© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2308.07372 · Physical review. B./Physical review. B

Classical nonrelativistic fractons

Abhishodh Prakash, Alain Goriely, Shivaji L. Sondhi

Published Feb 27, 2024• 10 citations

Abstract

We initiate the study of the classical mechanics of nonrelativistic fractons in its simplest setting—that of identical one-dimensional particles with local Hamiltonians characterized by a conserved dipole moment in addition to the usual symmetries of space and time translation invariance. We introduce a family of models and study the N -body problem for them. We find that locality leads to a “Machian” dynamics in which a given particle exhibits finite inertia only if within a specified distance of another particle. For well-separated particles, this dynamics leads to immobility, much as for quantum models of fractons discussed before. For two or more particles within inertial reach of each other at the start of motion, we obtain an interesting interplay of inertia and interactions. Specifically, for a solvable “inertia only” model of fractons, we find that two particles always become immobile at long times. Remarkably, three particles generically evolve to a late time state with one immobile particle and two oscillating about a common center of mass with generalizations of such “Machian clusters” for N > 3 . Interestingly, these Machian clusters exhibit physical limit cycles in a Hamiltonian system even though mathematical limit cycles are forbidden by Liouville's theorem. Published by the American Physical Society 2024

DOI: 10.1103/physrevb.109.054313

Related papers

Machian fractons, Hamiltonian attractors, and nonequilibrium steady states

Abhishodh Prakash, Ylias Sadki, Shivaji L. Sondhi

We study the N fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all N , despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to nonequilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. We provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of “order-by-disorder” borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a paradigm for Hamiltonian dynamics and nonequilibrium many-body physics. Published by the American Physical Society 2024

arXiv:2312.02271•Physical review. B./Physical review. B•Jul 3, 2024•4 citations

Large N Fractons

Kristan Jensen, Amir Raz

We consider theories of N-component fields with exotic spacetime symmetries, including a conserved dipole moment. Microscopic charges are immobile by symmetry, and so resemble fractons. Using collective fields we solve these models to leading order in large N. The large N solution reveals that these models are strongly correlated, and that interactions dress the microscopic charges so that they become mobile, long-lived quasiparticles. Dipole symmetry is spontaneously broken throughout the phase diagram of these models, leading to a low-energy Goldstone description.

Physical Review Letters•Feb 15, 2024•10 citations

Multi-Particle Quantum Walks in a Dipole-Conserving Bose-Hubbard Model

Sooshin Kim, Byungmin Kang, Perrin Segura +4 more

When particles move through a crystal or optical lattice, their motion can sometimes become frozen by strong external forces -- yet collective motion may still emerge through subtle many-body effects. In this work, we explore such constrained dynamics by realizing a dipole-conserving Bose-Hubbard model, where single atoms are immobile but pairs of particles can move cooperatively while preserving the system's center of mass, i.e. the overall dipole moment of the particle distribution. Starting from a one-dimensional chain of ultracold bosonic atoms in an optical lattice, we generate localized dipole excitations consisting of a hole and a doublon using site-resolved optical potentials and characterize their quantum walks and scattering dynamics. Our study provides a bottom-up investigation of a Hamiltonian with kinetic constraints, and paves the way for exploring low-energy phases of fractonic matter in existing experimental platforms.

cond-mat.quant-gasCross-listedarXiv:2511.02343•Nature Physics•Nov 5, 2025•0 citations

Relativistic quantum mechanics of particles with direct interactions

F. Coester, Wayne Nicholas Polyzou

For an arbitrary number of particles an explicit construction is given for direct interactions satisfying cluster separability (macrolocality). The Hamiltonian is in many ways similar to the nonrelativistic many-body Hamiltonian. Under conditions of physical interest the strength of the $N$-body interaction decreases rapidly with increasing $N$.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Sep 15, 1982•147 citations

Continuum fractons: Quantization and the few-body problem

Ylias Sadki, Abhishodh Prakash, Shivaji L. Sondhi

We formulate a continuum quantum mechanics for nonrelativistic, dipole-conserving fractons. Imposing symmetries and locality results in novel phenomena absent in ordinary quantum mechanical systems. A single fracton has a vanishing Hamiltonian, and thus its spectrum is entirely composed of zero modes. For the two-body problem, the Hamiltonian is perfectly described by Sturm-Liouville (SL) theory. The effective two-body Hamiltonian is an SL operator on ( − 1 , 1 ) whose spectral type is set by the edge behavior of the pair inertia function K ( x ) ∼ | x − x edge | θ . We identify a sharp transition at θ = 2 : for θ < 2 the spectrum is discrete and wave packets reflect from the edges, whereas for θ > 2 the spectrum is continuous and wave packets slow down and, dominantly, squeeze into asymptotically narrow regions at the edges. For three particles, the differential operator corresponding to the Hamiltonian is piecewise defined, requiring several “matching conditions” which cannot be analyzed as easily. We proceed with a lattice regularization that preserves dipole conservation and implicitly selects a particular continuum Hamiltonian that we analyze numerically. We find a spectral transition in the three-body spectrum and find evidence for quantum analogs of fracton attractors in both eigenstates and in the time evolution of wave packets. We provide intuition for these results which suggests that the lack of ergodicity of classical continuum fractons will survive their quantization for large systems.

arXiv:2510.00110•Physical review. B./Physical review. B•Jun 9, 2026•0 citations

On the nonrelativistic limit of action-at-a-distance relativistic classical mechanics

J. G. Wray

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B•Dec 1, 1972•1 citation

We initiate the study of the classical mechanics of nonrelativistic fractons in its simplest setting—that of identical one-dimensional particles with local Hamiltonians characterized by a conserved dipole moment in addition to the usual symmetries of space and time translation invariance. We introduce a family of models and study the N -body problem for them. We find that locality leads to a “Machian” dynamics in which a given particle exhibits finite inertia only if within a specified distance of another particle. For well-separated particles, this dynamics leads to immobility, much as for quantum models of fractons discussed before. For two or more particles within inertial reach of each other at the start of motion, we obtain an interesting interplay of inertia and interactions. Specifically, for a solvable “inertia only” model of fractons, we find that two particles always become immobile at long times. Remarkably, three particles generically evolve to a late time state with one immobile particle and two oscillating about a common center of mass with generalizations of such “Machian clusters” for N > 3 . Interestingly, these Machian clusters exhibit physical limit cycles in a Hamiltonian system even though mathematical limit cycles are forbidden by Liouville's theorem. Published by the American Physical Society 2024

We study the N fracton problem in classical mechanics, with fractons defined as point particles that conserve multipole moments up to a given order. We find that the nonlinear Machian dynamics of the fractons is characterized by late-time attractors in position-velocity space for all N , despite the absence of attractors in phase space dictated by Liouville's theorem. These attractors violate ergodicity and lead to nonequilibrium steady states, which always break translational symmetry, even in spatial dimensions where the Hohenberg-Mermin-Wagner-Coleman theorem for equilibrium systems forbids such breaking. We provide a conceptual understanding of our results using an adiabatic approximation for the late-time trajectories and an analogy with the idea of “order-by-disorder” borrowed from equilibrium statistical mechanics. Altogether, these fracton systems host a paradigm for Hamiltonian dynamics and nonequilibrium many-body physics. Published by the American Physical Society 2024

We consider theories of N-component fields with exotic spacetime symmetries, including a conserved dipole moment. Microscopic charges are immobile by symmetry, and so resemble fractons. Using collective fields we solve these models to leading order in large N. The large N solution reveals that these models are strongly correlated, and that interactions dress the microscopic charges so that they become mobile, long-lived quasiparticles. Dipole symmetry is spontaneously broken throughout the phase diagram of these models, leading to a low-energy Goldstone description.

When particles move through a crystal or optical lattice, their motion can sometimes become frozen by strong external forces -- yet collective motion may still emerge through subtle many-body effects. In this work, we explore such constrained dynamics by realizing a dipole-conserving Bose-Hubbard model, where single atoms are immobile but pairs of particles can move cooperatively while preserving the system's center of mass, i.e. the overall dipole moment of the particle distribution. Starting from a one-dimensional chain of ultracold bosonic atoms in an optical lattice, we generate localized dipole excitations consisting of a hole and a doublon using site-resolved optical potentials and characterize their quantum walks and scattering dynamics. Our study provides a bottom-up investigation of a Hamiltonian with kinetic constraints, and paves the way for exploring low-energy phases of fractonic matter in existing experimental platforms.

For an arbitrary number of particles an explicit construction is given for direct interactions satisfying cluster separability (macrolocality). The Hamiltonian is in many ways similar to the nonrelativistic many-body Hamiltonian. Under conditions of physical interest the strength of the

N

-body interaction decreases rapidly with increasing

N

.

We formulate a continuum quantum mechanics for nonrelativistic, dipole-conserving fractons. Imposing symmetries and locality results in novel phenomena absent in ordinary quantum mechanical systems. A single fracton has a vanishing Hamiltonian, and thus its spectrum is entirely composed of zero modes. For the two-body problem, the Hamiltonian is perfectly described by Sturm-Liouville (SL) theory. The effective two-body Hamiltonian is an SL operator on ( − 1 , 1 ) whose spectral type is set by the edge behavior of the pair inertia function K ( x ) ∼ | x − x edge | θ . We identify a sharp transition at θ = 2 : for θ < 2 the spectrum is discrete and wave packets reflect from the edges, whereas for θ > 2 the spectrum is continuous and wave packets slow down and, dominantly, squeeze into asymptotically narrow regions at the edges. For three particles, the differential operator corresponding to the Hamiltonian is piecewise defined, requiring several “matching conditions” which cannot be analyzed as easily. We proceed with a lattice regularization that preserves dipole conservation and implicitly selects a particular continuum Hamiltonian that we analyze numerically. We find a spectral transition in the three-body spectrum and find evidence for quantum analogs of fracton attractors in both eigenstates and in the time evolution of wave packets. We provide intuition for these results which suggests that the lack of ergodicity of classical continuum fractons will survive their quantization for large systems.