Lieb-Schultz-Mattis anomalies and web of dualities induced by gauging in quantum spin chains

TL;DR

<3-5 sentence high-level summary>This work treats Lieb-Schultz-Mattox anomalies as mixed 't Hooft anomalies between internal and crystalline symmetries in one-dimensional quantum spin chains. By gauging non-anomalous subgroups of internal symmetries, the authors construct KW and JW dualities that reveal nontrivial group extensions between dual internal and crystalline symmetries, using a triality of bond algebras as the organizing principle. They apply this framework to a spin-1/2 XYZ chain and its KW/JW duals, showing that deconfined quantum critical points map to conventional or topological transitions in the dual theories, and generalize the construction to $ frac{Z}{n}$ clock models with parity-dependent anomaly behavior. The results offer a lattice-diagnostic route to LSM anomalies and suggest broader implications for higher-dimensional systems and generalized symmetries.

Abstract

Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed 't Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal $\mathbb{Z}^{\,}_{2}\times\mathbb{Z}^{\,}_{2}$ symmetry with translation or reflection symmetry. We establish a triality of models by gauging a $\mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}\times\mathbb{Z}^{\,}_{2}$ symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 $XYZ$ chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to $\mathbb{Z}^{\,}_{n}$ clock models and provide a reinterpretation of the dual internal symmetries in terms of $\mathbb{Z}^{\,}_{n}$ charge and dipole symmetries.

Figures (6)

• Figure 1: Triality between the triplet of bond algebras (\ref{['eq:three trial bond algebras']}).
• Figure 2: Exactly soluble points in the phase diagram of Hamiltonian (\ref{['eq:def Hamiltonian b=0']}) in the reduced coupling space (\ref{['eq:reduced coupling space']}). The small squares at the lower left and right corners of the phase diagram each realize the antiferromagnetic nearest-neighbor Ising chain. The large squares at the upper left and right corners of the phase diagram each realize two identical and decoupled antiferromagnetic nearest-neighbor Ising chains. The ground states are long-range ordered, gapped, and two-fold (four-fold) degenerate for the lower (upper) corners. The line $(\Delta,J=0)$ realizes non-interacting fermions without fermion-number conservation, except when $\Delta=1$. The small circle at $(\Delta=1,J=0)$ realizes non-interacting spinless fermions at half filling with a nearest-neighbor uniform hopping amplitude. The ground state is gapless with a quantum criticality encoded by a $c=1$ conformal field theory in $(1+1)$-dimensional spacetime in the thermodynamic limit. The large circle at $(\Delta=1,J=\infty)$ realizes two decoupled chains of non-interacting spinless fermions at half filling with a nearest-neighbor uniform hopping amplitude. The ground state is gapless with a quantum criticality behavior encoded by a $c=2$ conformal field theory in $(1+1)$-dimensional spacetime in the thermodynamic limit. The diamonds at $(\Delta,J)=(0,1/2),(\infty,1/2)$ are first-order boundaries between the phases governed by the Ising fixed points (small and large squares) at $\Delta=0$ and $\Delta=\infty$, respectively. The open Majumdar-Ghosh (dashed) line at $(\Delta,1/2)$ with $0<\Delta<\infty$ realizes the dimer phase. The dimer ground states are gapped and two-fold degenerate along the Majumdar-Ghosh (dashed) line.
• Figure 3: Phase diagram of Hamiltonian (\ref{['eq:def Hamiltonian b=0']}) in the reduced coupling space (\ref{['eq:reduced coupling space']}) with $0\leq J\leq1/2$. There are three phases: the $\mathrm{Neel}^{\,}_{x}$, the $\mathrm{Neel}^{\,}_{y}$, and the dimer phase. Each one of these three phases corresponds to gapped and two-fold degenerate ground states in the thermodynamic limit. In each phase, a non-degenerate ground state is selected by spontaneous symmetry breaking of the symmetry group $\mathrm{G}^{\,}_{\mathrm{tot}}$ defined in Eq. (\ref{['eq:def Gtot for Hb']}). The dimer phase is found on both sides of the open MG line defined by $0<\Delta<\infty$ and $J=1/2$. All the phase boundaries with $0<\Delta<\infty$ and $J<1/2$ are continuous quantum phase transitions that realize deconfined quantum criticality Mudry19. The tricritical point (the large black circle) where the three phases meet realizes the $\mathrm{SU}(2)^{\,}_{1}$ conformal field theory in $(1+1)$-dimensional spacetime.
• Figure 4: Phase diagram of Hamiltonian (\ref{["eq:def Hamiltonian b'=0"]}) with the Hilbert space $\mathcal{H}^{\,\vee}_{b'=0}$ as domain of definition. The red boundaries realize a continuous quantum phase transition that separate two phases, one of which descends from the other through spontaneous symmetry breaking by which a symmetry-breaking local order parameter acquires a non-vanishing expectation value in the symmetry-broken phase, i.e., the Landau-Ginzburg paradigm of phase transitions. The blue boundary realizes a continuous topological quantum phase transition between two phases that are distinguished by a non-local order parameter. These phases are adiabatically connected to the ground states \ref{["eq:H b' paramagnet"]} and \ref{["eq:H b' spt"]} for $\Delta<1$ and $\Delta>1$, respectively.
• Figure 5: The phase diagram of Hamiltonian (\ref{['eq:def Hamiltonian b=0']}) restricted to the subspace $\mathcal{H}^{\,}_{b=0;+}$ of the Hilbert space $\mathcal{H}^{\,}_{b=0}$ is dual to the phase diagram of Hamiltonian (\ref{["eq:def Hamiltonian b'=0"]}) restricted to the subspace $\mathcal{H}^{\vee}_{b'=0;+}$ of the Hilbert space $\mathcal{H}^{\,\vee}_{b'=0}$. The pair of dual subspaces are to be found in the first line of Table \ref{["Table:Kramers-Wannier dualization b to b'"]}. The two states $| \mathrm{Neel}^{x}\rangle^{+}$ and $| \mathrm{Neel}^{y}\rangle^{+}$ are defined in Eqs. \ref{['eq:def Neelx+']} and \ref{['eq:def Neely+']}, while dimer doublet refers to the ground states \ref{['eq:Hb dimer GS']}. On the dual side, the paramagnetic states $| \mathrm{PM}\rangle$, and $| \mathrm{SPT}\rangle$ state are defined in Eqs. \ref{["eq:H b' paramagnet"]} and \ref{["eq:H b' spt"]}, respectively. By $\mathrm{D}^{+}_{8}$ doublet, we refer to the states $| \mathrm{Dimer}^{\vee}_{\mathrm{o}}\rangle$ and $| \mathrm{Dimer}^{\vee}_{\mathrm{e}}\rangle$, both of which are in the subspace $\mathcal{H}^{\vee}_{b'=0;+}$ and defined in Eq. \ref{['eq:D_8 doublet']}. The symbols $\rightharpoonup$ and $\leftharpoondown$ denote gauging the diagonal subgroups generated by $\widehat{U}^{\,}_{r^{z}_{\pi}}$ and by its dual $\widehat{U}^{\vee}_{r^{z}_{\pi}}$, respectively.

Theorems & Definitions (12)

• Theorem 1: Translation LSM
• Definition 1: Translation LSM anomaly
• Theorem 2: Reflection LSM
• Definition 2: Reflection LSM anomaly
• Remark : LSM anomaly versus mixed 't Hooft anomaly
• Theorem 3: Generalized translation LSM
• Definition 3: Generalized translation LSM anomaly
• Remark
• Conjecture 4: Generalized reflection LSM
• Definition 4: Reflection LSM anomaly