Locality of Temperature

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.

Popular Summary

An accurate measurement of temperature is crucial in numerous physics experiments and technological applications. With the relentless miniaturization of devices, the challenge of accurate thermometry is being extended into the nanoscale but it is not even clear whether temperature is a meaningful concept at such scales in the first place. We examine the properties of systems in thermal equilibrium and essentially solve the problem of how to assign a local temperature to a small subsystem of very general globally thermal quantum lattice systems at a sufficiently high temperature.

As a first result, we show that a consistent, local, and intensive definition of temperature is possible if and only if certain correlations in a system are sufficiently short ranged. In a second step, we prove a universal upper bound on the spatial decay of these correlations, which works whenever the global temperature is above a critical value that depends only on local properties of the system. Our results show that above this value, no phase transition involving long-range order is possible. This finding implies a universal upper bound on physically relevant critical temperatures such as the Curie temperature, which is remarkable since pinning down critical temperatures is a notoriously difficult problem, and analytical results only exist for a handful of special cases. In addition, our results show that at temperatures above the critical value, thermal states are locally stable against distant perturbations of the Hamiltonian, and that expectation values of local observables can be approximated efficiently, even in the thermodynamic limit.

Our mathematical tools allow one to exploit the Hamiltonian’s locality structure for the investigation of thermal states and open up new perspectives in the study of quantum systems in thermal equilibrium.

Figure 1

The locality-of-temperature problem: Subsystems of thermal states are themselves, in general, not in a state with a locally well-defined temperature. Down to which length scale can temperature be an intensive quantity?

Figure 2

A 2D square lattice: The boxes indicate subsystems $S\subset B\subset V$. The edges in $S$ are $E(S)$, boundary edges of $B$ are $\partial B$, and $F$ is a shortest path connecting $S$ and $\partial B$; hence, $d(S,\partial B)=|F|=2$. The set of edges $E(S)$ is an example for an animal of size $|E(S)|=7$, while $\partial B$ is not connected and hence not an animal.

Figure 3

The truncation from Corollary 2 and Implication 3: For $\beta <{\beta }^{*}$ and $d(S,\partial B)\ll \xi (\beta )$, Corollary 2 implies that $g^S(\beta )$, depicted on the left, and $g_{\upharpoonright B}^S(\beta )$, depicted on the right, are approximately equal.

Figure 4

One can obtain slightly tighter error bounds in Corollaries 2 and 5 by directly using Eq. (47). The plot shows this bound on the approximation error ${\Vert g^S(\beta )-g_{\upharpoonright B}^S(\beta )\Vert }_1$ for the case of $S$ being a single site on a 2D square lattice as a function of the inverse temperature $\beta$ in units of the critical temperature and the width of the buffer region $L$. This bound can be seen as an imaginary-time Lieb-Robinson “cone” with diverging width as $\beta \to {\beta }^{*}$.

Figure 5

A 2D square lattice. Three different subalphabets are indicated: Words that contain all letters in those alphabets are members of different sets ${\mathcal{C}}_{\ge L}(F)$.

Figure 6

The “multiple swap trick”: Eq. (40) as a tensor network.