Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2004.14393 · Physical review. B./Physical review. B

Odd fracton theories, proximate orders, and parton constructions

Michael Pretko, S. A. Parameswaran, Michael Hermele

Published Nov 6, 2020• 11 citations
arXivHTMLPDF

Abstract

The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy nontrivial conditions on their low-energy properties when a combination of lattice translation and $U(1)$ symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other subdimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that ``odd'' versions of X-cube fracton order can occur in systems at half-odd-integer filling, generalizing the notion of odd ${Z}_{2}$ gauge theory to the fracton setting. At half-odd-integer filling, exiting the X-cube phase by condensing fractional quasiparticles leads to symmetry breaking, thereby allowing us to identify a class of conventionally ordered phases proximate to phases with fracton order. We leverage a dual description of one of these ordered phases to show that its topological defects naturally have restricted mobility. Condensing pairs of these defects then leads to a fracton phase, whose excitations inherit these mobility restrictions.

DOI
10.1103/physrevb.102.205106

Related papers

Lieb-Schultz-Mattis–type constraints on fractonic matter

Huan He, Yizhi You, Abhinav Prem

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.

arXiv:1912.10520•Physical review. B./Physical review. B•Apr 30, 2020•23 citations

Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases

Shenghan Jiang, Meng Cheng, Yang Qi +1 more

We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.

arXiv:1907.08596•SciPost Physics•Aug 6, 2021•22 citations

Mechanisms of Fracton Phases

Han Ma

Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.

•Oct 9, 2018•0 citations

Lieb-Schultz-Mattis constraints for hyperbolic lattices

G. Shankar, Joseph Maciejko

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.

cond-mat.str-elCross-listedarXiv:2605.15974•Physical review. B./Physical review. B•May 18, 2026•0 citations

Quasiparticles in Exotic Phases of Matter

Daniel Sacco Shaikh

Phases of matter have always attracted great interest within the scientific community, as they represent emergent phenomena arising from the collective behavior of a large number of interacting degrees of freedom. Their macroscopic properties, often impossible to predict directly from the microscopic scale, can be effectively captured in terms of quasiparticles, \textit{i.e.} particle-like low-energy excitations that emerge from interactions among the underlying microscopic constituents and may exhibit properties entirely different from those of any fundamental particle, reflecting the rich structure of the many-body system. In this Thesis, we investigate two classes of quasiparticles that arise in distinct exotic phases of matter. The first class appears in antiferromagnetic quantum spin chains with discrete symmetries and frustrated boundary conditions, namely periodic boundary conditions with an odd number of sites. In this setting, the competition between global and local orders produces exotic gapless, non-relativistic excitations above the ground state, which can dramatically modify the physics even in the thermodynamic limit. In particular, we focus on how this specific choice of boundary conditions profoundly modifies the phase diagram of one of the most famous integrable models, the XY chain. The second class consists of subdimensional quasiparticles, excitations whose mobility is restricted to lower-dimensional subspaces and which characterize the low-energy sector of emerging fracton phases of matter. Quasiparticles with restricted mobility first appeared in exactly solvable quantum spin models, and it was later realized that they can also emerge from higher-rank gauge theories. In this part of the Thesis, we develop Lorentz-covariant gauge field theories for fractons in $2+1$ dimensions, starting from the first principles of Quantum Field Theory (fields and symmetries), with the aim of embedding the features observed in the condensed matter literature into a more formal field theoretical framework.

•May 13, 2026•0 citations

Dyonic Lieb-Schultz-Mattis theorem and symmetry protected topological phases in decorated dimer models

Xu Yang, Shenghan Jiang, Ashvin Vishwanath +1 more

Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.

arXiv:1705.05421•Physical review. B./Physical review. B•Sep 10, 2018•36 citations
The Lieb-Schultz-Mattis (LSM) theorem implies that gapped phases of matter must satisfy nontrivial conditions on their low-energy properties when a combination of lattice translation and U(1)U(1)U(1) symmetry are imposed. We describe a framework to characterize the action of symmetry on fractons and other subdimensional fractional excitations, and use this together with the LSM theorem to establish that X-cube fracton order can occur only at integer or half-odd-integer filling. Using explicit parton constructions, we demonstrate that ``odd'' versions of X-cube fracton order can occur in systems at half-odd-integer filling, generalizing the notion of odd Z2{Z}_{2}Z2​ gauge theory to the fracton setting. At half-odd-integer filling, exiting the X-cube phase by condensing fractional quasiparticles leads to symmetry breaking, thereby allowing us to identify a class of conventionally ordered phases proximate to phases with fracton order. We leverage a dual description of one of these ordered phases to show that its topological defects naturally have restricted mobility. Condensing pairs of these defects then leads to a fracton phase, whose excitations inherit these mobility restrictions.
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.
We propose and prove a family of generalized Lieb-Schultz-Mattis~(LSM) theorems for symmetry protected topological~(SPT) phases on boson/spin models in any dimensions. The ``conventional'' LSM theorem, applicable to e.g. any translation invariant system with an odd number of spin-1/2 particles per unit cell, forbids a symmetric short-range-entangled ground state in such a system. Here we focus on systems with no LSM anomaly, where global/crystalline symmetries and fractional spins within the unit cell ensure that any symmetric SRE ground state must be a non-trivial SPT phase with anomalous boundary excitations. Depending on models, they can be either strong or ``higher-order'' crystalline SPT phases, characterized by non-trivial surface/hinge/corner states. Furthermore, given the symmetry group and the spatial assignment of fractional spins, we are able to determine all possible SPT phases for a symmetric ground state, using the real space construction for SPT phases based on the spectral sequence of cohomology theory. We provide examples in one, two and three spatial dimensions, and discuss possible physical realization of these SPT phases based on condensation of topological excitations in fractionalized phases.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.
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.
Phases of matter have always attracted great interest within the scientific community, as they represent emergent phenomena arising from the collective behavior of a large number of interacting degrees of freedom. Their macroscopic properties, often impossible to predict directly from the microscopic scale, can be effectively captured in terms of quasiparticles, \textit{i.e.} particle-like low-energy excitations that emerge from interactions among the underlying microscopic constituents and may exhibit properties entirely different from those of any fundamental particle, reflecting the rich structure of the many-body system. In this Thesis, we investigate two classes of quasiparticles that arise in distinct exotic phases of matter. The first class appears in antiferromagnetic quantum spin chains with discrete symmetries and frustrated boundary conditions, namely periodic boundary conditions with an odd number of sites. In this setting, the competition between global and local orders produces exotic gapless, non-relativistic excitations above the ground state, which can dramatically modify the physics even in the thermodynamic limit. In particular, we focus on how this specific choice of boundary conditions profoundly modifies the phase diagram of one of the most famous integrable models, the XY chain. The second class consists of subdimensional quasiparticles, excitations whose mobility is restricted to lower-dimensional subspaces and which characterize the low-energy sector of emerging fracton phases of matter. Quasiparticles with restricted mobility first appeared in exactly solvable quantum spin models, and it was later realized that they can also emerge from higher-rank gauge theories. In this part of the Thesis, we develop Lorentz-covariant gauge field theories for fractons in 2+12+12+1 dimensions, starting from the first principles of Quantum Field Theory (fields and symmetries), with the aim of embedding the features observed in the condensed matter literature into a more formal field theoretical framework.
Lieb-Schultz-Mattis (LSM) theorem and its various generalizations provide powerful guidance towards the search for novel phases of matter. Here, the authors propose and prove a generalized LSM theorem suitable for 2+1D lattice models of interacting bosons or spins, with both magnetic flux and fractional spin in the unit cell. One outcome of this theorem is that, under certain conditions, the gapped ground states preserving all symmetries must be a nontrivial symmetry-protected topological (SPT) phase. Such symmetry-enforced SPTs display a dyonic character in that they associate charge with symmetry flux, which is demonstrated using quasiexactly solvable models constructed by decorating quantum dimer models.