Correlation Inequalities and the Lee-Yang Theorem

A. Brooks Harris

Taro Asano

It is shown that the zeros of the partition function of the anisotropic Heisenberg ferromagnet locate on the unit circle in the complex "fugacity" plane, and all the spin correlation functions are non-negative when no external magnetic field is present.

Chaithanya Rayudu, Jun Takahashi

The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.

Chaithanya Rayudu, Takahiro Misawa, Andrew Zhao +1 more

Lee-Yang theorems are a powerful tool for studying many-body systems, with applications ranging from analyzing phase transitions to proving the efficiency of certain classical and quantum algorithms. In this work, we prove a Lee-Yang zero-freeness theorem for the partition function of a broad class of interacting fermion models, implying the existence of a provably efficient quantum algorithm for estimating their ground-state energies. This class includes several well-known models such as the attractive Hubbard model, repulsive Hubbard model on bipartite graphs, and the interacting Hofstadter model. Our results also rigorously establish the nonexistence of phase transitions in these models in the presence of a nonzero local external field.

Taro Asano

Lee-Yang's theorem which states that all the roots of the partition function of the ferromagnetic Ising model of spin 1/2 lie on the unit circle in the complex `fugacity' plane is generalized to the case of S =1 and 3/2, where S is the magnitude of the spin.

V. A. Andreev