Yang and Lee investigated phase transitions in terms of zeros of partition functions, namely, Yang-Lee zeros [Phys. Rev. 87, 404 (1952)PHRVAO0031-899X10.1103/PhysRev.87.404; Phys. Rev. 87, 410 (1952)PHRVAO0031-899X10.1103/PhysRev.87.410]. We show that the essential singularity in the superconducting gap is directly related to the number of roots of the partition function of a BCS superconductor. Those zeros are found to be distributed on a semicircle in the complex plane of the interaction strength due to the Fermi-surface instability. A renormalization-group analysis shows that the semicircle theorem holds for a generic quantum many-body system with a marginal coupling, in sharp contrast with the Lee-Yang circle theorem for the Ising spin system. This indicates that the geometry of Yang-Lee zeros is directly connected to the Fermi-surface instability. Furthermore, we unveil the nonunitary criticality in BCS superconductivity that emerges at each individual Yang-Lee zero due to exceptional points and presents a universality class distinct from that of the conventional Yang-Lee edge singularity.
Yang and Lee investigated phase transitions in terms of zeros of partition functions, namely, Yang-Lee zeros [Phys. Rev. 87, 404 (1952)PHRVAO0031-899X10.1103/PhysRev.87.404; Phys. Rev. 87, 410 (1952)PHRVAO0031-899X10.1103/PhysRev.87.410]. We show that the essential singularity in the superconducting gap is directly related to the number of roots of the partition function of a BCS superconductor. Those zeros are found to be distributed on a semicircle in the complex plane of the interaction strength due to the Fermi-surface instability. A renormalization-group analysis shows that the semicircle theorem holds for a generic quantum many-body system with a marginal coupling, in sharp contrast with the Lee-Yang circle theorem for the Ising spin system. This indicates that the geometry of Yang-Lee zeros is directly connected to the Fermi-surface instability. Furthermore, we unveil the nonunitary criticality in BCS superconductivity that emerges at each individual Yang-Lee zero due to exceptional points and presents a universality class distinct from that of the conventional Yang-Lee edge singularity.
The Lee-Yang circle theorem revolutionized our understanding of phase transitions in ferromagnetic systems by showing that the complex zeros of partition functions lie on the unit circle, with criticality arising as these zeros approach the real axis in the thermodynamic limit. However, in frustrated systems such as antiferromagnets and spin glasses, the zeros deviate from this structure, making it challenging to extend the Lee-Yang theory to disordered systems. In this work, we establish a new circle theorem for two-dimensional Ising spin glasses, proving that the square of the partition function exhibits zeros densely packed along the unit circle. Numerical simulations on the square lattice confirm our theoretical predictions, demonstrating the validity of the circle law for quenched disorder. Furthermore, our results uncover a finite-temperature crossover in ±J spin glasses, characterized by the emergence of a spectral gap in the angular distribution of zeros. This result extends the Lee-Yang framework to disordered systems, offering new insights into spin-glass criticality.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
We study the Yang-Lee theory in quantum phase transitions from the perspective of quantum entanglement in one-dimensional many-body systems. We primarily focus on the distribution of Yang-Lee zeros and its associated Yang-Lee edge singularity of two prototypical models: the Su-Schrieffer-Heeger model and the XXZ spin chain. By taking the zero-temperature limit, we show how the Yang-Lee zeros approach the quantum phase transition points on the complex plane of parameters. To characterize the edge singularity induced by Yang-Lee zeros in a quantum phase transition, we introduce the entanglement entropy of the ground state to show that the edges of Yang-Lee zeros lead to the ground-state entanglement transition. We further show that our results are also applicable to the general noninteracting parity-time-symmetric Hamiltonians.
The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle theorem. Distinct features are observed at different temperatures---below, above, and at the critical temperature. Application of the tensor network approach is also made to the q-state Potts models in both two and three dimensions and a previous debate on whether, in the thermodynamic limit, the Yang-Lee zeros lie on a unit circle for q>2 is resolved: they clearly do not lie on a unit circle except at the zero temperature. For the Potts models (q=3,4,5,6) investigated in two dimensions, as the temperature is lowered the radius of the zeros at a fixed angle from the real axis shrinks exponentially towards unity with the inverse temperature.
We present a simple phenomenological scaling theory for the pairing instability of a quantum critical metal. It can be viewed as a minimal generalization of the classical Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (SC) for normal Fermi-liquid metals. We assume that attractive interactions are induced in the fermion system by an external ``bosonic glue'' that is strongly retarded. Resting on the small Migdal parameter, all the required information from the fermion system needed to address the superconductivity enters through the pairing susceptibility. Asserting that the normal state is a strongly interacting quantum critical state of fermions, the form of this susceptibility is governed by conformal invariance and one only has the scaling dimension of the pair operator as free parameter. Within this scaling framework, conventional BCS theory appears as the ``marginal'' case but it is now easily generalized to the (ir)relevant scaling regimes. In the relevant regime an algebraic singularity takes over from the BCS logarithm with the obvious effect that the pairing instability becomes stronger. However, it is more surprising that this effect is strongest for small couplings and small Migdal parameters, highlighting an unanticipated important role of retardation. Using exact forms for the finite-temperature pair susceptibility from 1+1D conformal field theory as models, we study the transition temperatures, finding that the gap to transition temperature ratios is generically large compared to the BCS case, showing, however, an opposite trend as a function of the coupling strength compared to the conventional Migdal-Eliashberg theory. We show that our scaling theory naturally produces the superconducting ``domes'' surrounding the quantum critical points (QCPs), even when the coupling to the glue itself is not changing at all. We argue that hidden relations will exist between the location of the crossover lines to the Fermi liquids away from the quantum critical points and the detailed form of the dome when the glue strength is independent of the zero-temperature control parameter. Finally, we discuss the behavior of the orbital-limited upper critical magnetic field as a function of the zero-temperature coupling constant. Compared to the variation in the transition temperature, the critical field might show a much stronger variation pending the value of the dynamical critical exponent.
Yang-Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions.However, Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical.A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters.However, how to extract the critical point and the critical exponent of Yang-Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive.Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang-Lee edge singularity.The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang-Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems.This finding provides a practical approach to studying the nature of Yang-Lee edge singularities of many-body systems.In 1952, Yang and Lee [1, 2] initiated the examination of phase transitions by studying the partition function zeros, termed Lee-Yang zeros, on the complex plane of a magnetic field h (or fugacity z for fluids).They found that for a system above its critical temperature T c , the partition function must be nonzero throughout some neighborhood of the real axis of the magnetic field within a gap determined by two edges at ( ) h T YL .Therefore the free energy is an analytic function of the real magnetic field.On the other hand, below the critical temperature T c , the Lee-Yang zeros will come arbitrarily to the real axis as the thermodynamic limit is taken, destroying the analyticity of the free energy in h for those real fields at which the Lee-Yang zeros accumulate.Kortman and Griffiths [3] pointed out that the edges of Lee-Yang zeros of a many-body system in the thermodynamic limit are singularity points, termed Yang-Lee edge singularities [4].Fisher [4] proposed that the Yang-Lee edge singularity could be considered as a new second order phase transition point with associated critical exponents.Renormalization group analyses [4] show that the Yang-Lee edge singularity is associated with an j i 3 theory and the crossover dimensionality of the Yang-Lee edge singularity is d c = 6.Later, the investigation of the Yang-Lee edge singularity has been extended to many other models, such as the classical nvector model [5], the quantum Heisenberg model [5], the spherical model [6], the quantum one-dimensional (1D) transverse Ising model [7], the hierarchical model [8], branched polymers [9], the directed-site animalsenumeration problem [10], Ising models on fractal lattices [11], Ising systems with correlated disorder [12], fluid models with repulsive-core interactions [13], the antiferromagnetic Ising model [14] etc.A comprehensive review of the study of Yang-Lee edge singularities can be found in [15, 16].Recently, Kibble-Zurek scaling was shown to appear at the Yang-Lee edge singularity [17], which supports that the Yang-Lee edge singularity is similar to conventional second order phase transitions.Yang-Lee edge singularity occurs in the ferromagnetic Ising models above its critical temperatures in a purely imaginary magnetic field h i (figure 1).For > ( ) h h T YL , the partition function acquires zeros, which becomes dense on the line with > ( ) Ih h T YL .The density of Lee-Yang zeros ( ) g T h , in the thermodynamic limit behaves as ~ -s ( ) | ( )| g T h h h T , YL when ( ) h h T YL from above, with σ being the universal critical