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Thermodynamics of one-dimensional spin-orbit-coupled bosonic model in deep insulating region

Bin Xi, Shijie Hu, Qiang Luo, Jize Zhao, Xiaoqun Wang

Published Jul 31, 2016• 0 citations
OpenAlex

Abstract

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

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

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(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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Phases and phase transitions in a dimerized spin- 1 2 XXZ chain

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We revisit the phase diagram of the dimerized XXZ spin-$\frac{1}{2}$ chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic $XY$ couplings which have a uniform value and $ZZ$ couplings which have a dimerized form, with strengths ${J}_{a}$ and ${J}_{b}$ on alternate bonds. We find a rich phase diagram in the region of positive ${J}_{a},\phantom{\rule{0.28em}{0ex}}{J}_{b}$. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called ${\mathrm{IPM}}_{0}$ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases ${\mathrm{IPM}}_{0}$ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge $c=1$. In contrast, there are two phase transition lines separating ${\mathrm{IPM}}_{0}$ or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by $c=\frac{1}{2}$ CFTs and correspond to quantum Ising transitions. In fact, the $c=1$ line bifurcates into the two $c=\frac{1}{2}$ lines at the point ${J}_{a}={J}_{b}=1$; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that ${\mathrm{IPM}}_{0}$ is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-$\frac{1}{2}$ states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.

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Quantum phase transitions in the spin-1 Kitaev-Heisenberg chain

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Recently, it has been proposed that higher-spin analogs of the Kitaev interactions $K>0$ may also occur in a number of materials with strong Hund's and spin-orbit coupling. In this work, we use Lanczos diagonalization and density matrix renormalization group methods to investigate numerically the $S=1$ Kitaev-Heisenberg model. The ground-state phase diagram and quantum phase transitions are investigated by employing local and nonlocal spin correlations. We identify two ordered phases at negative Heisenberg coupling $J<0$: a ferromagnetic phase with $\ensuremath{\langle}{S}_{i}^{z}{S}_{i+1}^{z}\ensuremath{\rangle}>0$ and an intermediate left-left-right-right phase with $\ensuremath{\langle}{S}_{i}^{x}{S}_{i+1}^{x}\ensuremath{\rangle}\ensuremath{\ne}0$. A quantum spin liquid is stable near the Kitaev limit, while a topological Haldane phase is found for $J>0$.

arXiv:2009.14159•Physical review. B./Physical review. B•Oct 23, 2020•15 citations
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.
We study the thermodynamics of a spin-1/2 XYZ Heisenberg chain with a Dzyaloshinskii-Moriya interaction. This model describes the low-energy behaviors of a one-dimensional two-component bosonic model with a synthetic spin-orbit coupling in the deep insulating region. In the limit ${U}^{\ensuremath{'}}/U\ensuremath{\rightarrow}\ensuremath{\infty}$, where UUU is the strength of the onsite intracomponent repulsion and ${U}^{\ensuremath{'}}$ is the intercomponent one, we solve our model exactly by Jordan-Wigner transformation, and thus provide a benchmark for our following numerical approach. In other cases, we calculate the entropy and the specific heat numerically by the transfer-matrix renormalization-group method. Their low-temperature behaviors depend crucially on the properties of the zero-temperature phases. A refined ground-state phase diagram is then deduced from their low-temperature behaviors. Our findings offer an alternative way to detect those distinguishable phases experimentally.
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.
(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.
We revisit the phase diagram of the dimerized XXZ spin-12\frac{1}{2}21​ chain with nearest-neighbor couplings which was studied numerically by Mondal et al. [Phys. Rev. B 106, L201106 (2022)]. The model has isotropic XYXYXY couplings which have a uniform value and ZZZZZZ couplings which have a dimerized form, with strengths Ja{J}_{a}Ja​ and Jb{J}_{b}Jb​ on alternate bonds. We find a rich phase diagram in the region of positive Ja,Jb{J}_{a},\phantom{\rule{0.28em}{0ex}}{J}_{b}Ja​,Jb​. We provide a detailed understanding of the different phases and associated quantum phase transitions using a combination of mean-field theory, low-energy effective Hamiltonians, renormalization group calculations employing the technique of bosonization, and numerical calculations using the density-matrix renormalization group (DMRG) method. The phase diagram consists of two Ising paramagnetic phases called IPM0{\mathrm{IPM}}_{0}IPM0​ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$, and a phase with Ising Neel (IN) order; all these phases are gapped. The phases IPM0{\mathrm{IPM}}_{0}IPM0​ and ${\mathrm{IPM}}_{\ensuremath{\pi}}$ are separated by a gapless phase transition line $0\ensuremath{\le}{J}_{a}={J}_{b}\ensuremath{\le}1$---the well-known Tomonaga Luttinger liquid captured by a conformal field theory (CFT) with central charge c=1c=1c=1. In contrast, there are two phase transition lines separating IPM0{\mathrm{IPM}}_{0}IPM0​ or the ${\mathrm{IPM}}_{\ensuremath{\pi}}$ from IN that are described by c=12c=\frac{1}{2}c=21​ CFTs and correspond to quantum Ising transitions. In fact, the c=1c=1c=1 line bifurcates into the two c=12c=\frac{1}{2}c=21​ lines at the point Ja=Jb=1{J}_{a}={J}_{b}=1Ja​=Jb​=1; the shape of the bifurcation is found analytically using RG calculations. A symmetry analysis shows that IPM0{\mathrm{IPM}}_{0}IPM0​ is a topologically trivial phase while ${\mathrm{IPM}}_{\ensuremath{\pi}}$ is a time-reversal symmetry-protected topological phase (SPT) with spin-12\frac{1}{2}21​ states at the two ends of an open system. A gap opens in the low-energy spectrum whenever one moves away from one of the phase transition lines; the scaling of the gap with the distance from a transition line is found analytically using the RG method. The numerical results obtained by the DMRG method are in good agreement with the analytical results. Further, we show that useful insights into the nature of the phase diagram of the above Hamiltonian, particularly the SPT as well as the phase transitions, is obtained via Kramers-Wannier duality to a deformed quantum Ashkin-Teller model. Finally, we propose experimental platforms for testing our results.
We consider the insulating spin-orbit coupled Bose atoms confined within one-dimensional optical lattices and explore their ground-state magnetic phase transitions. Under strong interactions, the charge degrees of atoms are frozen and the system can be described by an anisotropic XXZ Heisenberg chain with Dzyaloshinskii-Moriya interaction and transverse field. We apply the matrix product state method to obtain the ground states, and analyze spiral correlation functions, spin-spin correlation functions, and first energy gap. Under weakly transverse fields, the ground state is a ferromagnetic phase if the interspin sss-wave interaction strength is stronger than that of the intraspin ones, otherwise, it is a paramagnetic phase, an antiferromagnetic phase, or a gapless spiral phase with algebraic decaying spin correlations. When the transverse field is strengthened, the gapless spiral phase is broken and a new spiral phase with long-range spin-spin correlations emerges.
Recently, it has been proposed that higher-spin analogs of the Kitaev interactions K>0K>0K>0 may also occur in a number of materials with strong Hund's and spin-orbit coupling. In this work, we use Lanczos diagonalization and density matrix renormalization group methods to investigate numerically the S=1S=1S=1 Kitaev-Heisenberg model. The ground-state phase diagram and quantum phase transitions are investigated by employing local and nonlocal spin correlations. We identify two ordered phases at negative Heisenberg coupling J<0J<0J<0: a ferromagnetic phase with $\ensuremath{\langle}{S}_{i}^{z}{S}_{i+1}^{z}\ensuremath{\rangle}>0$ and an intermediate left-left-right-right phase with $\ensuremath{\langle}{S}_{i}^{x}{S}_{i+1}^{x}\ensuremath{\rangle}\ensuremath{\ne}0$. A quantum spin liquid is stable near the Kitaev limit, while a topological Haldane phase is found for J>0J>0J>0.