© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Semidefinite Pairing-Source Matrices Characterize Uniform Fermionic Lee-Yang Zeros

Piyush Sao

Published Sep 20, 2026• 0 citations

Abstract

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.

DOI: 10.5281/zenodo.22841846

Related papers

A Pairing-Source Lee–Yang Theorem for Attractive Hubbard Systems

Piyush Sao

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.

•Sep 15, 2026•0 citations

A Lee–Yang Circle Theorem for Equal-Time Fermionic Pairing Measurements

Piyush Sao

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.

•Sep 21, 2026•0 citations

Spectral gap of Lee-Yang Hamiltonians

Chaithanya Rayudu, Jun Takahashi

The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.

quant-pharXiv:2607.10765•Jul 14, 2026•0 citations

Zeros of the Partition Function for the Heisenberg, Ferroelectric, and General Ising Models

Masuo Suzuki, Michael Ellis Fisher

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.

Journal of Mathematical Physics•Feb 1, 1971•164 citations

Lee-Yang theorem for fermions

Chaithanya Rayudu, Takahiro Misawa, Andrew Zhao +1 more

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.

quant-pharXiv:2609.23942•Sep 22, 2026•0 citations

Lee–Yang Structure Does Not Imply Thermal Monotonicity

Piyush Sao

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.

•Sep 23, 2026•0 citations

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 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 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.

The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.

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.

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.

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.