Symmetry-Breaking Dimensional Expansion (SBDE)

• SBDE is a mechanism where symmetry breaking reassigns dynamically relevant dimensions, revealing lower-dimensional invariants.
• It spans contexts from condensed matter and matrix models to machine learning, influencing topological phases and optimization geometry.
• The concept unifies multiple frameworks by demonstrating how broken symmetry reorganizes effective dimensionality in physical and mathematical systems.

Searching arXiv for recent and relevant papers on symmetry-breaking dimensional expansion and closely related formulations. Symmetry-Breaking Dimensional Expansion (SBDE) is a cross-domain motif in which symmetry breaking reorganizes a system so that its physically or mathematically relevant structure is controlled by a reduced or redistributed notion of dimensionality. In the literature assembled under this label, SBDE does not denote a single universally standardized formalism. Instead, it names a recurring mechanism: a higher-symmetry or higher-dimensional description acquires nontrivial behavior only after symmetry breaking selects privileged sectors, leaves, coordinates, or vacua, and the resulting phase, effective theory, or optimization geometry is then characterized in a lower-dimensional subspace, a subset of extended directions, or an auxiliary expanded representation. In condensed-matter band theory this appears through partial-Brillouin-zone topology in a three-dimensional Dirac system (Habe et al., 2013); in matrix models it appears through spontaneous breaking of rotational symmetry that leaves only a subset of directions extended (Anagnostopoulos et al., 2020, Anagnostopoulos et al., 2013, Brahma et al., 2022); in lattice and graph models it appears through symmetry-breaking criteria governed by spectral rather than Euclidean dimension (Evnin, 10 Dec 2025); in machine learning it appears through deterministic input-dimension expansion that breaks translational or permutation-related degeneracies (Zhang et al., 2024, Bai et al., 20 Feb 2026). A broader mathematical and physical periphery includes covert symmetry breaking under dimensional reduction (Erickson et al., 2020), higher-form subsystem symmetry breaking in fracton constructions (Rayhaun et al., 2021), operator-theoretic symmetry-breaking families interpolating between discrete and continuous regimes (Kobayashi, 2023), and geometric symmetry reduction by passage to subgeometries (Fuchs et al., 2022).

1. Core concept and scope

Across the cited works, SBDE is best understood as a structural principle rather than a single axiomatized theory. The common pattern is that symmetry breaking does not merely remove symmetry; it changes which dimensions are dynamically relevant, topologically active, geometrically extended, or algorithmically vulnerable. In this sense, “dimensional expansion” may refer to at least three distinct but related operations.

First, it may denote a higher-dimensional system whose nontrivial classification is inherited from a lower-dimensional sector. In the three-dimensional Dirac setting, the full Hamiltonian becomes topologically nontrivial because a two-dimensional partial Brillouin zone supports a Chern number once a symmetry-breaking perturbation reorganizes the theory into decoupled sectors (Habe et al., 2013).

Second, it may denote a symmetric high-dimensional theory whose vacuum selects only a lower-dimensional extended subspace. This is the matrix-model usage: the microscopic theory has full rotational symmetry, but spontaneous symmetry breaking yields vacua such as SO(3), with three extended and seven compactified directions in the Euclidean IKKT model (Anagnostopoulos et al., 2020). Closely related Monte Carlo and Gaussian-expansion analyses of dimensionally reduced super Yang–Mills and BFSS models adopt the same logic (Anagnostopoulos et al., 2013, Brahma et al., 2022).

Third, it may denote an explicit enlargement of representation dimension that breaks a degeneracy-inducing symmetry. In neural networks, constant-valued inserted coordinates or pixels expand the input space and break translational or permutation-related symmetries, altering optimization and robustness geometry (Zhang et al., 2024, Bai et al., 20 Feb 2026).

This suggests that SBDE is unified less by one ontology of “dimension” than by a recurring mechanism: symmetry breaking changes the effective decomposition of degrees of freedom so that lower-dimensional invariants, reduced-dimensional vacua, or added auxiliary directions become decisive.

2. Partial-BZ topology in three-dimensional Dirac systems

A canonical condensed-matter realization is the three-dimensional Dirac Hamiltonian with a symmetry-breaking perturbation,

$$H_0 = a\alpha^\mu p_\mu + M\beta,\qquad M = (m-b\mathbf{p}^2),$$

with $\alpha^\mu=\sigma^\mu\tau^x$ and $\beta=\sigma^0\tau^z$, and full Hamiltonian

$H = H_0 + H_{\mathcal P}.$

In the formulation of “Three-dimensional symmetry breaking topological matters” (Habe et al., 2013), the crucial step is that $H_{\mathcal P}$ breaks the full symmetry content of $H_0$ while preserving a restricted symmetry in a partial Brillouin zone. In the $p_z=0$ plane, $H_0$ can be unitarily transformed into

$$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$

so that the problem decomposes into two $2\times2$ quantum Hall Hamiltonians of opposite chirality.

The topological invariant is then not a global three-dimensional symmetry-class invariant but a Chern number defined in the two-dimensional partial Brillouin zone. When $\alpha^\mu=\sigma^\mu\tau^x$0 for $\alpha^\mu=\sigma^\mu\tau^x$1, the partner block $\alpha^\mu=\sigma^\mu\tau^x$2 has $\alpha^\mu=\sigma^\mu\tau^x$3. The paper explicitly frames the resulting phases as topological even in the absence of time-reversal symmetry, particle-hole symmetry, or chiral symmetry in the usual global sense (Habe et al., 2013). The symmetry-breaking perturbations are instead organized by an Abelian group structure that preserves blockwise decoupling in the relevant slice.

This is an SBDE prototype because the nontrivial structure of the three-dimensional system is encoded in a lower-dimensional invariant embedded in momentum space. Gapless surface states on surfaces perpendicular to the chosen plane are explained by the nontrivial partial-BZ Chern number. The associated symmetry operators include

$\alpha^\mu=\sigma^\mu\tau^x$4

The principal physical illustration is the Zeeman perturbation

$\alpha^\mu=\sigma^\mu\tau^x$5

For a field along $\alpha^\mu=\sigma^\mu\tau^x$6,

$\alpha^\mu=\sigma^\mu\tau^x$7

The symmetric component $\alpha^\mu=\sigma^\mu\tau^x$8 and antisymmetric component $\alpha^\mu=\sigma^\mu\tau^x$9 affect the two quantum Hall sectors differently. Under strong field, when

$\beta=\sigma^0\tau^z$0

the topology changes qualitatively. The paper identifies a transition from a topological insulator to a Weyl semimetal for the symmetric field, whereas the antisymmetric field leads to a nodal semimetal (Habe et al., 2013). This establishes a concrete form of symmetry-breaking-induced topological reclassification.

3. Matrix models and dynamical compactification

In nonperturbative matrix models of string or M-theoretic type, SBDE refers to spontaneous rotational symmetry breaking that dynamically selects a lower-dimensional extended spacetime. The Euclidean IKKT matrix model provides the clearest explicit example (Anagnostopoulos et al., 2020). It is the zero-dimensional reduction of 10D $\beta=\sigma^0\tau^z$1 SU($\beta=\sigma^0\tau^z$2) super Yang–Mills theory, with bosonic matrices $\beta=\sigma^0\tau^z$3 interpreted as dynamical spacetime coordinates. The action is written as

$\beta=\sigma^0\tau^z$4

with

$\beta=\sigma^0\tau^z$5

$\beta=\sigma^0\tau^z$6

After Wick rotation, integrating out fermions yields a complex Pfaffian. The associated phase suppresses higher-dimensional configurations and is described as the origin of the mechanism that drives symmetry breaking (Anagnostopoulos et al., 2020).

The order parameters are the directional extents

$\beta=\sigma^0\tau^z$7

and the analysis introduces an explicit bosonic deformation

$\beta=\sigma^0\tau^z$8

with ordered masses $\beta=\sigma^0\tau^z$9, together with normalized ratios $H = H_0 + H_{\mathcal P}.$0. The numerical method is the complex Langevin method, stabilized against the singular-drift problem by the fermionic deformation

$H = H_0 + H_{\mathcal P}.$1

which breaks SO(10) to

$H = H_0 + H_{\mathcal P}.$2

As the deformation parameter is reduced, the spontaneous symmetry-breaking pattern changes: at $H = H_0 + H_{\mathcal P}.$3 the remaining SO(7) symmetry is not spontaneously broken; at $H = H_0 + H_{\mathcal P}.$4 it breaks to SO(4); at $H = H_0 + H_{\mathcal P}.$5 it breaks to SO(3) (Anagnostopoulos et al., 2020). The stated conclusion is that the original undeformed Euclidean IKKT model has an SO(3)-symmetric vacuum, so three directions remain large and seven are compactified dynamically.

The paper relates this to previous Gaussian expansion method results, including $H = H_0 + H_{\mathcal P}.$6, $H = H_0 + H_{\mathcal P}.$7, and the scaling relation

$H = H_0 + H_{\mathcal P}.$8

which expresses approximately conserved spacetime volume in the variational solutions (Anagnostopoulos et al., 2020). A plausible implication is that SBDE in this setting couples dimensional selection to a redistribution rather than a simple loss of extent.

Two related works reinforce this interpretation. Monte Carlo studies of dimensionally reduced super Yang–Mills in $H = H_0 + H_{\mathcal P}.$9 use the moment-of-inertia tensor

$H_{\mathcal P}$0

and its eigenvalues $H_{\mathcal P}$1 as order parameters, finding results consistent with $H_{\mathcal P}$2, a universal compactification scale, and the constant-volume property

$H_{\mathcal P}$3

(Anagnostopoulos et al., 2013). In the BFSS model, the Gaussian expansion method shows that the Euclidean high-temperature theory preserves SO(9) in the bosonic case but that fermion-induced gamma-matrix terms render an SO(9)-symmetric solution impossible in the full supersymmetric theory, thereby providing analytic evidence for spontaneous symmetry breaking only when fermions are present (Brahma et al., 2022).

4. Dimensional criteria on lattices, graphs, and subsystem constructions

A different SBDE strand reformulates symmetry breaking in terms of effective dimension determined by spectral geometry rather than ambient Euclidean dimension. In “Spontaneous symmetry breaking on graphs and lattices” (Evnin, 10 Dec 2025), the starting point is a free massless scalar with shift symmetry,

$H_{\mathcal P}$4

discretized to a coupled-oscillator network,

$H_{\mathcal P}$5

The local fluctuation width is

$H_{\mathcal P}$6

and the symmetry-breaking criterion is whether $H_{\mathcal P}$7 remains finite as $H_{\mathcal P}$8. On hypercubic lattices, the standard result is recovered: continuous symmetry cannot be spontaneously broken in one spatial dimension, while for $H_{\mathcal P}$9 the fluctuation width remains finite in the basic relativistic case $H_0$0 (Evnin, 10 Dec 2025).

The paper generalizes the control parameter from Euclidean dimension to spectral dimension $H_0$1, defined by

$H_0$2

near $H_0$3. Averaged fluctuations diverge whenever

$H_0$4

Thus the ordinary no-SSB threshold becomes a graph-theoretic condition governed by spectral dimension, with local structure encoded by fractional resistance distance and the Kirchhoff index. This is an SBDE variant because the relevant “dimension” controlling spontaneous symmetry breaking is not the embedding dimension of a manifold but the infrared spectral geometry of a discrete network (Evnin, 10 Dec 2025).

The paper also lists explicit examples. On random graphs of Erdős–Rényi or configuration-model type, the effective spectral dimension is typically infinite and symmetry breaking is therefore allowed. On the Sierpiński gasket,

$H_0$5

so a standard $H_0$6 scalar can break symmetry. For Dhar’s recursive lattices,

$H_0$7

which interpolates between 1 and 2 (Evnin, 10 Dec 2025). This suggests an engineerable SBDE threshold controlled by geometry.

A distinct but related dimensional mechanism appears in higher-form subsystem symmetry and fracton transitions. In “Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions” (Rayhaun et al., 2021), decoupled stacks of lower-dimensional models with higher-form subsystem symmetries are coupled or gauged to produce higher-dimensional fracton phases. The basic family

$H_0$8

unifies the TFIM, $H_0$9 gauge theory, plaquette Ising model, cubic Ising model, and X-cube model through different $p_z=0$0 choices. The order/disorder exchange under gauging maps symmetry-breaking transitions of lower-dimensional stacks to fracton or topological transitions in one higher spatial dimension (Rayhaun et al., 2021). In this setting, SBDE is a stacking-and-gauging principle: higher-dimensional phases emerge from lower-dimensional symmetry structures organized by foliations.

5. Dimensional reduction, covert symmetry breaking, and operator-theoretic generalizations

The phrase “dimensional expansion” can be paired with dimensional reduction when the lower-dimensional effective theory retains a hidden memory of the higher-dimensional symmetry structure. “Covert Symmetry Breaking” (Erickson et al., 2020) analyzes $p_z=0$1-dimensional scalar QED on

$p_z=0$2

with mixed Dirichlet/Robin boundary conditions. The key feature is that the lowest transverse mode of the gauge field is not constant: $p_z=0$3 Because the zero mode depends nontrivially on the reduction coordinate, the lower-dimensional theory contains a Stueckelberg structure already at level zero. The reduced theory looks Maxwellian at quadratic and cubic order, but the mismatch

$p_z=0$4

appears at quartic order in the vector-scalar interaction terms (Erickson et al., 2020). The resulting effect is called covert symmetry breaking: the higher-dimensional gauge symmetry survives in a nonlinearly realized form, and the lower-dimensional effective action departs from naïve scalar QED only at fourth order. This is an SBDE-adjacent mechanism because the effective lower-dimensional description is reshaped by the geometry and mode structure of the extra dimension.

A more abstract extension appears in operator theory. “Generating operators of symmetry breaking -- from discrete to continuous” (Kobayashi, 2023) starts from a countable family of differential symmetry-breaking operators $p_z=0$5 and packages them into a generating operator

$p_z=0$6

Pairing with the meromorphic distribution $p_z=0$7 yields a continuous family $p_z=0$8 whose residues recover the discrete operators: $p_z=0$9 This is not dimensional expansion in the geometric sense, but it is a discrete-to-continuous expansion of symmetry-breaking data. The construction generates invariant trilinear forms, Poisson transforms, Fourier transforms on $H_0$0, and embeddings of discrete series into principal series (Kobayashi, 2023). A plausible implication is that SBDE can also denote expansion across parameter spaces rather than only across physical dimensions.

An even looser periphery is provided by “Symmetry breaking in geometry” (Fuchs et al., 2022), which defines geometric symmetry breaking as passage from a geometry $H_0$1 to a subgeometry $H_0$2 via an absolute configuration $H_0$3 whose stabilizer is $H_0$4. This framework treats symmetry breaking as a route from larger ambient geometry to more specialized geometric regimes, with examples from Möbius, Lie sphere, and Euclidean similarity geometry (Fuchs et al., 2022). Although the paper does not formulate SBDE, it supports the broader interpretation that symmetry breaking can reorganize the effective geometric setting itself.

6. Neural-network SBDE: input expansion, degeneracy reduction, and robustness

In machine learning, SBDE is formulated explicitly as deterministic input-dimension expansion by constant-valued coordinates or pixels. “Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansion” (Zhang et al., 2024) advances the symmetry breaking hypothesis: excessive symmetry in model or objective induces degenerate minima, flat regions, or redundant optimization trajectories, and simple input expansion can reduce that degeneracy.

The paper motivates the method by analogy with the Ising model. A two-layer approximation is written as

$H_0$5

which is symmetric under permutations of hidden units. After adding a constant input coordinate $H_0$6,

$H_0$7

and the new term breaks the previous symmetry (Zhang et al., 2024). For images, the spatial grid is expanded by factor $H_0$8, original pixels are placed at regular intervals, and the remaining positions are filled with a constant such as $H_0$9. The paper reports improved convergence and test accuracy across diverse tasks and architectures, while preserving parameter counts via pooling design (Zhang et al., 2024).

The same work proposes a Wasserstein-distance-based Replica Distance metric to quantify the degree of symmetry breaking by training multiple replicas, reducing final weights with UMAP, and measuring pairwise Wasserstein distances (Zhang et al., 2024). It also interprets dropout, batch normalization, and equivariance as alternative symmetry-breaking mechanisms. A major caveat is explicit in the paper: more symmetry breaking is not automatically better; if the embedded symmetry is wrong, performance can degrade.

“A Geometric Probe of the Accuracy-Robustness Trade-off: Sharp Boundaries in Symmetry-Breaking Dimensional Expansion” (Bai et al., 20 Feb 2026) sharpens this neural SBDE formulation. There, SBDE expands an image

$$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$0

to

$$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$1

by interleaving original pixels with constant-valued auxiliary rows and columns. The coordinates are partitioned into $$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$2 and $$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$3, with

$$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$4

The paper reports that on CIFAR-10 with ResNet-18, clean accuracy improves from $$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$5 under natural training to $$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$6 under SBDE, but unprojected adversarial robustness collapses: reported PGD, AutoAttack, BIM, APGD, and APGDT accuracies are all $$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$7 for the SBDE model without projection (Bai et al., 20 Feb 2026).

The decisive verification tool is the test-time mask projection

$$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$8

With $$H_0^{U}= \begin{pmatrix} \mathbf d(M)\cdot\boldsymbol{\sigma} & 0\ 0 & \mathbf d(-M)\cdot\boldsymbol{\sigma} \end{pmatrix}, \qquad \mathbf d(M)=(ap_x,ap_y,M_{p_z=0}),$$9, the same SBDE model retains clean $2\times2$0 and recovers strong adversarial performance, including PGD $2\times2$1, AutoAttack $2\times2$2, BIM $2\times2$3, APGD $2\times2$4, APGDT $2\times2$5, and average robust accuracy $2\times2$6 (Bai et al., 20 Feb 2026). The paper interprets this as evidence that vulnerability is concentrated in the inserted auxiliary dimensions, where optimization erects sharp boundaries or steep loss gradients. The stated geometric explanation is that clean accuracy improves because the basin around natural data deepens, while robustness falls because steep walls arise along the auxiliary axes (Bai et al., 20 Feb 2026).

This neural-network usage differs from the matrix-model and condensed-matter usages, but the common logic remains recognizable: symmetry breaking plus dimension manipulation reorganizes the relevant geometry of the problem.

7. Related debates, misconceptions, and status of the term

A recurrent misconception is that symmetry breaking only destroys structure. The assembled literature shows several counterexamples. In three-dimensional topological matter, symmetry breaking can expose a lower-dimensional invariant rather than trivialize the phase (Habe et al., 2013). In matrix models, symmetry breaking is the mechanism by which an effectively lower-dimensional spacetime becomes dynamically selected (Anagnostopoulos et al., 2020, Anagnostopoulos et al., 2013, Brahma et al., 2022). In neural optimization, explicit symmetry breaking can reduce degeneracy and improve clean performance, even though it may simultaneously create off-manifold fragility (Zhang et al., 2024, Bai et al., 20 Feb 2026).

A second misconception is that “dimension” always means ordinary Euclidean spatial dimension. The graph-and-lattice formulation replaces it with spectral dimension and resistance geometry (Evnin, 10 Dec 2025). The fracton formulation organizes it through foliations, higher-form subsystem symmetries, and subdimensional critical objects (Rayhaun et al., 2021). The neural-network formulation treats it as representation dimension in input space (Zhang et al., 2024, Bai et al., 20 Feb 2026).

A third issue concerns terminology. Several cited papers do not themselves use “Symmetry-Breaking Dimensional Expansion” as a formal canonical label. Some are directly framed in terms of symmetry breaking plus lower-dimensional selection or expansion-like reorganization, while others support the concept only by close analogy. “Advanced Mathematical Approaches to Symmetry Breaking in High-Dimensional Field Theories” explicitly does not define SBDE as a named framework, though it advances a dimensional-expansion viewpoint through Laurent series, residues, and winding numbers in $2\times2$7 dimensions (Chen, 2024). “Symmetry breaking in geometry” likewise provides a related but distinct language of passage from larger ambient geometries to subgeometries (Fuchs et al., 2022). This suggests that SBDE presently functions more as an overview term spanning several research programs than as a universally standardized field-theoretic doctrine.

Within that synthesis, however, the unifying insight is stable: symmetry breaking can reorganize a system so that its decisive invariants, effective vacua, or optimization properties are controlled by a redistributed notion of dimension. Whether that redistribution appears as a partial Brillouin-zone Chern number, an SO(3)-symmetric vacuum emerging from SO(10), a spectral-dimension threshold on graphs, a gauged stack of lower-dimensional layers, or a constant-pixel auxiliary subspace in deep learning, the central SBDE logic is the same. Symmetry breaking is not merely subtractive; it can be the operation that makes the relevant dimensional structure manifest.