Bilinear-biquadratic Spin-1 Chain Research Articles

Superexchange Mechanism in Coupled Triangulenes Forming Spin-1 Chains.

We show that the origin of the antiferromagnetic coupling in spin-1 triangulene chains, which were recently synthesized and measured by Mishra et al. ( Nature 2021, 598, 287-292), originates from a superexchange mechanism. This process, mediated by intertriangulene states, opens the possibility to control parameters in the effective bilinear-biquadratic spin model. We start from the derivation of an effective tight-binding model for triangulene chains using a combination of tight-binding and Hartree-Fock methods fitted to hybrid density functional theory results. Next, correlation effects are investigated within the configuration interaction method. Our low-energy many-body spectrum for NTr = 2 and NTr = 4 triangulene chains agree well with the bilinear-biquadratic spin-1 chain antiferromagnetic model when indirect coupling processes and superexchange coupling between triangulene spins are taken into account.

Finite-size scaling at a topological transition: Bilinear-biquadratic spin-1 chain

We consider a finite size scaling function across a topological phase transition in 1D models. For models of non-interacting fermions it was shown to be universal for all topological symmetry classes and markedly asymmetric between trivial and topological sides of the transition (Gulden et al 2016). Here we verify its universality for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of $L/\xi$, where $L$ is the chain length and $\xi$ is the correlation length, coincides with that of three species of non-interacting massive Majorana fermions. The latter is known to be a proper description of the conformal critical theory with central charge $c=3/2$. We have shown that it still holds away from the conformal point, including the finite size corrections. We have also observed peculiar differences between even and odd size chains, which may be fully accounted for by residual interactions of the edge states.

Proposal of a spin-one chain model with competing dimer and trimer interactions

A new kind of spin-1 chain Hamiltonian consisting of competing dimer and trimer projection operators is proposed. As the relative strengths and signs of the interactions are varied, the model exhibits a number of different phases including the gapped dimer phase and the gapless trimer phase with critical correlations described by a conformal field theory with central charge $c=2$. A symmetry-protected topological phase also exists in this model, even though the microscopic interactions are not the simple adiabatic extensions of the well-known Heisenberg and the Affleck-Kennedy-Lieb-Tasaki model and contains both two- and three-particle permutations. A fourth phase is characterized by macroscopically degenerate ground states. While bearing almost a one-to-one resemblance to the phase diagram of the bilinear-biquadratic spin-1 chain Hamiltonian, our model is rooted on very different physical origin, namely the two competing tendencies of spin-1 particles to form singlets through either dimer or trimer formation.

Entanglement analysis of isotropic spin-1 chains

We investigate entanglement spectra of the SO(3) bilinear-biquadratic spin-1 chain, a model with phases exhibiting spontaneous symmetry breaking (both translation and spin rotation), points of enlarged symmetry, and a symmetry-protected topological phase (the Haldane phase). Our analysis reveals how these hallmark features are manifested in the entanglement spectra, and highlights the versatility of entanglement spectra as a tool to study one-dimensional quantum systems via small finite size realizations.

Anyonic quantum spin chains: Spin-1 generalizations and topological stability

There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism, occuring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)_k anyons, well-known deformations of SU(2), in which only the first k+1 angular momenta of SU(2) occur. In this manuscript, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S=1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter k when considering su(2)_k anyonic theories with k>4, as well as for the special case of the su(2)_4 theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-1/2 chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.

Bilinear-biquadratic spin-1 chain undergoing quadratic Zeeman effect

The Heisenberg model for spin-1 bosons in one dimension presents many different quantum phases, including the famous topological Haldane phase. Here we study the robustness of such phases in front of a SU(2) symmetry-breaking field as well as the emergence of unique phases. Previous studies have analyzed the effect of such uniaxial anisotropy in some restricted relevant points of the phase diagram. Here we extend those studies and present the complete phase diagram of the spin-1 chain with uniaxial anisotropy. To this aim, we employ the density-matrix renormalization group together with analytical approaches. The complete phase diagram can be realized using ultracold spinor gases in the Mott insulator regime under a quadratic Zeeman effect.

Critical intermediate phase and phase transitions in a triangular-lattice three-spin interaction model: Level-spectroscopy approach

We investigate infinite-order phase transitions like the Berezinskii–Kosterlitz–Thouless transition observed in a triangular-lattice three-spin interaction model. Based on a field theoretical description and the operator-production-expansion technique, we perform the renormalization-group analysis, and then clarify properties of marginal operators near the phase transition points. The results are utilized to establish criteria to determine the transition points and some universal relations among excitation levels to characterize the transitions. We verify these predictions via the numerical analysis on eigenvalue structures of the transfer matrix. Also, we discuss an enhancement of symmetry at the end points of a critical intermediate phase in connection with a transition observed in the ground state of the bilinear-biquadratic spin-1 chain.

Energy as Entanglement Witness in Bilinear-Biquadratic Spin-1Chain

Energy is introduced as an entanglement witness to describe theentanglement property of a quantum system. The thermal equilibriumsystem is guaranteed to be entangled when system is cooled downbelow the entanglement temperature TE. By virtue of thisconcept we exploit the minimum separable state energy andentanglement temperature TE of the bilinear-biquadraticantiferromagnetic spin-1 chain model. We numerically calculateTE for arbitrary values of the strength of biquadraticexchange interaction Q up to N = 7. We find TE decreaseswith Q for fixed N when Q is between -3 and 1/3 (J = 1).In this regime TE also decreases with N for fixed Q andvaries slowly for large N. While the thermal system is alwaysentangled when Q is smaller than -3.

Probable absence of a quadrupolar spin-nematic phase in the bilinear-biquadratic spin-1 chain

We study numerically the ground-state phase diagram of the bilinear-biquadratic spin-1 chain near the ferromagnetic instability point, where the existence of a gapped or gapless nondimerized quantum nematic phase has been suggested. Our results, obtained by a highly accurate density-matrix renormalization-group (DMRG) calculation are consistent with the view that the order parameter characterizing the dimer phase vanishes only at the point where the system becomes ferromagnetic, although the existence of a gapped or gapless nondimerized phase in a very narrow parameter range between the ferromagnetic and the dimerized regimes cannot be ruled out.

Incommensurability in the magnetic excitations of the bilinear-biquadratic spin-1 chain

Incommensurability in the magnetic excitations of the bilinear-biquadratic spin-1 chain

Incommensurability in the magnetic excitations of the bilinear-biquadratic spin-1 chain

We study the magnetic excitation spectrum of the S=1 quantum Heisenberg spin chain with Hamiltonian : H = sum_i cos(theta) S_i S_i+1 + sin(theta) (S_i S_i+1)^2. We focus on the range -pi/4 < theta < +pi/4 where the spin chain is in the gapped Haldane phase. The excitation spectrum and static structure factor is studied using direct Lanczos diagonalization of small systems and density-matrix renormalization group techniques combined with the single-mode approximation. The magnon dispersion has a minimum at q=pi until a critical value theta_c = 0.38 is reached at which the curvature (velocity) vanishes. Beyond this point, which is distinct from the VBS point and the Lifshitz point, the minimum lies at an incommensurate value that goes smoothly to 2pi/3 when theta approaches pi/4, the Lai-Sutherland point. The mode remains isolated from the other states: there is no evidence of spinon deconfinement before the point theta =+pi/4. These findings explain recent observation of the magnetization curve M approx (H -H_c)^1/4 for theta =theta_c.

Static and dynamic structure factors in the Haldane phase of the bilinear-biquadratic spin-1 chain

The excitation spectra of the $T=0$ dynamic structure factors for the spin, dimer, and trimer fluctuation operators as well as for the center fluctuation operator in the one-dimensional $S=1$ Heisenberg model with isotropic bilinear $(J\mathrm{cos}\ensuremath{\theta})$ and biquadratic $(J\mathrm{sin}\ensuremath{\theta})$ exchange are investigated via the recursion method for systems with up to $N=18$ sites over the predicted range, $\ensuremath{-}\ensuremath{\pi}/4l\ensuremath{\theta}\ensuremath{\lesssim}\ensuremath{\pi}/4,$ of the topologically ordered Haldane phase. The four static and dynamic structure factors probe the ordering tendencies in the various coupling regimes and the elementary and composite excitations which dominate the $T=0$ dynamics. At $\ensuremath{\theta}=\mathrm{arctan}\frac{1}{3}$ (valence-bond solid point), the dynamically relevant spectra in the invariant subspaces with total spin ${S}_{T}=0,1,2$ are dominated by a branch of magnon states ${(S}_{T}=1),$ by continua of two-magnon scattering states ${(S}_{T}=0,1,2),$ and by discrete branches of two-magnon bound states with positive interaction energy ${(S}_{T}=0,2).$ The dimer and trimer spectra at $q=\ensuremath{\pi}$ are found to consist of single modes with $N$-independent excitation energies ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{D}/|{e}_{0}|=5$ and ${\ensuremath{\omega}}_{\ensuremath{\lambda}}^{T}/|{e}_{0}|=6,$ where ${e}_{0}{=E}_{0}/N$ is the ground-state energy per site. The basic structure of the dynamically relevant excitation spectrum remains the same over a substantial parameter range within the Haldane phase. At the transition to the dimerized phase $(\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4),$ the two-magnon excitations turn into two-spinon excitations.

Density matrix renormalization group study of the correlation function of the bilinear-biquadratic spin-1 chain

Using the recently developed density matrix renormalization group approach, we study the correlation function of the spin-1 chain with quadratic and biquadratic interactions. This allows us to define and calculate the periodicity of the ground state which differs markedly from that in the classical analogue. Combining our results with other studies, we predict three phases in the region where the quadratic and biquadratic terms are both positive.

Search for the nondimerized quantum nematic phase in the spin-1 chain.

Chubukov's proposal concerning the possibility of a nondimerized quantum nematic phase in the ground-state phase diagram of the bilinear-biquadratic spin-1 chain is studied numerically. Our results do not support the existence of this phase, but they rather indicate a direct transition from the ferromagnetic into the dimerized phase.

Variational study of isotropic spin‐1 chains using matrix‐product states

AbstractWe study the phase diagram of the bilinear‐biquadratic spin‐1 chain. Using so‐called matrix‐product (MP) states we obtain upper bounds for the groundstate energy. The advantage of these MP states is their simplicity. Moreover they give an accurate description of the physics, especially in the Haldane phase.