Generalizations of the Lee-Yang theorem

James G. Glimm, Arthur M. Jaffe

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.

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.

J.G. Esteve, Fernando Falceto, Carlos A. García Canal

Vinod Mubayi, Chanchal K. Majumdar, Kanwar Krishan

The exact excitation spectrum of the 3 \ifmmode\times\else\texttimes\fi{} 3 square and the 3 \ifmmode\times\else\texttimes\fi{} 4 rectangular lattices interacting via the nearest-neighbor anisotropic Heisenberg interaction $H=J\ensuremath{\Sigma}{(\mathrm{ij})}^{}[S_{i}^{}{}_{}{}^{z}S_{j}^{}{}_{}{}^{z}+\ensuremath{\gamma}(S_{i}^{}{}_{}{}^{x}S_{j}^{}{}_{}{}^{x}+S_{i}^{}{}_{}{}^{y}S_{j}^{}{}_{}{}^{y})]$ has been obtained for various values of the anisotropy constant $\ensuremath{\gamma}$. For $0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}1$, the zeros of the partition function in the complex $\ensuremath{\mu}=\mathrm{exp}(\ensuremath{-}\frac{m{H}_{e}}{{k}_{B}T})$ plane obey the generalized Lee-Yang theorem for ferromagnetic coupling. The usual ferromagnetic transition seems to persist up to $\ensuremath{\gamma}\ensuremath{\simeq}0.6$. For $0.6\ensuremath{\lesssim}\ensuremath{\gamma}\ensuremath{\le}1$, nothing definite about a ferromagnetic transition can be inferred from the behavior of the zeros. When $\ensuremath{\gamma}>1$, the ferromagnetic zeros do not obey the generalized Lee-Yang theorem at sufficiently low temperatures. For antiferromagnetic coupling, the zeros lie on the negative real axis for all values of $\ensuremath{\gamma}$ and of the temperature of the studied lattices.

Josip Hendeković, Milica Pavlović, Franjo Sokolić