Occupancy Driven Phase Transitions in Gapped Systems

The gist: Occupancy-driven instabilities arise when interactions between thermal excitations cause their number to affect the size of an energy gap, leading to a transition between two states marked by a divergence in specific heat, even without an order parameter.

Mechanism of Instability

The paper demonstrates that once a critical number of excitations is reached, the excitation population becomes thermodynamically unstable and the system transitions between two states When the energy of an excitation is affected by the presence of other excitations, a point can be reached where the population of excitations becomes unstable against their number Bloch showed that in gapless magnets, the magnon population becomes unstable at a temperature close to the transition temperature J¨ackle demonstrated that a gapped system (superfluid helium) can exhibit a similar breakdown when excitations can form bound pairs, although the required binding energy was substantially larger than experimentally observed Here, the authors show that occupancy-driven instabilities are a generic consequence of interactions between excitations and do not require boundstate formation. In particular, they show that damping arising from Landau–Khalatnikov-type interactions can itself drive such an instability, which they identify as a fold instability where the free energy minimum and maximum solutions merge.

Mathematical Framework for Two-Level Systems

The analysis begins with a two-level system characterized by a temperature dependent energy gap ∆(T) The thermal occupancy n(T) of the upper level is given by n(T) = 1 / (1 + exp[∆(T)/(kBT)]). When the gap becomes a function of occupancy, ∆(n), they invert equation (1) to obtain t(n) = δ(n) ln[(1 − n)/n]. The energy shift βn(T) incorporates the Hartree term as well as the energy shift due to scattering by thermal excitations. They study a specific gap form δ(n) = p [1 + βn(T)] squared − [Γ(T)/∆0] squared with Γ(T) = αn(T). Setting β=0 and studying t(n) leads to different behaviors depending on whether α is less than, equal to, or greater than 2. For α > 2, the system becomes unstable before the gap closes at n = 1/α at a temperature given by dt(n)/dn = 0, which corresponds to a maximum of t(n).

Free Energy and Critical Points

The free energy is defined as F(n, T) = E(n) − T S(n), where the equilibrium condition ∂F/∂n = 0 is consistent with equation (1). For the gap form ∆(n) = ∆0 √[1 − α squared n 2], they obtain F(n, T) which they differentiate to yield equation (5). The equilibrium condition ∂F/∂n = 0 yields equation (3). A stable equilibrium state requires ∂2F/∂n2 > 0, and the endpoint of the low-temperature branch is given by the condition 1 = α / (2n squared c) (1 − nc) ln [1 − nc] / nc. This condition is identical to the one derived from dT/dn = 0, meaning that dT/dn = 0 implies ∂2F/∂n2 = 0.

Divergence of Specific Heat

For α > 2, this critical point is accompanied by a divergence of the specific heat. The specific heat is given by C = T dS/dT = T dS/dn dn/dT, and differentiating F along the equilibrium branch Fn(n, T) = 0 yields C = (T k squared / B) ln[1 − n / n 2] Fnn. The specific-heat singularity occurs because the curvature of the free-energy minimum vanishes, Fnn → 0 +. At the critical point, Fnn= 0, so that the leading free energy terms in Tc − T and nc − n are δF/δn = 0 ≈ FnT,c(T − Tc) + 1/2 Fnnn,c(n − nc) squared. This leads to the singular behavior of the specific heat C(T) ∝ (Tc − T) -1/2 and d∆/dT ∝ -(Tc − T) -1/2.

Implications for Real Systems

The two-level model establishes that an occupancy-dependent damping rate can destabilize a gapped excitation spectrum without requiring the gap to close. The authors argue that this mechanism is relevant to the superfluid transition in 4He at Tλ and the hidden-order transition (HO) in URu2Si2 at THO. Liquid 4He undergoes a second order phase transition into a superfluid state at 2.172 K This transition is marked by the characteristic λ-shaped curve of the specific heat, and below Tλ the ground state supports a Bose condensate. The damping mechanism goes by the name of Landau–Khalatnikov (LK) damping Upon raising the temperature close to Tλ, the roton excitation becomes very poorly defined, decaying back to equilibrium before it has propagated by one wavelength.

Experimental Comparison and Conclusion

The scaling of temperature by the relevant energy gap shows that both the entropy and damping rates of excitations have an identical temperature evolution, strongly suggesting that the transition in URu2Si2 is also driven by thermal occupancy. The authors conclude that in 4He, the occupancy drives the transition from superfluid to normal fluid. They expect occupancy-driven instabilities to be widespread in both gapped and gapless systems, whether the feedback arises from energy renormalization through the real part of the self-energy Σ or from damping through its imaginary part. In detail, whenever the self-consistency condition can be written as n = H(n, T) (with H containing all possible feedback mechanisms), a generic fold instability occurs when ∂H/∂n = 1. They demonstrate that feedback between thermal excitation populations on the one hand and both damping and energy shift on the other hand can itself destabilize a phase. The results show that the transition is occupancy driven with the LK-mechanism determining the transition temperature.

References

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[2] M. Bloch, J. Appl. Phys. 34, 1151 (1963)

[3] J. J¨ackle, Z. Phys. 257, 14 (1972)

[4] J. Ruvalds, Phys. Rev. Lett. 27, 1769 (1971)

[5] P. L. Kapitza, Nature 141, 74 (1938)

[6] F. London, Nature 141, 643 (1938)

[7] P. C. Hohenberg and P. C. Martin, Ann. Phys.(N.Y.) 34, 291 (1965) <ref:2610.

1. Occupancy Driven Instability Detection: The improved system can detect thermodynamic instabilities in gapped systems by monitoring the divergence of specific heat without requiring a traditional order parameter or full gap closure, as described by This transition is marked by a divergence in the specific heat, which does not have to be accompanied by an order parameter, nor by the full closure of the gap.

2. Hidden Order Transition Mapping: The AI can map phenomena like URu2Si2's hidden order transition to occupancy-driven instabilities in gapped systems where the hidden-order transition in URu2Si2 is an example of such an occupancy driven transition in the absence of an order parameter.

3. Damping Mechanism Analysis: The system can analyze how damping arising from Landau–Khalatnikov-type interactions can itself drive such an instability, specifically identifying this as a fold instability where the free energy minimum and maximum solutions merge.

4. Critical Population Estimation: The AI can calculate the critical excitation population nc for a given system by finding the point where the roton density equals the fluid density, which serves as an independent check for the fold instability.

Systems characterized by an energy gap between the ground state and excited states support a growing number of thermal excitations upon raising the temperature. Provided these excitations interact with each other, their number affects the size of the energy gap akin to a harmonic oscillator where the oscillation frequency is damping dependent. Here we show that once a critical number of excitations is reached, the excitation population becomes thermodynamically unstable and the system transitions between two states. This transition is marked by a divergence in the specific heat, which does not have to be accompanied by an order parameter, nor by the full closure of the gap. We illustrate the thermodynamics using a simple two level system and superfluid 4 He as examples, and we argue that the hidden order transition in URu 2 Si 2 is an example of such an occupancy driven transition in the absence of an order parameter.