Spontaneous breaking of multipole symmetries

Charles Stahl; Ethan Lake; Rahul Nandkishore

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Spontaneous breaking of multipole symmetries

Charles Stahl, Ethan Lake, Rahul Nandkishore

Statistical Mechanics · Submitted 15 Nov 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder.

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Multipole symmetries are of interest both as a window on fracton physics and as a crucial ingredient in realizing new universality classes for quantum dynamics. Here we address the question of whether and when multipole symmetries can be spontaneously broken, both in thermal equilibrium and at zero temperature. We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder. We present both general results and explicit examples. Our results should be directly applicable to quantum dynamics with multipolar symmetries and also provide a useful stepping stone to understanding the robustness of fracton phases to thermal fluctuations, quantum fluctuations, and disorder.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Multipole symmetries are of interest both as a window on fracton physics and as a crucial ingredient in realizing new universality classes for quantum dynamics.	background 1.00	0.04	0.03	asserted 1.00
s1	Here we address the question of whether and when multipole symmetries can be spontaneously broken, both in thermal equilibrium and at zero temperature.	contribution 0.45	0.29	0.05	unclear 0.88
s2	We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder.	contribution 0.71	0.96	0.25	asserted 1.00
s3	We present both general results and explicit examples.	contribution 0.96	0.87	0.05	asserted 0.95
s4	Our results should be directly applicable to quantum dynamics with multipolar symmetries and also provide a useful stepping stone to understanding the robustness of fracton phases to thermal fluctuations, quantum fluctuations, and disorder.	interpretation 0.94	0.35	0.21	hedged possibility 0.63

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Original abstract

Multipole symmetries are of interest both as a window on fracton physics and as a crucial ingredient in realizing new universality classes for quantum dynamics. Here we address the question of whether and when multipole symmetries can be spontaneously broken, both in thermal equilibrium and at zero temperature. We derive generalized Mermin-Wagner arguments for the total or partial breaking of multipolar symmetry groups and generalized Imry-Ma arguments for the robustness of such multipolar symmetry breaking to disorder. We present both general results and explicit examples. Our results should be directly applicable to quantum dynamics with multipolar symmetries and also provide a useful stepping stone to understanding the robustness of fracton phases to thermal fluctuations, quantum fluctuations, and disorder.

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References

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Partially Massless Superconductor
hep-th 2025-07 conditional novelty 8.0 of 10

A covariant Higgs mechanism turns the partially massless graviton into a fully massive spin-2 field by condensing a fractonic matter field with dipole symmetry.
Subsystem Non-Invertible Symmetry Operators and Defects
cond-mat.str-el 2023-04 conditional novelty 7.0 of 10

A subsystem Z2 gauging on a 2D lattice yields non-invertible duality operators and defects whose fusion rules mix operators of different codimensions, and whose defects are mobile in both spatial directions.
Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
cond-mat.str-el 2022-01 accept novelty 6.0 of 10

Compact Lifshitz theory and its U(1) and Z_N dipole gauge versions have non-unique continuum limits, and their defect mobility is controlled by time-like global symmetries that act only in the presence of defects.
Generalized Symmetries in Condensed Matter
cond-mat.str-el 2022-04 conditional novelty 2.0 of 10

Generalized symmetries and their anomalies may extend the Landau classification to cover topological order, fracton phases, symmetry-protected topological phases and Fermi liquids, according to this review.
Space-dependent symmetries and fractons
hep-th 2021-12 accept novelty 2.0 of 10

Coordinate-dependent symmetries and elastic dualities provide a unified effective description of gapless fractons, their defects, and their subdiffusive hydrodynamics.

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24 arXiv papers cite this in 34 sentences

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Papers citing it · 19 sentences without a confident label

Cites it as support 1 paper

Despite being a continuous symmetry, and despite being in one dimension, dipole symmetry is actually spontaneously broken in the BEI at $T=0$ [ref] : indeed, correlations of the dipole order parameter $e^i_xϕ$ go as C_e^i_x ϕ(0,x) ∼e^-∫_q,øq^2 1-cos(qx)/ø^2 K_τ+q^4 K_D x ∞ 1,as the integral in the exponential is IR-finite (c.f. the power law behavior in the DC phase [ref]), implying a nonzero expectation value $|d_i | ≠0$.

Dipole condensates in tilted Bose-Hubbard chains · Finite size effects and the Bose Einstein Insulator · 0.98

Cites it as background 12 papers

This global symmetry and the current conservation have been studied in [ref] .

Global Dipole Symmetry, Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons · Introduction · 0.80
So the mobile subspace is the $t-x$ plane, and the defect with respect to the $t-x$ plane is of codimension one.] [ref] .[There are also earlier works on subsystem symmetry from algebraic duality approach [ref] .] Here and in Tab. [ref] we only present a highly incomplete list of references.[Another line of development of unconventional symmetry, motivated by commutant algebra in the study of quantum many body scar, was explored in [ref] .

Subsystem Non-Invertible Symmetry Operators and Defects · Subsystem non-invertible symmetry · 0.80
They encompass the higher-form or higher-group symmetries [ref] , non-invertible symmetries [ref] , subsystem symmetries [ref] , etc.

Web of 4D Dualities, Supersymmetric Partition functions and SymTFT · Introduction and Conclusion · 0.99

What it cites · 3 sentences without a confident label

Cites as background 3 papers

A detailed discussion of the phase diagram and phenomenology of this model is given in [ref] ; in the following we will simply discuss the symmetry breaking patterns realized in various different limits.

The dipolar Bose-Hubbard model · Explicit lattice example · 0.93
They are known to arise in `fracton' phases of quantum matter [ref] , the key dynamical properties of which are known to descend from conservation laws on multipole moments of charge [ref] .

Generalized $U(1)$ Gauge Field Theories and Fractal Dynamics · Introduction · 0.85
Another setting for generalized Mermin-Wagner-type arguments is higher form global symmetries [ref] .

Higher-form symmetries and spontaneous symmetry breaking · Discussion · 0.89

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