Symmetry-Enhanced Pseudo-First-Order Transition

• The paper demonstrates that competing Dirac masses and emergent U(1) symmetry in the SSH-Holstein model produce finite-size behavior mimicking a first-order jump that later resolves into two continuous transitions.
• Researchers employ joint order-parameter distributions, Binder cumulants, and entanglement entropy to diagnose pseudo-first-order behavior versus true first-order transitions.
• This phenomenon is observed across various systems—including 3D loop, checkerboard J-Q, and O(N)×O(M) models—showing that suppressed anisotropies and RG flows can yield emergent symmetry with modified critical scaling.

Searching arXiv for recent and directly relevant papers on symmetry-enhanced pseudo-first-order transitions and related emergent-symmetry weak/first-order behavior. Symmetry-enhanced pseudo-first-order transition denotes a class of transition phenomena in which finite-size or intermediate-scale data display first-order-like behavior together with an enlarged emergent symmetry, even though the asymptotic thermodynamic behavior is either weaker than a conventional discontinuity or organized by a different long-distance structure. The expression is used explicitly for a one-dimensional SSH-Holstein problem with competing charge-density-wave and bond-order-wave masses, where a transition that looks like a direct first-order jump with emergent chiral $U(1)$ symmetry resolves in the thermodynamic limit into two continuous transitions around a narrow coexistence phase (Weber, 2 Sep 2025). Closely related behavior appears in deconfined and near-deconfined settings with approximate $O(4)$ or $SO(5)$ superspin symmetry, in symmetry-enhanced but genuinely first-order order-order transitions, and in renormalization-group scenarios where enlarged symmetry or symmetry-permitted couplings suppress the microscopic anisotropies that eventually control the transition (Serna et al., 2018, Deng et al., 2024, Zhao et al., 2018, Sorokin, 2021).

1. Definition and taxonomy

A precise usage of the term must separate three nearby but non-identical notions. In the narrowest sense, a symmetry-enhanced pseudo-first-order transition is a transition that appears to be a direct first-order transition with emergent symmetry over a broad finite-size window, but asymptotically turns into something else. This is the explicit meaning in the competing-Dirac-mass study of the SSH-Holstein model: the low-energy Dirac theory predicts a direct first-order CDW–BOW transition with emergent $U(1)$, whereas the lattice model ultimately splits this into two continuous transitions around a very narrow coexistence phase (Weber, 2 Sep 2025).

A second category consists of weakly first-order transitions that are not merely pseudo-first-order, but whose phenomenology is strongly softened by emergent symmetry. The 3D loop model with a Néel–twofold-VBS transition shows approximate emergent $O(4)$ symmetry, a regime resembling spontaneous breaking of that emergent symmetry, and nonzero coexistence order parameters at the transition, even though the transition is asymptotically first order (Serna et al., 2018). The checkerboard $J$-$Q$ quantum magnet provides an even sharper example: the AFM and plaquette-singlet-solid order parameters form an $O(4)$ vector at a strongly discontinuous transition, yet the coexistence lacks the usual barrier phenomenology, and the order parameter rotates between sectors rather than tunneling between well-separated minima (Zhao et al., 2018).

A third category consists of systems that are highly relevant conceptually but do not establish the phenomenon directly. The $J\!-\!Q_3$ model is argued to exhibit finite-size emergent $SO(5)$ symmetry while being weakly first-order in the thermodynamic limit, diagnosed through smooth-boundary entanglement entropy rather than conventional coexistence analysis (Deng et al., 2024). The high-dimensional $O(4)$0 model shows that product symmetry can allow pseudo-first-order behavior in principle, though the paper’s explicit four-dimensional lattice examples are distinct first-order transitions and the pseudo-first-order regime in $O(4)$1 is presented only as a possibility (Sorokin, 2021).

2. Mechanisms that produce the phenomenon

One mechanism is competition between anticommuting Dirac masses with an emergent continuous rotation in order-parameter space. In the SSH-Holstein chain, the adiabatic low-energy Hamiltonian takes the form

$O(4)$2

and near the Fermi points the two masses combine into a vector $O(4)$3 with $O(4)$4. At $O(4)$5, the continuum Dirac theory has a chiral $O(4)$6 symmetry that rotates CDW into BOW. The continuum theory therefore predicts a direct first-order transition with emergent $O(4)$7, but the full lattice weakly breaks that symmetry and stabilizes a narrow coexistence phase. The pseudo-first-order regime arises because the $O(4)$8-breaking energy splitting on the chiral manifold is tiny compared with the gap scale (Weber, 2 Sep 2025).

A second mechanism is pseudocritical RG flow toward an enlarged-symmetry ordered manifold, followed only later by flow away from it due to weak anisotropy. In the 3D loop model, the long-distance theory is described by an $O(4)$9 nonlinear sigma model with a topological term plus anisotropies

$SO(5)$0

The RG flow first suppresses the leading $SO(5)$1-breaking anisotropy $SO(5)$2, so the system enters an $SO(5)$3-symmetric regime in which the superspin $SO(5)$4 behaves as though it were spontaneously ordered on $SO(5)$5; only on still longer scales does the residual anisotropy force an asymptotic first-order selection between the Néel and VBS sectors (Serna et al., 2018). This suggests a general template: emergent symmetry can dominate intermediate scales even when it is not the true asymptotic symmetry.

A third mechanism is symmetry-permitted coupling-space geometry. In the $SO(5)$6 model, the product symmetry allows two quartic invariants $SO(5)$7 and $SO(5)$8, with stability region

$SO(5)$9

Because the stable quartic region is a wedge rather than a half-line, RG trajectories can leave the quartic stability region and later return to the Gaussian fixed point. The paper interprets this as a possible pseudo-first-order scenario in $U(1)$0, with higher-order terms needed to stabilize the potential once the quartic sector becomes unstable (Sorokin, 2021). Here the role of “symmetry enhancement” is not emergent rotational symmetry of order parameters, but the richer invariant structure created by the product group.

A fourth mechanism is symmetry fractionalization. In the toric-code confinement problem, projective symmetry action on the condensing $U(1)$1-anyon forces the low-energy theory into an emergent $U(1)$2 structure. Near the weakly first-order transition between two ordered phases, the leading $U(1)$3 anisotropy vanishes and only higher-order $U(1)$4 anisotropy ultimately destroys the would-be $U(1)$5 structure. The finite-size spectrum then resembles an approximate $U(1)$6 rotor rather than a conventional first-order coexistence problem (Schuler et al., 2022).

3. Diagnostics

The central diagnostic is the joint order-parameter distribution. At a symmetry-enhanced pseudo-first-order point, histograms frequently become nearly rotationally invariant rather than resolving into the sharply separated lobes of ordinary first-order coexistence. In the SSH-Holstein example, the histogram of $U(1)$7 becomes a nearly perfect circle at the apparent direct transition, reflecting the approximate $U(1)$8 of the mass vector $U(1)$9 (Weber, 2 Sep 2025). In the 3D loop model, the single-component distribution approaches the semicircle

$O(4)$0

the hallmark of a uniform distribution on the $O(4)$1-ordered sphere $O(4)$2 (Serna et al., 2018).

Anisotropy moments provide a more quantitative version of the same test. Both the loop-model work and the SSH-Holstein study use

$O(4)$3

or equivalent low-order moments such as $O(4)$4 and $O(4)$5, which vanish when the projected order-parameter distribution is $O(4)$6-symmetric (Serna et al., 2018, Weber, 2 Sep 2025). These observables are especially valuable because simple histogram bimodality can be misleading: in the generalized 2D $O(4)$7 model with higher harmonics, the $O(4)$8 equal-coupling case shows double-peaked energy histograms but ultimately scales like a BKT transition, whereas $O(4)$9 shows true first-order behavior through stable peak separation and $J$0 (Žukovič, 10 Feb 2025).

Binder cumulants behave in a distinctly nonstandard way in symmetry-enhanced settings. Ordinary first-order transitions often exhibit strong negative Binder dips associated with barrier-separated minima. By contrast, the AFM–PSS transition in the checkerboard $J$1-$J$2 model is first-order, yet negative Binder peaks are absent because the two order parameters form an emergent $J$3 vector at the transition and interconvert without an energy barrier (Zhao et al., 2018). This absence of conventional first-order signatures is one reason why pseudo-first-order and symmetry-enhanced first-order phenomena can be difficult to distinguish.

Entanglement entropy offers a complementary diagnostic when conventional finite-size observables are ambiguous. In the $J$4 model, the second Rényi entropy with a smooth cut is fitted to

$J$5

and yields $J$6, consistent with spontaneous symmetry breaking $J$7. Because a smooth-boundary critical point in $J$8 dimensions is not expected to show this logarithm, the result is interpreted as evidence for a weak first-order transition hidden behind finite-size $J$9 symmetry (Deng et al., 2024).

4. Representative case studies

The 3D classical loop model furnishes one of the clearest archetypes. The transition is between a Néel phase and a twofold-degenerate VBS, so the soft fields combine into a four-component superspin $Q$0. Numerically, both order parameters extrapolate to nonzero values at the critical coupling, while the order-parameter distribution and cumulants match those of an ordered $Q$1 sigma model over a broad range of sizes. The paper interprets this as approximate emergent $Q$2 symmetry and an intermediate-scale regime in which that emergent symmetry is effectively spontaneously broken, even though the transition is ultimately first order (Serna et al., 2018).

The checkerboard $Q$3-$Q$4 magnet realizes a different limit: not pseudo-first-order in the strict sense, but a genuinely first-order transition with barrierless symmetry enhancement. The AFM order parameter $Q$5 and the PSS order $Q$6 form

$Q$7

and the transition point is diagnosed by rotationally symmetric $Q$8 and $Q$9 histograms, finite coexistence values of both order parameters, and absence of negative Binder anomalies. The result is an $O(4)$0-organized first-order transition whose finite-size phenomenology overlaps strongly with pseudo-first-order behavior even though the discontinuity is genuine (Zhao et al., 2018).

The square-lattice $O(4)$1 model sharpens the relation between emergent symmetry and weak first order. Here the five-component superspin combining the three Néel and two VBS components is argued to exhibit finite-size $O(4)$2 symmetry, while entanglement scaling points instead to spontaneous symmetry breaking $O(4)$3 with four Goldstone modes. The authors conclude that the putative deconfined critical point is a weak first-order transition whose finite-size data are masked by the emergent superspin symmetry (Deng et al., 2024).

The SSH-Holstein chain is presently the most explicit realization of the term itself. The paper identifies a broad regime where the transition shows discontinuity-fixed-point scaling with

$O(4)$4

together with emergent $O(4)$5 order-parameter symmetry. Yet the exact adiabatic solution shows that the thermodynamic system instead passes through a narrow CDW+BOW coexistence phase. Increasing the phonon frequency $O(4)$6 shrinks this coexistence region and eventually restores a genuine deconfined critical point, making the phenomenon the “conceptual counterpart” of deconfined pseudocriticality (Weber, 2 Sep 2025).

5. Neighboring phenomena and boundary cases

Not every emergent-symmetry transition with unusual finite-size behavior is a symmetry-enhanced pseudo-first-order transition. The mixed-coupling 3D Potts models provide a cautionary example. Their upper transition shows exponents and order-parameter histograms consistent with emergent $O(4)$7 symmetry, while lower-temperature transitions for $O(4)$8 are unmistakably first-order. The authors explicitly note that for $O(4)$9 one cannot exclude weak first order because cubic anisotropy should be relevant for $J\!-\!Q_3$0, but they do not present pseudo-first-order diagnostics at the emergent-$J\!-\!Q_3$1 critical point (Ding et al., 2015). The system is therefore highly relevant conceptually, but not a direct demonstration.

The generalized 2D $J\!-\!Q_3$2 model with higher harmonics provides an analogous but mechanistically different route to pseudo-first-order behavior. For equal couplings, the transition evolves from BKT to pseudo-first-order at $J\!-\!Q_3$3 and to true first-order at $J\!-\!Q_3$4; with increasing couplings toward higher harmonics, a five-term model can already become first-order. The paper interprets the crossover not as emergent enlarged global symmetry but as narrowing of the effective angular potential well, which suppresses gradual vortex proliferation (Žukovič, 10 Feb 2025). This case shows that pseudo-first-order behavior need not be symmetry-enhanced in the superspin sense.

The SPT-transition framework based on a nonlocal $J\!-\!Q_3$5 self-duality is another important boundary case. The interpolating Hamiltonian

$J\!-\!Q_3$6

has an exact extra nonlocal symmetry at $J\!-\!Q_3$7, which exchanges the trivial and nontrivial SPTs. The paper proves that the transition point must then be either gapless, first-order through spontaneous $J\!-\!Q_3$8 breaking, or split by an intermediate symmetry-breaking phase (Tsui et al., 2015). This does not diagnose pseudo-first-order behavior directly, but it supplies a general symmetry-enhanced framework in which such behavior could naturally arise.

A distinct thermodynamic consequence appears in symmetry-enhanced first-order quantum phase transitions. For anisotropic pseudospin models such as XZ and XXZ systems, the order parameter jumps at the $J\!-\!Q_3$9 transition, yet the Grüneisen ratio diverges because the enhanced symmetry at the transition point produces a soft mode. The result is a critical-like thermodynamic singularity,

$SO(5)$0

coexisting with a first-order ground-state transition (Beneke et al., 2021). This broadens the notion of pseudo-first-order phenomenology from finite-size histograms to low-energy thermodynamics.

6. Conceptual significance and unresolved issues

The main conceptual lesson is that emergent symmetry is not, by itself, evidence for a continuous critical point. Rotationally symmetric order-parameter histograms, vanishing anisotropy moments, large apparent correlation lengths, and even clean finite-size scaling collapses can arise in regimes whose true thermodynamic behavior is weakly first-order, split by an intermediate phase, or governed by an ordered superspin manifold rather than a conformal critical point (Weber, 2 Sep 2025, Deng et al., 2024, Serna et al., 2018).

A second lesson is that ordinary first-order diagnostics may fail or become misleading in the presence of symmetry enhancement. The checkerboard $SO(5)$1-$SO(5)$2 transition is first-order without barrier-separated histograms or negative Binder peaks because the AFM and PSS sectors are unified into an $SO(5)$3 vector (Zhao et al., 2018). Conversely, the generalized $SO(5)$4 model shows that energy-histogram bimodality can occur in a pseudo-first-order regime that is not asymptotically discontinuous (Žukovič, 10 Feb 2025). The distinction therefore requires multi-observable analysis: histogram evolution with size, anisotropy moments, susceptibilities, torus spectra, entanglement scaling, and direct thermodynamic extrapolation.

A third lesson is that the microscopic source of the enlarged symmetry matters. In some systems the key ingredient is competing Dirac masses and weak lattice anisotropy; in others it is slow RG flow near an ordered $SO(5)$5 manifold, product-group invariant structure, or symmetry fractionalization of condensing anyons. These mechanisms are not equivalent, but they all support the broader inference that enlarged symmetry can strongly reorganize the finite-scale appearance of a transition.

Several issues remain open. In some models, such as the mixed 3D Potts $SO(5)$6 case, theory suggests a relevant anisotropy while finite-size numerics remain almost perfectly $SO(5)$7-symmetric (Ding et al., 2015). In the $SO(5)$8 problem, pseudo-first-order behavior in $SO(5)$9 is an RG possibility rather than a numerically established phase-transition class (Sorokin, 2021). In deconfined settings, the distinction between an extremely weak first-order transition and a long crossover into a different asymptotic regime continues to depend sensitively on the observable and scale window used (Deng et al., 2024, Serna et al., 2018).

In this sense, symmetry-enhanced pseudo-first-order transition is best understood not as a single universality class but as a recurring pattern: microscopic anisotropies are suppressed so efficiently by an enlarged symmetry structure that finite systems mimic a direct discontinuous transition, while the true long-distance outcome is revealed only at much larger scales or through diagnostics that directly probe the ordered manifold, the anisotropy field, or the thermodynamic splitting of the competing phases.