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:1803.08665 · Physical review. B./Physical review. B

Topological phase transition of the anisotropic XY model with Dzyaloshinskii-Moriya interaction

T. Farajollahpour, S. A. Jafari

Published Aug 22, 2018• 14 citations
arXivHTMLPDF

Abstract

Within the real space renormalization group we obtain the phase portrait of the anisotropic quantum $XY$ model on square lattice in presence of Dzyaloshinskii-Moriya (DM) interaction. The model is characterized by two parameters, $\ensuremath{\lambda}$ corresponding to $XY$ anisotropy, and $D$ 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 $D=0$. 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 $p$-wave pairing of strength $\ensuremath{\lambda}$ of Jordan-Wigner fermions coupled with Chern-Simons gauge fields.

DOI
10.1103/physrevb.98.085136

Related papers

The XYZ chain with Dzyaloshinsky–Moriya interactions: from spin–orbit-coupled lattice bosons to interacting Kitaev chains

Sebastiano Peotta, Leonardo Mazza, Ettore Vicari +3 more

Using the density-matrix renormalization-group algorithm (DMRG) and a finite-size scaling analysis, we study the properties of the one-dimensional completely-anisotropic spin-1/2 XYZ model with Dzyaloshinsky-Moriya (DM) interactions. The model shows a rich phase diagram: depending on the value of the coupling constants, the system can display different kinds of ferromagnetic order and Luttinger-liquid behavior. Transitions from ferromagnetic to Luttinger-liquid phases are first order. We thoroughly discuss the transition between different ferromagnetic phases, which, in the absence of DM interactions, belongs to the XX universality class. We provide evidence that the DM exchange term turns out to split this critical line into two separated Ising-like transitions and that in between a disordered phase may appear. Our study sheds light on the general problem of strongly-interacting spin-orbit-coupled bosonic gases trapped in an optical lattice and can be used to characterize the topological properties of superconducting nanowires in the presence of an imposed supercurrent and of interactions.

arXiv:1403.4568•Journal of Statistical Mechanics Theory and Experiment•Sep 8, 2014•38 citations

Quantum entanglement and quantum phase transition in the XY model with staggered Dzyaloshinskii-Moriya interaction

Fu-Wu Ma, Shengxin Liu, Xiang-Mu Kong

We study the quantum entanglement and quantum phase transition (QPT) of the anisotropic spin-$1/2$ $XY$ model with staggered Dzyaloshinskii-Moriya (DM) interaction by means of the quantum renormalization group method. The scaling of coupling constants and the critical points of the system are obtained. It is found that when the number of renormalization group iterations tends to infinity, the system exhibit a QPT between the spin-fluid and N\'eel phases which correspond with two saturated values of the concurrence for a given value of the strength of DM interaction. The DM interaction can enhance the entanglement and influence the QPT of the system. To gain further insight, the first derivative of the entanglement exhibit a nonanalytic behavior at the critical point and it directly associates with the divergence of the correlation length. This shows that the correlation length exponent is closely related to the critical exponent, i.e., the scaling behaviors of the system.

arXiv:1106.0901•Physical Review A•Oct 3, 2011•87 citations

Phase boundaries in an alternating-field quantum XY model with Dzyaloshinskii-Moriya interaction: Sustainable entanglement in dynamics

Saptarshi Roy, Titas Chanda, Tamoghna Das +3 more

We report all phases and corresponding critical lines of the quantum anisotropic transverse XY model with uniform and alternating transverse magnetic fields (ATXY) in the presence of the Dzyaloshinskii-Moriya (DM) interaction by using appropriately chosen order parameters. We prove that when DM interaction is weaker than the anisotropy parameter, it has no effect at all on the zero-temperature states of the XY model with uniform transverse magnetic field (UXY), which is not the case for the ATXY model. However, when DM interaction is stronger than the anisotropy parameter, we show the appearance of a new gapless chiral phase---in the XY model with uniform as well as alternating field. We further observe that first derivatives of nearest neighbor two-site entanglement with respect to magnetic fields can detect all the critical lines present in the system. We also find that the factorization surface at zero temperature present in this model without DM interaction becomes a volume on the introduction of the latter. Moreover, DM interaction turns out to be good to generate bipartite entanglement sustainable at large times, leading to a proof of ergodic nature of bipartite entanglement in this system.

arXiv:1710.11037•Physical review. B./Physical review. B•Feb 19, 2019•18 citations

Quantum coherence of XY model with Dzyaloshinskii-Moriya interaction

Yi Tian-Cheng, Ding Yue-Ran, Jie Ren +2 more

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

Exact phase boundaries and topological phase transitions of the XYZ spin chain

Somaye Jafari

Within the block spin renormalization group, we give a very simple derivation of the exact phase boundaries of the XYZ spin chain. First, we identify the Ising order along x[over ̂] or y[over ̂] as attractive renormalization group fixed points of the Kitaev chain. Then, in a global phase space composed of the anisotropy λ of the XY interaction and the coupling Δ of the Δσ^{z}σ^{z} interaction, we find that the above fixed points remain attractive in the two-dimesional parameter space. We therefore classify the gapped phases of the XYZ spin chain as: (1) either attracted to the Ising limit of the Kitaev-chain, which in turn is characterized by winding number ±1, depending on whether the Ising order parameter is along x[over ̂] or y[over ̂] directions; or (2) attracted to the charge density wave (CDW) phases of the underlying Jordan-Wigner fermions, which is characterized by zero winding number. We therefore establish that the exact phase boundaries of the XYZ model in Baxter's solution indeed correspond to topological phase transitions. The topological nature of the phase transitions of the XYZ model justifies why our analytical solution of the three-site problem that is at the core of the present renormalization group treatment is able to produce the exact phase boundaries of Baxter's solution. We argue that the distribution of the winding numbers between the three Ising phases is a matter of choice of the coordinate system, and therefore the CDW-Ising phase is entitled to host appropriate form of zero modes. We further observe that in the Kitaev-chain the renormalization group flow can be cast into a geometric progression of a properly identified parameter. We show that this new parameter is actually the size of the (Majorana) zero modes.

arXiv:1703.01420•Physical review. E•Jul 31, 2017•6 citations

Phase Diagram of spin 1/2 XXZ Model With Dzyaloshinskii-Moriya Interaction

Rouhollah Jafari, A. Langari

We have studied the phase diagram of the one dimensional XXZ model with Dzyaloshinskii-Moriya (DM) interaction. We have applied the quantum renormalization group (QRG) approach to get the stable fixed points, critical point and the scaling of coupling constants. This model has three phases, ferromagnetic, spin-fluid and Néel phases which are separated by a critical line which depends on the DM coupling constant. We have shown that the staggered magnetization is the order parameter of the system and investigated the influence of DM interaction on the chiral ordering as a helical magnetic order.

arXiv:0812.1862•Dec 10, 2008•4 citations
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, \ensuremathλ\ensuremath{\lambda}\ensuremathλ 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 (\ensuremathλ=\ifmmode±\else\textpm\fi1,D=0)(\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}1,D=0)(\ensuremathλ=\ifmmode±\else\textpm\fi1,D=0) and a repeller line, \ensuremathλ=0\ensuremath{\lambda}=0\ensuremathλ=0. This line itself is characterized by an attractor at D=0D=0D=0. Renormalization flow of concurrence suggests that the \ensuremathλ=0\ensuremath{\lambda}=0\ensuremathλ=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 \ensuremathλ\ensuremath{\lambda}\ensuremathλ of Jordan-Wigner fermions coupled with Chern-Simons gauge fields.
Using the density-matrix renormalization-group algorithm (DMRG) and a finite-size scaling analysis, we study the properties of the one-dimensional completely-anisotropic spin-1/2 XYZ model with Dzyaloshinsky-Moriya (DM) interactions. The model shows a rich phase diagram: depending on the value of the coupling constants, the system can display different kinds of ferromagnetic order and Luttinger-liquid behavior. Transitions from ferromagnetic to Luttinger-liquid phases are first order. We thoroughly discuss the transition between different ferromagnetic phases, which, in the absence of DM interactions, belongs to the XX universality class. We provide evidence that the DM exchange term turns out to split this critical line into two separated Ising-like transitions and that in between a disordered phase may appear. Our study sheds light on the general problem of strongly-interacting spin-orbit-coupled bosonic gases trapped in an optical lattice and can be used to characterize the topological properties of superconducting nanowires in the presence of an imposed supercurrent and of interactions.
We study the quantum entanglement and quantum phase transition (QPT) of the anisotropic spin-1/21/21/2 XYXYXY model with staggered Dzyaloshinskii-Moriya (DM) interaction by means of the quantum renormalization group method. The scaling of coupling constants and the critical points of the system are obtained. It is found that when the number of renormalization group iterations tends to infinity, the system exhibit a QPT between the spin-fluid and N\'eel phases which correspond with two saturated values of the concurrence for a given value of the strength of DM interaction. The DM interaction can enhance the entanglement and influence the QPT of the system. To gain further insight, the first derivative of the entanglement exhibit a nonanalytic behavior at the critical point and it directly associates with the divergence of the correlation length. This shows that the correlation length exponent is closely related to the critical exponent, i.e., the scaling behaviors of the system.
We report all phases and corresponding critical lines of the quantum anisotropic transverse XY model with uniform and alternating transverse magnetic fields (ATXY) in the presence of the Dzyaloshinskii-Moriya (DM) interaction by using appropriately chosen order parameters. We prove that when DM interaction is weaker than the anisotropy parameter, it has no effect at all on the zero-temperature states of the XY model with uniform transverse magnetic field (UXY), which is not the case for the ATXY model. However, when DM interaction is stronger than the anisotropy parameter, we show the appearance of a new gapless chiral phase---in the XY model with uniform as well as alternating field. We further observe that first derivatives of nearest neighbor two-site entanglement with respect to magnetic fields can detect all the critical lines present in the system. We also find that the factorization surface at zero temperature present in this model without DM interaction becomes a volume on the introduction of the latter. Moreover, DM interaction turns out to be good to generate bipartite entanglement sustainable at large times, leading to a proof of ergodic nature of bipartite entanglement in this system.
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.
Within the block spin renormalization group, we give a very simple derivation of the exact phase boundaries of the XYZ spin chain. First, we identify the Ising order along x[over ̂] or y[over ̂] as attractive renormalization group fixed points of the Kitaev chain. Then, in a global phase space composed of the anisotropy λ of the XY interaction and the coupling Δ of the Δσ^{z}σ^{z} interaction, we find that the above fixed points remain attractive in the two-dimesional parameter space. We therefore classify the gapped phases of the XYZ spin chain as: (1) either attracted to the Ising limit of the Kitaev-chain, which in turn is characterized by winding number ±1, depending on whether the Ising order parameter is along x[over ̂] or y[over ̂] directions; or (2) attracted to the charge density wave (CDW) phases of the underlying Jordan-Wigner fermions, which is characterized by zero winding number. We therefore establish that the exact phase boundaries of the XYZ model in Baxter's solution indeed correspond to topological phase transitions. The topological nature of the phase transitions of the XYZ model justifies why our analytical solution of the three-site problem that is at the core of the present renormalization group treatment is able to produce the exact phase boundaries of Baxter's solution. We argue that the distribution of the winding numbers between the three Ising phases is a matter of choice of the coordinate system, and therefore the CDW-Ising phase is entitled to host appropriate form of zero modes. We further observe that in the Kitaev-chain the renormalization group flow can be cast into a geometric progression of a properly identified parameter. We show that this new parameter is actually the size of the (Majorana) zero modes.
We have studied the phase diagram of the one dimensional XXZ model with Dzyaloshinskii-Moriya (DM) interaction. We have applied the quantum renormalization group (QRG) approach to get the stable fixed points, critical point and the scaling of coupling constants. This model has three phases, ferromagnetic, spin-fluid and Néel phases which are separated by a critical line which depends on the DM coupling constant. We have shown that the staggered magnetization is the order parameter of the system and investigated the influence of DM interaction on the chiral ordering as a helical magnetic order.