The correlation density matrix: new tool for analyzing exact diagonalizations

When a lattice model of strongly interacting spins or fermions is studied numerically, it may be unclear {\em a priori} what kind of correlation will be dominant -- especially in cases where exotic order or disorder has been suggested (e.g. Kagom\'e spin-1/2 antiferromagnet, or doped square-lattice Hubbard model). To resolve this, consider two small clusters of a few sites $A$ and $B$, offset by a vector $\bf r$. Let $\hat{\rho}_{AB}$ be the many-body density matrix for the disconnected cluster $A\cup B$, constructed from the whole system's ground state wavefunction by tracing out all sites except those in $A$ or $B$, with $\hat{\rho}_A$ and $\hat{\rho}_B$ similarly defined. Then all possible correlations between $A$ and $B$ are contained in the ``correlation density matrix'' $\hat{\rho}_{\rm corr}(\bf r) \equiv \hat{\rho}_{AB} -\hat{\rho}_A \otimes \hat{\rho}_B$. Using singular-value decomposition we can write $\hat{\rho}_{\rm corr} = \sum _i \lambda_i \hat{\Phi}_i(A) \hat{\Phi}'_i(B)$, where $\hat{\Phi}_i$ and $\hat{\Phi}'_i$ are normalized operators on the respective clusters; the terms represent different correlation functions, which are naturally ordered by the magnitudes $|\lambda_i|$. This permits a systematic, unbiased numerical method to identify the important correlations, given the ground state wavefunction. The procedure will be tested on ladders of spinless fermions with infinite nearest-neighbor repulsion, which are expected to have Luttinger liquid behavior.