Representations in Thermodynamic Limits

Two infinite-system states may look like ordinary density matrices on every finite region and still generate disjoint Hilbert-space representations of the full observable algebra. A concrete diagnostic is a spatial average whose strong limit is a different scalar in the two representations. This page proves that mechanism for independent spins; it does not assume a phase transition or identify inequivalence with spontaneous symmetry breaking.

Required background. The GNS construction turns a state into a cyclic representation. Thermodynamic limits of local states supplies the quasi-local algebra. The CCR representation page treats the distinct Shale implementability problem for bosonic canonical transformations.

Product states on one fixed spin algebra

Consider the quasi-local algebra of spin-$1/2$ sites on $\mathbb N$. For $0<p<1$ put

$$\rho_p=p|\uparrow\rangle\langle\uparrow| +(1-p)|\downarrow\rangle\langle\downarrow|.$$

The compatible finite-region product density matrices define a state $\omega_p$. Write its GNS triple as $(\pi_p,\mathcal H_p,\Omega_p)$. Every finite-region density matrix is faithful. In particular, no finite measurement has an outcome with positive probability for one $p$ and exactly zero probability for another $q$ in this open interval.

Let $Z_j$ be $\sigma_z$ at site $j$, and define bounded local averages

$$M_N=\frac1N\sum_{j=1}^N Z_j, \qquad m_p=2p-1.$$

Independence gives

$$\omega_p(M_N)=m_p, \qquad \omega_p((M_N-m_pI)^2) =\frac{1-m_p^2}{N}.$$

Thus the fluctuations vanish in the cyclic vector, but a representation-level conclusion requires convergence on every vector.

The averages converge strongly in each representation

For a local observable $A$ supported on a finite set $S$,

$$\|[M_N,A]\| \leq\frac{2|S|}{N}\|A\|.$$

Indeed, only summands with $j\in S$ can fail to commute with $A$. Therefore

$$\|(\pi_p(M_N)-m_pI)\pi_p(A)\Omega_p\| \leq\|[M_N,A]\| +\|A\| \sqrt{\frac{1-m_p^2}{N}}.$$

The right side tends to zero. Local vectors $\pi_p(A)\Omega_p$ are dense, and $\|\pi_p(M_N)-m_pI\|\leq2$ uniformly. Approximation of an arbitrary vector by local vectors proves

$$\operatorname*{s-lim}_{N\to\infty}\pi_p(M_N)=m_p I.$$

This is a law of large numbers expressed inside the representation. It is stronger than convergence of a single expectation and uses both the variance estimate and asymptotic commutation with local observables.

The convergence is not norm convergence in the abstract algebra. Each $M_N$ has both $+1$ and $-1$ in its spectrum, so

$$\|M_N-m_pI\| =\max\{|1-m_p|,|-1-m_p|\}.$$

The nonzero norm obstruction explains how one algebra sequence can acquire different strong limits after representation.

Different macroscopic limits imply disjointness

Suppose a bounded operator $T:\mathcal H_p\to\mathcal H_q$ intertwines the representations:

$$T\pi_p(A)=\pi_q(A)T \qquad(A\in\mathcal A).$$

Apply this to $M_N$ and take the strong limits. For every vector,

$$m_pT=m_qT.$$

When $p\ne q$, this forces $T=0$. Hence the representations have no unitarily equivalent nonzero subrepresentations: they are disjoint. This proves more than the failure of a unitary mapping one cyclic vector to another.

For comparison, two orthogonal vectors in a single irreducible representation define different states but do not define disjoint representations. Vanishing overlap of selected vectors is therefore not the decisive criterion. Here the same labeled observables force incompatible macroscopic values on every represented vector.

Normal states preserve the macroscopic sector

A state normal in $\pi_p$ extends to a normal functional on $\pi_p(\mathcal A)''$. Uniformly bounded strong convergence of self-adjoint operators implies ultraweak convergence, so every such state $\varphi$ satisfies

$$\varphi(\pi_p(M_N))\longrightarrow m_p.$$

Since $\omega_q(M_N)=m_q$, the state $\omega_q$ cannot be normal in the $p$ representation when $q\ne p$. Normal mixtures and local excitations of the $p$ background cannot change this macroscopic value.

In the disjoint sum representation $\pi_p\oplus\pi_q$, the strong limit is

$$m_pP_p+m_qP_q.$$

The sector projections belong to the von Neumann algebra by spectral calculus of this limit and commute with represented local observables. The mixture $(\omega_p+\omega_q)/2$ can therefore carry a classical sector label in its center. The same abstract quasi-local algebra has acquired different von Neumann closures through different states.

These observations motivate the use of representations and their normal-state families, often called folia, when comparing thermodynamic phases. Establishing phase coexistence additionally requires that the candidate states satisfy the relevant equilibrium condition for the same interaction and temperature. The independent-spin family above does not establish such coexistence.

Exercises

Finite modifications. Replace $\rho_p$ on a fixed finite set of sites by arbitrary density matrices. Show that the limiting average is still $m_p$.

Solution

The changed expectations contribute at most twice the number of modified sites divided by $N$. Their variance contribution also vanishes because only finitely many bounded terms changed. Repeating the local-vector argument yields the same strong limit. Macroscopic density is insensitive to this finite modification.

A mixture retains fluctuations. For $\omega=(\omega_p+\omega_q)/2$, calculate the limiting variance of $M_N$ about its mean.

Solution

The mean is $(m_p+m_q)/2$. The within-product variances vanish, but the variance of the two limiting means remains: $\lim_N\omega((M_N-\omega(M_N)I)^2)=(m_p-m_q)^2/4$. The surviving fluctuation is the uncertainty of the sector label, rather than a failure of the law of large numbers within either component.

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.