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arXiv:cond-mat/9503139 · Physical review. B, Condensed matter

Singularity of the density of states in the two-dimensional Hubbard model from finite-size scaling of Yang-Lee zeros

E. Abraham, Ian M. Barbour, P.H. Cullen, Elyakum G. Klepfish, Edward R. Pike, Sarben Sarkar

Published Mar 15, 1996• 11 citations
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Abstract

A finite-size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the two-dimensional Hubbard model in the complex chemical potential plane. The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential. Our analysis points to a second-order phase transition with critical exponent 1/\ensuremath{\delta}=1/2\ifmmode\pm\else\textpm\fi{}1/3, which leads to a divergence of the electronic susceptibility. We interpret these results as reflecting the singular behavior of the density of states in the quasiparticle spectrum. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.7704

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A finite-size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the two-dimensional Hubbard model in the complex chemical potential plane. The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential. Our analysis points to a second-order phase transition with critical exponent 1/\ensuremath{\delta}=1/2\ifmmode\pm\else\textpm\fi{}1/3, which leads to a divergence of the electronic susceptibility. We interpret these results as reflecting the singular behavior of the density of states in the quasiparticle spectrum. \textcopyright{} 1996 The American Physical Society.
We present a technique for a numerical analysis of the phase structure of the two-dimensional Hubbard model as a function of the hole chemical potential. The grand-canonical partition function for the model is obtained via Monte Carlo simulations. The dependence of the hole occupation number on the chemical potential and the temperature is evaluated. These calculations, together with a study of the Yang-Lee zeros of the grand-canonical partition function, show evidence of a phase transition at zero temperature and particle density below half filling. The binding energy of a pair of holes is calculated in the low-temperature regime and the possibility for pairing is explored.
We use finite-size scaling of Lee-Yang partition function zeros to study the critical behavior of the two-dimensional step or sgn O(2) model. We present evidence that, like the closely related XY model, this has a phase transition from a disordered high-temperature phase to a low-temperature massless phase where the model remains critical. The critical parameters (including logarithmic corrections) are compatible with those of the XY model indicating that both models belong to the same universality class. \textcopyright{} 1996 The American Physical Society.
Using large-scale dynamical cluster quantum Monte Carlo simulations, we study the Lifshitz transition of the two-dimensional Hubbard model with next-nearest-neighbor hopping (${t}^{\ensuremath{'}}$), chemical potential, and temperature as control parameters. At ${t}^{\ensuremath{'}}\ensuremath{\le}0$, we identify a line of Lifshitz transition points associated with a change in the Fermi surface topology at zero temperature. In the overdoped region, the Fermi surface is complete and electron-like; across the Lifshitz transition, the Fermi surface becomes hole-like and develops a pseudogap. At (or very close to) the Lifshitz transition points, a van Hove singularity in the density of states crosses the Fermi level. The van Hove singularity occurs at finite doping due to correlation effects and becomes more singular when ${t}^{\ensuremath{'}}$ becomes more negative. The resulting temperature dependence on the bare ddd-wave pairing susceptibility close to the Lifshitz points is significantly different from that found in the traditional van Hove scenarios. Such unambiguous numerical observation of the Lifshitz transition at ${t}^{\ensuremath{'}}\ensuremath{\le}0$ extends our understanding of the quantum critical region in the phase diagram and shines lights on future investigations of the nature of the quantum critical point in the two-dimensional Hubbard model.
Lee-Yang theory is central to the analysis of thermal phase transitions. However, the underlying mechanism of the theory and the nature of Lee-Yang zeros in quantum many-body systems remains elusive. Here, we develop a unified framework for understanding quantum phase transitions from fidelity zeros induced by symmetry breaking. These zeros, arising from transitions between symmetry sectors, obey the Lee-Yang theorem and give rise to fidelity edges near critical points. Quantum criticality is further characterized through the finite-size scaling of fidelity zeros. As concrete examples, we analytically and numerically investigate fidelity zeros in one- and two-dimensional ferromagnetic Ising models under a complex magnetic field. Our results provide new insights into the mechanism of Lee-Yang theory and open avenues for exploring unexplored landscapes of phase transitions in quantum many-body systems.
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 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.