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arXiv:2504.01880 · Physical review. E

Partition function zeros of quantum many-body systems

B. Sriram Shastry

Published Jul 16, 2025• 2 citations
arXivHTMLPDF

Abstract

We present a method for calculating the Yang-Lee partition function zeros of a translationally invariant model of lattice fermions, exemplified by the Hubbard model. The method rests on a theorem involving the usual single-electron self-energy ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i{\ensuremath{\omega}}_{n}|\ensuremath{\mu})$ with chemical potential $\ensuremath{\mu}$, in the imaginary time Matsubara formulation. The theorem maps the Yang-Lee zeros to a set of wave vector and spin labeled virtual energies ${\ensuremath{\xi}}_{\stackrel{P\vec}{k}\ensuremath{\sigma}}$. These, thermodynamically derived virtual energies, are solutions of the set of equations ${\ensuremath{\xi}}_{k\ensuremath{\sigma}}={\ensuremath{\varepsilon}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k})\ensuremath{-}\frac{1}{2}U+{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i\frac{\ensuremath{\pi}}{\ensuremath{\beta}}|{\ensuremath{\xi}}_{k\ensuremath{\sigma}}+\frac{1}{2}U\ensuremath{-}i\frac{\ensuremath{\pi}}{\ensuremath{\beta}})=0$. Examples of the method in simplified situations are provided.

DOI
10.1103/3zm3-wbjk

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We present a method for calculating the Yang-Lee partition function zeros of a translationally invariant model of lattice fermions, exemplified by the Hubbard model. The method rests on a theorem involving the usual single-electron self-energy {\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i{\ensuremath{\omega}}_{n}|\ensuremath{\mu}) with chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ, in the imaginary time Matsubara formulation. The theorem maps the Yang-Lee zeros to a set of wave vector and spin labeled virtual energies {\ensuremath{\xi}}_{\stackrel{P\vec}{k}\ensuremath{\sigma}}. These, thermodynamically derived virtual energies, are solutions of the set of equations {\ensuremath{\xi}}_{k\ensuremath{\sigma}}={\ensuremath{\varepsilon}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k})\ensuremath{-}\frac{1}{2}U+{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i\frac{\ensuremath{\pi}}{\ensuremath{\beta}}|{\ensuremath{\xi}}_{k\ensuremath{\sigma}}+\frac{1}{2}U\ensuremath{-}i\frac{\ensuremath{\pi}}{\ensuremath{\beta}})=0. Examples of the method in simplified situations are provided.
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.
A systematic method to find the Yang-Lee partition function zeros of quantum many-body systems based on perturbation theory at finite temperatures was recently introduced in arXiv:2504.01880. This method identifies wave-vector and temperature-dependent complex virtual energies obtainable from the thermal electronic Greens function. The collection of virtual energies over all k⃗\vec{k}k yield the Yang-Lee zeroes. We apply this approach to the one-dimensional Hubbard model for different boundary conditions. We compare the results obtained by this method up to second-order in perturbation theory with the results found by exact diagonalization. We also propose a quantity that could be used for experimental detection of these zeros of a Hubbard ring. An example of the detection method is presented using exact diagonalization of the 8-site Hubbard ring.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
The Lee-Yang theorem for the zeros of the partition function of a ferromagnetic Ising model with real pair spin interactions is extended to general Ising models with complex many-spin interactions (satisfying appropriate ``ferromagnetic'' and spin inversion symmetry conditions). When many-spin interactions are present, all zeros lie on the imaginary Hz-axis for sufficiently low (but fixed) T, but, in general, some leave the imaginary axis as T → ∞. The extended Ising theorem is used to prove the same result for a Heisenberg system of arbitrary spin with the real anisotropic pair interaction Hamiltonian Hij=−(JijzSizSjz+JijxSixSjx+JijySiySjy)in an arbitrary transverse field (Hx, Hy) under the ``ferromagnetic'' condition Jijz≥|Jijx| and Jijz≥|Jijy|. The analyticity of the limiting free energy of such a Heisenberg ferromagnet and the absence of a phase transition are thereby established for all (real) nonzero magnetic fields Hz. The Ising theorem is also applied to hydrogen-bonded ferroelectric models to prove, in particular, that the zeros for the KDP model lie on the imaginary electric field axis for all T below the transition temperature Tc.
Abstract To understand the distribution of the Yang–Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the 2D Yang–Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature β , can be computed in terms of the thermodynamics Bethe Ansatz based on its exact S -matrix. We extract the Yang–Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories. There is though an important difference between the interactive theory and the free theories, for the radius of the zeros in the interactive theory goes continuously to zero in the high-temperature limit β → 0 while in the free theories it remains close to 1 even for small values of β , jumping to 0 only at β = 0 .
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.