Imaginary pairing-source zeros at every temperature do not force the scaled polarization β⟨M⟩ to increase as the inverse temperature β increases. We prove this with a fermionic atom coupled to a neutral auxiliary degree of freedom. Its zero-field ground state is simple, and all its source zeros are imaginary for every β > 0. At fixed field h = 1/100, exact rational inequalities certify Dβ(h) := ∂β[β⟨M⟩] < −9/1000 throughout |β − 20 log 2| ≤ 1/100. These properties persist in a neighborhood within the same parameter family. On this interval, the scaled polarization within the even-parity sector increases. Cooling nevertheless shifts enough probability to the odd sector, on which the pair operator vanishes, to make the total decrease. The sector derivative identity separates this probability shift from the change within each sector. The same sector analysis gives the exact boundary J ≤ 4U/3 for growth of the zero-field curvature at every temperature in a two-site onsite-attraction and exchange family. Further examples distinguish this curvature from inverse-zero moments and from the unscaled polarization, which can decrease while the scaled polarization increases. For auxiliary-field parity comparisons, we identify the corresponding configuration and redistribution terms and show why a proof must control their sum.
Imaginary pairing-source zeros at every temperature do not force the scaled polarization β⟨M⟩ to increase as the inverse temperature β increases. We prove this with a fermionic atom coupled to a neutral auxiliary degree of freedom. Its zero-field ground state is simple, and all its source zeros are imaginary for every β > 0. At fixed field h = 1/100, exact rational inequalities certify Dβ(h) := ∂β[β⟨M⟩] < −9/1000 throughout |β − 20 log 2| ≤ 1/100. These properties persist in a neighborhood within the same parameter family. On this interval, the scaled polarization within the even-parity sector increases. Cooling nevertheless shifts enough probability to the odd sector, on which the pair operator vanishes, to make the total decrease. The sector derivative identity separates this probability shift from the change within each sector. The same sector analysis gives the exact boundary J ≤ 4U/3 for growth of the zero-field curvature at every temperature in a two-site onsite-attraction and exchange family. Further examples distinguish this curvature from inverse-zero moments and from the unscaled polarization, which can decrease while the scaled polarization increases. For auxiliary-field parity comparisons, we identify the corresponding configuration and redistribution terms and show why a proof must control their sum.
Proving a quantum-response inequality at every temperature requires control beyond a finite temperature grid or a truncated series. We give a sufficient finite certificate for strict growth of the pairing-source curvature 𝜅2(𝛽)=𝜕2ℎ log Tr 𝑒−𝛽(𝐻−ℎ𝑀)|ℎ=0, which equals 𝛽 times the physical source susceptibility. For finite Hermitian systems with source reversal, a positive spectral shift produces nonnegative two-insertion temperature coefficients. Ordering their ratios to the partition coefficients implies 𝜅′2(𝛽)> 0 for every 𝛽 > 0. A simple ground state coupled to the source forces these ratios toward a line with positive slope; an explicit spectral error bound proves every increment beyond a computable cutoff. The complete certificate therefore requires a finite coefficient prefix and finitely many spectral bounds. For the half-filled uniform-attraction Hubbard dimer, two ordered temperature-series quotients and the atomic formula prove curvature growth over all couplings, with connected- cluster consequences near gapped dimer products. For a connected four-site attractive chain, integer and rational arithmetic verifies 3199 increments and closes the infinite tail. A uniform perturbation bound also gives an explicit rectangle in hopping and attraction parameters. A self-contained finite-field certificate shows that the same chain’s unscaled polarization can decrease while i
Lee-Yang theorems constrain the analytic response of a system to an external field. In attractive Hubbard systems, local pairing sources are known to have imaginary-axis zeros. However, it remains unknown which spatially extended pairing profiles retain this property uniformly as hopping, onsite attraction, and temperature vary. We address this gap by classifying real symmetric singlet source matrices for systems with conjugate-spin Hermitian hopping. We prove that a source matrix has the uniform Lee-Yang property if and only if it is positive or negative semidefinite, including singular matrices. To prove sufficiency, we show that the partition function is nonzero for complex symmetric source matrices with positive-definite Hermitian part while retaining the physical interaction basis. We then preserve this domain under arbitrary nonnegative onsite attractions by combining a transfer inequality with a nonzero full-degree coefficient. To prove necessity, we show that every indefinite source has eigenmodes of both signs. A free two-mode hopping matrix can mix opposite-sign modes, producing off-axis zeros that persist under small parameter perturbations. This mechanism distinguishes uniform admissibility from fixed-model admissibility, for which commuting free Hamiltonians can retain imaginary-axis zeros even for indefinite sources. We further derive zero-free domains for simultaneous pairing sources and collinear Zeeman fields, together with preservation and classification results for additional interaction channels. For the centered half-filled dimer, a complete count of complex-plane zeros determines the low-temperature first-loss radius. The boundary is set by two competing mechanisms—an exceptional-point collision and a high-zero tail—which exchange dominance at U/t = 1/3.
We locate the complex zeros of an equal-time pairing measurement in a finite fermion system. Let P_i = c_{i↓}c_{i↑} annihilate an on-site pair and let M = Σ_i (P_i + P_i†) be its uniform pairing quadrature. For complex-conjugate hopping matrices in the two spin sectors, nonnegative onsite attraction, and nonnegative couplings between these quadratures, the polynomial z^n ⟨e^{(log z)M}⟩_β has all 2n zeros on |z| = 1 at every finite inverse temperature β > 0. We reach the measurement generating function by concentrating a time-dependent source into a shrinking imaginary-time interval with fixed integrated strength. If e^{±iθ₁} are the zeros nearest z = 1, then π²/(4θ₁²) ≤ ⟨M²⟩β ≤ 9n/(2√7 θ₁). For charge-conserving Gibbs states and P = Σ_i P_i, these bounds make lim inf_n ⟨P†P⟩{β_n}/n² > 0 equivalent to lim sup_n nθ₁^(n) < ∞. The criterion allows arbitrary finite positive β_n and unrestricted hopping geometry. Convergence of the scaled log generating functions on any positive-real set with an interior accumulation point determines a unique limiting angle measure and its origin density. Independent growing ordered blocks show why positive origin density and whole-system macroscopic fluctuations have different content. The folded zero product also gives positive inverse-root sums and finite cumulant recurrences. An exact free four-site calculation and an interacting rational certificate separate directional cumulant signs from individual-component signs. The pulse implication also yields a finite-temperature Ising application.
We prove a Lee–Yang theorem for onsite pairing sources in finite attractive Hubbard systems. For spin-conserving hopping with an arbitrary Hermitian up-spin matrix and its complex conjugate for down spin, real onsite chemical potentials, and nonnegative onsite attractions, the partition function is nonzero whenever the real parts of all local pairing fields have the same strict sign. In particular, its zeros in a uniform pairing field lie on the imaginary axis at every positive inverse temperature. This time-reversal-invariant class includes real spin-independent hopping and opposite orbital fluxes for the two spins. The proof has two steps. After a particle–hole transformation, the free Fock-space trace becomes a determinant. An indefinite-form argument shows that this determinant is nonzero throughout the source domain. We then represent the onsite attraction as a local operation on a quadratic source polynomial and prove that this operation preserves the same zero-free domain. The highest coefficient remains nonzero, so we can iterate the operation and take a locally uniform Trotter product limit to obtain the interacting theorem. For a nonzero uniform real pairing field, the same transfer argument gives a zero-free spin-fugacity annulus, positivity of the parity-twisted trace, and a lower bound on the induced parity gap. The uniform-source zero theorem also yields bounds on pair response and on the first zero. Finally, for uniformly local sequences at fixed temperature, we relate the first zero to uniform equal-time pair order.
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 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 .