Large Finite Systems

M. P. Nightingale

Hidetsugu Kitatani, Takehiko Oguchi

A new method is presented which can deal with large finite quantum spin systems (which are so large that direct diagonalization is not possible) by calculating exactly Trotter formula for each finite Trotter number n , and extrapolating them to infinite n . This method is powerful for calculating spin systems which contain frustration. A simple example is also discussed.

M. P. Nightingale

Yu-Chen Cheng, Wenning Wang, Zhiyue Lu +1 more

The theory of probability shows that, as the fraction $X_n/Y\to 0$, the conditional probability for $X_n$, given $X_n+Y \in h_δ:=[h, h+δ]$, has a limit law $f_{X_n}(x)e^{-ψ_n(h_δ)x}$, where $ψ_n(h_δ) $ equals to $[\partial \ln P(Y \in y_δ)/\partial y]_{y=h}$ plus an additional term, contributed from the correlation between $X_n$ and bath $Y$. By applying this limit law to an isolated composite system consisting of two strongly coupled parts, a system of interest and a large but finite bath, we derive the generalized Boltzmann distribution law for the system of interest in the exponential form of a redefined Hamiltonian and corrected Boltzmann temperature that reflects the modification due to strong system-bath coupling and the large but finite bath.

Chen, Xiaosong S., V. Dohm

We study the large-distance behavior of the bulk order-parameter correlation function for T > T c within the lattice version of the theory including lattice effects. We also study the large-L behavior of the susceptibility for T > T c of the confined lattice system of linear size L with periodic boundary conditions. We find that the structure of the large-L behavior of of the confined system is closely related to the structure of the large-distance behavior of of the bulk system. Explicit results are derived in the spherical (large-n) limit and in one-loop order for general dimensions d > 2. For the lattice model with cubic symmetry we find that finite-size scaling must be formulated in terms of the anisotropic bulk correlation length (exponential correlation length) that governs the exponential decay of for large r rather than in terms of the ordinary isotropic bulk correlation length defined via the second moment of . We show that it is the exponential bulk correlation length in the direction of the cubic axes that determines the exponential finite-size scaling behavior of lattice systems in a rectangular geometry. This result modifies a recent interpretation concerning an apparent violation of finite-size scaling in terms of the second-moment correlation length . Exact results for the one-dimensional Ising model illustrate our conclusions. Furthermore we show for general d>2 that a description of finite-size effects for finite n in the entire region requires different perturbative approaches that are applicable either to the region or , respectively. In particular we show that the exponential finite-size behavior for above T c is not captured by the standard perturbation approach that separates the homogeneous lowest mode from the inhomogeneous higher modes. Consequences for the theory of finite-size effects above four dimensions are discussed. We show that the two-variable finite-size scaling form predicts an exponential approach to the bulk critical behavior above T c whereas the reduction to a single-variable scaling form implies a power-law approach .