Momentum-Dependent Exchange Interaction

• Momentum-dependent exchange interactions are effective coupling terms in quantum systems that vary explicitly with momentum, reflecting nonlocality and internal symmetry effects.
• They are pivotal in nuclear forces, magnetic materials, and ultracold atomic setups, explaining phenomena such as spin observables, magnon dispersion, and synthetic pairing.
• Controlled via experimental tuning and theoretical modeling, these interactions enable refined predictions and engineering of complex many-body states.

A momentum-dependent exchange interaction is a class of effective microscopic interaction in quantum many-body systems, wherein the strength, symmetry, or operator structure of the exchange coupling depends explicitly on the wave vector(s) or momentum of the interacting degrees of freedom. Such interactions arise in a wide range of physical contexts including nuclear systems, correlated electrons, magnetic materials, ultracold atomic gases, and composite quasiparticles (e.g., excitons, magnons). Their form, physical consequences, and methods of determination are system-dependent but share the unifying motif that spatial nonlocality, relativity, or internal symmetries project onto a nontrivial dependence in momentum space.

1. General Principles and Theoretical Background

Momentum-dependent exchange interactions refer to two-body (or higher-body) coupling terms in an effective Hamiltonian or Lagrangian, whose operator structure or coupling constants vary as functions of the momentum variables conjugate to real-space coordinates. This stands in contrast to simple contact (momentum-independent), local, or nearest-neighbor exchanges.

In the most general effective field-theory or tight-binding setting, one expands the interaction in powers of the relative, center-of-mass, or transferred momentum between constituents: $$V(\mathbf{p}', \mathbf{p}) = V^{(0)} + V^{(2)}[\mathbf{p}, \mathbf{p}'] + V^{(4)}[\mathbf{p}, \mathbf{p}', \mathbf{P}] + \dots$$ such that at higher order, explicit dependence on the total momentum $\mathbf{P}$ or other composite momenta ($\mathbf{k}$, $\mathbf{Q}$, etc.) emerges.

In systems with spin or internal structure, the momentum-dependence can occur in both the scalar coupling and in the coefficients of operator products linking momenta and internal degrees of freedom, such as $$(\mathbf{p}' \times \mathbf{p}) \cdot (\boldsymbol{\sigma}_1 - \boldsymbol{\sigma}_2)$$ for spin-orbit-type exchange.

Momentum-dependence typically reflects:

• Nonlocality or finite range of the microscopic interaction,
• Relativistic corrections or boosts to nonrelativistic potential models,
• Band structure, spin-orbit coupling, or symmetry in crystals,
• Coupling of center-of-mass and internal degrees of freedom (e.g., through laser-induced processes or projection to effective pseudospin).

2. Nuclear Effective Interactions and P-Dependent Operators

In chiral effective field theory for nucleons, momentum-dependent exchange manifests through systematically constructed contact terms in the nucleon-nucleon (NN) potential. Relativistic invariance constrains the available operator structures, but allows for specific momentum-dependent contact interactions.

Explicitly, a relativistic Lagrangian for parity- and time-reversal-even NN contacts up to $O(p^4)$ yields 54 Lorentz-invariant operators built from pairs of Dirac bilinears and up to four derivatives. Foldy–Wouthuysen reduction and matching project these to a nonrelativistic basis comprising exactly 26 independent operators: $2$ at leading $O(p^0)$, $7$ at $O(p^2)$, $17$ at $O(p^4)$ (Filandri et al., 2023).

Among these, two operators at $O(p^4)$ depend explicitly on the total pair momentum $\mathbf{P} = \mathbf{p}_1 + \mathbf{p}_2$: $$\begin{aligned} O''_{16}(p',p) &= i\,(\mathbf{k}\cdot\mathbf{Q})\;[(\mathbf{Q} \times \mathbf{P}) \cdot (\boldsymbol{\sigma}_1 - \boldsymbol{\sigma}_2)],\ O''_{17}(p',p) &= (\mathbf{k}\cdot\mathbf{Q})\;[(\mathbf{k} \times \mathbf{P}) \cdot (\boldsymbol{\sigma}_1 \times \boldsymbol{\sigma}_2)], \end{aligned}$$ where $\mathbf{k} = \mathbf{p}' - \mathbf{p}$ and $\mathbf{Q} = (\mathbf{p}' + \mathbf{p})/2$ are interchange variables. Their low-energy constants (LECs), $D_{16}$ and $D_{17}$, are free parameters unconstrained by two-nucleon observables in the center-of-mass frame (where $\mathbf{P}=0$), but essential to describe observables in moving two-body subsystems, such as spin observables in nucleon-deuteron or proton-deuteron scattering. Specifically, a modest nonzero combination of these LECs generates the requisite boost–spin interaction in three-body systems, resolving the $A_y$ vector analyzing power discrepancy by $\mathcal{O}(10-20\%)$ at $E_\text{lab}\lesssim 10$ MeV (Filandri et al., 2023). All other $P$-dependent contact terms at $O(p^4)$ are algebraically fixed by lower-order LECs via the constraints of Lorentz invariance.

3. Momentum-Space Exchange in Magnetic Solids

Momentum-dependent exchange is central to the description and computational modeling of quantum and classical magnetism in solids. The Heisenberg Hamiltonian, typically formulated in real space as $$H = -\frac{1}{2}\sum_{i\neq j} J_{ij} \mathbf{M}_i\cdot\mathbf{M}_j$$, transforms in reciprocal space to capture dispersion via the momentum-resolved exchange: $$J(\mathbf{q}) = \frac{1}{N}\sum_{i,j} J_{ij} e^{i\mathbf{q} \cdot (\mathbf{R}_j - \mathbf{R}_i)}.$$

In density functional theory (DFT) and dynamical mean-field theory (DMFT), $J(\mathbf{q})$ is directly obtained by constraining the magnetization (e.g., via frozen-magnon or spin-spiral configurations) and mapping total energy changes onto the spin Hamiltonian. If weak local moments are present, analytic elimination (integration over their quadratic fluctuations) yields a renormalized $J^{\rm R}(\mathbf{q})$ valid at all temperatures (Lezaic et al., 2012).

In practice, momentum dependence is extracted by evaluating the total energy difference for a family of constrained spin structures parameterized by a variable $\mathbf{q}$, exploiting symmetries for computational efficiency. The resulting $J(\mathbf{q})$, possibly tensorial in multi-sublattice models, governs the magnon spectrum, Curie temperature, and response functions such as susceptibility $$\chi(\mathbf{q}) = \beta M^2/[J^{\mathrm{R}}(0)-J^{\mathrm{R}}(\mathbf{q})]$$ (Lezaic et al., 2012, Belozerov et al., 2017).

In itinerant magnets, momentum-resolved anisotropic exchange tensors $J_{\alpha\beta}(\mathbf{q})$ and Dzyaloshinskii–Moriya vectors $D(\mathbf{q})$ arise from magnetic representation analysis, spin-orbit coupling, and symmetry classification. Six specific symmetry rules determine presence and orientation of these components for a given space group and $\mathbf{q}$ (Yambe et al., 2022).

4. Exchange Interactions in Composite Quasiparticles

Momentum-dependent exchange plays a nuanced role in the fine structure of composite excitations such as excitons and magnons in insulators and semiconductors.

For Wannier excitons in Cu$_2$O, analytic $K$-dependent exchange interaction terms can in principle originate from higher-order expansion in $\mathbf{K}$ (exciton total momentum) via $\mathbf{k}\cdot\mathbf{p}$ perturbation theory. However, these terms are shown to be microscopically negligible (≤neV–μeV for 1S ortho excitons), establishing that experimental $K$-dependent excitonic splittings are dominated not by exchange but by anisotropic kinetic (center-of-mass) dispersion (Schweiner et al., 2016).

In quantum magnets, the magnon excitation spectrum inherits momentum-dependent spin texture from bond- and momentum-dependent exchange parameters. For example, in the Kitaev-Heisenberg-$\Gamma$ model, the momentum-dependent fictitious Zeeman field $\mathbf{h}(\mathbf{k})$ on the magnon Bloch sphere determines vortex or antivortex spin textures, controlled by the original exchange couplings and their momentum dependence (Kawano et al., 2019). Such structure can enable pure spin currents in insulators and be exploited in magnon spintronics.

5. Artificial and Tunable Momentum-Dependent Exchange in Cold Atoms

Ultracold atom platforms provide mechanisms for engineering novel classes of momentum-dependent exchanges, with no analog in solid-state materials.

A paradigmatic example is the realization of a center-of-mass momentum-dependent two-body interaction via a laser-modulated magnetic Feshbach resonance (MFR). Here, two-photon (Raman) transitions between molecular bound states, driven by lasers in different directions, introduce a Doppler effect such that the closed-channel resonance detuning depends explicitly on the two-body total momentum $\mathbf{P}$: $$a(\mathbf{P}) \simeq a_{\rm bg} - \frac{(\delta\mu) a_{\rm bg} \Delta_B}{(\delta\mu)(B - B_0) - \frac{|\Omega_\alpha|^2}{\Delta_{2p}(\mathbf{P})}},$$ with $$\Delta_{2p}(\mathbf{P}) = \Delta_{2p}^{(0)}+(\mathbf{k}_\alpha-\mathbf{k}_\beta)\cdot\mathbf{P}/M$$ (Jie et al., 2016). The effective two-body potential $$V_{\rm eff}(\mathbf{P};\mathbf{r}) = (4\pi\hbar^2/m)a(\mathbf{P})\delta(\mathbf{r})$$ now couples center-of-mass and relative motion.

This technique enables, for instance, momentum-space pairing, the stabilization of Fulde–Ferrell–Larkin–Ovchinnikov-type superfluids without population imbalance, and synthetic anyonic statistics in 1D. Control over $a(\mathbf{P})$ leads to qualitatively new many-body states unavailable in systems with simple contact or range-limited interactions (Jie et al., 2016).

6. Relativistic and Off-Shell Momentum-Dependent Exchange Channels

A comprehensive classification of spin- and momentum-dependent exchange interactions for fermions mediated by bosonic fields involves deriving all operators consistent with the relevant symmetries. For light spin-0 boson exchange between spin-1/2 fermions, the full operator basis includes sixteen terms encoding scalar, spin-spin, tensor, and spin-orbit couplings, each with coefficients depending nontrivially on both the momentum transfer $\mathbf{q}$ and the average momentum $\mathbf{P}$: $$V(\mathbf{q},\mathbf{P}) = \mathcal{P}(q^2)\sum_{i=1}^{16} \mathcal{O}_i^{(1)}(\mathbf{q},\mathbf{P})\,f_i(q^2,P^2,\mathbf{q}\cdot\mathbf{P}) + \text{h.c.}$$ (Zhong et al., 13 Oct 2024).

Relativistic (NLO) and off-shell contributions emerge in coefficient functions $f_i$ that involve powers of $P^2$, $q^2$, and $\mathbf{q}\cdot\mathbf{P}$, many of which vanish in the strict nonrelativistic limit but are nonzero in actual atomic and macroscopic settings. Transforming the operator basis to forms more suitable for coordinate space enables precise mapping to observables and experimental constraints. Importantly, long-range (Yukawa-type) and short-range (contact or derivative delta) contributions can be separated in the Fourier transform from momentum to position representation.

7. Magnetic Anisotropy, Noncollinearity, and Momentum-Dependent Kinetic Exchange

In systems with unquenched orbital angular momentum and strong magnetic anisotropy, exchange interactions acquire explicit dependence on the quantum numbers labeling the orbital momentum projection ($m$). In noncollinear arrangements of local axes, new second-order kinetic exchange pathways emerge between single- and double-occupied orbitals, leading to momentum-dependent Ising-like exchange terms. Specifically, the coefficient of the $|J,\pm J\rangle$ pseudospin projections in the effective Hamiltonian

$$\hat{H}_\text{ex} = -\mathcal{J}\, \tilde{S}_{1z} \tilde{S}_{2z}, \quad \mathcal{J} = \mathcal{J}_{ss} + \mathcal{J}_{sd}$$

(cf. $s$–$s$ and $s$–$d$ virtual hopping), is momentum-dependent through the $t_{mn}$ matrix elements and their angular content (Iwahara et al., 2015).

This mechanism enables strong, and in some cases ferromagnetic, exchange in lanthanide and actinide complexes, as validated by ab initio DFT plus downfolding calculations. The magnitude and sign of the effective exchange are sensitive to the angle between anisotropy axes, with noncollinearity suppressing the antiferromagnetic $s$–$s$ contribution and enhancing or revealing the ferromagnetic $s$–$d$ term.

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Momentum-dependent exchange interactions are thus a pervasive, multifaceted feature of many-body quantum systems, synthesizing symmetry, relativity, internal structure, and external field effects into precise operator forms. Their correct identification, classification, and inclusion is critical for accurate modeling of nuclear reactions, correlated electron behavior, composite quasiparticle responses, quantum magnetism, and engineered many-body states in synthetic platforms.