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Physical Review B

Emergent quantum criticality from fractionalizing one-dimensional SO(5) symmetric valence-bond solid states

Wen-Jia Rao, Kang Cai, Xin Wan, Guang-Ming Zhang

Published Dec 21, 2015• 1 citation
Publisher

Abstract

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.

DOI
10.1103/physrevb.92.214430

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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 LA\ensuremath−\ensuremathν{L}_{A}^{\ensuremath{-}\ensuremath{\nu}}LA\ensuremath−\ensuremathν​ with \ensuremathν\ensuremath≃1.978\ensuremath{\nu}\ensuremath{\simeq}1.978\ensuremathν\ensuremath≃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 study the Rényi entanglement entropy (EE) of the two-dimensional J-Q model, the emblematic quantum spin model of deconfined criticality at the phase transition between antiferromagnetic and valence-bond-solid ground states. State-of-the-art quantum Monte Carlo calculations of the EE reveal critical corner contributions that scale logarithmically with the system size, with a coefficient in remarkable agreement with the form expected from a large-N conformal field theory with SO(N=5) symmetry. However, details of the bipartition of the lattice are crucial in order to observe this behavior. If the subsystem for the reduced density matrix does not properly accommodate valence-bond fluctuations, logarithmic contributions appear even for cornerless bipartitions. We here use a 45° tilted cut on the square lattice. Beyond supporting an SO(5) deconfined quantum critical point, our results for both the regular and tilted cuts demonstrate important microscopic aspects of the EE that are not captured by conformal field theory.
Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki valence-bond solid (VBS) states for the integer spin-SSS Haldane gapped phase, we can create an array of fractionalized spin-SSS/2 edge states with the super unit cell lll in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with even lll corresponds to the quantum antiferromagnetic Heisenberg spin-SSS/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberg half-odd integer-spin model with spinon excitations, characterizing the critical point separating the topological Haldane phase from the trivial gapped phase. Our results thus demonstrate that the topological bulk property not only determines its fractionalized edge states but also the quantum criticality associated with the topological phase, where the elementary excitations are precisely those fractionalized edge degrees of freedom confined in the bulk of the topological phase.
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 show numerically that the "deconfined" quantum critical point between the Néel antiferromagnet and the columnar valence-bond solid, for a square lattice of spin 1/2, has an emergent SO(5) symmetry. This symmetry allows the Néel vector and the valence-bond solid order parameter to be rotated into each other. It is a remarkable (2+1)-dimensional analogue of the SO(4)=[SU(2)×SU(2)]/Z(2) symmetry that appears in the scaling limit for the spin-1/2 Heisenberg chain. The emergent SO(5) symmetry is strong evidence that the phase transition in the (2+1)-dimensional system is truly continuous, despite the violations of finite-size scaling observed previously in this problem. It also implies surprising relations between correlation functions at the transition. The symmetry enhancement is expected to apply generally to the critical two-component Abelian Higgs model (noncompact CP(1) model). The result indicates that in three dimensions there is an SO(5)-symmetric conformal field theory that has no relevant singlet operators, so is radically different from conventional Wilson-Fisher-type conformal field theories.
Under an appropriate symmetric extensive bipartition in a one-dimensional symmetry-protected topological (SPT) phase, a bulk critical-entanglement spectrum can be obtained, resembling the excitation spectrum of the critical point separating the SPT phase from the trivial (vacuum) state. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry-breaking phase transitions. For the S=1S=1S=1 SPT (Haldane) phase with the Affleck-Kennedy-Lieb-Tasaki exact wave function, the resulting critical-entanglement spectrum has a residual entropy per lattice site sr=0.67602{s}_{r}=0.67602sr​=0.67602, showing a delocalized version of the edge excitations in the SPT phase. From the wave function corresponding to the lowest entanglement energy level, the central charge of the critical point can be extracted $c\ensuremath{\approx}1.01\ifmmode\pm\else\textpm\fi{}0.01$. The critical theory can be identified as the same effective field theory as the spin-1/2 antiferromagnetic Heisenberg chain or the spin-1/2 Haldane-Shastry model with inverse-square long-range interaction.
Deconfined quantum criticality with emergent SO(5) symmetry in correlated systems remains elusive. Here, by performing numerically-exact state-of-the-art quantum Monte Carlo (QMC) simulations, we show convincing evidences of deconfined quantum critical points (DQCP) between antiferromagnetic and valence-bond-solid phases in the extended Hubbard model of fermions on the honeycomb lattice with large system sizes. We further demonstrate evidences of the SO(5) symmetry at the DQCP. It is important to note that the critical exponents obtained by finite-size scaling at the DQCP here are consistent with the rigourous conformal bounds. Consequently, we established a promising arena of DQCP with emergent SO(5) symmetry in interacting systems of fermions. Its possible experimental relevances in correlated systems of Dirac fermions will be discussed briefly.