Spontaneous Symmetry Breaking (SSB)

• Spontaneous symmetry breaking is the process where symmetric equations lead to asymmetric states, crucial in molecular, field, and nonequilibrium physics.
• It demonstrates how collective interactions and external fields, such as pressure or magnetic fields, trigger transitions from symmetric to localized states.
• Its applications range from explaining chiral molecule behavior and catalyzing mass generation in fermion systems to inducing asymmetry in boundary-driven models.

Spontaneous symmetry breaking (SSB) refers to the phenomenon whereby the underlying equations of motion, Lagrangians, or system Hamiltonians possess a particular symmetry, yet the actual ground state or stationary state of the system exhibits lower symmetry. Manifesting across molecular, condensed matter, particle, and statistical physics, SSB plays a central role in diverse physical contexts, including the existence of chiral molecules, mass generation in relativistic field theories, catalysis of symmetry breaking by external fields, and emergence of asymmetry in nonequilibrium steady states. Variations of SSB mechanisms often illuminate the interplay between microscopic interactions, collective effects, and macroscopic observables, as well as offer insight into deep analogies connecting distinct domains.

1. Spontaneous Symmetry Breaking in Molecular Physics

A quintessential illustration of SSB in molecular systems is provided by the case of chiral molecules—specifically, the resolution of Hund’s paradox. For molecules described by a symmetric double-well potential (such as pyramidal molecules like ammonia), quantum mechanics predicts delocalized eigenstates with definite parity due to tunneling between the wells. These eigenstates would manifest as superpositions of left- and right-handed configurations, preserving mirror symmetry. However, experimentally, molecules are observed as localized in one well, yielding optically active (chiral) states.

This paradox is resolved by considering weak but collectively significant intermolecular (dipole–dipole) interactions. Modeling each molecule as a two-state system with Hamiltonian

$h_0 = -\frac{\Delta E}{2} \sigma^x,$

an interaction term of the form $-G \sigma^z \langle \sigma^z \rangle$ is introduced to account for the mean-field effect of surrounding molecules. The self-consistent mean-field Hamiltonian becomes

$$h(\lambda) = -\frac{\Delta E}{2} \sigma^x - G \sigma^z \langle \lambda | \sigma^z | \lambda \rangle,$$

where $\sigma^z$'s eigenstates correspond to localized chiral states $|L\rangle$ and $|R\rangle$. When the mean-field strength exceeds a critical value $G_{cr} = \Delta E/2$, the ground state bifurcates into two degenerate, symmetry-broken chiral states, thus spontaneously breaking the original parity symmetry. Experimentally, under conditions such as high pressures that enhance collective interactions, the tunneling-induced inversion line disappears and molecules remain permanently localized.

2. Magnetic Catalysis and SSB in Fermion Systems

In relativistic field theory, SSB typically arises when a nonlinear interaction (such as in the Nambu–Jona-Lasinio (NJL) model) exceeds a threshold, resulting in dynamical mass generation and chiral symmetry breaking. Remarkably, in the presence of an external homogeneous magnetic field, even arbitrarily weak interactions suffice to induce SSB—a phenomenon termed magnetic catalysis.

An explicit construction in 2+1 dimensions, using operator methods rather than the path-integral approach, reveals how the Dirac field vacuum is fundamentally reorganized by the magnetic field. The analysis proceeds in two stages:

• First, a Bogolyubov transformation is applied to the free Dirac field (with the magnetic field off), pairing creation and annihilation operators for particles and antiparticles, leading to a new Fock vacuum characterized by altered correlations.
• Second, the field is expanded in terms of Landau level eigenmodes. A further Bogolyubov transformation, mapping new operators $a_{n p_1}$, $b_{n,-p_1}$ onto hatted operators, reflects the vacuum's double-pairing nature, both in the Dirac spectrum and in Landau levels.

The presence of the magnetic field thus catalyzes symmetry breaking via a robust pairing mechanism, even in the absence of a critical nonlinearity. The dynamical mass gap and associated chiral symmetry breaking are present for any nonzero coupling, with the magnetic field altering the system's infrared properties by reducing its effective dimensionality. This scenario, established by Gusynin, Miransky, and Shovkovy and explicitly analyzed here in operator language, demonstrates the universal role of fluctuations and external fields in driving SSB phenomena.

3. SSB in Stationary Nonequilibrium States

While equilibrium statistical mechanics predicates SSB upon systems with infinite degrees of freedom and dimensionality constraints (for instance, precluding SSB in 1D short-range models), nonequilibrium stationary states can exhibit symmetry breaking even in one dimension. Exemplified by boundary-driven models—such as the so-called bridge model with two particle species injected and removed at opposing boundaries—nonequilibrium driving can break the symmetry between species in the stationary state, despite symmetric microscopic rules.

In such systems, although microscopic dynamics may be symmetric under charge conjugation and spatial reflection ($CP$), the steady-state densities and currents become asymmetric due to imposed boundary conditions. This nonequilibrium SSB is relevant for understanding molecular chirality in biological systems and asymmetry phenomena in cosmology, including the matter/antimatter imbalance.

4. Mathematical Formulation and Mechanisms

The underlying mechanism in these instances of SSB is the bifurcation or instability of a symmetric ground state under weak symmetry-breaking perturbations or collective interactions. This mechanism is well captured in a mean-field or operator framework. In the chiral molecule case, the mean-field Hamiltonian

$$h(\lambda) = -\frac{\Delta E}{2}\sigma^x - G \sigma^z \langle \sigma^z \rangle$$

exhibits a transition at $G_{cr} = \Delta E/2$ where the symmetric ground state splits into two degenerate chiral states. For Dirac fermions in a magnetic field, the double-pairing structure introduced by successive Bogolyubov transformations leads to an effectively symmetry-broken vacuum for any weak nonlinearity, with the explicit forms of the Landau level wavefunctions and their pairing coefficients providing operator-level clarity of the process.

5. Implications and Unifying Insights

Across these examples, pairing mechanisms and collective effects emerge as unifying elements in SSB. In molecular systems, environmental couplings select among symmetry-equivalent configurations, resulting in chiral order even when the intrinsic Hamiltonian is fully symmetric. In relativistic fermion systems, a magnetic field reorganizes the structure of the vacuum, erasing the threshold for SSB and enforcing chiral order through catalysis. In nonequilibrium low-dimensional models, boundary-induced flows and collective interactions yield broken-symmetry stationary states in contradiction to equilibrium-based expectations.

These insights highlight that SSB is not confined to conventional thermodynamic or field-theoretic settings and can be engineered or induced by leveraging environment interactions, external fields, or nonequilibrium drive. Furthermore, SSB operates not merely as a formal mathematical phenomenon, but as a central organizing principle bridging microscopic laws and macroscopic emergent behavior.

6. Broader Context and Applications

The paper of SSB in molecular, field-theoretic, and nonequilibrium models demonstrates the phenomenon's universality. The mechanism elucidated for chiral molecules addresses fundamental questions in molecular spectroscopy and optical activity. Magnetic catalysis provides a paradigm for mass generation and phase transitions in gauge and fermionic systems, with implications for condensed matter (e.g., quantum Hall effects) and high-energy physics. The observation of SSB in nonequilibrium systems suggests new avenues for understanding the spontaneous emergence of chirality and asymmetry in natural and synthetic systems, including potential pathways for homochirality in biological molecules and for addressing fundamental asymmetries in cosmological models.