Spin-Resolved Momentum Distribution

• Spin-resolved momentum distribution is a quantum measure defining how momentum states are occupied based on spin, incorporating both statistical and dynamic correlations.
• It distinguishes between Pauli exclusion-induced exchange effects and interparticle interaction-driven dynamic correlations using density matrix factorization.
• The framework uncovers signatures of anomalous phases and topological transitions, such as biconnected Fermi surfaces and renormalized quasi-particle behavior.

Spin-resolved momentum distribution is a fundamental quantity in quantum many-body physics, characterizing how momentum eigenstates are occupied as a function of the spin and, where present, isospin degrees of freedom. It encodes the interplay of Fermi statistics, interparticle interactions, and possible phase transitions in a wide range of systems including liquid $^3$He and nuclear matter at nonzero temperature. Its theoretical treatment requires separating statistical (exchange) effects, enforced by the Pauli exclusion principle and dependent on particle spin, from dynamic (or direct) correlations arising from interparticle interactions. The correlated density matrix framework enables this separation and facilitates the analysis of normal and anomalous Fermi liquid phases as well as the identification of spin-dependent features in the momentum distribution.

1. Factorization of the One-Body Reduced Density Matrix

The starting point for quantifying spin-resolved momentum distributions in interacting Fermi systems is the one-body reduced density matrix (OBDM), $n(12)$, which depends on spatial coordinates and the spin (and possibly isospin) variables of two particles. In the correlated density matrix (CDM) approach at finite temperature, this matrix is factorized as

$n(12) = N_0(12) \exp\left[-Q(12)\right],$

where:

• $N_0(12)$ is the statistical distribution factor, capturing exchange (Pauli) correlations. For spinful systems, $N_0(12)$ is nonzero only for identical (parallel) spin configurations.
• $Q(12)$, the phase-phase correlation function, encompasses all effects from dynamic (direct) correlations due to interparticle forces. For a system without spin dependence, this reduces to $n(r) = N_0(r) \exp[-Q(r)]$.

This decomposition permits a clear distinction between the effects of Fermi statistics—entirely spin-dependent and manifesting as exchange holes in the density matrix—and interaction-driven modifications, which may display more complex spatial and spin structures.

2. Spin and Isospin Decomposition via Projection Techniques

To extend the formalism to systems with spin ($v=2$ for unpolarized $^3$He; $v=4$ for symmetric nuclear matter), a projection procedure is employed to separate spin-parallel and spin-antiparallel contributions. For $v=2$, any function $L(12)$ defined on two-particle coordinates and spins is decomposed as

$L(12) = L'(r)\, \delta_0 + L^a(r)\, [1-\delta_0],$

with $\delta_0$ projecting onto parallel spins. Equivalently, using spin projection operators,

$L(12) = L(r) + L^\circ(r)\, \sigma_{012},$

where $L(r)$ locates spin-independent or parallel-spin parts and $L^\circ(r)$ signals contributions from antiparallel spins ($\sigma_{012}$ serves as the projective index).

Both $N_0(12)$ and $Q(12)$ are thus decomposed, and the exchange factor $N_0(12)$ is nonvanishing only for parallel-spin components (by Pauli exclusion), reinforcing the strong spin-resolution in $n(k)$. Analytical expressions involve combinations of so-called nodal and exchange circulant functions, with contributions labeled by HNC (hypernetted-chain) diagrammatic indices, e.g.,

$$N_0(12) = T_{cc}(12) - N_{0,cc}(12) - E_{QQ,cc}(12),$$

$-Q(12) = N_{0,da}(12) + E_{0,da}(12),$

where the subscripts distinguish various circular (exchange) or dynamic (correlation) terms.

3. Momentum Distribution and Correlation Effects

The spin-resolved single-particle momentum distribution, $n(k)$, is computed as the Fourier transform of the OBDM: $$n(k) = \int n(12) \, e^{i {\bf k} \cdot {\bf r}} \, d{\bf r}.$$ For a system with spin only, in the absence of interactions, $N_0(k)$ is the Fermi–Dirac distribution and $Q(12)=0$. When interactions are accounted for, dynamic correlations encoded in $Q(12)$ modify the distribution: $$n(k) = \int N_0(12) \exp\left[-Q(12)\right] e^{i {\bf k} \cdot {\bf r}} d{\bf r}.$$ The statistical factor $N_0(12)$ maintains the strict spin-resolution of the distribution by Pauli exclusion, while $Q(12)$, which can be both spin-independent and spin-dependent, imparts further structure—oscillations or nonmonotonicities—depending on the strength and nature of interactions and correlations.

These modifications can have significant implications:

• If $Q(12)$ (in momentum space) develops periodic (condensate-like) structure at $k_0$, this signals the emergence of anomalous phases, possibly precursor to density wave or superfluid ordering.
• If the effective single-particle energy becomes nonmonotonous, the chemical potential may cross the dispersion at multiple $k$, leading to multi-connected or “biconnected” Fermi surfaces, a hallmark of topological phase transitions.

4. Anomalous Phases and Topological Transitions

Enhanced spin or isospin correlations can drive the system into regimes where traditional Fermi liquid theory fails. A periodic dynamic correlation $Q(r)$ (or corresponding $Q(k)$) may create structure in $n(k)$ (e.g., a sharp peak at a finite $k_0$), suggesting condensation at nonzero momentum or new broken symmetry—analogous to FFLO-like states but rooted in two-body correlations.

Furthermore, if the effective single-particle spectrum (arising from $N_0(k)$ in the factorization) exhibits maxima and minima (Landau maxon/roton-like features), the Fermi surface itself may become multiply connected, corresponding to abrupt changes or even discontinuities in $n(k)$. In this context, the spin-resolved distribution serves not only as a diagnostic for the phase, but as an order parameter for the underlying topological transition.

5. Renormalized Fermion Description and Thermal Boundaries

The analysis introduces “renormalized fermions”: quasi-particle excitations whose energies and chemical potentials are dressed by the full interactions. The momentum distribution for such renormalized fermions reads

$$n_{\rm qp}(k) = T_{\rm qp}(k)\left[1+I_{\rm qp}(k)\right]^{-1},$$

with

$$T_{\rm qp}(k) = \exp\left[\beta(\mu_{\rm qp} - \tilde{\epsilon}_{\rm qp}(k))\right],$$

where $\tilde{\epsilon}_{\rm qp}(k)$ indicates the renormalized dispersion, and $I_{\rm qp}(k)$ encodes residual exchange corrections. The utility of this construction is that deviations between the exact $n(k)$ and $n_{\rm qp}(k)$ quantify non-Fermi liquid behavior. The method allows tracing the effective mass, excitation spectrum, and their evolution with temperature, yielding insight into the thermal boundaries of the normal Fermi liquid regime and its breakdown toward anomalous phases.

6. Summary and Implications

The spin-resolved momentum distribution, as derived via the factorization $n(12)=N_0(12) \exp[-Q(12)]$, provides a precise and highly structured view of the effects of exchange and direct correlations in Fermi liquids at finite temperature. Spin and isospin projections naturally separate the Pauli-principle–enforced statistical correlations from the dynamic correlations that can lead to novel phase behavior. Through the explicit computation of $n(k)$, the approach enables the quantitative study of:

• Spin- (and isospin-) dependent effects in the momentum distribution, fundamentally constrained by the symmetry (Pauli exclusion), with $N_0(12)$ nonzero only for parallel spin.
• Correlation-induced modifications due to $Q(12)$, whose structure—periodic or spatially oscillatory—serves as the origin for phase transitions and the formation of anomalous phases.
• The emergence of phenomena such as condensate formation at finite momentum, or the realization of biconnected Fermi surfaces and corresponding topological transitions.
• The definition of “renormalized fermions” as a baseline enables deviations in $n(k)$ to be parsed into those due to exchange-only effects and those due to further dynamic correlations, clarifying the nature and boundaries of Fermi liquid phases.

This framework, via correlated density matrix theory, thus provides essential theoretical infrastructure for systematically analyzing spin-dependent momentum distributions and connecting their features to emergent macroscopic phases in correlated Fermi systems at nonzero temperature (Serhan et al., 2010).