Quantum belief propagation: An algorithm for thermal quantum systems

We present an accurate numerical algorithm, called quantum belief propagation, for simulation of one-dimensional quantum systems at nonzero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at nonzero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-$1∕2$ Heisenberg chain. Even a very modest calculation, requiring diagonalizing only ten-by-ten matrices, reproduces the peak susceptibility with a relative error of less than ${10}^{-5}$, while more elaborate calculations further reduce the error.

Figure 1

(Color online) Specific heat against temperature for $l_0=3$ (dashed line), 5 (dotted line), 7 (solid line). Curves that go negative are from $-3{\beta }^2{\partial }_{\beta }⟨S_i^z{S}_{i+1}^z⟩$, while those that diverge positively are from ${\beta }^2{\partial }_{\beta }^2\ln \left(Z\right)$. Inset: $l_0=9$ and Bethe ansatz.