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The XYZ chain with Dzyaloshinsky-Moriya interaction and the magnetic properties of spin-orbit-coupled lattice bosons

Sebastiano Peotta, Leonardo Mazza, Ettore Vicari, Marco Polini, Rosario Fazio, Davide Rossini, Largo B. Pontecorvo

Published Jan 1, 2014• 0 citations
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Abstract

(1) Department of Physics, University of California, San Diego, La Jolla, CA 92093,USA(2) NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa,Italy(3) Dipartimento di Fisica dellUniversita di Pisa and INFN, Largo Pontecorvo 3,I-56127 Pisa, Italy(4) NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa(5) Center for Quantum Technologies, National University of Singapore, SingaporeAbstract. Using the density-matrix renormalization-group algorithm (DMRG)and nite-size-scaling, we study the properties of the one-dimensional completely-anisotropic spin-1/2 XYZ model with Dzyaloshinsky-Moriya (DM) interactions. Themodel shows a rich phase diagram: depending on the value of the coupling constants,the system can display di erent kinds of ferromagnetic order and Luttinger-liquidbehavior. Transitions from ferromagnetic to Luttinger-liquid phases are rst order.We thoroughly discuss the transition between di erent ferromagnetic phases, which,in the absence of DM interactions, belongs to the XX universality class; we provideevidence that in presence of DM it is split into two Ising-like transitions. In between,an interesting disordered phase appears. Our study sheds light on the general problemof strongly-interacting spin-orbit-coupled systems and is particularly useful to guidefuture experimental work on cold bosons loaded in optical lattices.

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

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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.

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We study the thermodynamics of an XYZ Heisenberg chain with Dzyaloshinskii-Moriya interaction, which describes the low-energy behaviors of a one-dimensional spin-orbit-coupled bosonic model in the deep insulating region. The entropy and the specific heat are calculated numerically by the quasi-exact transfer-matrix renormalization group. In particular, in the limit $U^\prime/U\rightarrow\infty$, our model is exactly solvable and thus serves as a benchmark for our numerical method. From our data, we find that for $U^\prime/U>1$ a quantum phase transition between an (anti)ferromagnetic phase and a Tomonaga-Luttinger liquid phase occurs at a finite $\theta$, while for $U^\prime/U<1$ a transition between a ferromagnetic phase and a paramagnetic phase happens at $\theta=0$. A refined ground-state phase diagram is then deduced from their low-temperature behaviors. Our findings provide an alternative way to detect those distinguishable phases experimentally.

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The ground-state phase diagram of a spin $S=\frac{1}{2}\phantom{\rule{4pt}{0ex}}\text{XXZ}$ Heisenberg chain with spatially modulated Dzyaloshinskii-Moriya interaction $\mathcal{H}={\ensuremath{\sum}}_{n}{J\left({S}_{n}^{x}{S}_{n+1}^{x}+{S}_{n}^{y}{S}_{n+1}^{y}\right)+{J}_{z}{S}_{n}^{z}{S}_{n+1}^{z}+[{D}_{0}+{(\ensuremath{-}1)}^{n}{D}_{1}]({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x})}$ is studied using the continuum-limit bosonization approach and extensive density-matrix renormalization group computations. It is shown that the effective continuum-limit bosonized theory of the model is given by the double-frequency sine-Gordon model (DSG) where the frequencies, i.e., the scaling dimensions of the two competing cosine perturbation terms, depend on the effective anisotropy parameter ${\ensuremath{\gamma}}^{*}={J}_{z}/\sqrt{{J}^{2}+{D}_{0}^{2}+{D}_{1}^{2}}$. Exploring the ground-state properties of the DSG model we show that the zero-temperature phase diagram contains the following four phases: (i) the ferromagnetic phase at ${\ensuremath{\gamma}}^{*}\ensuremath{\le}\ensuremath{-}1$; (ii) the gapless Luttinger-liquid (LL) phase at $\ensuremath{-}1<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C1}}^{*}=\ensuremath{-}1/\sqrt{2}$; (iii) the gapped composite (C1) phase characterized by coexistence of the long-range-ordered (LRO) dimerization pattern $\ensuremath{\epsilon}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}({\mathbf{S}}_{n}{\mathbf{S}}_{n+1})$ with the LRO alternating spin chirality pattern $\ensuremath{\kappa}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}\left({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x}\right)$ at ${\ensuremath{\gamma}}_{\text{C1}}^{*}<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C2}}^{*}$; and (iv) at ${\ensuremath{\gamma}}^{*}>{\ensuremath{\gamma}}_{\text{C2}}^{*}>1$ the gapped composite (C2) phase characterized in addition to the coexisting spin dimerization and alternating chirality patterns, by the presence of LRO antiferromagnetic order. The transition from the LL to the C1 phase at ${\ensuremath{\gamma}}_{\text{C1}}^{*}$ belongs to the Berezinskii-Kosterlitz-Thouless universality class, while the transition at ${\ensuremath{\gamma}}^{*}={\ensuremath{\gamma}}_{\text{C2}}^{*}$ from C1 to C2 phase is of the Ising type.

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Incommensurability and phase transitions in two-dimensional XY models with Dzyaloshinskii-Moriya interactions

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The Dzyaloshinskii-Moriya (DM) interaction in magnetic models is the result of a combination of superexchange and spin-orbital coupling, and it can give rise to rich phase-transition behavior. In this paper, we study ferromagnetic XY models with the DM interaction on two-dimensional L×L square lattices using a hybrid Monte Carlo algorithm. To match the incommensurability between the resultant spin structure and the lattice due to the DM interaction, a fluctuating boundary condition is adopted. We also define a different kind of order parameter and use finite-size scaling to study the critical properties of this system. We find that a Kosterlitz-Thouless-like phase transition appears in this system and that the phase-transition temperature shifts toward higher temperature with increasing DM interaction strength.

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We have studied the phase diagram and entanglement of the one-dimensional Ising 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 two phases: antiferromagnetic and saturated chiral ones. We have shown that the staggered magnetization is the order parameter of the system and DM interaction produces the chiral order in both phases. We have also implemented the exact diagonalization (Lanczos) method to calculate the static structure factors. The divergence of structure factor at the ordering momentum as the size of systems goes to infinity defines the critical point of the model. Moreover, we have analyzed the relevance of the entanglement in the model which allows us to shed insight on how the critical point is touched as the size of the system becomes large. Nonanalytic behavior of entanglement and finite-size scaling have been analyzed which is tightly connected to the critical properties of the model. It is also suggested that a spin-fluid phase has a chiral order in terms of spin operators which are defined by a nonlocal transformation.

arXiv:0804.4579•Physical Review B•Dec 10, 2008•162 citations
(1) Department of Physics, University of California, San Diego, La Jolla, CA 92093,USA(2) NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa,Italy(3) Dipartimento di Fisica dellUniversita di Pisa and INFN, Largo Pontecorvo 3,I-56127 Pisa, Italy(4) NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa(5) Center for Quantum Technologies, National University of Singapore, SingaporeAbstract. Using the density-matrix renormalization-group algorithm (DMRG)and nite-size-scaling, we study the properties of the one-dimensional completely-anisotropic spin-1/2 XYZ model with Dzyaloshinsky-Moriya (DM) interactions. Themodel shows a rich phase diagram: depending on the value of the coupling constants,the system can display di erent kinds of ferromagnetic order and Luttinger-liquidbehavior. Transitions from ferromagnetic to Luttinger-liquid phases are rst order.We thoroughly discuss the transition between di erent ferromagnetic phases, which,in the absence of DM interactions, belongs to the XX universality class; we provideevidence that in presence of DM it is split into two Ising-like transitions. In between,an interesting disordered phase appears. Our study sheds light on the general problemof strongly-interacting spin-orbit-coupled systems and is particularly useful to guidefuture experimental work on cold bosons loaded in optical lattices.
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 thermodynamics of an XYZ Heisenberg chain with Dzyaloshinskii-Moriya interaction, which describes the low-energy behaviors of a one-dimensional spin-orbit-coupled bosonic model in the deep insulating region. The entropy and the specific heat are calculated numerically by the quasi-exact transfer-matrix renormalization group. In particular, in the limit U′/U→∞U^\prime/U\rightarrow\inftyU′/U→∞, our model is exactly solvable and thus serves as a benchmark for our numerical method. From our data, we find that for U′/U>1U^\prime/U>1U′/U>1 a quantum phase transition between an (anti)ferromagnetic phase and a Tomonaga-Luttinger liquid phase occurs at a finite θ\thetaθ, while for U′/U<1U^\prime/U<1U′/U<1 a transition between a ferromagnetic phase and a paramagnetic phase happens at θ=0\theta=0θ=0. A refined ground-state phase diagram is then deduced from their low-temperature behaviors. Our findings provide an alternative way to detect those distinguishable phases experimentally.
The ground-state phase diagram of a spin S=12XXZS=\frac{1}{2}\phantom{\rule{4pt}{0ex}}\text{XXZ}S=21​XXZ Heisenberg chain with spatially modulated Dzyaloshinskii-Moriya interaction $\mathcal{H}={\ensuremath{\sum}}_{n}{J\left({S}_{n}^{x}{S}_{n+1}^{x}+{S}_{n}^{y}{S}_{n+1}^{y}\right)+{J}_{z}{S}_{n}^{z}{S}_{n+1}^{z}+[{D}_{0}+{(\ensuremath{-}1)}^{n}{D}_{1}]({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x})}$ is studied using the continuum-limit bosonization approach and extensive density-matrix renormalization group computations. It is shown that the effective continuum-limit bosonized theory of the model is given by the double-frequency sine-Gordon model (DSG) where the frequencies, i.e., the scaling dimensions of the two competing cosine perturbation terms, depend on the effective anisotropy parameter ${\ensuremath{\gamma}}^{*}={J}_{z}/\sqrt{{J}^{2}+{D}_{0}^{2}+{D}_{1}^{2}}$. Exploring the ground-state properties of the DSG model we show that the zero-temperature phase diagram contains the following four phases: (i) the ferromagnetic phase at ${\ensuremath{\gamma}}^{*}\ensuremath{\le}\ensuremath{-}1$; (ii) the gapless Luttinger-liquid (LL) phase at $\ensuremath{-}1<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C1}}^{*}=\ensuremath{-}1/\sqrt{2}$; (iii) the gapped composite (C1) phase characterized by coexistence of the long-range-ordered (LRO) dimerization pattern $\ensuremath{\epsilon}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}({\mathbf{S}}_{n}{\mathbf{S}}_{n+1})$ with the LRO alternating spin chirality pattern $\ensuremath{\kappa}\ensuremath{\sim}{(\ensuremath{-}1)}^{n}\left({S}_{n}^{x}{S}_{n+1}^{y}\ensuremath{-}{S}_{n}^{y}{S}_{n+1}^{x}\right)$ at ${\ensuremath{\gamma}}_{\text{C1}}^{*}<{\ensuremath{\gamma}}^{*}<{\ensuremath{\gamma}}_{\text{C2}}^{*}$; and (iv) at ${\ensuremath{\gamma}}^{*}>{\ensuremath{\gamma}}_{\text{C2}}^{*}>1$ the gapped composite (C2) phase characterized in addition to the coexisting spin dimerization and alternating chirality patterns, by the presence of LRO antiferromagnetic order. The transition from the LL to the C1 phase at ${\ensuremath{\gamma}}_{\text{C1}}^{*}$ belongs to the Berezinskii-Kosterlitz-Thouless universality class, while the transition at ${\ensuremath{\gamma}}^{*}={\ensuremath{\gamma}}_{\text{C2}}^{*}$ from C1 to C2 phase is of the Ising type.
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.
The Dzyaloshinskii-Moriya (DM) interaction in magnetic models is the result of a combination of superexchange and spin-orbital coupling, and it can give rise to rich phase-transition behavior. In this paper, we study ferromagnetic XY models with the DM interaction on two-dimensional L×L square lattices using a hybrid Monte Carlo algorithm. To match the incommensurability between the resultant spin structure and the lattice due to the DM interaction, a fluctuating boundary condition is adopted. We also define a different kind of order parameter and use finite-size scaling to study the critical properties of this system. We find that a Kosterlitz-Thouless-like phase transition appears in this system and that the phase-transition temperature shifts toward higher temperature with increasing DM interaction strength.
We have studied the phase diagram and entanglement of the one-dimensional Ising 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 two phases: antiferromagnetic and saturated chiral ones. We have shown that the staggered magnetization is the order parameter of the system and DM interaction produces the chiral order in both phases. We have also implemented the exact diagonalization (Lanczos) method to calculate the static structure factors. The divergence of structure factor at the ordering momentum as the size of systems goes to infinity defines the critical point of the model. Moreover, we have analyzed the relevance of the entanglement in the model which allows us to shed insight on how the critical point is touched as the size of the system becomes large. Nonanalytic behavior of entanglement and finite-size scaling have been analyzed which is tightly connected to the critical properties of the model. It is also suggested that a spin-fluid phase has a chiral order in terms of spin operators which are defined by a nonlocal transformation.