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arXiv:1011.1265 · Physical Review B

Quantum criticality in the SO(5) bilinear-biquadratic Heisenberg chain

Fabien Alet, Sylvain Capponi, Heloise Nonne, P. Lecheminant, Ian P. McCulloch

Published Feb 22, 2011• 26 citations
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Abstract

The zero-temperature properties of the SO(5) bilinear-biquadratic Heisenberg chain are investigated by means of a low-energy approach and large-scale numerical calculations. In sharp contrast to the spin-1 SO(3) Heisenberg chain, we show that the SO(5) Heisenberg chain is dimerized with a twofold degenerate ground state. On top of this gapful phase, we find the emergence of a nondegenerate gapped phase with hidden (${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}){}^{2}$ symmetry and spin-3/2 edge states that can be understood from a SO(5) AKLT wave function. We derive a low-energy theory describing the quantum critical point which separates these two gapped phases. It is shown and confirmed numerically that this quantum critical point belongs to the SO(5)${}_{1}$ universality class.

DOI
10.1103/physrevb.83.060407

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The zero-temperature properties of the SO(5) bilinear-biquadratic Heisenberg chain are investigated by means of a low-energy approach and large-scale numerical calculations. In sharp contrast to the spin-1 SO(3) Heisenberg chain, we show that the SO(5) Heisenberg chain is dimerized with a twofold degenerate ground state. On top of this gapful phase, we find the emergence of a nondegenerate gapped phase with hidden (Z2\ifmmode×\else\texttimes\fiZ2)2{Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}){}^{2}Z2​\ifmmode×\else\texttimes\fiZ2​)2 symmetry and spin-3/2 edge states that can be understood from a SO(5) AKLT wave function. We derive a low-energy theory describing the quantum critical point which separates these two gapped phases. It is shown and confirmed numerically that this quantum critical point belongs to the SO(5)1{}_{1}1​ universality class.
Higher-spin materials can host exotic quantum phases with no counterparts in familiar spin-\textonehalf{} systems. Using density matrix renormalization group calculations, the authors chart here the phase diagram of a spin-1 bilinear-biquadratic-Kitaev chain and reveal two Kitaev-driven phases. A Kitaev nematic phase emerges from a delicate biquadratic-dimer background through an Ising quantum critical point, which further evolves into a Kitaev dimer phase that spontaneously breaks screw symmetry and crystallizes into a gapped order of alternating \ensuremath{\mathbb{Z}}2{}_{2}2​ bond parities.
We study the phase diagram of the SOq(3){\text{SO}}_{q}(3)SOq​(3) quantum group invariant spin-1 bilinear-biquadratic spin chain for real values of q>1q>1q>1. Numerical computations suggest that the chain has at least three clearly distinguished phases: A chiral analog of the Haldane phase, a dimerized phase, and a ferromagnetic phase. In contrast, the counterpart of the extended critical region that is known to exist for q=1q=1q=1 remains elusive. Our results show that the Haldane phase fails to exhibit a twofold degeneracy in the entanglement spectrum but that the degeneracy is restored upon a suitable qqq deformation of the entanglement Hamiltonian, which can be interpreted as a Zeeman field. The structure of the phase diagram is confirmed through analytical calculations in the extreme anisotropic limit $q\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results suggest that symmetries of the form Uq[su(2)]{\mathcal{U}}_{q}\left[su(2)\right]Uq​[su(2)] for distinct choices of qqq should be interpreted as one single family instead of separate symmetries when defining SPT phases, leading naturally to the notion of a qqq-SPT phase.
We present a low-energy effective field theory to describe the SO(n) bilinear-biquadratic spin chain. We start with n=6 and construct the effective theory by using six Majorana fermions. After determining various correlation functions, we characterize the phases and establish the relation between the effective theories for SO(6) and SO(5). Together with the known results for n=3 and 4, a reduction mechanism is proposed to understand the ground state for arbitrary SO(n). Also, we provide a generalization of the Lieb-Schultz-Mattis theorem for SO(n). The implications of our results for entanglement and correlation functions are discussed.
A common feature of topological phases of matter is the fractionalization of the quantum number in their low-energy excitations. Such information is encoded in their ground state wave functions, but emerges in the bipartite entanglement spectra. The symmetric extensive bipartition is an effective novel method to create deconfined fractionalized edge particles in the reduced subsystem, which lead to quantum critical behavior associated with the transition from the topological phase to its adjacent trivial phase. Here we report the interesting results revealed by applying this method to the one-dimensional SO(5) symmetric valence-bond solid state being a spin-2 symmetry protected topological phase. From the finite-size entanglement spectrum, we find the lowest level to be an SO(5) singlet with a logarithmic entanglement entropy. Surprisingly, the first excited level is also an SO(5) singlet and the spectral gap scales with the subsystem size as ${L}_{A}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\simeq}1.978$. In the thermodynamic limit, a novel quantum criticality emerges with SO(5) spinons and their four-body singlet bound states as elementary excitations, hence ruling out the possibility of being described by a conformal field theory. Moreover, the entanglement Hamiltonian can be determined as an SO(5) symmetric nearest neighbor spin-3/2 quadruple-quadruple interaction with a negative coupling. Our work thus demonstrates the power of this new method in the study of quantum criticality encoded in the topological ground states.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.
We investigate the ground-state phase diagram and critical properties of the S=1/2S=1/2S=1/2 two-leg Heisenberg spin ladder system with a negative four-spin interaction using a numerical exact diagonalization method. Using the perturbation theory in the strong negative rung-coupling limit, we derive an S=1S=1S=1 bilinear-biquadratic chain as an effective model. We discuss the ground-state phase diagram in this limit. Next we numerically determine a phase boundary between the rung singlet phase and the columnar dimer (CD) phase by the phenomenological renormalization group method, and one between the CD phase and the Haldane phase by the twisted boundary condition method. We confirm that the phase transition between the CD phase and the Haldane phase is of second order and this universality class is described by the k=2k=2k=2 SU(2)SU(2)SU(2) Wess-Zumino-Novikov-Witten nonlinear $\ensuremath{\sigma}$ model, calculating the central charge and scaling dimensions.