Spin and Pseudospin Symmetries

• Spin and pseudospin symmetries are inherent SU(2) symmetries arising in the Dirac equation when specific conditions on scalar and vector potentials yield near-degenerate doublet structures.
• Supersymmetric quantum mechanics and perturbative expansions provide a rigorous framework to analyze the symmetry limits, breaking mechanisms, and relativistic corrections.
• These symmetries have practical implications in nuclear shell evolution, resonant state analysis, and analogous phenomena in atomic and condensed matter systems.

Spin and pseudo-spin symmetries are emergent SU(2) symmetries in the spectrum of the Dirac equation with central scalar and vector potentials. These symmetries manifest in various physical contexts, most notably in nuclear structure physics, where they explain critical features of shell evolution, single-particle level patterns, and observed quasi-degeneracies in nuclei. Spin symmetry (SS) and pseudo-spin symmetry (PSS) reflect the suppression of spin–orbit or pseudo–spin–orbit couplings under specific relations between the central potentials. Their dynamical breaking, partial realization, and interplay with relativistic mean-field (RMF) theory underpin substantial aspects of single-nucleon dynamics.

1. Fundamental Definitions and Algebraic Structure

Spin symmetry in the Dirac equation is realized when the difference of the local Lorentz scalar $S(r)$ and vector $V(r)$ potentials is a constant, i.e., $\Delta = V(r) - S(r) \approx \text{const}$. Pseudospin symmetry is realized if the sum $\Sigma = V(r) + S(r) \approx \text{const}$. These conditions simplify the matrix structure of the Dirac equation such that either the upper or lower radial component (depending on the symmetry) satisfies a Schrödinger-like equation without matrix coupling.

In the SS limit, the Dirac Hamiltonian commutes with the standard spin SU(2) algebra generated by

$$\vec{S} = \left(\begin{array}{cc} \vec{s} & 0 \ 0 & U_p \vec{s} U_p \end{array}\right)$$

where $\vec{s}$ are the spin-1/2 generators and $U_p$ is the helicity operator; whereas in the PSS limit, the system is invariant under the pseudospin SU(2) algebra generated by an operator of the form

$$\vec{\tilde{S}} = \left(\begin{array}{cc} U_p \vec{s} U_p & 0 \ 0 & \vec{s} \end{array}\right)$$

This operator mixes spin and momentum, reflecting the entanglement characteristic of the lower large components of the Dirac spinor (Ginocchio, 2014).

In three dimensions, the pseudospin doublets correspond to pairs of orbitals with quantum numbers $(n, \ell, j = \ell + 1/2)$ and $(n-1, \ell+2, j' = \ell + 3/2)$, while spin doublets refer to $(n, \ell, j = \ell \pm 1/2)$. The suppression of the respective (pseudo)spin–orbit couplings ensures the near-degeneracy of these doublets in the respective symmetry limits (Ginocchio, 2010, Castro et al., 2012, Heitz et al., 31 Jan 2025).

2. Supersymmetric Factorization and Analytical Approaches

The application of methods from supersymmetric quantum mechanics (SUSY QM) has provided a rigorous algebraic framework for understanding the structure and breaking of (pseudo)spin symmetries. The Dirac Hamiltonian, when subject to suitable similarity transformations, can be factorized via shift (ladder) operators $A$, $A^\dagger$ constructed from superpotentials $W(r)$ as

$$A = \frac{d}{dr} + W(r), \qquad A^\dagger = -\frac{d}{dr} + W(r).$$

This factorization produces SUSY partner Hamiltonians $H_1 = A^\dagger A$ and $H_2 = AA^\dagger$, which are isospectral apart from possible zero-modes.

For specific central potentials:

• Coulomb: The spectrum in the spin symmetry limit is

$E_n = m \left[1 - \frac{v^2}{(k_d + n)^2}\right],$

with $k_d$ related to angular momentum (Hall et al., 2010).

• Kratzer: The SUSY shift operators acquire $r$-dependent modifications, and the exact spectrum is recovered via shape invariance.

Under SUSY, collaboration with the similarity renormalization group (SRG) permits a Hermitian diagonalization of the Dirac Hamiltonian. The resulting Schrödinger-like equation features a pseudo-centrifugal barrier $\propto \kappa(\kappa-1)/r^2$—essential for manifesting PSS within the SUSY partner Hamiltonian $\tilde{H}$. The near-degeneracy of pseudospin doublets is thus deeply rooted in this algebraic structure (see (Liang et al., 2012), Table below for structural differences):

Term in Schrödinger Eq.	$H$ (normal)	$\tilde{H}$ (SUSY partner)
Angular barrier	$\kappa(\kappa+1)/2Mr^2$	$\kappa(\kappa-1)/2Mr^2$
Relevant symmetry limit	Spin	Pseudospin

The SUSY approach quantitatively explains the smallness of pseudospin–orbit splittings and tracks their systematic decrease with increasing energy in realistic nuclei (Liang et al., 2012, Shen et al., 2013).

3. Perturbative Expansion, Relativistic Effects, and Non-relativistic Limits

Spin symmetry breaking arises from relativistic spin–orbit coupling, whereas pseudospin symmetry breaking is subtler and cannot be uniquely attributed to relativistic effects alone (Chen et al., 2012, Heitz et al., 31 Jan 2025). The covariant Dirac equation in standard units features a term $\propto 1/r (d\Delta/dr)$ responsible for spin–orbit splitting; this term vanishes in the non-relativistic limit, restoring spin symmetry (Chen et al., 2012).

A perturbative expansion around a symmetry-conserving reference (such as a relativistic harmonic oscillator) systematizes the analysis of deviation from perfect symmetry. The first-order correction to doublet splittings, expressed in terms of upper ($g$) and lower ($f$) Dirac components, is (Heitz et al., 31 Jan 2025): $$\Delta E^{(1)} = \int dr\, r^2 \left\{ [g_a^2(r) - g_b^2(r)]\, [\Sigma(r) - \Sigma_{HO}(r)] + [f_a^2(r) - f_b^2(r)]\, [d_0 - \Delta(r)] \right\}.$$ For spin symmetry, splitting is determined solely by the lower component and the difference potential; for pseudospin symmetry, both components are involved. Second-order corrections involve sums over intermediate states, further refining the theoretical understanding of shell evolution and magicity.

Non-relativistic limits ($\lambda \rightarrow 0$ for reduced Compton wavelength) suppress relativistic couplings and erase spin–orbit splittings. However, pseudospin splitting behaviors can vary—sometimes rising, sometimes remaining, or even decreasing—as $\lambda$ decreases, indicating their origin is not exclusively relativistic (Chen et al., 2012).

4. Physical Realizations: Atomic, Nuclear, and Condensed Matter Systems

Nuclear single-particle spectra: The near degeneracy of pseudospin doublets underlies identical band structures, low-lying $0^+$ states, and systematic patterns across isotopic chains (Ginocchio, 2014, Heitz et al., 31 Jan 2025). The "intruder-state puzzle"—the absence of a partner for $j=l+1/2$ orbitals with zero radial nodes—can be accounted for by exact node counting in the lower component and is resolved within confining potentials (Liang et al., 2014).

Resonant and deformed states: In both continuum and deformed nuclei, respective (pseudo)spin symmetries manifest in the energy spectra and wave function structure, as observed in Nilsson diagrams and density distributions for axially-deformed rare earth nuclei (Sun et al., 2023).

Anti-nucleon and hyperon spectra: Charge conjugation exchanges sign between scalar and vector potentials, mapping nucleon PSS to anti-nucleon/anti-hyperon spin symmetry, resulting in near degeneracy of spin doublets (very small spin–orbit splitting) in anti–baryonic bound states (Ginocchio, 2010, Ginocchio, 2014, Liang et al., 2014).

Condensed matter applications: The formalism underlying (pseudo)spin symmetry has analogues in Dirac-like electron systems such as graphene, where tailored potentials can induce or break the symmetry, affecting the emergent energy landscape (Alberto et al., 2015, Alberto et al., 2017).

5. Influence of Tensor and Coulomb Interaction Terms

Extended potential models—HeLLMann, Kratzer, Cornell, hyperbolic, and diatomic molecular potentials—have been analyzed under (pseudo)spin symmetry constraints. Tensor interactions (e.g. Coulomb-like terms) systematically break doublet degeneracy by modifying centrifugal (or pseudo-centrifugal) barriers, generating observable splittings particularly relevant in nuclear and molecular spectroscopy (Rajabi et al., 2012, 1212.5349). Inclusion of Hund’s coupling and pair-hopping terms in spin–orbit coupled electron systems leads to explicit PSS breaking and is crucial for understanding magnetic anisotropies in materials such as Sr$_2$IrO$_4$ (Mohapatra et al., 2020).

6. Advanced Aspects: Exceptional Polynomials, Dynamical and Partial Symmetry

Recent developments have extended solvable Dirac systems to those involving exceptional orthogonal polynomials (e.g. exceptional Hermite polynomials) under symmetry constraints, leading to rational potential models with energy dependence and nontrivial spectral properties (Yeşiltaş et al., 2021).

Partial dynamical pseudospin symmetry arises when not all but certain (near Fermi surface) single-particle states satisfy the symmetry relations. These are characterized by certain matching conditions on radial wave functions or on the commutator $[\hat{H}, \tilde{S}]$ acting on specific eigenstates (Ginocchio, 2014). Partial restoration mechanisms are relevant given the realistic shapes of mean-field RMF potentials in finite nuclei.

7. Summary Table of Key Properties

Symmetry	Condition	SU(2) Generators	Physical Effect	Energy Splitting
Spin	$\Delta = V-S$~const	$\vec{S}$	Suppresses spin–orbit splitting	Controlled by $d\Delta/dr$; $\propto A^{-2/3}$
Pseudo-spin	$\Sigma = V+S$~const	$\vec{\tilde{S}}$	Pseudospin doublet quasi-degeneracy	Competing $d\Sigma/dr$ and $d\Delta/dr$ contributions

($A$: Mass number; $g$, $f$: Dirac large/small components)

8. Conclusions and Outlook

Spin and pseudo-spin symmetries, arising from precise algebraic properties of the Dirac equation with specific central potentials, offer a unifying scheme for understanding diverse spectroscopic phenomena in nuclei, atomic, and even condensed matter systems. Supersymmetric methods reveal their hidden structure, justify near-degeneracies, and provide analytic protocols for systematic expansions away from exact symmetry via physically transparent perturbative corrections. The manifestation and breaking of these symmetries through tensor, Coulomb, and deformation effects are now quantitatively understood in the framework of covariant mean-field theory, with precise predictions for energy splittings and shell evolution throughout the nuclear landscape. These insights have cemented (pseudo)spin symmetry as a central organizing principle in modern quantum many-body physics.