Continuous Bogoliubov Coefficient Formalism

• Continuous Bogoliubov Coefficient Formalism is a unified operator-based method that encodes thermal and quantum correlations in many-body fermionic and bosonic systems.
• It employs temperature-dependent Bogoliubov transformations to mix original and tilde operators, ensuring canonical commutation, continuity, and detailed balance.
• The approach extends QPM beyond mean-field approximations, enabling accurate predictions of pairing gaps, transition strengths, and resonance fragmentation in hot nuclei.

The continuous Bogoliubov coefficient formalism provides a unified, operator-based approach for encoding thermal and quantum correlations in many-body fermionic and bosonic systems, particularly within the context of finite-temperature nuclear structure models and quantum field theory at nonzero temperature. Through temperature-dependent Bogoliubov transformations, this formalism introduces continuously varying coefficients that govern the structure of quasiparticle and collective excitations, as well as their interactions, in a rigorous microscopic framework that remains canonical and maintains detailed balance. These features enable descriptions beyond mean-field and standard small-amplitude approximations, making the formalism a powerful tool for understanding and predicting the properties of hot nuclei and related systems.

1. Thermal Bogoliubov Transformation and Its Role in Thermo Field Dynamics

The thermal Bogoliubov transformation constitutes the foundation of the real-time operator approach to finite-temperature quantum field theory known as thermo field dynamics (TFD). In TFD, every original operator is doubled by introducing a tilde-conjugate partner, enabling a construction of the thermal vacuum state |0(T)) that correctly reproduces grand canonical ensemble averages as expectation values. The thermal Bogoliubov transformation mixes the ordinary annihilation operators with their tilde counterparts via a rotation parameterized by temperature-dependent coefficients. For fermions, the transformation is

$$\begin{aligned} B_{jm} &= x_j\, a_{jm} - i y_j\, \tilde{a}_{jm}^\dagger, \ \tilde{B}_{jm} &= x_j\, \tilde{a}_{jm} + i y_j\, a_{jm}^\dagger, \ x_j^2 + y_j^2 &= 1, \end{aligned}$$

where $x_j$ and $y_j$ are continuous functions of temperature, and $y_j^2$ directly yields the Fermi-Dirac occupation factors: $y_j^2 = \frac{1}{1 + \exp(-E_j/T)}.$ This transformation ensures that the resulting thermal quasiparticle operators inherit all necessary canonical commutation relations and that the vacuum |0(T)) is annihilated by all thermal quasiparticle annihilation operators. The explicit temperature dependence of the Bogoliubov coefficients guarantees a smooth interpolation from zero to finite temperature.

2. Formulation within the Quasiparticle-Phonon Model (QPM) at Finite Temperature

Extending the zero-temperature Quasiparticle-Phonon Model (QPM) to finite temperatures using the TFD formalism requires a two-step transformation. The first transformation (the standard $u_j$, $v_j$ Bogoliubov transformation) diagonalizes the BCS mean-field Hamiltonian, yielding static quasiparticles. The second, thermal Bogoliubov transformation mixes these quasiparticles with their tilde partners to construct the actual thermal quasiparticle basis: $$\begin{aligned} u_j^2 &= \frac{1}{2}\left[1 + \frac{E_j - X}{\sqrt{(E_j - X)^2 + \Delta^2}}\right], \ v_j^2 &= 1 - u_j^2, \ y_j^2 &= \frac{1}{1 + \exp(-E_j/T)}, \quad x_j^2 = 1 - y_j^2, \end{aligned}$$ where $E_j$ is the BCS quasiparticle energy and $X$ is the chemical potential. The corresponding diagonalization shows that the energy spectrum for the thermal quasiparticles consists of both $+E_j$ (ordinary) and $-E_j$ (tilde) branches, capturing thermal de-excitations.

When extending collective degrees of freedom, such as phonons, thermal Bogoliubov transformations are also applied to phonon operators: $$\tilde{Q}_i = X_\xi Q_i - Y_\xi Q_i^\dagger, \quad X_\xi^2 - Y_\xi^2 = 1,$$ with $X_\xi$ and $Y_\xi$ varying smoothly with temperature, implementing the Bose–Einstein thermal factors into the phonon sector.

3. Continuity and Detailed Balance in the Thermal Bogoliubov Formalism

The temperature dependence of the Bogoliubov coefficients embodies the "continuity" within this formalism: as temperature changes, the coefficients $x_j$, $y_j$ (for quasiparticles) or $X_\xi$, $Y_\xi$ (for phonons) change smoothly, ensuring that the occupation numbers and transition matrix elements evolve continuously without unphysical discontinuities. This continuity secures thermodynamic consistency and detailed balance, as ensured by the thermal state condition: $A|0(T)) = o e^{H/2T} \tilde{A}^\dagger |0(T)),$ where $A$ is a general operator, $o=1$ for bosons and $o=-i$ for fermions, and $H$ is the system Hamiltonian. This property is essential to maintain correct relations between absorption and emission strengths, as encoded in relations like

$D_R = \exp(-\omega_\xi/T)\, \Phi,$

which connects upward and downward transition probabilities in the thermal ensemble.

4. Generalization Beyond Thermal Hartree–Fock and Thermal RPA

The QPM–TFD formalism systematically incorporates correlations and couplings that are inaccessible in standard thermal Hartree–Fock (THF) or thermal Random Phase Approximation (TRPA) approaches, both of which treat only mean-field or small-amplitude fluctuations. In QPM–TFD, the entire coupling between thermal quasiparticles and collective phonons is realized. The procedure consists of:

• Diagonalizing the thermal BCS Hamiltonian in the space of thermal quasiparticle operators;
• Applying the thermal QRPA (TQRPA) to construct collective states on top of thermal quasiparticles;
• Including couplings to more complex configurations, such as two-phonon admixtures, through the extension of the thermal Bogoliubov transformation to the phonon sector.

Characteristic equations, such as the secular equation for mode energies (see Eq. (20)), incorporate both single- and two-phonon states, enabling the description of resonance fragmentation and spreading widths at finite temperature.

5. Implications for the Description of Hot Nuclei and Experimental Observables

The continuous Bogoliubov coefficient formalism enables the computation and interpretation of thermal pairing gaps, occupation numbers, and collective transition strengths in excited nuclei. For example, the finite-temperature BCS gap and number equations are

$$\Delta = G \sum_j (2j+1)(1-2y_j^2) u_j v_j, \quad N = \sum_j (2j+1)\left(v_j^2 + y_j^2(u_j^2 - v_j^2)\right),$$

demonstrating how pairing correlations are modified by the Fermi–Dirac factors. With the inclusion of residual particle–hole correlations and coupling to phonons, the model predicts the redistribution of nuclear transition strengths and the fragmentation of collective resonances as observed in electromagnetic response functions for hot nuclei.

The ability to incorporate the continuous, temperature-dependent evolution of the coefficients also guarantees that the theoretical framework is applicable from the ground state up to high excitation energy, thus providing a comprehensive microscopic description of nuclear matter under varying thermal conditions.

6. Mathematical Framework and Core Expressions

The formalism employs two-step transformations, summarized as:

Transformation	Operators	Coefficient Relations
Static Bogoliubov	$a$, $a^\dagger$ to $b$, $b^\dagger$	$u_j^2$, $v_j^2$, $u_j^2 + v_j^2=1$
Thermal Bogoliubov (TFD)	$b$, $b^\dagger$ and tilde partners to $B$	$x_j^2$, $y_j^2$, $x_j^2 + y_j^2=1$

Key expressions: $$\begin{aligned} &\text{For quasiparticles:} \ &y_j^2 = [1 + \exp(-E_j/T)]^{-1},\quad x_j^2 = 1 - y_j^2,\ &\text{For phonons:} \ &X_\xi^2 - Y_\xi^2 = 1, \quad \text{continuous, temperature-dependent}. \end{aligned}$$

These coefficients map directly onto the correct thermal occupation numbers for both fermionic and bosonic sectors.

7. Significance within Nuclear Structure Theory and Beyond

The continuous Bogoliubov coefficient formalism within TFD, as implemented in the extension of QPM to finite temperatures, provides a tractable method to capture both statistical and dynamical effects of finite-temperature nuclear matter. By encoding quantum statistical correlations and allowing the description of thermally induced resonance fragmentation and strength redistribution, it offers a route to systematically study and predict experimental observables in hot nuclei—such as transition strength distributions and resonance widths—at a level of theoretical rigor and completeness that surpasses the standard mean-field or small-amplitude approximations.

This approach's flexibility ensures its relevance for a wide class of quantum systems where temperature, pairing, and collective phenomena interplay, including applications in quantum field theory, quantum optics, and other finite-temperature many-body settings.