Symmetry-Broken Correlated Quantum States

• Symmetry-broken correlated states are many-body quantum states that exhibit spontaneous long-range order by breaking an underlying symmetry through interactions, fluctuations, or entanglement.
• They emerge in diverse systems such as ultracold gases, moiré superlattices, and quantum magnets, displaying both classical and topological orders.
• Advanced theoretical frameworks and experimental probes, including Bogoliubov theory and STM/STS, enable precise detection and control of these emergent phases.

Symmetry-broken correlated states are a central concept in contemporary quantum many-body physics, arising when interactions, entanglement, or fluctuations induce long-range order that spontaneously breaks an underlying symmetry of the system. Such states frequently appear in a rich variety of physical platforms, from ultracold atomic mixtures to correlated electrons in moiré superlattices, quantum magnets, and programmable quantum simulators. Importantly, beyond classical paradigms, symmetry-broken order may also involve topological character, non-onsite symmetries, or manifest in mixed (non-pure) quantum states in open systems. Below, key theoretical frameworks, mechanisms, mathematical structures, and experimental realizations are described.

1. Definitions and Mechanisms of Symmetry-Broken Correlated States

Symmetry-broken correlated states describe ground or stationary states of many-body systems in which spontaneous symmetry breaking (SSB) is established by nontrivial quantum correlations, often beyond mean-field or single-particle descriptions. In quantum systems, SSB is characterized by the emergence of a macroscopic order parameter that does not respect a symmetry of the underlying microscopic Hamiltonian or density matrix, even though the Hamiltonian itself remains invariant.

Correlations play a pivotal role:

• Quantum fluctuations (e.g., zero-point fluctuations) can favor one ground state over others, lifting degeneracies not resolved at the mean-field level, as in spinor Bose gases (Ge et al., 2014).
• Entanglement-driven order is possible even in globally symmetric quantum wavefunctions, where projection or strong correlations induce long-range order without explicit symmetry breaking in the variational ansatz (Kaneko et al., 2015).
• Collective dynamical effects, such as the spontaneous partitioning of oscillator networks into coherent and incoherent domains (chimera states), exemplify symmetry breaking in correlated dynamical systems (Jiang et al., 2016).

Mathematically, the phenomenon is encoded via anomalous propagators, long-range correlators, or structure factors that develop nonzero values (or scaling) only in the symmetry-broken phase.

2. Correlated Quantum Mixtures and Fluctuation-Induced Locking

A prototypical example is the mixture of two species of spin-1 Bose gases, each described by a nematic (spin-ordered) condensate. At the mean-field level, the joint ground state is degenerate with respect to the relative orientation of the two spin directors $\mathbf{n}_a$ and $\mathbf{n}_b$; the Hamiltonian does not fix the angle $\theta$ defined by $\cos\theta = \mathbf{n}_a \cdot \mathbf{n}_b$.

However, incorporating zero-point quantum fluctuations via a Bogoliubov expansion and proper regularization generates an effective potential $E_0(\theta)$ (arising from summing zero-point energies over fluctuation modes):

$$E_0(\theta) = \frac{1}{2} \sum_k [\omega_{x+,k} + \omega_{x-,k}](\theta)$$

This effective potential is minimized at $\theta = 0$, locking the spin directors together. The SSB here is genuinely fluctuation-induced, and the ground state becomes a product of macroscopic nematic states, both pointing in the same direction. The locking is robust to harmonic trapping and persists for weak Zeeman fields, making it possible to observe via oscillations in Zeeman sublevel populations and optical probes. The locking gap (oscillation frequency) is tunable and can be enhanced in optical lattices (Ge et al., 2014).

3. Correlated Phases in Moiré Superlattices: Twisted Bilayer and Double Bilayer Graphene

Correlated symmetry-broken states abound in magic-angle twisted bilayer graphene (TBG) and twisted double bilayer graphene (tDBG):

• Dominance of Electron-Electron Interactions: Flat bands (small bandwidth $W$ from suppressed kinetic energy) yield a large $U/W$ ratio, amplifying the effect of Coulomb repulsion and promoting a variety of symmetry-broken orders (Bhowmik et al., 2021, Li et al., 2023).
• Emergence of Ordered Phases at Fractional Fillings: Experiment reveals robust states at half-integer fillings ($\nu \approx 0.5, 3.5$ per moiré cell), described by a spin/charge density wave that doubles the supercell, inferred from zone folding and transport signatures (Hall conductance resets, thermoelectric sign reversals) (Bhowmik et al., 2021).
• Symmetry-Broken Chern Insulators: At certain fillings and magnetic fields, translational symmetry breaking and strong correlations stabilize Chern insulators with large Chern numbers (e.g., $C = -4$ at $\nu \approx -0.5$) (Bhowmik et al., 2021). Analogous states arise in tDBG in a range of twist angles, exhibiting spontaneous time-reversal symmetry breaking (zero-field anomalous Hall effect), moiré unit cell enlargement (doubling/tripling), and a sequence of symmetry-broken Chern insulators (He et al., 2021).
• Correlated Stripe/Nematic Orders Beyond the Magic Angle: In large-angle TBG (e.g., $3.45^\circ$ on MoS$_2$), tuning the dielectric environment and interlayer coupling can bring $U/W > 1$, yielding symmetry-broken phases with giant van Hove singularity splitting and stripy charge order—demonstrated by STM/STS (Li et al., 2023).

These phenomena connect to a quantum critical landscape in which band topology, interactions, and symmetry breaking intermix, providing direct analogies to high-$T_c$ materials and kagome superconductors (Zhao et al., 2021).

4. Quantum Hall Ferromagnetism, Lattice-Scale Orders, and Topological Order

Strong correlations and symmetry breaking also manifest as:

• Kekulé bond, charge-density-wave (CDW), and valley/spin-ordered phases in the $n=0$ Landau level of graphene, tunable via Coulomb interaction screening. Scanning tunneling microscopy directly reveals transitions between these orders as functions of magnetic field and dielectric environment (Coissard et al., 2021). Coexisting and competing orders (Kekulé + CDW) suggest the possibility of a rich, theoretically underexplored phase diagram.
• SO(8) Dynamical Symmetry in Monolayer Graphene: Quantum Hall states in graphene can be mathematically described by SO(8) symmetry, unifying antiferromagnetism, spin/valley polarization, and pairing, in close analogy to high-$T_c$ superconductivity and collective nuclear structure (Wu et al., 2016).

5. Theoretical Foundations: Anomalous Green Functions, Mixed-State Correlators, and Non-Onsite Symmetries

Symmetry-broken correlated states are characterized by the emergence of anomalous propagators—off-diagonal elements in Green’s functions that represent transient processes such as pairing or spin-flip events (Janiš et al., 9 Oct 2024). A consistent treatment of SSB in quantum many-body systems requires two-particle propagators to include only even powers of anomalous quantities, ensuring conservation laws and the continuous connection between ordered and normal phases (e.g., in DMFT descriptions of Hubbard antiferromagnets).

In open systems or mixed states, strong and weak symmetries are distinguished (Ando et al., 7 Nov 2024, Liu et al., 12 Oct 2024):

• Weak symmetry breaking is detected by standard two-point functions.
• Strong (doubly acted) symmetry breaking is visible only via Renyi-2 or Wightman correlators, often formulated via thermofield double purifications. A non-vanishing value of

$$C_2(r) = \frac{\mathrm{Tr}(\rho \sigma^z_i \sigma^z_{i+r} \rho \sigma^z_i \sigma^z_{i+r})}{\mathrm{Tr}(\rho^2)}$$

as $r \to \infty$ uniquely signals SSB of a strong symmetry in a mixed state (Ando et al., 7 Nov 2024).

SSB of non-onsite symmetries (not factorizing over local degrees of freedom) gives rise to ground states with long-range entanglement, multiple locally-indistinguishable sectors, and topologically nontrivial properties (e.g., SPT order, SPT “soup” in 2D, algebraic correlations at quantum criticality) (Zhang et al., 7 Nov 2024). Measurement–feedback protocols can be designed to prepare such fragile long-range entangled states with constant probability in the thermodynamic limit.

6. Experimental and Numerical Detection

Experimental evidence for symmetry-broken correlated states relies on a wide array of probes, including:

• STM/STS mapping of local density of states and detection of quasiparticle interference and charge order modulations (Li et al., 2023, Zhao et al., 2021, Coissard et al., 2021).
• Magnetotransport and thermoelectric measurements tracking Hall conductivity, Landau fan diagrams, and thermopower resets (Bhowmik et al., 2021, He et al., 2021).
• Measurements of energy gaps, plateau values in two-point correlators, coherence oscillations (e.g., spin population oscillations in spinor mixtures (Ge et al., 2014), or spatial correlators in programmable Rydberg arrays (Chen et al., 2022)).
• Quantum simulation algorithms for entangled and symmetry-projected states; in quantum computing, symmetry-broken correlated states can be engineered via the quantum phase estimation algorithm and subsequent symmetry restoration (Lacroix, 2020).

Numerical and analytical tools include Bogoliubov theory, variational Monte Carlo with Jastrow–Slater ansatze, coupled-cluster methods built on symmetry-projected references, dynamical mean-field theory, and mappings to classical statistical mechanics for effective potential landscapes (Kaneko et al., 2015, Hermes et al., 2017, Janiš et al., 9 Oct 2024).

7. Generalization and Outlook

Symmetry-broken correlated states have expanded beyond conventional contexts (magnetism, charge order) into systems exhibiting:

• Multimode and intertwined orders (e.g., coexistence of charge, spin, orbital, and nematic order),
• Topological symmetry breaking, including SSB of higher-form or non-onsite symmetries with robust long-range entanglement and quantum criticality distinct from classical paradigms (Zhang et al., 7 Nov 2024),
• Mixed-state and open-system criticality, where decoherence and quantum channels can generate, preserve, or transform symmetry-broken orders (Ando et al., 7 Nov 2024, Liu et al., 12 Oct 2024).

These advances set the foundation for new methods of state preparation (quantum computers, adaptive measurement–feedback circuits), enable direct observation and control of order in programmable quantum hardware, and establish a framework for the classification of phases of matter that includes both traditional symmetry breaking and more exotic, topologically-rich quantum phenomena.