Heat Capacity Anomalies and Thermal Crossovers in Finite Su-Schrieffer-Heeger Chains

The gist: The analysis reveals a double-peak anomaly in the heat capacity for non-dimerized configurations, which is identified as a Schottky-type thermal crossover governed by two distinct energy scales, providing insights into the interplay between topology, finite-size effects, and thermal fluctuations in one-dimensional topological systems>

Model and Topological Context

The study investigates the thermodynamic properties of finite Su-Schrieffer-Heeger (SSH) chains in thermal equilibrium at fixed temperature and chemical potential using both canonical and grand canonical ensembles. The SSH model, which describes a dimerized chain with alternating hopping amplitudes, is the quintessential example of a 1D topological insulator, capturing features like quantized Berry phases and topologically protected edge states. The Hamiltonian is defined by intra-cell hopping amplitude v and inter-cell hopping amplitude w, where the relative strength determines the topological phase (trivial when v > w, topological when v < w).

Thermodynamic Analysis and Anomalies

The thermodynamic description is developed using the grand canonical ensemble, yielding a grand canonical potential that allows for the calculation of energy density, particle number density, entropy, and heat capacity. A central finding is a double-peak anomaly in the heat capacity of non-dimerized chains, which consists of two maxima separated by a local minimum. This feature is identified as a Schottky-type thermal crossover between two energy scales rather than a phase transition and is governed by the widening separation of the two energy scales and the growing density of states as hopping asymmetry increases and chain length grows.

Dependence on System Parameters

The anomaly's behavior is highly sensitive to system parameters:

1. Hopping Asymmetry (θ): The two maxima separate further and the intervening minimum deepens as the hopping asymmetry increases.

2. Chain Length (N): The anomaly becomes more pronounced as the chain length grows, reflecting the increasing density of states.

3. Chemical Potential (µ): The double-peak structure is most pronounced when the chemical potential is comparable to the band scale, specifically around µ˜ ∼ 1.

The analysis also shows that while topological properties are determined by boundary states, the bulk thermodynamic behavior exhibits a rich crossover structure tunable through the hopping parameter ratio.

Ensemble Comparison and Interpretation

The study compares results from both ensembles:

(i) Grand Canonical Ensemble (Fixed Chain Length):

In this scenario, the heat capacity is calculated as Cµ = (∂E/∂T ¯)µ − µ(∂N /∂T ¯)µ, where the mean particle number N¯ varies with temperature. The anomaly becomes more pronounced as chain length increases from N = 5 to N = 400, indicating a gradual sharpening of a crossover rather than finite-size scaling toward criticality.

(ii) Canonical Ensemble (Fixed Particle Number):

In this scenario, the anomaly persists but with reduced magnitude compared to the open boundary case, suggesting that particle number fluctuations, which are present in the grand canonical ensemble but suppressed here, play a role in amplifying the thermodynamic signatures during particle exchange.

The double-peak anomaly is interpreted as a two-scale Schottky crossover, where the first maximum sits near kBT ∼ ∆1 and the second near kBT ∼ ∆2, with the minimum in the window ∆1 ≲ kBT ≲ ∆2. The density of states analysis reveals that for intermediate asymmetries, van Hove singularities emerge at both the inner and outer band edges, producing four distinct peaks which corresponds to a double-peak structure.

Experimental Relevance

The identified thermodynamic signatures are experimentally accessible in platforms such as LC circuit arrays, ultracold atoms in optical lattices, and topoelectrical circuits. The control parameter of the hopping asymmetry offers a way to engineer the thermodynamic response and the position of the crossover, potentially useful for thermal management in nanoscale topological devices. The work demonstrates that conventional thermodynamic quantities like heat capacity can also reveal rich crossover structure, albeit one that is distinct from the topological classification and highlights that the anomaly is most pronounced away from the topological critical point.

Conclusion

The research concludes that finite SSH chains possess rich thermodynamic structure beyond their topological classification, characterized by double-peak thermal crossovers that reflect bulk properties encoded in the full density of states. The findings emphasize that the anomaly is a bulk effect, distinct from the topological phase transition at θ = 0, and its growth with system size reflects a crossover sharpening rather than critical scaling. The hopping parameter ratio provides an experimentally accessible control parameter for tuning the system through different thermodynamic phases. The anomaly persists in both ensembles but is enhanced when particle exchange is permitted, confirming that coupling to a particle reservoir amplifies these signatures.

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The double-peak anomaly should also be compared directly with the single-peak CN¯ obtained for the SSH model at µ = 0 in Ref. [25]: the two results are consistent once it is recognized that the second peak requires a chemical potential comparable to the band scale, so the µ = 0 case sits in the singlescale regime where only one peak survives. The anomaly similarly reflects a purely microscopic mechanism—the existence of a finite set of accessible low-energy states separated by distinct gaps—rather than spontaneous symmetry breaking or collective critical behavior. The growth of the anomaly with system size [Fig. 3, panels (a)-(e)] reflects the sharpening of a crossover as the density of states increases, not finite-size scaling toward a critical point.

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The double-peak anomaly should also be compared directly with the single-peak CN¯ obtained for the SSH model at µ = 0 in Ref. [25]: the two results are consistent once it is recognized that the second peak requires a chemical potential comparable to the band scale, so the µ = 0 case sits in the singlescale regime where only one peak survives. The anomaly similarly reflects a purely microscopic mechanism—the existence of a finite set of accessible low-energy states separated by distinct gaps—rather than spontaneous symmetry breaking or collective critical behavior. Finally, the growth of the anomaly with system size [Fig. 3, panels (a)-(e)] reflects the sharpening of a crossover as the density of states increases, not finite-size scaling toward a critical point. The hopping parameter ratio v/w (or equivalently θ) provides an experimentally accessible control parameter for tuning the system through different thermodynamic phases. The anomaly persists in both ensembles but is more pronounced when particle exchange is permitted, indicating that coupling to a particle reservoir enhances thermodynamic signatures.

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In conclusion, our work demonstrates that finite topological systems possess thermodynamic complexity beyond their topological classification. The double-peak thermal crossovers and heat capacity anomalies we have identified represent experimentally accessible signatures that could be observed in quantum simulator platforms. More broadly, our results highlight the importance of considering thermodynamic properties alongside topological invariants for a complete understanding of topological materials, particularly in the finite-size regime relevant to nanoscale devices and quantum technologies. By comparing grand canonical and canonical ensembles, we have demonstrated that particle number fluctuations play an important role in shaping thermodynamic behavior.

1. Improved AI systems can perform real-time thermal management in nanoscale topological devices by predicting the double-peak anomaly in heat capacity for non-dimerized configurations as a function of temperature and chemical potential. This allows for engineering thermal properties by tuning the hopping parameter ratio, which is presented as a control parameter for engineering the thermodynamic response.

2. Improved AI systems can accurately model quantum simulator platforms (like cold atoms or topoelectrical circuits) by predicting the double-peak anomaly in the heat capacity and its dependence on chain length and hopping asymmetry. This capability allows for the design of experimental setups that reveal experimental signatures of bulk properties encoded in the density of states, distinguishing them from topological invariants.

3. Improved AI systems can optimize material design for topological functionality by predicting how the anomaly becomes more pronounced as the chain length increases and grows, reflecting the widening gap between the two energy scales and the increasing density of states. This provides a predictive tool for understanding how bulk thermodynamic behavior can be tuned through the hopping parameter ratio.

4. Improved AI systems can differentiate between topological classification and thermodynamic response during material characterization by analyzing data where the anomaly is most pronounced when the chemical potential is comparable to the band scale, thereby providing a clear distinction that topological classification and the thermodynamic behavior are distinct.