Lee-Yang theory, based on the study of zeros of the partition function, is widely regarded as a powerful and complimentary approach to the study of critical phenomena and forms a foundational part of the theory of phase transitions. Its widespread use, however, is complicated by the fact that it requires introducing complex-valued fields that create an obstacle for many numerical methods, especially in the quantum case where very limited studies exist beyond one dimension. Here we present a simple and statistically exact method to compute partition function zeros with general complex-valued external fields in the context of large-scale quantum Monte Carlo simulations. We demonstrate the power of this approach by extracting critical exponents from the leading Lee-Yang zeros of 2D quantum antiferromagnets with a complex staggered field, focusing on the Heisenberg bilayer and square-lattice $J$-$Q$ models. The method also allows us to introduce a complex field that couples to valence bond solid order, where we observe extended rings of zeros in the $J$-$Q$ model with purely imaginary staggered and valence bond solid fields.
Lee-Yang zeros at O(3) and deconfined quantum critical points
Lee-Yang theory, based on the study of zeros of the partition function, is widely regarded as a powerful and complimentary approach to the study of critical phenomena and forms a foundational part of the theory of phase transitions. Its widespread use, however, is complicated by the fact that it requires introducing complex-valued fields that create an obstacle for many numerical methods, especially in the quantum case where very limited studies exist beyond one dimension. Here we present a simple and statistically exact method to compute partition function zeros with general complex-valued external fields in the context of large-scale quantum Monte Carlo simulations. We demonstrate the power of this approach by extracting critical exponents from the leading Lee-Yang zeros of 2D quantum antiferromagnets with a complex staggered field, focusing on the Heisenberg bilayer and square-lattice J-Q models. The method also allows us to introduce a complex field that couples to valence bond solid order, where we observe extended rings of zeros in the J-Q model with purely imaginary staggered and valence bond solid fields.
Square-lattice J-Q models provide a sign-problem-free setting for probing the quantum phase transition between Néel antiferromagnet and columnar valence-bond solid. We analyze this transition through the scaling of Lee--Yang zeros, computed within stochastic series expansion quantum Monte Carlo by reweighting configurations sampled near criticality in the presence of complex source fields. Benchmark studies of the dimerized Heisenberg model and the checkerboard J-Q model validate the method, yielding stable O(3) critical scaling in the former and clear spacetime-volume scaling in the latter, as expected for a first-order transition. Applying the same analysis to the J-Q models, we find a pronounced and systematic drift of the leading-zero scaling with increasing system size, consistent with an extended pseudocritical regime. The Lee--Yang scaling implies an effective scaling dimension Δϕ of the SO(5) order-parameter field that decreases with size and is consistent with vanishing in the thermodynamic limit. Such behavior lies below the scalar unitarity bound of any unitary relativistic conformal field theory in 2+1 dimensions and enforces inverse spacetime-volume scaling of the zeros, the hallmark of a first-order transition. These results support a weakly first-order interpretation of the Néel--VBS transition and establish the finite-size Lee--Yang zeros as a sensitive, symmetry-resolved diagnostic of pseudocriticality and transition order in the J-Q family.
We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice and the more tractable mean-field model corresponding to infinite-ranged coupling between all sites. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, k, with the leading term ∼k^{1/2}. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities (akin to sum rules) for the nearest-neighbor case. These identities are verified numerically by computing the exact partition function for a two-dimensional square lattice of size 16×16. For the mean-field model, we write down the partition function (termed the mean-field polynomials) for the ferromagnetic (FM) and antiferromagnetic (AFM) cases and derive from them the mean-field equations. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as ∼k^{1/2} as well. Using a simple numerical method, we find the roots lie on certain curves (the root curves), in the thermodynamic limit for the mean-field polynomials for the AFM case as well as for the FM one. Our results show a new root curve that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and nonzero complex staggered magnetization, and thus depict a complex phase boundary.
Abstract Since the landmark work of Lee and Yang, locating the zeros of the partition function in the complex magnetic-field plane has become a powerful method for studying phase transitions. Fisher later extended this approach to complex temperatures, and subsequent generalizations introduced other control parameters, such as the crystal field. In previous works (Moueddene et al 2024 J. Stat. Mech. 023206; 2024 Phys. Rev. E 110 064144) we applied this framework to the two- and three-dimensional Blume–Capel model–a system with a rich phase structure where a second-order critical line meets a first-order line at a tricritical point. We showed that the scaling of Lee-Yang, Fisher, and crystal-field zeros yields accurate critical exponents even for modest lattice sizes. In the present study, we extend this analysis and demonstrate that simulations need not be performed exactly at the nominal transition point to obtain reliable exponent estimates. Strikingly, small system sizes are sufficient, which not only improves methodological efficiency but also advances the broader goal of reducing the carbon footprint of large-scale computational studies.
The phase diagram of the two- and three-state Potts model with infinite-range interactions in the external field is analyzed by studying the partition function zeros in the complex field plane. The tricritical point of the three-state model is observed as the approach of the zeros to the real axis at the nonzero field value. Different regimes, involving several first- and second-order transitions of the complicated phase diagram of the three-state model, are identified from the scaling properties of the zeros closest to the real axis. The critical exponents related to the tricritical point and the Yang-Lee edge singularity are well reproduced. Calculations are extended to the negative fields, where the exact implicit expression for the transition line is derived.
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
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 .