We investigate how spontaneous symmetry breaking and symmetry-protected topology compete and cooperate in one-dimensional quantum spin-1/2 systems. We introduce two families of local Hamiltonians with
Z2×Z2×Z2T symmetry which host a zoo of broken-symmetry phases with and without gapless boundary spins, namely, six long-range-ordered gapped phases that differ by their
Z2-symmetry-breaking patterns (FM
x,y,z and Neel
x,y,z), their partner symmetry-protected topological phases (FM
x,ySPT and Neel
x,ySPT), and two gapless phases (gFM
z and gNeel
z) with incommensurate
Z2T-breaking magnetic long-range order and central charge
c=1. Direct transitions between competing gapped phases can be continuous or weakly first order. We demonstrate that continuous phase transition lines have exotic critical properties. For example, a line of deconfined quantum critical points (between the Neel
x and Neel
y phases) described by Tomonaga-Luttinger liquid theory is terminated by the gFM
z phase at a Lifshitz tricritical point which exhibits spontaneous breaking of the continuous U(1) symmetry despite the locality and one-dimensionality of the Hamiltonian, contrary to a naive expectation from the Hohenberg-Mermin-Wagner-Coleman theorem. Furthermore, critical lines between symmetry-protected topological phases with different
Z2-symmetry breaking (e.g., Neel
xSPT and Neel
ySPT) additionally retain low-energy boundary degrees of freedom, realizing symmetry-enriched criticality (or a gapless SPT phase). Our results reveal unconventional mechanisms by which competing symmetry breaking and topology generate ordered critical phases and multicriticality.
We continue recent efforts to discover examples of deconfined quantum criticality in one-dimensional models. In this work we investigate the transition between a
Z3 ferromagnet and a phase with valence bond solid (VBS) order in a spin chain with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ global symmetry. We study a model with alternating projective representations on the sites of the two sublattices, allowing the Hamiltonian to connect to an exactly solvable point having VBS order with the character of
SU(3)-invariant singlets. Such a model does not admit a Lieb-Schultz-Mattis theorem typical of systems realizing deconfined critical points. Nevertheless, we find evidence for a direct transition from the VBS phase to a
Z3 ferromagnet. Finite-entanglement scaling data are consistent with a second-order or weakly first-order transition. We find in our parameter space an integrable lattice model apparently describing the phase transition, with a very long, finite, correlation length of 190878 lattice spacings. Based on exact results for this model, we propose that the transition is extremely weakly first order and is part of a family of deconfined quantum critical points described by walking of renormalization group flows.
The phase diagram of the quantum spin-
21 antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XXZ chain was obtained by Haldane using bosonization techniques [Haldane, Phys. Rev. B 25, 4925 (1982); Phys. Rev. B 26, 5257 (1982)]. It supports three distinct phases for $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$, i.e., a gapless algebraic spin-liquid phase, a gapped long-range ordered Neel phase, and a gapped long-range ordered dimer phase. Even though the Neel and dimer phases are not related hierarchically by a pattern of symmetry breaking, it was shown that they meet along a line of quantum critical points with a U(1) symmetry and central charge
c=1. Here, we extend the analysis made by Haldane on the quantum spin-
21 antiferromagnetic ${J}_{1}^{\phantom{\rule{0.16em}{0ex}}}\text{\ensuremath{-}}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}$ XYZ chain using both bosonization and numerical techniques. We show that there are three Neel phases and the dimer phase that are separated from each other by six planes of phase boundaries realizing Gaussian criticality when $0\ensuremath{\le}{J}_{2}^{\phantom{\rule{0.16em}{0ex}}}/{J}_{1}^{\phantom{\rule{0.16em}{0ex}}}<\frac{1}{2}$. We also show that each long-range ordered phase harbors topological point defects (domain walls) that are dual to those across the phase boundary in that a defect in one ordered phase locally binds the other type of order around its core. By using the bosonization approach, we identify the critical theory that describes simultaneous proliferation of these dual point defects, and show that it supports an emergent U(1) symmetry that originates from the discrete symmetries of the XYZ model. To confirm this numerically, we perform exact diagonalization and density-matrix renormalization-group calculations and show that the critical theory is characterized by the central charge
c=1 with critical exponents that are consistent with those obtained from the bosonization approach. Furthermore, we generalize the field-theoretic description of direct continuous phase transition to higher dimensions, especially in
d=3, by using a nonlinear sigma model (NLSM) with a topological term. In particular, we propose the $\ensuremath{\pi}$-flux phase on the cubic lattice with local quartic interactions as a platform for direct continuous phase transition and deconfined criticality. We discuss possible phase diagrams for the $\ensuremath{\pi}$-flux phase on the cubic lattice with these quartic interactions from the renormalization flow of NLSMs.
Abstract We study the spin-1/2 Kitaev chain with additional XX and symmetric off-diagonal Gamma interactions. By combining the Jordan–Wigner transformation with density matrix renormalization group (DMRG) numerical simulations, we obtain the exact solution of the model and map out the phase diagram containing six distinct phases. The four gapped phases display ferromagnetic and antiferromagnetic magnetic orders along the (1,1,0)- and (1,-1,0)-spin directions, whereas in the gapless phases, the low-energy spectrum consists of two branches of helical Majorana fermions with unequal velocities. The transition lines separating different phases include deconfined quantum critical lines with dynamical critical exponent z = 1 and quadratic critical lines with z = 2. Our work reveals the rich interplay among symmetry, magnetic order, and quantum criticality in the Kitaev–XX–Gamma chain.
We use quantum Monte Carlo simulations to study an S = 1/2 spin model with competing multi-spin interactions. We find a quantum phase transition between a columnar valence-bond solid (cVBS) and a Néel antiferromagnet (AFM), as in the scenario of deconfined quantum-critical points, as well as a transition between the AFM and a staggered valence-bond solid (sVBS). By continuously varying a parameter, the sVBS–AFM and AFM–cVBS boundaries merge into a direct sVBS–cVBS transition. Unlike previous models with putative deconfined AFM–cVBS transitions, e.g., the standard J–Q model, in our extended J–Q model with competing cVBS and sVBS inducing terms the transition can be tuned from continuous to first-order. We find the expected emergent U(1) symmetry of the microscopically Z 4 symmetric cVBS order parameter when the transition is continuous. In contrast, when the transition changes to first-order, the clock-like Z 4 fluctuations are absent and there is no emergent higher symmetry. We argue that the confined spinons in the sVBS phase are fracton-like. We also present results for an SU(3) symmetric model with a similar phase diagram. The new family of models can serve as a useful tool for further investigating open questions related to deconfined quantum criticality and its associated emergent symmetries.
Deconfined criticality and gapless topological states have recently attracted growing attention, as both phenomena go beyond the traditional Landau paradigm. However, the deep connection between these two critical states, particularly in lattice realization, remains insufficiently explored. In this Letter, we reveal that certain deconfined criticality can be regarded as an intrinsically gapless topological state without gapped counterparts in a one dimensional lattice model. Using a combination of field-theoretic arguments and large-scale numerical simulations, we establish the global phase diagram of the model, which features deconfined critical lines separating two distinct spontaneous symmetry breaking ordered phases. More importantly, we unambiguously demonstrate that the mixed anomaly inherent to deconfined criticality enforces topologically robust edge modes near the boundary, providing a general mechanism by which deconfined criticality manifests as a gapless topological state. Our findings not only offer a new perspective on deconfined criticality but also deepen our understanding of gapless topological phases of matter.