Phase diagram of the massless Gross-Neveu model with two components

Abstract: Recently, Benini, Mamroud, Reis and Serone have presented exact results for the Gross-Neveu model at finite density, both at finite N and in the large N limit. Generalizing previous studies, they introduce a chemical potential acting only on a subset of fermion flavors - the two-component Gross-Neveu model. Here, we take up this idea and extend the semiclassical study to finite temperature. The full (large N) phase diagram including inhomogeneous phases is constructed without solving the thermal Hartree-Fock problem. All phase boundaries can be found on the basis of various kinds of stability analyses. As a result, we find a qualitative change of the phase diagram at some critical filling fraction.

• The paper introduces a tunable filling fraction ν that distinguishes between connected and disconnected phase boundaries.
• It employs both perturbative Ginzburg-Landau theory and variational analyses to delineate homogeneous and inhomogeneous condensate phases.
• The study reveals novel bifurcations and critical endpoints, providing new benchmarks for nonperturbative quantum field theory explorations.

Authoritative Summary of "Phase diagram of the massless Gross-Neveu model with two components" (2606.21246)

Introduction and Motivation

The work investigates the phase diagram of the massless Gross-Neveu (GN) model, particularly the large $N$ limit, where the chemical potential acts selectively on a subset of fermion flavors, thereby generalizing previous studies where the chemical potential was universal across all flavors. This two-component model introduces a tunable filling fraction $\nu \in [0,1]$ which governs the degree of occupation of valence bands in an ensemble of Dirac fermions. The model is of interest due to its renormalizability, asymptotic freedom, chiral symmetry breaking, bound states, and integrability—all features lending it to study nonperturbative phenomena in low-dimensional QFT.

Recent advances using both integrability and lattice approaches for finite $N$ have validated semiclassical results but also delineated their limitations. The selective chemical potential approach proposed by Benini et al. uncovers new ground-state features—namely, crystals composed of Dashen-Hasslacher-Neveu (DHN) baryons with arbitrary fermion number. This generalization is crucial for uncovering partially filled valence bands in dense matter, and the present work extends these results to finite temperature, circumventing the explicit solution of the thermal Hartree-Fock (HF) problem in favor of efficient stability analyses.

Homogeneous Phase Diagram

The analysis begins with the thermodynamic potential for homogeneous condensates, expressed for the two-component case by weighting the standard one-component GN model potential according to the occupation fraction $\nu$. The perturbative phase boundary between chirally symmetric ($M=0$) and broken ($M>0$) phases is determined analytically and mapped using Ginzburg-Landau (GL) theory, leveraging expansions in powers of the condensate $M$. Notably, for filling fractions below a critical value $\nu_0 \approx 0.8271$, the tricritical point vanishes due to a bifurcation, and the first and second order phase boundaries decouple. This is quantitatively determined by tracking zeros of the quartic coefficient in the GL expansion and its derivatives.

The homogeneous calculations show:

• For $\nu \geq \nu_0$: Second and first order boundaries connect via a tricritical point.
• For $\nu < \nu_0$: Phase boundaries are disconnected, with the first order line terminating inside the massive region at a critical endpoint.

The procedure enables precise determination of phase boundaries and endpoint coordinates for a range of $\nu \in [0,1]$0, providing a clear topological classification (connected vs. disconnected) contingent on occupation fraction—a result that recapitulates and extends $\nu \in [0,1]$1 results.

Inhomogeneous Phase Structure: Connected Region ($\nu \in [0,1]$2)

With inhomogeneous solutions allowed, the phase diagram for $\nu \in [0,1]$3 mirrors the classic one-component scenario but generalizes it for arbitrary filling. Perturbative and non-perturbative stability analyses are employed:

• Horizontal (massless–crystal) boundaries: Derived from instability analysis in the massless phase using spatially modulated mean fields with variable wavenumber $\nu \in [0,1]$4. The envelope of critical lines generated per $\nu \in [0,1]$5 yields the phase boundary.
• Vertical (massive–crystal) boundaries: Determined by analyzing the thermodynamics of single baryons with occupation fraction $\nu \in [0,1]$6, utilizing a variational approach that tracks the onset of solitonic instability.

Strong numerical results demonstrate a smooth evolution of all phase boundaries with $\nu \in [0,1]$7, with key features including:

• The crystal region enclosed by a cusp structure formed from two second order lines.
• The connection of vertical and horizontal boundaries at the tricritical point.
• The critical wave number $\nu \in [0,1]$8 along phase boundaries converging to $\nu \in [0,1]$9 asymptotically.

The analysis confirms consistency with previous results for the massive GN model and elucidates the mechanism for nucleation of DHN baryons with fractional occupancy.

Inhomogeneous Phase Structure: Disconnected Region ($N$0)

For $N$1, the phase diagram topology shifts; the tricritical point is replaced by a critical endpoint, and stability analyses must be adapted:

• Horizontal boundary: Instability of the massive homogeneous phase toward periodic modulations is analyzed using formulas from the massive GN model, generalized to the two-component case.
• Vertical boundary: Instability against formation of single baryons tracked through minimization of a variational ansatz, connecting the critical endpoint to the $N$2 baryon mass/occupation.

A novel feature appears for intermediate values near $N$3: the horizontal boundary formed via massive-phase instability intersects the perturbative boundary at a new bifurcation point $N$4 (not present in the homogeneous phase diagram), requiring dual perturbative analyses for both the massless and massive phases.

Numerical results show:

• The crystal region rapidly shrinks with decreasing $N$5,
• The bifurcation point $N$6 migrates to higher $N$7 as $N$8 decreases,
• For small $N$9, the crystal region becomes negligible, paralleling the effect of increasing bare mass in the massive GN model.

These observations indicate that the topological distinction between connected and disconnected diagrams does not persist once inhomogeneous condensates are allowed, with cusp and bifurcation structures arising for $\nu$0.

Theoretical and Practical Implications

The methodology demonstrates that phase structures in the GN model can be classified and computed without explicit HF solutions, relying instead on analytic and numeric stability analyses. This approach is robust and extensible, as shown by recovery of all critical features for both massless and massive regimes, and is particularly impactful for models where analytic HF solutions are intractable.

The implications span:

• Nonperturbative QFT: The selective chemical potential opens new avenues for studying crystalline condensates and partially filled valence bands, with possible relevance to condensed matter analogues and cold atom systems.
• Thermal phase transitions: The exact mapping of all phase boundaries using variational and perturbative techniques highlights the utility of the GN model as a prototype for nontrivial phase transition structures.
• Numerical simulations: The analytic classification informs lattice and Bethe ansatz calculations at finite $\nu$1, providing benchmarks and qualitative insights for future studies.

Speculatively, future directions may involve:

• Exploring dynamical effects of fluctuating occupation fractions,
• Extending to higher dimensions or other symmetry classes,
• Investigating relations to integrable systems and finite-gap potentials.

Conclusion

The study provides a comprehensive and precise characterization of the phase diagram for the massless Gross-Neveu model with two components, elucidating the dependence on occupation fraction and mapping all phase boundaries using stability analyses rather than explicit HF solutions. The findings clarify the interplay between homogeneous and inhomogeneous phases, demonstrate new bifurcation phenomena, and establish a framework for understanding partially filled bound-state crystals at finite temperature and density. The methods and insights furnish a foundation for both practical calculations and further theoretical explorations in the domain of nonperturbative quantum field theory.