Structure, topology, and differential surface tension of inhomogeneous chiral condensates in the chiral limit of QCD from constrained simulations

In the three-dimensional O(2) model, lattice simulations with a constrained value of the average order parameter showed that spontaneous breaking of a continuous symmetry is realized by inhomogeneous, spin-wave-like configurations, classified by an integer winding number and carrying an excess energy parameterized by a newly introduced differential surface tension; the authors explicitly proposed extending this constrained-ensemble approach to QCD in the chiral limit. The open problem is to carry out this program: simulate QCD with massless light quarks in 3+1 dimensions with the spatially averaged chiral condensate constrained to values inside the coexistence disk, and determine the specific structure, the topological classification, and the associated differential surface tension(s) of the inhomogeneous chiral condensates that dominate the constrained path integral. This includes the methodological prerequisites the sources identify: a controlled implementation of the Grassmann-valued condensate constraint at physical flavor count, sampling across topological sectors, and stable extraction of the surface tension near the coexistence edge, where its defining ratio becomes of 0/0 type. A 2025 follow-up constructed and tested the fermionic constraint machinery in the two-dimensional chiral Gross-Neveu model at large flavor number but explicitly leaves the QCD application open.

Background

Spontaneous symmetry breaking (SSB) of a continuous symmetry underlies superconductivity, the Higgs mechanism, and the chiral symmetry breaking of the strong interactions. In the standard lattice treatment one adds an explicit symmetry-breaking field $h$ (a magnetic field for a spin model, a small quark mass for QCD), works at $h\ne 0$, and takes the infinite-volume limit before sending $h\to 0$; the order parameter is defined only through this double limit. The constraint effective potential, introduced by O'Raifeartaigh, Wipf and Yoneyama, offers an alternative: one restricts the Euclidean path integral to field configurations whose spatially averaged order parameter equals a prescribed value $\bar\phi$ and defines $\Omega(\bar\phi)=-(1/V)\ln Z_{\rm constrained}$. In the thermodynamic limit $\Omega$ coincides with the (convex) quantum effective potential, which is exactly flat in the broken phase on the coexistence disk $|\bar\phi|<\bar\phi_{\min}$, where $\bar\phi_{\min}$ is the spontaneous order parameter; in finite volume $\Omega$ is generally concave there, and the mechanism by which flatness is recovered was, until recently, described only abstractly.

Endrődi, Kovács and Markó (Phys. Rev. Lett. 127, 232002 (2021)) simulated the three-dimensional O(2) model (the lattice XY model of two-component classical spins) in this constrained ensemble. Inside the coexistence disk they found the path integral dominated by inhomogeneous, spin-wave-like configurations rather than homogeneous ones: the slice-averaged spin vector $\Sigma(x_1)$ precesses along one spatial direction while its length stays close to $\bar\phi_{\min}$. Each configuration carries an integer winding number $w$ counting how many times $\Sigma$ wraps the internal circle as $x_1$ traverses the periodic lattice; the $w=0$ (non-winding) and $w=1$ (winding) sectors dominate, with a sharp sector transition at an intermediate $\bar\phi_c$ that their algorithms could not equilibrate. The excess free energy of these inhomogeneities is captured by the differential surface tension $\sigma$, defined through

$\Omega(\bar\phi)=\sigma\,\frac{E_{\Sigma}(\bar\phi)-E_{\Sigma}(\bar\phi_{\min})}{\bar\phi_{\min}^{2}},$

where $E_{\Sigma}$ is the slice-gradient energy of $\Sigma$; this generalizes the domain-wall surface tension of discretely broken symmetries (bubble walls at first-order transitions) to continuously broken ones. They measured $\sigma=0.427(8)$, independent of $\bar\phi$ and of the winding sector, showed that $\Omega$ vanishes as $1/L^2$ so that flatness is recovered in the thermodynamic limit, and noted that their setup provides a counterexample to a conjecture of Splittorff, Son and Stephanov that spontaneous breaking of translational invariance requires a path integral with a sign problem.

In QCD with massless light quarks, the chiral condensate $\langle\bar\psi\psi\rangle$ is the order parameter of the spontaneously broken chiral symmetry, with the light quark mass playing the role of the explicit breaking field $h$. The chiral limit is notoriously difficult to simulate: the double limit requires extremely light quarks at very large volumes, and the nature of the finite-temperature chiral transition in the chiral limit remains one of the few open questions of zero-density QCD. Inhomogeneous chiral condensates have a long history in effective models — chiral spirals and inhomogeneous phases in Gross-Neveu and Nambu-Jona-Lasinio type theories, reviewed by Buballa and Carignano, and in lattice studies of the chiral Gross-Neveu model by Lenz, Mandl and Wipf — but not in full QCD. The source paper proposes that inside the coexistence disk of the chiral order parameter the QCD path integral should likewise be dominated by configurations with inhomogeneous chiral condensates, whose specific structure, topological properties, and associated differential surface tensions could then be determined with the methods introduced for the O(2) model. This requires constraining a fermionic condensate — an order parameter built from Grassmann-valued (anticommuting) fermion fields, $\bar\psi\sigma_a\psi$ with $\sigma_a$ the flavor Pauli matrices — a problem the same authors solved in the large-$N_f$ limit (the limit of a large number of fermion flavors, where saddle-point expansions become exact) of the two-dimensional chiral Gross-Neveu model (a four-fermion theory with a spontaneously broken continuous chiral symmetry) in 'Constraining fermionic condensates' (Phys. Rev. D 111, 054512 (2025)), via a saddle-point (large-volume) expansion of the exact constraint. There they reproduced the flattening of the constraint potential and the winding/non-winding phenomenology, and stated that implementing such constrained simulations in QCD is left for the future.

Problem Statement

Apply the constrained-ensemble approach to QCD with massless light quarks in $3+1$ spacetime dimensions in the chirally broken phase — that is, perform (or rigorously analyze) simulations in which the spatially averaged chiral condensate is constrained to values $\bar\phi$ inside the coexistence disk $|\bar\phi|<\bar\phi_{\min}$ (here $\bar\phi$ denotes the constrained spatial average of the renormalized chiral condensate and $\bar\phi_{\min}$ its spontaneous chiral-limit value), implementing the fermionic constraint beyond the large-$N_f$, two-dimensional Gross-Neveu test — and determine: (1) the specific structure of the inhomogeneous chiral condensate configurations that dominate the constrained path integral inside the disk (their spatial profile, characteristic wavelengths, and symmetry-breaking pattern, as analogues of the O(2) spin waves); (2) their topological classification — whether an analogue of the O(2) winding number exists for the chiral order-parameter manifold, which sectors dominate, and where sector transitions occur as a function of $\bar\phi$; and (3) the associated differential surface tension(s): their value, their dependence on $\bar\phi$, on the topological sector, and on the number of flavors, and their behavior at and arbitrarily close to the coexistence edge $\bar\phi_{\min}$, where the defining ratio becomes of 0/0 type. The determination must be statistically controlled, which requires resolving the obstacles identified in the source: sampling across topological sectors (updates changing the winding-type sector are strongly suppressed) and controlled finite-volume and continuum extrapolations of the sector-resolved constraint potential and condensate profiles. The completion boundary is a study of chiral-limit QCD itself that reports the three determinations above; studies of bosonic or lower-dimensional surrogate models, however instructive, do not by themselves complete the problem.

References

1. Gergely Endrődi, Tamás György Kovács, Gergely Markó, Spontaneous Symmetry Breaking via Inhomogeneities and the Differential Surface Tension, Phys. Rev. Lett. 127 (2021) 232002, DOI: 10.1103/PhysRevLett.127.232002, arXiv:2109.03668, https://arxiv.org/abs/2109.03668
2. Gergely Endrődi, Tamás G. Kovács, Gergely Markó, Laurin Pannullo, Constraining fermionic condensates, Phys. Rev. D 111 (2025) 054512, DOI: 10.1103/PhysRevD.111.054512, arXiv:2412.12973, https://arxiv.org/abs/2412.12973
3. Laurin Pannullo, Gergely Endrődi, Tamás G. Kovács, Gergely Markó, Flattening of the quantum effective potential in fermionic theories, Proceedings of Science, PoS LATTICE2023 (2024) 379, DOI: 10.22323/1.453.0379, arXiv:2312.01960, https://arxiv.org/abs/2312.01960
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