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arXiv:2012.00700 · Physical review. B./Physical review. B

Gapless to gapless phase transitions in quantum spin chains

Shi Feng, Gonzalo Alvarez, Nandini Trivedi

Published Jan 27, 2022• 9 citations
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Abstract

We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field $h$. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor $S(q,\ensuremath{\omega})$ as a function of $h$. At $h=0$, the envelope of the lowest-energy excitations goes soft at two points, ${q}_{1}=2\ensuremath{\pi}/3$ and ${q}_{2}=4\ensuremath{\pi}/3$, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total four soft modes, and combine to form a new set of excitations that soften at a single point $q=\ensuremath{\pi}$ at ${h}_{c1}\ensuremath{\approx}0.94$. Beyond ${h}_{c1}$ the system enters another gapless B phase until the transition at ${h}_{c2}=4$ to the fully polarized phase. We compare the ULS model results with those for the Affleck-Kennedy-Lieb-Tasaki model as a representative of the gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.

DOI
10.1103/physrevb.105.014435

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Phases of the Bilinear-biquadratic Spin-1 Chains

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Measurements done on LiVGe2O6 by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)] on phases of the Singlet ground state of the Spin-1 Chain can be described by the Bilinear-biquadratic spin-1 chain. I will discuss these results, among the results of others and talk about theoretical calculations done by A. Lauchli [Phys. Rev. B 74, 144426 (2006)] that describe the phase diagram of the bilinear-biquadratic model. At a particular point in the phase diagram the model becomes SU(3) globally symmetric and is known as the Lai-Sutherland model, it is Bethe ansatz solvable as shown in [A Schmitt et al., J. Phys. A: Math. Gen. 29 (1996) 3951-3962]. Finally I will describe the thermodynamics of the model as where calculated by Lou et al. [Phys. Rev. Lett. 85, 11 (2000)].

•Jan 1, 2013•0 citations
We investigate spin chains with bilinear-biquadratic (BLBQ) spin interactions as a function of an applied magnetic field hhh. At the Uimin-Lai-Sutherland (ULS) critical point we find a gapless to gapless transition revealed by the dynamical structure factor S(q,\ensuremathω)S(q,\ensuremath{\omega})S(q,\ensuremathω) as a function of hhh. At h=0h=0h=0, the envelope of the lowest-energy excitations goes soft at two points, q1=2\ensuremathπ/3{q}_{1}=2\ensuremath{\pi}/3q1​=2\ensuremathπ/3 and q2=4\ensuremathπ/3{q}_{2}=4\ensuremath{\pi}/3q2​=4\ensuremathπ/3, dubbed the phase A. With increasing field, the spectral peaks at each of the gapless points bifurcate, making in total four soft modes, and combine to form a new set of excitations that soften at a single point q=\ensuremathπq=\ensuremath{\pi}q=\ensuremathπ at hc1\ensuremath≈0.94{h}_{c1}\ensuremath{\approx}0.94hc1​\ensuremath≈0.94. Beyond hc1{h}_{c1}hc1​ the system enters another gapless B phase until the transition at hc2=4{h}_{c2}=4hc2​=4 to the fully polarized phase. We compare the ULS model results with those for the Affleck-Kennedy-Lieb-Tasaki model as a representative of the gapped Haldane phase. We explain the mechanism of the gapless to gapless transition in the ULS model using its conserved charges and a spinon band picture. We also discuss the universality of central charges of the BLBQ family of models subjected to a magnetic field.
We investigate spin chains with bilinear-biquadratic spin interactions as a function of an applied magnetic field hhh. At the Uimin-Lai-Sutherland (ULS) critical point we find a remarkable hierarchy of fractionalized excitations revealed by the dynamical structure factor S(q,ω)S(q,\omega)S(q,ω) as a function of magnetic field yielding a transition from a gapless phase to another gapless phase before reaching the fully polarized state. At h=0h=0h=0, the envelope of the lowest energy excitations goes soft at two points q1=2π/3q_1=2\pi/3q1​=2π/3 and q2=4π/3q_2=4\pi/3q2​=4π/3, dubbed the A-phase. With increasing field, the spectral peaks at each of the gapless points bifurcate and combine to form a new set of fractionalized excitations that soften at a single point q=πq=\piq=π at hc1≈0.94h_{c1}\approx 0.94hc1​≈0.94. Beyond hc1h_{c1}hc1​ the system remains in this phase dubbed the B-phase until the transition at hc2=4h_{c2}=4hc2​=4 to the fully polarized phase. We discuss the central charge of these two gapless phases and contrast the behavior with that of the gapped Haldane phase in a field.
Using a matrix product state algorithm with infinite boundary conditions, we compute high-resolution dynamic spin and quadrupolar structure factors in the thermodynamic limit to explore the low-energy excitations of isotropic bilinear-biquadratic spin-1 chains. Haldane mapped the spin-1 Heisenberg antiferromagnet to a continuum field theory, the nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$). We find that the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ fails to capture the influence of the biquadratic term and provides only an unsatisfactory description of the Haldane phase physics. But several features in the Haldane phase can be explained by noninteracting multimagnon states. The physics at the Uimin-Lai-Sutherland point is characterized by multisoliton continua. Moving into the extended critical phase, we find that these excitation continua contract, which we explain using a field-theoretic description. New excitations emerge at higher energies and, in the vicinity of the purely biquadratic point, they show simple cosine dispersions. Using block fidelities, we identify them as elementary one-particle excitations and relate them to the integrable Temperley-Lieb chain.
We investigate the properties of the Heisenberg S=1S=1S=1 chain with bilinear and biquadratic interactions in a magnetic field using the density- matrix renormalization-group, Bethe ansatz, and field-theoretical considerations. In a large region of the parameter space, we identify a magnetized ferroquadrupolar Luttinger liquid consisting of a quasicondensate of bound magnon pairs. This liquid undergoes a continuous pair-unbinding transition to a more conventional Luttinger liquid region obtained by polarizing the system above the Haldane gap region. This pair-unbinding transition is shown to be in the Ising universality class on top of a Luttinger liquid, leading to an effective central charge 3/2. We also revisit the nature of the partially polarized Luttinger liquid around and above the Uimin-Lai-Sutherland point. Our results confirm that this is a two-component liquid and rule out the formation of a single-component vector-chiral phase.
The S=12S=\frac{1}{2}S=21​ chain-boundary excitations occurring in the Haldane phase of S=1S=1S=1 antiferromagnetic spin chains are investigated. The bilinear-biquadratic Hamiltonian is used to study these excitations as a function of the strength of the biquadratic term, \ensuremath{\beta}, between $\ensuremath{-}1<~\ensuremath{\beta}<~1.$ At the Affleck-Kennedy-Lieb-Tasaki point, $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3},$ we show explicitly that these excitations are localized at the boundaries of the chain on a length scale equal to the correlation length $\ensuremath{\xi}=1/\mathrm{ln}3$, and that the on-site magnetization for the first site is 〈S1z〉=23.〈{S}_{1}^{z}〉=\frac{2}{3}.〈S1z​〉=32​. Applying the density matrix renormalization group we show that the chain-boundary excitations remain localized at the boundaries for $\ensuremath{-}1<\ensuremath{\beta}<1.$ As the two critical points \ensuremath{\beta}=\ifmmode\pm\else\textpm\fi{}1 are approached the size of the S=12S=\frac{1}{2}S=21​ objects diverges and their amplitude appears to vanish.
Measurements done on LiVGe2O6 by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)] on phases of the Singlet ground state of the Spin-1 Chain can be described by the Bilinear-biquadratic spin-1 chain. I will discuss these results, among the results of others and talk about theoretical calculations done by A. Lauchli [Phys. Rev. B 74, 144426 (2006)] that describe the phase diagram of the bilinear-biquadratic model. At a particular point in the phase diagram the model becomes SU(3) globally symmetric and is known as the Lai-Sutherland model, it is Bethe ansatz solvable as shown in [A Schmitt et al., J. Phys. A: Math. Gen. 29 (1996) 3951-3962]. Finally I will describe the thermodynamics of the model as where calculated by Lou et al. [Phys. Rev. Lett. 85, 11 (2000)].