Moriya (DM) interaction along the z direction. The limiting cases such as γ=0 and 1 reduce to the isotropic XX model and the Ising model, respectively. We use the Jordan-Winger transform to map explicitly spin operators into spinless fermion operators, and then adopt the discrete Fourier transform and the Bogoliubov transform to solve the Hamiltonian Eq.(8) analytically. When the DM interactions appear, the excitation spectrum becomes asymmetric in the momentum space and is not always positive, and thus a gapless chiral phase is induced. Based on the exact solutions, three phases are identified by varying the parameters:antiferromagnetic phase, paramagnetic phase, and gapless chiral phase. The antiferromagnetic phase is characterized by the dominant x-component nearest correlation function, while the paramagnetic phase can be characterized by the z component of spin correlation function. The two-site correlation functions Grxy and Gryx (r is the distance between two sites) are nonvanishing in the gapless chiral phase, and they act as good order parameters to identify this phase. The critical lines correspond to h=1, γ=2D, and h=√4D2 -γ2 + 1 for γ>0. When γ=0, there is no antiferromagnetic phase. We also find that the correlation functions undergo a rapid change across the quantum critical points, which can be pinpointed by the first-order derivative. In addition, Grxy decreases oscillatingly with the increase of distance r. The correlation function Grxy for γ=0 oscillates more dramatically than for γ=1. The upper boundary of the envelope is approximated as Grxy~r-1/2, and the lower boundary is approximately Grxy~r-3/2, so the long-range order is absent in the gapless chiral phase. Besides, we study various quantum coherence measures to quantify the quantum correlations of Eq.(8). One finds that the relative entropy CRE and the Jensen-Shannon entropy CJS are able to capture the quantum phase transitions, and quantum critical points are readily discriminated by their first derivative. We conclude that both quantum coherence measures can well signify the second-order quantum phase transitions. Moreover, we also point out a few differences in deriving the correlation functions and the associated density matrix in systems with broken reflection symmetry.