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arXiv:2308.16241 · Physical review. B./Physical review. B

Filling constraints on translation invariant dipole conserving systems

F. J. Burnell, Sanjay Moudgalya, Abhinav Prem

Published Sep 18, 2024• 10 citations
arXivHTMLPDF

Abstract

Systems with conserved dipole moment have drawn considerable interest in light of their realization in recent experiments on tilted optical lattices. An important issue regarding such systems is delineating the conditions under which they admit a unique gapped ground state that is consistent with all symmetries. Here, we study one-dimensional translation-invariant lattices that conserve $\mathrm{U}(1)$ charge and ${\mathbb{Z}}_{L}$ dipole moment, where discreteness of the dipole symmetry is enforced by periodic boundary conditions, with $L$ the system size. We show that in these systems a symmetric, gapped, and nondegenerate ground state requires not only integer charge filling, but also a fixed value of the dipole filling, while other fractional dipole fillings enforce either a gapless or symmetry-breaking ground state. In contrast with prior results in the literature, we find that the dipole filling constraint depends both on the charge filling as well as the system size, emphasizing the subtle interplay of dipole symmetry with boundary conditions. We support our results with numerical simulations and exact results.

DOI
10.1103/physrevb.110.l121113

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Systems with conserved dipole moment have drawn considerable interest in light of their realization in recent experiments on tilted optical lattices. An important issue regarding such systems is delineating the conditions under which they admit a unique gapped ground state that is consistent with all symmetries. Here, we study one-dimensional translation-invariant lattices that conserve U(1)\mathrm{U}(1)U(1) charge and ZL{\mathbb{Z}}_{L}ZL​ dipole moment, where discreteness of the dipole symmetry is enforced by periodic boundary conditions, with LLL the system size. We show that in these systems a symmetric, gapped, and nondegenerate ground state requires not only integer charge filling, but also a fixed value of the dipole filling, while other fractional dipole fillings enforce either a gapless or symmetry-breaking ground state. In contrast with prior results in the literature, we find that the dipole filling constraint depends both on the charge filling as well as the system size, emphasizing the subtle interplay of dipole symmetry with boundary conditions. We support our results with numerical simulations and exact results.
The Lieb-Schultz-Mattis (LSM) theorem and its descendants impose strong constraints on the low-energy behavior of interacting quantum systems. In this paper, we formulate LSM-type constraints for lattice translation invariant systems with generalized U(1) symmetries which have recently appeared in the context of fracton phases: U(1) polynomial shift and subsystem symmetries. Starting with a generic interacting system with conserved dipole moment, we examine the conditions under which it supports a symmetric, gapped, and nondegenerate ground state, which we find requires that both the filling fraction and the bulk charge polarization take integer-values. Similar constraints are derived for systems with higher moment conservation laws or subsystem symmetries, in addition to lower bounds on the ground-state degeneracy when certain conditions are violated. Finally, we discuss the mapping between LSM-type constraints for subsystem symmetries and the anomalous symmetry action at boundaries of subsystem symmetric topological (SSPT) states.
Nonperturbative constraints on many-body physics, such as the famous Lieb-Schultz-Mattis theorem, are valuable tools for studying strongly correlated systems. To this end, we present a number of nonperturbative results that constrain the low-energy physics of systems having conserved dipole moments. We find that for these systems, a unique translationally invariant gapped ground state is only possible if the polarization of the system is integer. Furthermore, if a lattice system also has U(1)\text{U}(1)U(1) subsystem charge conservation symmetry, a unique gapped ground state is only possible if the particle filling along these subsystems is integer. We also apply these methods to spin systems, and determine criteria for the existence of a different type of magnetic response plateau in the presence of a nonuniform magnetic field. Finally, we formulate a version of Luttinger's theorem for one-dimensional systems consisting of dipoles.
The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.
Interest in finite-size systems has risen in the last decades, due to the focus on nanotechnological applications and because they are convenient for numerical treatment that can subsequently be extrapolated to infinite lattices.Independently of the envisioned application, special attention must be given to boundary condition, which may or may not preserve the symmetry of the infinite lattice.Here we present a detailed study of the compatibility between boundary conditions and conservation laws.The conflict between open boundary conditions and momentum conservation is well understood, but we examine other symmetries, as well: we discuss gauge invariance, inversion, spin, and particle-hole symmetry and their compatibility with open, periodic, and twisted boundary conditions.In the interest of clarity, we develop the reasoning in the framework of the one-dimensional half-filled Hubbard model, whose Hamiltonian displays a variety of symmetries.Our discussion includes analytical and numerical results.Our analytical survey shows that, as a rule, boundary conditions break one or more symmetries of the infinite-lattice Hamiltonian.The exception is twisted boundary condition with the special torsion Θ = πL/2, where L is the lattice size.Our numerical results for the ground-state energy at half-filling and the energy gap for L = 2-7 show how the breaking of symmetry affects the convergence to the L → ∞ limit.We compare the computed energies and gaps with the exact results for the infinite lattice drawn from the Bethe-Ansatz solution.The deviations are boundary-condition dependent.The special torsion yields more rapid convergence than open or periodic boundary conditions.For sizes as small as L = 7, the numerical results for twisted condition are very close to the L → ∞ limit.We also discuss the ground-state electronic density and magnetization at half filling under the three boundary conditions.
Systems with dipole moment conservation have been of recent interest, as they realize both novel quantum dynamics and exotic ground state phases. In this work, we study some generic properties of 1-D and 2-D dipole-conserving fermionic models at integer fillings. We find that a dipolar symmetry-breaking phase can result in a mean-field band insulator whose topological indices can strongly affect the low-energy physics of the dipolar Goldstone modes. We study the 2-D topological phase transition of the mean-field ground states in the presence of the Goldstone modes. The critical theory resembles the 2+1d quantum electrodynamics coupled to massless Dirac fermions with some crucial differences and shows a novel quantum critical point featuring a nontrivial dynamical exponent. We also discuss the analogous case of 1-D dipole-conserving models and the role of topological invariants.
Past studies of fractional quantum Hall systems have found that the charge gap dominates the neutral gap for all relevant parameter choices. We report a wide-ranging proof that this domination is in fact a universal property of any Hamiltonian that satisfies a few simple structural properties: translation-invariance, charge conservation, dipole conservation, and a fractionally filled ground state. The result applies to both fermions and bosons. Our main tool is a new mathematical scheme, the gap comparison method, which provides a sequence of inequalities that relate the spectral gaps in successive particle number sectors. Our finding sheds new light on dipole conservation's profound effects on many-body physics.