Thermodynamic Limits of Local States

A thermodynamic-limit state can exist even when there is no limiting density matrix on a chosen infinite tensor-product Hilbert space. For a countable lattice with finite-dimensional sites, finite-volume states always have subsequences converging on every local observable. This page proves that compactness statement and separates it from uniqueness, convergence of dynamics, and convergence of thermodynamic functions. The physical thermodynamic-limit treatment explains the choices of density, shape, boundary conditions, and intensive quantities.

Required background. Algebraic states supplies positive normalized functionals. Infinite tensor products explains why choosing one reference-vector Hilbert space is additional structure.

Local algebras provide a common comparison space

Let $\Gamma$ be a countable site set, with $\mathcal H_x=\mathbb C^{d_x}$ and $d_x<\infty$. For finite $\Lambda\subset\Gamma$ define

$$\mathcal A_\Lambda =\mathcal B\!\left(\bigotimes_{x\in\Lambda}\mathcal H_x\right).$$

The embedding for $\Lambda\subset\Lambda'$ is $A\mapsto A\otimes I_{\Lambda'\setminus\Lambda}$. The quasi-local C*-algebra is

$$\mathcal A=\overline{\bigcup_{\Lambda\Subset\Gamma} \mathcal A_\Lambda}^{\,\|\cdot\|}.$$

Choose nested finite volumes $\Lambda_n$ exhausting $\Gamma$. A density matrix $\rho_n$ on $\Lambda_n$ defines expectations of every observable whose support lies inside $\Lambda_n$. To regard it as a state on all of $\mathcal A$, choose any state extension $\widetilde\omega_n$. One concrete choice tensors $\rho_n$ with fixed product states outside $\Lambda_n$.

The eventual local limit is independent of this extension choice. For a fixed local $A$, all sufficiently large volumes contain its support, and every extension gives the same value $\operatorname{Tr}(\rho_n A)$. Extensions let us compare states on one algebra without imposing physical conditions at infinity.

Subsequence compactness has an elementary proof

For each fixed finite $\Lambda_m$, the restrictions of $\widetilde\omega_n$ are density matrices in a finite-dimensional compact set. Extract a subsequence convergent on $\mathcal A_{\Lambda_1}$, then one convergent on $\mathcal A_{\Lambda_2}$, and continue. The diagonal subsequence converges on every finite-volume algebra.

The limits are consistent under restriction, because partial trace is continuous in finite dimension. They define a positive normalized functional $\omega$ on the local union. Its bound

$$|\omega(A)|\leq\|A\|$$

extends it uniquely to a state on $\mathcal A$.

Convergence also extends from local observables to every quasi-local observable. Given $A\in\mathcal A$ and a local approximation $A_{\mathrm{loc}}$,

$$|\widetilde\omega_{n_j}(A)-\omega(A)| \leq2\|A-A_{\mathrm{loc}}\| + |\widetilde\omega_{n_j}(A_{\mathrm{loc}}) -\omega(A_{\mathrm{loc}})|.$$

First take the subsequence limit, then improve the approximation. This is weak-star convergence in the algebraic state space. Countability and finite-dimensional sites make the algebra separable, so sequences suffice here. In a general nonseparable C*-algebra the compactness theorem is naturally stated using subnets.

No Hamiltonian, locality estimate, translation invariance, or equilibrium assumption entered this proof.

Compactness does not imply a unique thermodynamic state

An alternating sequence of product states, all spins up for even $n$ and all spins down for odd $n$, has two distinct limit points. The one-site observable $\sigma_z^{(0)}$ already distinguishes them. Compactness guarantees extraction, not convergence of the entire sequence.

Likewise, different boundary conditions for an interacting model can select different limit states. Proving this requires analysis of that model; the existence of two arbitrary sequences is not a proof of a phase transition. To show convergence without passing to subsequences, one can prove that every cluster point is the same state. Compactness then forces convergence.

For independent spins with single-site Hamiltonian $h$, the finite-volume Gibbs state is

$$\rho_\Lambda= \bigotimes_{x\in\Lambda} \frac{e^{-\beta h}}{\operatorname{Tr}e^{-\beta h}}.$$

Its restriction to any fixed finite region is exactly constant once that region is included. The limiting state therefore exists directly, independently of the volume shape. A global infinite product of partition functions is neither required nor generally finite.

Dynamics requires a uniform locality estimate

Suppose now $\Gamma=\mathbb Z^d$, the on-site dimension is uniformly finite, and the interaction is uniformly bounded and finite range. The finite-volume Hamiltonians define

$$\alpha_t^\Lambda(A)=e^{itH_\Lambda/\hbar}Ae^{-itH_\Lambda/\hbar}.$$

The Lieb–Robinson theorem gives the volume-uniform commutator estimate used to prove

$$\sup_{|t|\leq T} \|\alpha_t^{\Lambda_n}(A)-\alpha_t(A)\| \longrightarrow0$$

for each local $A$ and finite $T$. The limiting maps extend to a point-norm continuous automorphism group on $\mathcal A$. The proof and the broader decaying-interaction hypotheses are developed in that article.

This norm limit of observables is a separate theorem from state compactness. It also does not imply that $H_{\Lambda_n}$ converges in norm: its norm typically grows with volume. An infinite Hamiltonian may be represented by a generator in a chosen GNS Hilbert space, while the algebra dynamics is already well-defined independently of that choice.

When states and dynamics both vary, one must control both limits to preserve equilibrium. KMS equilibrium limits supplies that argument. A pressure limit is another statement again: local-state compactness alone does not estimate the extensive quantity $\log\operatorname{Tr}e^{-\beta H_\Lambda}$.

Exercises

Why local trace norm is sufficient. Suppose every fixed-region reduced density matrix converges in trace norm. Prove convergence on every quasi-local observable.

Solution

For local $A$ supported in $\Lambda$, trace duality bounds the expectation error by $\|\rho_{n,\Lambda}-\rho_\Lambda\|_1\|A\|$. This tends to zero. The two-approximation estimate above extends convergence to the norm completion. It asserts no trace-norm convergence of full density matrices on increasing Hilbert spaces.

A local perturbation and a macroscopic one. Compare changing one spin at the origin with changing the state of every spin.

Solution

Both can alter a local limit: a one-site observable at the origin detects the first modification. Spatially averaged observables, however, suppress a single changed spin by the inverse volume, while a uniform change can alter the limiting average by order one. Local convergence and convergence of averaged observables ask different questions; their relation requires additional uniform estimates.

References

• O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States, Models in Quantum Statistical Mechanics, 2nd ed., Springer, 1997, doi:10.1007/978-3-662-03444-6.
• B. Nachtergaele, R. Sims, and A. Young, “Quasi-locality bounds for quantum lattice systems. Part I. Lieb–Robinson bounds, quasi-local maps, and spectral flow automorphisms,” Journal of Mathematical Physics 60, 061101, 2019, doi:10.1063/1.5095769, author text.