Transfer-matrix construction in 1D

Rewrite 1D Boltzmann weights as powers of a matrix to compute partition functions, free energies, and correlations.

Transfer-matrix construction in 1D

For one-dimensional, nearest-neighbor lattice models, the transfer matrix turns the sum over all microstates into linear algebra. It provides an exact route to the canonical partition function , the free energy , and decay of two-point correlations in 1D.

Nearest-neighbor form and the transfer matrix

Consider a 1D chain of length $N$ with spins $\sigma_i\in S$ and a nearest-neighbor energy of the form

H(\sigma_1,\dots,\sigma_N)=\sum_{i=1}^{N} U(\sigma_i,\sigma_{i+1}), \qquad \sigma_{N+1}=\sigma_1

(periodic boundary conditions). Fix inverse temperature $\beta$ .

Define the transfer matrix $T$ (a matrix indexed by $S$ ) by

T_{ab}=\exp\!\big(-\beta\,U(a,b)\big).

More generally, if the Hamiltonian includes on-site terms $V(\sigma_i)$ , one often splits them symmetrically into neighboring factors so that $T_{ab}=\exp(-\beta[U(a,b)+\tfrac12 V(a)+\tfrac12 V(b)])$ ; this keeps the representation exact and symmetric.

Partition function as a trace

The periodic-chain partition function is

Z_N=\sum_{\sigma_1,\dots,\sigma_N}\exp\!\big(-\beta H(\sigma_1,\dots,\sigma_N)\big) =\mathrm{Tr}\big(T^N\big),

where $\mathrm{Tr}$ is the trace . (For open boundary conditions, one obtains a bilinear form $Z_N=\langle \ell, T^{N-1} r\rangle$ with boundary vectors encoding the end weights.)

This representation is a concrete instance of “compute thermodynamics from $\log Z$ ,” as in extracting observables from $\log Z$ .

Thermodynamic limit and dominant eigenvalue

Let the eigenvalues of $T$ satisfy $|\lambda_0|\ge |\lambda_1|\ge\cdots$ . Under mild positivity assumptions (typical for physical transfer matrices), $\lambda_0>0$ is the unique largest eigenvalue. Then, in the thermodynamic limit $N\to\infty$ ,

\frac{1}{N}\log Z_N\longrightarrow \log \lambda_0, \qquad f = -\frac{1}{\beta}\log \lambda_0,

where $f$ is the free energy per site (a 1D analogue of the thermodynamic density encoded by statistical free energy ).

Correlations and correlation length

Transfer matrices also control spatial correlations. For suitable local observables $A$ and $B$ at separation $r$ , one typically finds asymptotic decay

\langle A_0 B_r\rangle - \langle A_0\rangle\langle B_r\rangle \;\propto\; \left(\frac{\lambda_1}{\lambda_0}\right)^r \quad\text{as } r\to\infty,

so the associated connected correlation decays exponentially. The corresponding correlation length is

\xi^{-1} = -\log\left|\frac{\lambda_1}{\lambda_0}\right|.

Physical interpretation

The transfer matrix is a “propagator” that advances the system by one lattice step. Because $Z_N$ is governed by a largest eigenvalue, thermodynamic quantities in 1D with finite-range interactions are typically analytic in parameters for $T$ with strictly positive entries—one reason phase transitions are absent in many 1D short-range models.