Locality of temperature (1309.0816v4)

Abstract: This work is concerned with thermal quantum states of Hamiltonians on spin and fermionic lattice systems with short range interactions. We provide results leading to a local definition of temperature, thereby extending the notion of "intensivity of temperature" to interacting quantum models. More precisely, we derive a perturbation formula for thermal states. The influence of the perturbation is exactly given in terms of a generalized covariance. For this covariance, we prove exponential clustering of correlations above a universal critical temperature that upper bounds physical critical temperatures such as the Curie temperature. As a corollary, we obtain that above the critical temperature, thermal states are stable against distant Hamiltonian perturbations. Moreover, our results imply that above the critical temperature, local expectation values can be approximated efficiently in the error and the system size.

• The paper introduces a perturbation formula that captures the response of thermal states to Hamiltonian perturbations via generalized covariance.
• It demonstrates exponential clustering of correlations above a universal critical temperature, ensuring thermal stability in quantum lattice systems.
• Numerical results using MPO approximations confirm that local expectation values efficiently scale with system size at small inverse temperatures.

Locality of Temperature in Quantum Lattice Systems

This paper offers a detailed examination of thermal quantum states of Hamiltonians on spin and fermionic lattice systems characterized by short-range interactions. It provides significant insights into the local definition of temperature in such systems. The authors extend the notion of "intensivity of temperature," traditionally perceived as a macroscopic attribute, into the domain of interacting quantum models.

Key Contributions

The central contribution of the paper is the derivation of a perturbation formula for thermal states. This formula characterizes the response of thermal states to perturbations in terms of a generalized covariance, providing a robust framework for exploring temperature localization at the quantum level. One of the strong results proved is the exponential clustering of correlations beyond a universal critical temperature, which serves as an upper bound on physical critical temperatures such as the Curie temperature. This result is crucial as it implies the stability of thermal states against distant Hamiltonian perturbations above the critical temperature.

Numerical Results and Claims

The paper establishes a universal critical inverse temperature $\beta^\ast$ and thermal correlation length $\xi(\beta)$. Notably, it demonstrates that when the inverse temperature $|\beta|$ is below $\beta^\ast$, correlations decay exponentially with distance, quantified by the correlation length $\xi(\beta)$. The scaling with system size and approximation error achieved by matrix product operator (MPO) approximations for thermal states at small enough $\beta$ are paradigmatic.

Implications

Practical Implications

The results have several practical implications for quantum systems. Above the universal critical temperature, thermal states become stable with respect to local perturbations, meaning that system properties can be predicted and managed more effectively in high-temperature regimes.

For computational quantum physics, the efficient approximation of local expectation values, independent of system size, represents a significant advancement. Particularly, for quantum Monte Carlo simulations, the results guide the determination of sufficient system sizes for correct partition function sampling and the identification of observables indicating long-range correlations.

Theoretical Implications

Theoretically, the paper enhances the understanding of thermal quantum states, providing a unifying view of high-temperature phases where exponential clustering of correlations is expected. This view aligns with known results about the clustering of correlations in classical systems and quantum gases and expands them into the quantum lattice systems framework.

The locality of temperature problem is conclusively addressed through the equivalence of temperature intensivity and correlation decay across length scales. This result could help redefine perspectives on how temperature can be assigned to small subsystems in quantum statistical mechanics.

Future Directions

The implications of this work suggest several directions for future research. One area of interest is the refinement of the presented methods to achieve tighter critical temperature bounds by incorporating specific model properties. Additionally, exploring the connections between these results and thermalization behaviors in closed quantum systems presents intriguing possibilities for understanding equilibration dynamics and eigenstate thermalization in complex quantum systems.

Overall, while the paper's results primarily pertain to theoretical foundations, they lay the groundwork for substantial advancements in both our theoretical understanding and practical capabilities in quantum lattice systems and quantum informatics.