Correlation length

Consider an equilibrium state (for instance the canonical ensemble ) on a $d$-dimensional system such as a $\mathbb{Z}^d$ lattice or a continuum. For two (typically local) observables $A_x$ and $B_0$, the two-point correlation function is

where $\langle\cdot\rangle$ denotes an ensemble average . The physically meaningful quantity for “independent fluctuations at long distance” is the connected correlation function

Exponential correlation length

If the state is translation invariant and correlations cluster, one often observes an exponential envelope

in the thermodynamic limit . When the limit exists, the (exponential) correlation length $\xi\in(0,\infty]$ can be defined by

Equivalently, one may define an upper correlation length via a bound of the form

• $\xi<\infty$ indicates short-ranged correlations (typical of disordered/gapped phases).
• $\xi=\infty$ indicates long-ranged or critical correlations (e.g. algebraic decay at a critical point).

Second-moment correlation length

A widely used alternative is the second-moment correlation length, which is well-defined when $G_c$ is nonnegative and summable (common in ferromagnets above criticality):

This version is especially convenient numerically and in finite systems.

Link to susceptibility and integrated correlations

For many order parameters, the susceptibility is an integral/sum of connected correlations, so a growing $\xi$ typically produces a growing response. Concretely, when an extensive observable is a spatial sum of local fields, the integrated $G_c$ controls the linear response and becomes large when the decay length $\xi$ becomes large.

Transfer-matrix interpretation in 1D

In one-dimensional models constructed via the transfer matrix , connected correlations often behave like a ratio of eigenvalues:

so the exponential correlation length is

Here $\lambda_0$ is the leading eigenvalue and $\lambda_1$ the subleading eigenvalue controlling the slowest decay mode.