Strongly First-Order Phase Transition

• Strongly first-order phase transitions are defined by a discontinuous jump in the order parameter, release of latent heat, and distinct phase coexistence.
• They involve metastability and bubble nucleation processes that result in sharp non-analytic behavior and gravitational wave signatures in cosmological settings.
• Analytical and numerical methods such as effective potential techniques and partition function decomposition quantify transition strength and predict observable phenomena.

A strongly first-order phase transition is characterized by a discontinuous change in an order parameter and the presence of a latent heat during the transition. In statistical mechanics, quantum field theory, and condensed matter physics, such transitions are distinguished by metastability, phase coexistence, nucleation processes, and sharp non-analytic behavior at the transition point. Strongly first-order transitions play crucial roles in electroweak baryogenesis, the cosmology of the early universe, correlated electron phenomena, and the statistical mechanics of lattice and liquid models.

1. Definition and General Features

A first-order phase transition is said to be strongly first order when the barrier between the competing minima in the (effective) potential is large, with well-separated phases and substantial discontinuity in the order parameter, typically accompanied by exponentially suppressed tunneling and a sharply defined coexistence temperature or field. Quantitatively, for thermal or quantum field theoretic transitions, the “strength” is often expressed by the ratio of the order parameter jump to the critical temperature (such as $v(T_c)/T_c$), or by the large free energy difference between minima compared to the ambient thermal scale.

Typical features include:

• Discontinuous change in the order parameter at $T_c$ (or critical field).
• Release of latent heat.
• Phase coexistence with mixed-phase configurations during the transition.
• Metastability and nucleation: the system transitions via bubble formation.
• In field-theoretic contexts, exponential suppression of sphaleron (non-perturbative) transitions in the broken phase if the barrier is large.

2. Formalism and Criteria for Strong First-Order Transitions

2.1 Energetic and Thermodynamic Conditions

The generic condition for a strong first-order transition at temperature $T_c$ can often be cast as

$\frac{\Delta \phi_c}{T_c} > 1$

where $\phi_c$ is the vacuum expectation value (VEV) of the relevant field(s) in the broken phase and $T_c$ is the critical temperature. For electroweak baryogenesis, this translates into

$\frac{v(T_c)}{T_c} > 1 \tag{1}$

as used, for example, in the $U(1)$-extended MSSM (UMSSM) (Ahriche, 2010) and many subsequent models.

An explicit analytic criterion can be constructed in effective models. For example, in singlet scalar extensions, the critical temperature is determined by equating effective potentials at the nontrivial and trivial minima, with strong strength ensured if the curvature at the minima is large and the order parameter jump is sizable (Ghorbani et al., 2018).

2.2 Partition Function Decomposition

For statistical systems, strong first-order transitions are realized when the partition function near coexistence can be decomposed as

$$Z = \sum_{n=1}^\mathcal{N} \alpha_n \exp[-\beta V f_n]$$

where $\mathcal{N}$ is the number of coexisting phases, $\alpha_n$ are degeneracy factors, $f_n$ are free energies per volume, and $\beta = 1/(kT)$. The condition for a strong transition is $\beta\Delta_{n'n} \gg 1$ for all pairs of phases, i.e., free energy barriers much larger than thermal fluctuations (Fiore et al., 2012).

3. Realizations in Quantum Field Theory and Beyond the Standard Model

3.1 Electroweak Phase Transitions and Baryogenesis

A central context for strongly first-order transitions is electroweak symmetry breaking in models beyond the Standard Model (SM):

• UMSSM and $U(1)'$ models: The addition of a singlet and $Z'$-boson modifies the tree-level Higgs sector and introduces new minima and exotica. The coupling between singlet and doublets relaxes the effective potential, greatly enhancing $v(T_c)/T_c$ beyond 1, robustly accommodating a strong first-order EWPT (Ahriche, 2010).
• Singlet-extended Higgs and Higgs portal models: Both tree-level (via cubic and quartic portal couplings) and loop-level baryers can be engineered. For instance, in models with a singlet fermionic DM candidate and scalar portal, the parameter space with $m_H\sim30-350$ GeV, $m_\psi\sim15-350$ GeV, and $\alpha\lesssim28^\circ$ can achieve very strong transitions satisfying $v_c/T_c>1$ (Li et al., 2014).
• Classically scale-invariant and conformal extensions: Dynamical symmetry breaking via the Coleman-Weinberg mechanism endows the Higgs sector with flat directions at tree-level, leading to radiatively generated, strongly first-order transitions once thermal corrections are included, with predictive correlations among scalar masses, mixing angles, and heavy fermion/DM masses (Farzinnia et al., 2014, Ayazi et al., 2019).
• New charged/multiscalar sectors: Extended Higgs sectors, extra triplets (Jangid et al., 15 Apr 2025), and vector-like (VLQ) matter often produce enlarged barrier heights via both thermal and zero-temperature quantum effects, admitting strong first-order transitions even for new states at the TeV scale, with strong constraints from perturbativity and unitarity up to high scales (Kanemura et al., 2022).

3.2 Mechanisms beyond Thermal Fluctuations

Strong first-order transitions may also be realized in non-thermal settings:

• Photoinduced structural transitions (e.g., in GeTe): Optical excitation can close electronic gaps and induce abrupt phase coexistence without phonon softening, with a strongly first-order character confirmed by free energy calculations and ultrafast X-ray diffraction (Furci et al., 30 Apr 2024).
• Quantum phase transitions in correlated electron systems: Fermi surface reconstruction involving topologically distinct Fermi surface sheets, characterized by abrupt changes in momentum distribution and competing quantum minima, can occur as g or T is varied (e.g., from a quasiparticle halo to a hole pocket), producing a first-order transition evidenced by phase coexistence and discontinuity in ground-state observables (Pankratov et al., 2011).

4. Dynamics, Metastability, and Cosmological Signatures

4.1 Bubble Nucleation and Runaway Walls

Strongly first-order transitions proceed via nucleation of critical bubbles, driven by the interfacial free energy $\mu$ and the vacuum energy difference $\Delta V$. In quantum field theory or holographic models, nucleation is described in the thin-wall regime as:

$F(R,T) = 4\pi R^2\mu - \frac{4\pi}{3}R^3\Delta V$

with the critical radius $R_c = 2\mu/\Delta V$ (Shavrin, 19 May 2025). For sufficiently strong transitions, the released latent heat greatly exceeds dissipative friction (runaway regime), accelerating bubble walls to ultrarelativistic speeds and favoring gravitational wave (GW) production dominated by bubble collisions.

4.2 Gravitational Wave Production

The violent bubble dynamics in strong first-order transitions naturally produce stochastic GW backgrounds. The GW energy density spectrum can be written (in a representative parametrization specific to runaway regimes) as:

$$\Omega_{GW} \sim (\beta/H)^{-2} \kappa^2 \left[\frac{\alpha^2}{(1+\alpha)^2}\right] \mathcal{S}(f, \beta/H)$$

where $\alpha$ is the transition strength, $\beta/H$ the inverse duration, $\kappa$ the efficiency, and $\mathcal{S}$ the spectral shape function. LISA and BBO are projected to be sensitive to such signals for phase transitions with $\alpha \gtrsim 0.1$ and $\beta/H \lesssim 100$ (Jangid et al., 15 Apr 2025, Shavrin, 19 May 2025). In multistep transitions or models with spontaneous discrete symmetry breaking, additional low-frequency peaks may result from domain wall decay (Zhou et al., 2020).

4.3 Primordial Black Holes and Cosmological Probes

Strong first-order transitions can yield over-dense regions via delayed bubble nucleation as described by the rate $\Gamma(t) = \Gamma_0 e^{\beta t}$, with spatial fraction of remaining metastable regions $F(t)$. For density contrasts exceeding $\delta_c \sim 0.45$, horizon-sized PBHs can form at the transition epoch, with expected masses set by $M_{PBH} \sim 4\pi H^{-1}(t_{PBH})$ (e.g., $10^{-5} M_\odot$ for the electroweak epoch). The abundance and mass spectrum of such PBHs constitute uniquely accessible signatures for microlensing surveys (Hashino et al., 2021).

5. Analytical and Numerical Methods for Strong First-Order Transitions

5.1 Semi-Analytical and Scaling Techniques

For lattice/finite-size systems, strong first-order character permits analytical treatment via decomposition of the partition function and the thermodynamic function $W$:

$$W \approx \frac{b_1 + \sum_{i=2}^\mathcal{N} b_i \exp[-a_i y]}{1 + \sum_{i=2}^\mathcal{N} c_i \exp[-a_i y]}$$

where $y = \xi - \xi^*$ (the deviation from the transition field), and only $a_i$ depend on the system size. Universal crossing points among finite-size curves at $\xi^*$ provide precise determination of transition points and response functions (Fiore et al., 2012). This requires that $\beta \Delta_{n'n} \gg 1$ to ensure non-overlapping order parameter distributions—a condition always realized for strongly first-order transitions.

5.2 Effective Potentials and Daisy Resummations

In quantum and thermal field theories, the full effective potential,

$$V_{eff}(\phi, T) = V_0(\phi) + V_{1-loop}(\phi) + V_{thermal}(\phi, T) + V_{Daisy}$$

must include both one-loop corrections (for quantum barrier formation) and Daisy (ring) resummations to capture the correct behavior of scalar, gauge, and fermion modes, particularly near the transition point. The formation of a large barrier is closely tied to these effects and may be controlled by tuning portal coupling, field content, and quantum numbers of new degrees of freedom.

6. Physical Implications, Observational Signatures, and Model Constraints

6.1 Collider and Astrophysical Probes

• Triple Higgs coupling ($hhh$): Large deviations from the SM value in the $hhh$ self-coupling ($$\Delta\lambda_{hhh}/\lambda_{hhh}^{SM} \gtrsim 60\%$$) are predicted in strongly first-order scenarios, particularly when heavy new scalar bosons are involved (Kanemura et al., 2022, Florentino et al., 7 Oct 2024).
• $h \rightarrow \gamma\gamma$: In models where charged new scalar loops drive both $hhh$ and $h\gamma\gamma$ deviations, a correlated measurement can serve as an indirect probe of a strongly first-order EWPT. Constraints on these operators are synergistic between the HL-LHC and Future Linear Colliders (Florentino et al., 7 Oct 2024).
• Direct detection of DM and PBHs: In scenarios where the new sector includes dark matter candidates, constraints from direct and indirect detection (e.g., LUX, XENON1T) impact the parameter region where SFOPT is realized (Li et al., 2014, Ghorbani et al., 2018, Hall et al., 2019). PBH detection in microlensing searches provides a cosmological probe (Hashino et al., 2021).

6.2 Theoretical Constraints

Allowed parameter space for strong first-order transitions is typically delimited by:

• Vacuum stability and unitarity bounds (up to the Planck scale as in triplet-extended models (Jangid et al., 15 Apr 2025)).
• Electroweak precision observables and collider signatures (Higgs mixing angles, invisible decays).
• Sphaleron decoupling and completion conditions for baryogenesis viability (met via e.g., $v_n / T_n \gtrsim 1$).
• The effectiveness of new physics sectors (charged scalars, Loryons [Editor's term], etc.) in generating large enough cubic terms without spoiling stability.

7. Cross-disciplinary and Model-Independent Insights

Strongly first-order phase transitions are a multifaceted phenomenon:

• In condensed matter and quantum systems, they manifest as abrupt topological transitions (e.g., Fermi surface reconstruction (Pankratov et al., 2011)) and are tractable via general statistical mechanical scaling and partition function methods (Fiore et al., 2012).
• In particle cosmology, achieving a SFOPT is tightly linked with realizing baryogenesis, generating detectable gravitational wave backgrounds and potentially primordial black holes, as well as linking with new physics in scalar, gauge, and DM sectors.
• Model-independently, necessary and often sufficient conditions involve large discontinuities in order parameters, well-separated minima in the constrained or effective potential, βΔ ≫ 1 (free energy scales large compared to temperature), and robust kinetic barriers as seen in system-independent formulations.

The paper of strongly first-order phase transitions informs the search for new physics at colliders and in cosmic probes, unifying concepts across condensed matter, statistical physics, and high-energy theory.