Quantum Tsallis Distribution Framework

• Quantum Tsallis distribution is a q-deformed quantum statistics framework that generalizes Fermi–Dirac and Bose–Einstein distributions using the Tsallis entropy formula.
• It introduces a deformation parameter q to model nonextensive entropy effects, soft collective modes, and anomalous transport observed in strange metals.
• The approach leverages corrections via Schwarzian reparametrization and memory-matrix formalism to interpolate between Fermi-liquid and non-Fermi-liquid behavior.

The Quantum Tsallis Distribution is a generalization of the standard Boltzmann–Gibbs quantum statistics within the framework of nonextensive statistical mechanics. It introduces a deformation parameter $q$ into the quantum density matrix and occupation functions, capturing effects beyond conventional Fermi–Dirac or Bose–Einstein behavior. This approach enables an explicit connection between entropic nonextensivity, soft collective modes (such as Schwarzian reparametrization fields), and experimentally observed transport anomalies in strongly correlated quantum materials, particularly strange metals (Ge, 15 Sep 2025).

1. Mathematical Definition and Reduction to Fermi–Dirac

The Tsallis entropy is defined for a density matrix $\rho$ as

$$S_q^{\text{Tsallis}} = \frac{1 - \operatorname{Tr}(\rho^q)}{q-1}.$$

The maximization of %%%%2%%%% under normalization and energy constraints,

$$\operatorname{Tr}(\rho) = 1, \quad \operatorname{Tr}(\rho H) = \langle E \rangle,$$

yields the quantum Tsallis density matrix: $$\rho = \frac{1}{Z_q}\, [1 - (1-q)\beta H]_+^{1/(1-q)},$$ where $$Z_q = \operatorname{Tr}\{[1 - (1-q)\beta H]_+^{1/(1-q)}\}$$. The notation $[X]_+$ enforces positivity in the spectral decomposition of $H$.

In the limit $q \to 1$, this construction recovers the standard Gibbs (or Fermi–Dirac/Bose–Einstein for indistinguishable particles) distribution via

$$\lim_{q\to1} [1-(1-q)\beta H]^{\frac{1}{1-q}} = e^{-\beta H}.$$

The quantum generalization of the occupation number then takes the form

$$n_q(\epsilon) = \frac{1}{\left[1-(1-q)\beta(\epsilon-\mu)\right]_+^{1/(1-q) + a}},$$

with $a = +1$ for fermions, $a = -1$ for bosons, and $a = 0$ for the classical Maxwell–Boltzmann case. This reproduces the Fermi–Dirac occupation $n_{FD}(\epsilon) = 1/(e^{\beta(\epsilon-\mu)} + 1)$ as $q\rightarrow1$.

2. Schwarzian Correction: q-Deformation and Soft Reparametrization Modes

Small deviations from $q=1$ lead to corrections in the occupation function,

$$n_q(x) = n_{FD}(x) + (q-1)\,\varphi(x) + O((q-1)^2),\quad x\equiv\beta(\epsilon-\mu),$$

where

$$\varphi(x) = \frac{x^2}{2}\, n_{FD}(x)[1-n_{FD}(x)].$$

By associating $x$ with imaginary time derivatives ($x \leftrightarrow -i\partial_\tau$), these corrections can be translated into a functional of the soft mode $f(\tau)$ in systems with emergent reparametrization symmetry. The corresponding local functional is

$$\Phi[f](\tau) = [u'(\tau)]^2 - 2 u''(\tau),\quad u(\tau) = \log f'(\tau).$$

The q-deformed effective action then becomes

$$S_q[f] = -C \int d\tau\, \{f, \tau\} + (q-1) \lambda \int d\tau\, \Phi[f](\tau) + O((q-1)^2),$$

with $\{f, \tau\}$ the Schwarzian derivative,

$$\{f, \tau\} = \frac{f'''(\tau)}{f'(\tau)} - \frac{3}{2} \left(\frac{f''(\tau)}{f'(\tau)}\right)^2.$$

The term $\lambda (q-1)\Phi[f]$ effectively tunes the rigidity of the soft mode spectrum, providing a controlled way to interpolate between Fermi-liquid (FL) and non-Fermi-liquid (nFL) regimes.

3. Electron–Soft Mode Coupling and Quantum Scattering

Electrons in the presence of soft Schwarzian modes couple via a symmetry-imposed vertex. Under $\tau \mapsto f(\tau)$ (dropping SL$(2,\mathbb{R})$ zero modes),

$\psi(\tau) \to \sqrt{f'(\tau)} \psi(f(\tau)),$

the fermion stress tensor couples to the derivative of the collective field: $$S_\mathrm{int} = g \int_0^\beta d\tau\, \epsilon'(\tau)\, \psi^\dagger(\tau)\, i\partial_\tau \psi(\tau).$$ In frequency space, the interaction vertex is linear in energy: $$\Gamma(\epsilon, \omega; \epsilon-\omega) = g\,\epsilon + O(\omega).$$

The finite-temperature scattering rate then emerges from the Fermi Golden Rule integrated over the spectral density $A_\epsilon(\omega)$ of the soft mode: $$\Gamma(\epsilon) = \int_0^\infty \frac{d\omega}{2\pi}\, g^2 A_\epsilon(\omega) [n_B(\omega) + n_{FD}(\omega - \epsilon)].$$ The resultant temperature dependence for the total rate is

$$\Gamma(T) = \frac{\pi^2 \tilde{g}^2}{4} \cdot \frac{T^2}{T_0 + T},$$

exhibiting $\Gamma \sim T^2$ in the FL limit ($T \ll T_0$) and $\Gamma \sim T$ in the nFL regime ($T \gg T_0$).

4. Memory-Matrix Approach and Magnetotransport Observables

The memory-matrix formalism allows a systematic evaluation of transport in systems where certain slow modes (momentum, energy) are only weakly broken. The conductivity tensor is

$$\sigma_{ij} = \sum_{\alpha\beta} \chi_{J_iA_\alpha}\, [M(0)+N]^{-1}_{\alpha\beta}\, \chi_{A_\beta J_j}$$

where the $A_\alpha$ are nearly conserved operators (e.g., $Q$ for energy, $P$ for momentum), $\chi_{J_i A_\alpha}$ are static susceptibilities, $M$ encodes dissipative effects, and $N$ implements the effect of an external magnetic field via Lorentz-force-induced offdiagonal entries.

This formalism reveals that energy relaxation, set by $\Gamma_Q \sim T$, dominates longitudinal transport ($\rho_{xx} \propto T$), while momentum relaxation, characterized by a slower rate $\Gamma_P \sim T^2$, controls the Hall angle ($\cot\theta_H \propto T^2$). Coupling to a magnetic field modifies the soft spectral density,

$$A(\omega; B) \simeq \frac{\kappa_b}{\bar{C}} \cdot \frac{|B|\,|\omega|}{T_0 + T},$$

resulting in a scattering rate and resistivity linear in $|B|$: $$\Gamma(B) \propto |B|T^2/(T_0 + T),\qquad \rho_{xx}(B) \propto |B|.$$

5. Interpolation between Fermi-Liquid and Strange Metal Regimes

The theoretical framework provides a quantitative interpolation between Fermi-liquid and non-Fermi-liquid regimes via the single parameter $q$ and the associated strength of soft-mode fluctuations. The following transport characteristics emerge:

• For $q \approx 1$, the system is close to a Fermi liquid with quadratic ($T^2$) resistivity and Hall response.
• For larger $q-1$, the FL quadratic regime gives way to $T$-linear resistivity and a $T^2$ Hall angle, consistent with Anderson's two-lifetime scenario and experimental strange-metal phenomenology.
• Linear-in-field magnetoresistance emerges from the $|B|$ scaling of the soft-mode spectral density, a feature observed in various quantum-critical compounds. The q-deformation thus acts as a knob tuning the system from coherent quasiparticle transport to regimes dominated by incoherent, quantum-critical fluctuations.

6. Physical Implications and Outlook

This approach links Tsallis nonextensive statistics—originally formulated for classical systems—to the quantum dynamics of strongly correlated materials by mapping the entropic deformation onto a reparametrization (Schwarzian) sector. The resultant quantum Tsallis distribution, its occupation function corrections, and the associated effective actions directly control both single-electron scattering and collective transport properties. Coupling between the electron sector and soft collective modes modifies both thermodynamic and transport responses, providing a theoretically controlled path from conventional metals to the anomalous, quantum-critical behavior of strange metals.

This unified perspective integrates quantum statistical mechanics, dynamical soft-mode fluctuations, and transport theory to offer a promising analytic approach to the longstanding puzzle of non-Fermi-liquid behavior in correlated electron systems (Ge, 15 Sep 2025).