We study the phase diagram of a one-dimensional spin quantum breakdown model, which has an exponential U(1) symmetry with charge unit decaying as 2^{-j} with site position j. By exact diagonalization and density matrix renormalization group, we show that the model with spin S≥2 exhibits an exponential U(1) spontaneous symmetry-breaking (SSB) phase dubbed a quantum breakdown condensate. It exhibits a bulk gap violating the Goldstone theorem and an edge mode only on the left edge if in open boundary condition. In a length L lattice, the condensate has O(2^{L}) number of SSB ground states originating from the O(2^{L}) number of exponential U(1) charge sectors, leading to a finite entropy density ln2. This enforces a first-order SSB phase transition into this phase, as observed numerically and verified in the large S limit on an exactly solvable Rokhsar-Kivelson line. The condensate has an SSB order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli (dyadic) map and thus is effectively random. Moreover, we show the condensate exhibits nondecaying local autocorrelations and does not have an off-diagonal long-range order. The quantum breakdown condensate thus behaves as a disorder-free quantum glass and is beyond the existing classifications of phases of matter.
We propose a bosonic quantum breakdown Hubbard model, which generalizes the Bose-Hubbard model by adding an asymmetric breakdown interaction turning one boson into two between adjacent sites. When the normal hopping is zero, this model has a global exponential U(1) symmetry, and we show that the ground state undergoes a first-order phase transition from a Mott insulator (MI) to a spontaneously symmetry breaking (SSB) breakdown condensate as the breakdown interaction increases. Surprisingly, the SSB breakdown condensate does not have a gapless Goldstone mode, which invalidates the Mermin-Wagner theorem and leads to stable SSB in one dimension. Moreover, we show that the quench dynamics of a boson added to MI exhibits a dynamical transition from dielectric to breakdown phases, which happens at a larger breakdown interaction than the ground state phase transition. Between these two transitions, the MI (dielectric) state is a false vacuum stable against dynamical breakdown. Our results reveal that quantum models with unconventional symmetries such as the exponential symmetry can exhibit unexpected properties.
We study symmetric gapped ground states of one-dimensional spin
S chains with a continuous exponential symmetry
U(1)⋊Z2 with integer base
q, where the U(1) charge has a modulated unit of charge
qL−j decaying exponentially with site
j. The simplest model is a quantum breakdown model that can be viewed as a
q-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on
S and
q. For integer
S, odd
q allows a nontrivial
Z2 Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even
q allows no nontrivial SPT phase. For half-odd integer
S, odd
q prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even
q enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.
We study spontaneous particle-hole symmetry breaking in a disordered two-site quantum breakdown model. The model is solvable in the limit of a large number of fermionic modes on each site. For a small ratio of the numbers of modes on the two sites, we derive the free energy analytically and find a finite-temperature symmetry-breaking transition. This transition changes from continuous to first order as the number of fermions produced in each conversion increases. Numerical results show that both types of transition persist beyond this regime, with an order-parameter exponent close to
1/2 for continuous transitions.
We show that a gapped spin-
S chain with antiferromagnetic (AFM) order exhibits, in the thermodynamic limit, exponentially localized fractional $\ifmmode\pm\else\textpm\fi{}\frac{S}{2}$ edge modes when the system possesses U(1) symmetry. We show this for integrable and nonintegrable spin chains, both analytically and numerically. Through exact analytical solutions, we show that an AFM spin-
21 chain with explicitly broken
Z2 symmetry and an integrable AFM spin-1 chain with spontaneously broken
Z2 symmetry have $\ifmmode\pm\else\textpm\fi{}\frac{1}{4}$ and $\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$ fractionalized edge modes, respectively. Furthermore, employing the density matrix renormalization group technique, we extend this analysis to generic $XXZ\text{\ensuremath{-}}S$ chains with $S\ensuremath{\le}3$ and demonstrate that these fractional spins are robust quantum observables, substantiated by the observation of variance of the associated fractional spin operators that is consistent with a vanishing functional form in the thermodynamic limit. Moreover, we find that the edge modes are robust to a disorder that couples to the N\'eel order parameter.
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.
We explore the states of matter arising from the spontaneous symmetry breaking (SSB) of
Z2 non-onsite symmetries. In one spatial dimension, we construct a frustration-free lattice model exhibiting SSB of a non-onsite symmetry, which features the coexistence of two ground states with distinct symmetry-protected topological (SPT) orders. We analytically prove the two-fold ground-state degeneracy and the existence of a finite energy gap. Fixing the symmetry sector yields a long-range entangled ground state that features long-range correlations among non-invertible charged operators. We also present a constant-depth measurement-feedback protocol to prepare such a state with a constant success probability in the thermodynamic limit, which may be of independent interest. Under a symmetric deformation, the SSB persists up to a critical point, beyond which a gapless phase characterized by a conformal field theory emerges. In two spatial dimensions, the SSB of 1-form non-onsite symmetries leads to a long-range entangled state (SPT soup) - a condensate of 1d SPT along any closed loops. On a torus, there are four such locally indistinguishable states that exhibit algebraic correlations between local operators, which we derived via a mapping to the critical
O(2) loop model. This constitutes an intriguing example of `topological quantum criticality'.