Motional Jahn-Teller Distortion in Correlated Systems

• Motional Jahn-Teller Distortion is a dynamic lattice distortion driven by electron-phonon couplings that lifts orbital degeneracy in correlated systems.
• It couples electronic, magnetic, orbital, and lattice degrees of freedom, resulting in competing phases such as ferromagnetism and structural distortions.
• Theoretical models like the Hubbard-Jahn-Teller framework with a spectral density approach reveal threshold behaviors and complex temperature-dependent phase transitions.

Motional Jahn-Teller Distortion refers to the dynamic interplay between electronic degeneracy and symmetry-lowering nuclear displacements driven by vibronic couplings, resulting in a time-dependent or fluctuating structural distortion rather than a statically ordered symmetry breaking. In strongly correlated electron systems, especially those with orbital degeneracies, this phenomenon introduces complex coupling between lattice, orbital, magnetic, and electronic degrees of freedom, often leading to nontrivial coexistence or competition between magnetic order and structural distortions. The term “motional” (sometimes also referred to as “dynamic” Jahn-Teller distortion) specifically denotes situations in which the system either tunnels between multiple symmetry-equivalent minima on the potential energy surface or, as in solid-state applications, where the order-parameter associated with the lattice distortion develops a nontrivial temperature or field-dependent behavior rather than freezing into a static configuration.

1. Theoretical Model and Formalism

The minimal model encapsulating motional Jahn-Teller distortion in correlated systems is the two-fold degenerate Hubbard model with explicit Jahn-Teller (J-T) coupling. The Hamiltonian contains three key contributions,

$H = H_s + H_{JT} + H_L$

where:

• $H_s$ is the Hubbard term for two bands (label α), with local Coulomb repulsion $U$:

$$H_s = \sum_{\alpha, ij, \sigma} (T_{ij} - \mu \delta_{ij}) c^{\dagger}_{\alpha i \sigma} c_{\alpha j \sigma} + \frac{1}{2}U \sum_{\alpha,i,\sigma} n_{\alpha i \sigma} n_{\alpha i -\sigma}$$

• $H_{JT}$ is the Jahn-Teller electron-lattice coupling,

$$H_{JT} = G e \sum_{i,\sigma} (n_{1i\sigma} - n_{2i\sigma}),$$

with $G$ the JT coupling constant and $e$ the lattice strain, self-consistently determined via

$$e = \frac{G}{N C_0} \sum_{i,\sigma} (\langle n_{1i\sigma} \rangle - \langle n_{2i\sigma} \rangle),$$

and $C_0$ the elastic constant.

• $H_L$ is an elastic lattice term.

The electronic correlations are treated via the spectral density approach (SDA), which uses a two-peak ansatz for the single-particle spectral density (lower and upper Hubbard bands), and the self-energy is given by: $$\Sigma_{\alpha \sigma}(E) = U n_{\alpha -\sigma} + \frac{U^2 n_{\alpha -\sigma}(1 - n_{\alpha -\sigma})}{E + \mu - B_{\alpha -\sigma} - U(1 - n_{\alpha -\sigma})},$$ with $B_{\alpha \sigma}$ the spin-dependent band shift. Green's function formalism yields the density of states and sub-band occupations.

The JT interaction splits the degenerate bands through

$$\epsilon_{\alpha}(\mathbf{k}) = \epsilon(\mathbf{k}) + (-1)^{\alpha} G e,$$

so the onset of a nonzero $e$ (and hence motional J-T distortion) occurs only for $G > G_c$. For $G < G_c$, the system remains undistorted.

2. Coupling Between Jahn-Teller Distortion and Magnetic Order

A central insight is the mutual competition and coexistence between magnetic (ferromagnetic) order and structural (J-T) distortion. In the absence of $G$, for sufficiently large $U$ and appropriate band filling $n$, a spontaneous spin polarization (magnetization) emerges equally in both bands: $$m_{\alpha} = \langle n_{\alpha \uparrow} \rangle - \langle n_{\alpha \downarrow} \rangle.$$ As the Jahn-Teller coupling $G$ increases past $G_c$, the system undergoes a simultaneous phase transition characterized by:

• The development of a static lattice strain $e \neq 0$,
• An abrupt differentiation between bands: one band (e.g., band 1) becomes predominantly occupied and magnetized ($m_1 \neq 0$), while the other loses both occupation and magnetization ($m_2 \approx 0$).

This is a hallmark of motional J-T distortion in a correlated context: the order parameter for structural distortion (e.g., $e$) and that for magnetic order (e.g., $m_\alpha$) interact nontrivially, and their interplay determines the character of the ground and excited states.

The following table summarizes the regime structure at $T = 0$:

Regime	Spontaneous Magnetization	Lattice Strain (e)	Band Population
$G < G_c$	Yes (both $m_1, m_2 \neq 0$)	No ($e = 0$)	Equal
$G > G_c$	Asymmetric ($m_1 \neq 0, m_2 \approx 0$)	Yes ($e \neq 0$)	Unequal, band 1 dominant

The onset of strain and associated redistribution in the band occupations are continuous past the threshold $G_c$; the physical origin is the energetic competition between minimizing Coulomb exchange via parallel spin alignment and minimizing the JT energy by breaking orbital degeneracy via lattice distortion.

3. Critical Parameters Governing Motional Jahn-Teller Distortion

The existence and interplay of motional J-T distortion and magnetic order depend sharply on three main parameters:

• Electron correlation strength $U$: A threshold value is required for magnetic order; for instance, $U = 3$ yields $T_C \approx 290\,\text{K}$ at $n = 1.4$ and $G = 0$.
• J-T coupling strength $G$: Spontaneous JT distortion requires $G > G_c$. Numerically, $G_c \approx 0.462$ for $n = 1.4$, $U = 3$.
• Band occupation $n$: The phase diagram is sensitive to filling; only within certain $n$ does robust ferromagnetism and/or JT distortion develop. The electronic redistribution and hence lattice strain are dependent on $n$.

Only for combinations of $(U, G, n)$ in a suitable window is the coexistence phase (ferromagnetic and distorted) accessible. For example, if the electron-phonon coupling is marginal ($G \approx G_c$), the system may show weak distortion and unusual thermal response.

4. Temperature Dependence and Phase Transitions

The temperature evolution is a decisive diagnostic of motional JT distortion in correlated systems:

• For $G > G_c$, raising $T$ from zero gradually decreases both strain $e$ and the dominant magnetization ($m_1$).
• At the Curie temperature $T_C$ (magnetization collapse), a discontinuous increase in strain is seen, corresponding to an abrupt reordering of electronic states upon loss of magnetic order; above $T_C$, $m_\alpha = 0$.
• Strain only vanishes at a higher temperature $T_s$, a second phase boundary marking the restoration of the undistorted (symmetric) phase.
• For $G \approx G_c$, non-monotonic transitions (thermal enhancement of magnetization, abrupt jumps in $e$) occur over finite $T$ intervals, reflecting complex entropy-driven reoccupation and reordering.
• For $G < G_c$, strain is absent in the magnetically ordered phase and appears only above $T_C$, indicative of a first-order-like transition boundary in the regime near the JT threshold.

This intricate thermally driven evolution of order parameters demonstrates the richness of the motional JT distortion, particularly in the proximity of phase boundaries.

5. Implications for Correlated Materials and Functional Applications

Motional Jahn-Teller distortion, especially when entangled with itinerant magnetism, is relevant for a broad class of transition metal systems including:

• Manganites (e.g., $\text{La}_{1-x}\text{Ca}_x\text{MnO}_3$): The suppression of magnetic order by JT coupling and the emergence of complex orbital/lattice-magnetic phases underlie the phenomena of colossal magnetoresistance (CMR).
• Heusler Alloys: Experimental observations of coexisting magnetism and JT distortion align with the model's phase diagram.
• Orbitronics and Multiferroics: The possibility to tailor both orbital (structural) and magnetic order by tuning $G, U, n$ offers design strategies for materials with coupled order parameters and tunable phase transitions, e.g., for spintronics or functional device applications.

Moreover, the spectral density approach and its embedding of both electronic and lattice degrees of freedom provide a platform for extending such studies to more complex situations (multi-orbital, non-trivial lattice topologies, or disorder).

6. Methodological Significance and Outlook

The description and calculation of motional Jahn-Teller distortion in the presence of strong electron correlations benefits from techniques capable of self-consistently incorporating both band splitting (by JT instability) and magnetic order, as done using the spectral density approach. This dual treatment:

• Captures the critical nature of JT coupling (threshold behavior);
• Uncovers the nontrivial feedback of magnetic and structural order parameters;
• Predicts observable (and observed) discontinuities in thermodynamic and transport behavior near phase boundaries.

These methodological features can be systematically generalized to other strongly correlated models where competing interactions drive complex intertwined phases, as in systems at the frontier of correlated electron physics and materials design.

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In summary, motional Jahn-Teller distortion in correlated electron systems is characterized by the competition and coexistence of lattice strain (lifting orbital degeneracy) and spontaneous spin polarization, controlled by the interplay of electron-electron and electron-phonon couplings and sensitive to band filling. Its thermodynamic signatures, parameter dependence, and implications for functional materials are quantitatively captured by correlated Hubbard-JT models solved at the level of the spectral density approach, laying a foundational framework for understanding and engineering quantum cooperative phenomena in solids (Haritha et al., 2010).