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arXiv:1101.1268 · Physical Review B

Quantum phase transitions in a strongly entangled spin-orbital chain: A field-theoretical approach

A. A. Nersesyan, Gia-Wei Chern, Natalia Borisovna Perkins

Published May 26, 2011• 5 citations
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Abstract

Motivated by recent experiments on quasi-one-dimensional vanadium oxides, we study quantum phase transitions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain locally coupled by the relativistic spin-orbit interaction. By employing a field-theoretical approach, we analyze the topology of the ground-state phase diagram and identify the nature of the phase transitions. In the strong coupling limit, a long-range N\'eel order of entangled spin and orbital angular momenta appears in the ground state. We find that, depending on the relative scales of the spin and orbital gaps, the linear chain follows two distinct routes to reach the N\'eel state. First, when the orbital exchange is the dominating energy scale, a two-stage ordering takes place in which the magnetic transition is followed by melting of the orbital Ising order; both transitions belong to the two-dimensional Ising universality class. In the opposite limit, the low-energy orbital modes undergo a continuous reordering transition which represents a line of Gaussian critical points. On this line the orbital degrees of freedom form a Tomonaga--Luttinger liquid. We argue that the emergence of the Gaussian criticality results from merging of the two Ising transitions in the strong hybridization region where the characteristic spin and orbital energy scales become comparable. Finally, we show that, due to the spin-orbit coupling, an external magnetic field acting on the spins can induce an orbital Ising transition.

DOI
10.1103/physrevb.83.205132

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Motivated by recent experiments on quasi-one-dimensional vanadium oxides, we study quantum phase transitions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain locally coupled by the relativistic spin-orbit interaction. By employing a field-theoretical approach, we analyze the topology of the ground-state phase diagram and identify the nature of the phase transitions. In the strong coupling limit, a long-range N'eel order of entangled spin and orbital angular momenta appears in the ground state. We find that, depending on the relative scales of the spin and orbital gaps, the linear chain follows two distinct routes to reach the N'eel state. First, when the orbital exchange is the dominating energy scale, a two-stage ordering takes place in which the magnetic transition is followed by melting of the orbital Ising order; both transitions belong to the two-dimensional Ising universality class. In the opposite limit, the low-energy orbital modes undergo a continuous reordering transition which represents a line of Gaussian critical points. On this line the orbital degrees of freedom form a Tomonaga--Luttinger liquid. We argue that the emergence of the Gaussian criticality results from merging of the two Ising transitions in the strong hybridization region where the characteristic spin and orbital energy scales become comparable. Finally, we show that, due to the spin-orbit coupling, an external magnetic field acting on the spins can induce an orbital Ising transition.
We study quantum phase transitions induced by the on-site spin-orbit interaction $\ensuremath{\lambda}\mathbf{L}\ensuremath{\cdot}\mathbf{S}$ in a toy model of vanadium chains. In the $\ensuremath{\lambda}\ensuremath{\rightarrow}0$ limit, the decoupled spin and orbital sectors are described by a Haldane and an Ising chain, respectively. The gapped ground state is composed of a ferro-orbital order and a spin liquid with finite correlation lengths. In the opposite limit, strong spin-orbital entanglement results in a simultaneous spin and orbital-moment ordering, which can be viewed as an orbital liquid. Using a combination of analytical arguments and density-matrix renormalization-group calculation, we show that an intermediate phase, where the ferro-orbital state is accompanied by a spin N\'eel order, is bounded on both sides by Ising transition lines. Implications for vanadium compounds CaV2O4{\text{CaV}}_{2}{\text{O}}_{4}CaV2​O4​ and ZnV2O4{\text{ZnV}}_{2}{\text{O}}_{4}ZnV2​O4​ are also discussed.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interaction emerges from the interplay between the Kitaev ( K ) and Heisenberg ( J ) exchanges, and is accompanied by an additional bond-dependent interaction (Γ), leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J - K -Γ model with four-fold screw symmetry in a spin-orbit-entangled chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic ordered structures, the transition remains in the Ising universality class with central charge c = 1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to Ising materials such as chains of Co 2+ ions is also discussed.
We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling ααα. This bond-anisotropic parameter continuously interpolates among the Ising-like limit α=0α=0α=0, the isotropic point α=1α=1α=1, and the XY-dominated regime α≫1α\gg 1α≫1. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge c≃1c \simeq 1c≃1, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe c≃1/2c \simeq 1/2c≃1/2, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point α=1α= 1α=1.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interactions arise alongside bond-dependent couplings such as Kitaev (KKK) and ΓΓΓ terms in addition to the Heisenberg (JJJ) interaction, leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J−K−ΓJ-K-ΓJ−K−Γ model with four-fold screw symmetry in a spin-orbit-entangled pseudospin-1/2 chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic order structures, the transition remains in the Ising universality class with central charge c=1/2c=1/2c=1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to antiferromagnetic Ising materials such as BaCo2_22​V2_22​O8_88​ is also discussed.
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The phase diagram of the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge c=1c=1c=1. Here, we extend the analysis made by Haldane on the quantum spin-12\frac{1}{2}21​ antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge c=1c=1c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in d=3d=3d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.