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arXiv:2605.10387

Exact Gaussian pinning of a relative-phase mode and interaction-driven detuning from the pinned frequency

Yogeshwar Prasad

Published May 11, 2026• 0 citations
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Abstract

Hybridization between two identical fermionic layers or components produces a rigid bonding--antibonding splitting $2t_h$. We show that in a paired state this scale enters the collective dynamics exactly within Gaussian fluctuation theory: the copy-odd relative-phase kernel vanishes at $ω=2t_h$, with the underlying parity-pair identity holding pointwise in momentum. Within a broad class of conventionally paired systems, the result is independent of chemical potential, band structure, orbital content, and dimensionality. It follows from copy-pseudospin Larmor precession generated by intercopy hopping together with the Goldstone--Ward condition imposed by superconducting self-consistency. Pairing also isolates $2t_h$ from quasiparticle excitations, with thermal scattering below and pair creation above it whenever the gap is finite. At particle-hole symmetry, where the copy-odd amplitude decouples, the kernel zero becomes an isolated relative-phase pole at $Ω_L=2t_h$, and the layer-imbalance response is a single Gaussian mode exhausting its Gaussian first moment; the corresponding full-Hamiltonian first moment obeys an exact operator sum rule. This pinning is not protected beyond Gaussian order. An exact interaction torque destroys the Larmor closure, while an exactly solvable rung and exact diagonalization of the attractive Hubbard model on a periodic $N_s=12$, $z=3$ bilayer honeycomb cluster show strong detuning. At $|U|=4t$, the dominant layer-imbalance excitation occurs at $0.467(2t_h)$ while retaining $98.6\%$ of the zeroth-moment spectral weight but only $92.4\%$ of the exact first moment. Thus the Gaussian result provides an exact reference frequency whose many-body renormalization can be substantial while a dominant layer-odd excitation survives.

DOI
10.48550/arxiv.2605.10387

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Hybridization between two identical fermionic layers or components produces a rigid bonding--antibonding splitting 2th2t_h2th​. We show that in a paired state this scale enters the collective dynamics exactly within Gaussian fluctuation theory: the copy-odd relative-phase kernel vanishes at ω=2thω=2t_hω=2th​, with the underlying parity-pair identity holding pointwise in momentum. Within a broad class of conventionally paired systems, the result is independent of chemical potential, band structure, orbital content, and dimensionality. It follows from copy-pseudospin Larmor precession generated by intercopy hopping together with the Goldstone--Ward condition imposed by superconducting self-consistency. Pairing also isolates 2th2t_h2th​ from quasiparticle excitations, with thermal scattering below and pair creation above it whenever the gap is finite. At particle-hole symmetry, where the copy-odd amplitude decouples, the kernel zero becomes an isolated relative-phase pole at ΩL=2thΩ_L=2t_hΩL​=2th​, and the layer-imbalance response is a single Gaussian mode exhausting its Gaussian first moment; the corresponding full-Hamiltonian first moment obeys an exact operator sum rule. This pinning is not protected beyond Gaussian order. An exact interaction torque destroys the Larmor closure, while an exactly solvable rung and exact diagonalization of the attractive Hubbard model on a periodic Ns=12N_s=12Ns​=12, z=3z=3z=3 bilayer honeycomb cluster show strong detuning. At ∣U∣=4t|U|=4t∣U∣=4t, the dominant layer-imbalance excitation occurs at 0.467(2th)0.467(2t_h)0.467(2th​) while retaining 98.6%98.6\%98.6% of the zeroth-moment spectral weight but only 92.4%92.4\%92.4% of the exact first moment. Thus the Gaussian result provides an exact reference frequency whose many-body renormalization can be substantial while a dominant layer-odd excitation survives.
Collective excitations can survive profound changes in the microscopic degrees of freedom that generate them. We track a layer-odd collective response from an exact weak-coupling hybridisation anchor into the paired regime of an attractive Hubbard bilayer and find that its Hamiltonian sensitivities evolve in a manner consistent with a crossover from single-particle interlayer hybridisation towards collective pair superexchange. At weak coupling, an exact identity for identical tunnel-coupled layers fixes a relative-phase Gaussian kernel zero at 2th2t_h2th​, independent of the intralayer hopping. Sign-free determinant quantum Monte Carlo reveals an interaction-driven softening of the many-body spectral scale without analytic continuation, while Gaussian response theory supplies the continuous pole interpretation and shows a shift from layer-density to relative-pair-phase character. In the paired regime, the characteristic response scale acquires a growing fractional sensitivity to the intralayer hopping and an interaction dependence partway towards the 1/U1/U1/U law. These results show how a collective excitation can be tracked across a fermion-to-composite-boson crossover while constraining changes in its microscopic drive.
Beyond the frozen-pairing contribution, the superfluid weight of a multiband superconductor contains a correction from the self-consistent relaxation of the pairing amplitudes under a phase twist. We show that this correction has a quantum-geometric origin, not in the space of Bloch states but in the space of collective pairing fields. For an attractive multiband Hubbard model at the mean-field level, the thermodynamic Hessian governing the pairing response coincides with the static Gaussian pair-fluctuation kernel at zero momentum at every temperature below TcT_cTc​. Since gauge invariance relates a phase twist to a finite center-of-mass momentum of the pair field, the superfluid weight is set by the small-momentum curvature of the eigenvalue branch that evolves from the Goldstone mode, and the relaxation correction is a stiffness-weighted quantum metric of the corresponding soft eigenvector on the manifold of pairing configurations. The same construction yields the Ginzburg-Landau gradient coefficient with the identical geometric correction, hence the coherence-length tensor, and, with the low-frequency dynamics of the kernel, the pair-mass tensor. Lieb-lattice calculations verify the equivalence of the thermodynamic, fluctuation-kernel, and soft-mode formulations and the independence of the superfluid weight from the orbital embedding.
In lattice models, destructive interference can quench single-particle motion and produce (nearly) dispersionless Bloch bands, known as flat bands. Suppressing kinetic energy enhances the role of interactions and can lead to pairing, constrained dynamics, and localization-like behavior even in translationally invariant settings. This thesis studies attractive Hubbard-type interactions in such flat-band systems using an isolated-band (projected) description, which yields an effective Hamiltonian in a Wannier-orbital basis. The π-flux dice lattice is a convenient focus because its two lowest bands are perfectly flat and degenerate, and their subspace admits compactly localized Wannier functions. This makes the projection straightforward to implement and yields a projected Hamiltonian with a simple, short-range structure. A one-dimensional reference model is used alongside the dice lattice to separate and benchmark the effects of different processes. In Publication I, we construct a class of exact many-body eigenstates of the projected Hamiltonian of generic lattice models with flat bands in which localized unpaired quasiparticles coexist with a background of paired particles in the zero-momentum state. This general result motivates the study of a specific model, the π-flux dice-lattice. Using exact diagonalization and realtime evolution from few-particle initial states, we show that two-body bound states behave ergodically. At the same time, the density and spin distribution retain strong memory of the initial quasiparticle positions at long times. These results demonstrate localization-like dynamics in a translationally invariant flat-band model, where memory arises from projection-induced constraints rather than quenched disorder. In Publication II, we introduce a one-dimensional on-site/bond singlet model whose terms mirror those of the projected dice-lattice Hamiltonian, with tunable couplings controlling their relative strengths. Exact diagonalization shows that, for some couplings, the spectrum and dynamics of the on-site/bond-singlet model closely reproduce those of the projected dice lattice. We then use spectral statistics to probe how the different dynamical regimes manifest in energy-level correlations. In Publication III, we apply spectral statistics to the projected dice lattice using higher-order spacing and gap-ratio diagnostics. We find a robust parity dependence: even particle-number sectors show multi-block GOE behavior within fixed symmetry sectors, while odd particle-number sectors exhibit exact doublets and inter-doublet correlations that are GUE at even orders. The appearance of distinct random-matrix ensembles (GOE and GUE) within the same physical model is an unusual and interesting feature, which calls for a deeper understanding of the hidden symmetry structure of the Hamiltonians obtained by projecting onto the flat-band subspace in the dice lattice and other lattice models with flat bands realized with ultracold gases in optical lattices and in the newly discovered class of moiré materials.
We study a quantum Hall bilayer system of bosons at total filling factor $\ensuremath{\nu}=1$, and study the phase that results from short-ranged pair tunneling combined with short-ranged interlayer interactions. We introduce two exactly solvable model Hamiltonians which both yield the coupled Moore-Read state [Phys. Rev. Lett. 108, 256809 (2012)] as a ground state, when projected onto fixed particle numbers in each layer. One of these Hamiltonians describes a gapped topological phase, while the other is gapless. However, on introduction of a pair-tunneling term, the second system becomes gapped and develops the same topological order as the gapped Hamiltonian. Supported by the exact solution of the full zero-energy quasihole spectrum and a conformal field-theory approach, we develop an intuitive picture of this system as two coupled composite fermion superconductors. In this language, pair tunneling provides a Josephson coupling of the superconducting phases of the two layers, and gaps out the Goldstone mode associated with particle transport between the layers. In particular, this implies that quasiparticles are confined between the layers. In the bulk, the resulting phase has the topological order of the Halperin 220 phase with U(1)${}_{2}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1)2{}_{2}2​ topological order, but it is realized in the symmetric/antisymmetric basis of the layer index. Consequently, the edge spectrum at a fixed particle number reveals an unexpected U(1)${}_{4}\ifmmode\times\else\texttimes\fi{}\mathrm{U}$(1) structure.
The bilayer Hubbard model with an intralayer hopping ttt and an interlayer hopping ${t}_{\ensuremath{\perp}}$ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half filling, if the splitting between the bonding and antibonding bands ${t}_{\ensuremath{\perp}}/t$ is small, the gap has B1g{B}_{1g}B1g​ (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave) symmetry, but when the splitting becomes larger, A1g{A}_{1g}A1g​ (${s}^{\ifmmode\pm\else\textpm\fi{}}$-wave) pairing is favored. We also find that in the ${s}^{\ifmmode\pm\else\textpm\fi{}}$ pairing region, the pairing is driven by interlayer spin fluctuations and that Tc{T}_{c}Tc​ is enhanced.
The bilayer Hubbard model with an intra-layer hopping ttt and an inter-layer hopping t⊥t_\perpt⊥​ provides an interesting testing ground for several aspects of what has been called unconventional superconductivity. One can study the type of pair structures which arise when there are multiple Fermi surfaces. One can also examine the pairing for a system in which the structure of the spin-fluctuation spectral weight can be changed. Using a dynamic cluster quantum Monte Carlo approximation, we find that near half-filling, if the splitting between the bonding and anti-bonding bands t⊥/tt_\perp/tt⊥​/t is small, the gap has B1gB_{1g}B1g​ (dx2−y2d_{x^2-y^2}dx2−y2​-wave) symmetry but when the splitting becomes larger, A1gA_{1g}A1g​ (s±s^\pms±-wave) pairing is favored. We also find that in the s±s^\pms± pairing region, the pairing is driven by inter-layer spin fluctuations and that TcT_cTc​ is enhanced.