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arXiv:2309.07737 · New Journal of Physics

Eleven competing phases in the Heisenberg-Gamma ( JΓ ) ladder

Sebastien J. Avakian, Erik Schwartz Sørensen

Published Jan 1, 2024• 2 citations
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Abstract

Abstract The spin-orbit generated Γ interaction is known to induce strong frustration and to be significant in realistic models of materials. To gain an understanding of the possible phases that can arise from this interaction, it is of considerable interest to focus on a limited part of parameter space in a quasi one-dimensional model where high precision numerical results can be obtained. Here we study the Heisenberg–Gamma ( <?CDATA $J\Gamma$?> J Γ ) ladder, determining the complete zero temperature phase diagram by analyzing the entanglement spectrum (ES) and energy susceptibility. A total of 11 different phases can be identified, among them the well known rung-singlet (RS) phase and 5 other phases, FM, FM-Z, FM-XY, AF and AF-Z, with conventional long-range magnetic order. The 3 ferromagnetic phases, FM, FM-Z and FM-XY simultaneously have non-zero scalar chirality. Two other phases, the antiferromagnetic Gamma (AΓ) and ferromagnetic Gamma (FΓ) phases, have previously been observed in the Kitaev–Gamma ladder, demonstrating that the AΓ-phase is a symmetry protected topological phase (SPT) protected by TR <?CDATA $\times \mathcal{R}_{b}$?> × R b symmetry, the product of time-reversal (TR) and π rotation around the b-axis ( <?CDATA $\mathcal{R}_{b}$?> R b ), while the FΓ-phase is related to the RS phase through a local unitary transformation. The 3 remaining phases, ϒ, Ω and δ show no conventional order, a doubling of the ES and for the ϒ and Ω-phases a gap is clearly present. The δ-phase has a significantly smaller gap and displays incommensurate correlations, with a peak in the static structure factor, S(k) continuously shifting from <?CDATA $k/\pi=2/3$?> k / π = 2 / 3 to <?CDATA $k=\pi$?> k = π . In the Ω-phase we find pronounced edge-states consistent with a SPT phase protected by the same TR <?CDATA $\times \mathcal{R}_{b}$?> × R b symmetry as the AΓ-phase. The precise nature of the ϒ and δ-phases is less clear.

DOI
10.1088/1367-2630/ad1d0f

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Abstract Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma (Γ) interaction, causes magnetic frustration and its interplay with the Kitaev (K) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined KΓ model, quasi-one-dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic Γ limit (denoted by AΓ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the AΓ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of KΓ ladder is also reported.
We study an asymmetric ${J}_{1}\ensuremath{-}{J}_{2}$ zigzag ladder consisting of two different spin-12\frac{1}{2}21​ antiferromagnetic (AFM; ${J}_{2}, \ensuremath{\gamma}{J}_{2}>0$) Heisenberg legs coupled by zigzag-shaped ferromagnetic (FM; J1<0{J}_{1}<0J1​<0) interleg interaction. On the basis of density-matrix renormalization group based calculations, the ground-state phase diagram is obtained as functions of $\ensuremath{\gamma}$ and J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣. It contains four kinds of frustration-induced ordered phases except a trivial FM phase. Two of the ordered phases are valence bond solid (VBS) with spin-singlet dimerization, which is a rather conventional order by disorder. Still, it is interesting to note that the VBS states possesses an Affleck-Kennedy-Lieb-Tasaki--type topological hidden order. The remaining two phases are ferrimagnetic orders, each of which is distinguished by commensurate or incommensurate spin-spin correlation. It is striking that the ferrimagnetic orders are not associated with geometrical symmetry breaking; instead, the global spin-rotation symmetry is broken. In other words, the system lowers its energy via the FM interleg interaction by polarizing both of the AFM Heisenberg legs. This is a rare type of order by disorder. Besides, the incommensurate ferrimagnetic state appears as a consequence of the competition between a polarization and a critical Tomonaga-Luttinger--liquid behavior in the AFM Heisenberg legs.
A spin-1/2 Heisenberg model with a chirality-chirality interaction (CCI) on a two-leg ladder provides a minimal setup to explore an interplay between spin and chirality degrees of freedom. This model is potentially relevant for understanding the effects of a four-spin ring exchange and a phonon-mediated interaction on a ladder. By applying the Abelian bosonization and two types of spin-chirality duality, we derive a comprehensive ground-state phase diagram that includes previously unexplored cases of a ferromagnetic rung exchange J⊥<0J_\perp<0J⊥​<0 or a negative CCI Kch<0K_{\rm ch}<0Kch​<0. The phase diagram consists of the rung singlet (RS) phase, the Haldane phase, and three non-magnetic ordered phases. Moreover, the RS and Haldane phases consist of several regions with different dominant correlations in terms of spin, dimer, or chirality. We perform numerical simulations to confirm the predicted phase structure and critical properties.
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.
We study a spin-12\frac{1}{2}21​ XXZ model with a four-spin interaction on a two-leg ladder. By means of effective field theory and matrix product state calculations, we obtain rich ground-state phase diagrams that consist of eight distinct gapped phases. Four of them exhibit spontaneous symmetry breaking with either a magnetic or valence-bond-solid (VBS) long-range order. The other four are featureless, i.e., the bulk ground state is unique and does not break any symmetry. The featureless phases include the rung singlet (RS) and Haldane phases as well as their variants, the RS* and Haldane* phases, in which twisted singlet pairs $(|\ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\rangle}+|\ensuremath{\downarrow}\ensuremath{\uparrow}\ensuremath{\rangle})/\sqrt{2}$ are formed between the two legs. We argue and demonstrate that Gaussian transitions with the central charge c=1c=1c=1 occur between the featureless phases and between the ordered phases while Ising transitions with c=1/2c=1/2c=1/2 occur between the featureless and ordered phases. The two types of transition lines cross at the SU(2)\mathrm{SU}(2)SU(2)-symmetric point, where the criticality is described by the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Witten theory with c=3/2c=3/2c=3/2. The RS-Haldane* and RS*-Haldane transitions give examples of topological phase transitions. Interestingly, the RS* and Haldane* phases, which have highly anisotropic nature, appear even in the vicinity of the isotropic case. We demonstrate that all the four featureless phases are distinguished by topological indices in the presence of certain symmetries.
We study the interplay between the Kitaev and Ising interactions on a ladder geometry. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The nature of the SPT phase is confirmed by degeneracy of the entanglement spectrum. Nonlocal order parameters that indirectly measure phase factors (inequivalent projective representations of the symmetries) of the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry explicitly show the protection of the SPT phase under the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. We derive the effective theory to describe the topological phase transition on the ladder geometry, which is given by a transverse field Ising model with/without next-nearest-neighbor coupling based on the primary Ising configurations. The ladder has three phases, namely, the Kitaev SPT, symmetry-broken ferro/antiferromagnetic order, and classical spin liquid. The nonzero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point and change of the entanglement spectrum degeneracy. The central charge of the critical points are either c=1c=1c=1 or c=2c=2c=2, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. The transition from the classical spin-liquid phase of the frustrated Ising ladder to the Kitaev SPT phase is mediated by a floating phase, which shows strong finite entanglement scaling.