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arXiv:2605.25862

Bargmann Zeros as a Diagnostic of the Tunneling Transition in Double-Well Quantum Systems

Tughanbulut Kurtulush, Maciej Janowicz

quant-phSubmitted May 25, 2026• Announced May 26, 2026• 0 citations
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Abstract

Complex zeros of wavefunctions represented as entire functions in Bargmann--Fock space encode structural information about the underlying quantum state. Prior work employed zero galleries of randomly generated polynomial superpositions of Fock states as visual fingerprints suitable for classification. Here we examine whether Bargmann zeros of physically realized eigenstates of one-dimensional anharmonic and double-well Hamiltonians carry a recognizable signature of the tunneling transition in the symmetric double well. Ground and first-excited eigenstates are obtained from a variational ansatz consisting of a physically motivated symbolic envelope multiplied by a small flexible correction network, trained by Rayleigh--Ritz minimization of the finite-difference Hamiltonian expectation value and validated to reproduce energies to within $\sim 10^{-5}\,\mathrm{Ha}$. The resulting wavefunctions are projected onto the harmonic-oscillator basis and the complex zeros of the truncated Bargmann polynomial are located by numerical root-finding. For harmonic and quartic-anharmonic potentials the zeros show no preferred orientation. For double-well eigenstates, by contrast, the zeros condense onto the imaginary axis. A sweep of the barrier parameter $a$ from $0.5$ to $2.3$ reveals a continuous migration of zeros toward the imaginary axis, concurrent with the exponential collapse of the tunneling splitting $Δ(a) = E_1 - E_0$ over $3.5$ decades. This condensation is traced to a sign-alternation pattern in the Fock-coefficient spectrum that is characteristic of bimodally localized wavefunctions. The complex zero set of the Bargmann-represented wavefunction thereby provides a compact, purely analytic diagnostic for the tunneling regime of one-dimensional double-well Hamiltonians, extending the random-polynomial zero-image framework to physical eigenstates.

Comments
22 pages, 6 figures

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Complex zeros of wavefunctions represented as entire functions in Bargmann--Fock space encode structural information about the underlying quantum state. Prior work employed zero galleries of randomly generated polynomial superpositions of Fock states as visual fingerprints suitable for classification. Here we examine whether Bargmann zeros of physically realized eigenstates of one-dimensional anharmonic and double-well Hamiltonians carry a recognizable signature of the tunneling transition in the symmetric double well. Ground and first-excited eigenstates are obtained from a variational ansatz consisting of a physically motivated symbolic envelope multiplied by a small flexible correction network, trained by Rayleigh--Ritz minimization of the finite-difference Hamiltonian expectation value and validated to reproduce energies to within ∼10−5 Ha\sim 10^{-5}\,\mathrm{Ha}∼10−5Ha. The resulting wavefunctions are projected onto the harmonic-oscillator basis and the complex zeros of the truncated Bargmann polynomial are located by numerical root-finding. For harmonic and quartic-anharmonic potentials the zeros show no preferred orientation. For double-well eigenstates, by contrast, the zeros condense onto the imaginary axis. A sweep of the barrier parameter aaa from 0.50.50.5 to 2.32.32.3 reveals a continuous migration of zeros toward the imaginary axis, concurrent with the exponential collapse of the tunneling splitting Δ(a)=E1−E0Δ(a) = E_1 - E_0Δ(a)=E1​−E0​ over 3.53.53.5 decades. This condensation is traced to a sign-alternation pattern in the Fock-coefficient spectrum that is characteristic of bimodally localized wavefunctions. The complex zero set of the Bargmann-represented wavefunction thereby provides a compact, purely analytic diagnostic for the tunneling regime of one-dimensional double-well Hamiltonians, extending the random-polynomial zero-image framework to physical eigenstates.
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We present a systematic study of the Rayleigh--Ritz variational method for quantum oscillators in the Segal--Bargmann space. We rigorously derive the normalizability condition ∣α∣<12|α| < \tfrac{1}{2}∣α∣<21​ for generalized Gaussian trial functions ψ(z)=eαz2+βzψ(z) = e^{αz^2 + βz}ψ(z)=eαz2+βz through convergence analysis of Gaussian integrals in the complex plane. Applications to the harmonic oscillator demonstrate exact recovery of the ground state in Segal--Bargmann space when the trial family contains the true solution. For the quartic anharmonic oscillator (H^=−12∂x2+12x2+λx4\hat{H} = -\tfrac{1}{2}\partial_x^2 + \tfrac{1}{2}x^2 + λx^4H^=−21​∂x2​+21​x2+λx4), adaptive Gaussian ansätze in position space yield a cubic stationarity equation and perturbative energy expansions beyond first order, capturing anharmonic wavefunction narrowing. In contrast, monomial trial functions (ψn(z)=znψ_n(z) = z^nψn​(z)=zn) in the Segal--Bargmann space -- while providing rigorous upper bounds En=n+12+3λ4(2n2+2n+1)E_n = n + \tfrac{1}{2} + \tfrac{3λ}{4}(2n^2 + 2n + 1)En​=n+21​+43λ​(2n2+2n+1) for excited states -- lack width adaptability and are limited to first-order accuracy for ground-state calculations. We further analyze displaced Gaussians and displaced monomials for asymmetric potentials (e.g., x3+x4x^3 + x^4x3+x4), showing that displacement parameters are essential to capture parity breaking and stabilization effects (E0≈12+3μ4−9λ24+⋯E_0 \approx \tfrac{1}{2} + \tfrac{3μ}{4} - \tfrac{9λ^2}{4} + \cdotsE0​≈21​+43μ​−49λ2​+⋯).
We show that variational method for harmonic oscillator in Bargmann representation breaks down since we always end up with one of the exact eigenvalues of Hamiltonian using any normalized monomials in the complex plane. Using the holomorphic representation of pure spin systems , we also encounter the same result found earlier for harmonic oscillators in Bargmann representation. We conclude that the exact energy eigenvalues of these systems are unavoidable starting from any normalized monomials of the complex phase-space coordinate. Since {znn!}\{\frac{z^{n}}{\sqrt{n!}}\}{n!​zn​} form an orthonormal basis, one could in principle expand any complex function f(z)f(z)f(z) in powers of zzz and use this expansion as energy eigenstates. Such expansion is valid since the magnitude of zzz carry no physical information only their non-negative exponents nnn are physically important. These results may lead to new selection principle regarding the quantized energy spectrum of physical systems and enhance our understanding of quantum world.
The far-from-equilibrium dynamics of certain interacting quantum systems still defy precise understanding. One example is the so-called quantum many-body scars (QMBSs), where a set of energy eigenstates evade thermalization to give rise to long-lived oscillations. Despite the success of viewing scars from the perspectives of symmetry, commutant algebra, and quasiparticles, it remains a challenge to elucidate the mechanism underlying all QMBS and to distinguish them from other forms of ergodicity breaking. In this work, we introduce an alternative route to detect and diagnose QMBS based on Fisher zeros, i.e., the patterns of zeros of the analytically continued partition function Z on the complex β (inverse temperature) plane. For systems with scars, a continuous line of Fisher zeros will appear off the imaginary β axis and extend upward, separating the β plane into regions with distinctive thermalization behaviors. This conjecture is motivated from interpreting the complex Z as the return amplitude of the thermofield double state, and it is validated by analyzing two models with QMBS, the P[over ¯]XP[over ¯] model and the Ising chain in external fields. These models also illustrate the key difference between QMBS and strong ergodicity breaking including their distinctive renormalization group flows on the complex β plane. This "statistical mechanics" approach places QMBS within the same framework of thermal and dynamical phase transitions. It has the advantage of spotting scars without exhaustively examining each individual quantum state.
Abstract Quantum computing has recently been emerging in theoretical chemistry as a realistic avenue meant to offer computational speedup to challenging eigenproblems in the context of strongly-correlated molecular systems or extended materials. Most studies so far have been devoted to the quantum treatment of electronic structure, for which the transformation of fermionic operators into the qubit space is quite transparent. In contrast, only a few were directed to the quantum treatment of vibrational structure, which at the moment remains not devoid of unknowns. When simulating an anharmonic vibrational mode under bosonic second quantization, the biggest problem, which we analyze in detail in the present work, is the disruption of the resolution of the identity when truncating the infinite harmonic-oscillator basis set of primitive modals and how this alters the canonical commutation relation. This may lead to serious incorrectness in the evaluation of the Hamiltonian matrix elements when assembling them from finite matrix products, which eventually entail a nonvariational behavior with respect to the finite size of the computational basis. As we show here, a simple cure occurs to be obtained upon using Wick's normal order, for reasons that are not standardly evoked. We also provide a detailed comparison between the boson-to-qubit unary and binary mappings under two different representations: one for the bosonic ladder operators within the harmonic-oscillator primitive basis, and one for the so-called n-mode representation within any type of computational basis. In addition, we discuss the impact of choosing an adequate primitive basis set in terms of quantum computing with respect to its variational convergence efficiency (number of basis functions, hence of qubits) and as regards the magnitude of the 1-norm of the encoded Hamiltonian (a measure of the computational complexity of the quantum algorithm). Such fundamental aspects are illustrated numerically on a one-dimensional anharmonic Hamiltonian model corresponding to a symmetric double-well potential, of interest both for vibrational spectroscopy and chemical reactivity, and which is a challenging situation for numerical convergence due to fine tunneling splitting.
Motivated by the advance of dynamical quantum phase transitions (DQPTs), we analyze the zeros of the complex-time survival (Loschmidt) amplitude in finite quantum systems and develop a general framework for their approximation based on the stability of zeros of holomorphic functions. We show that the large-scale properties of the distribution of zeros are governed by the envelope of the energy distribution of the initial state and can be constructed from chains of periodic zeros associated with its dominant contributions. In this picture, zeros reach the real-time axis when two or more eigenstates become equally populated at the maximum of the envelope, providing a finite-size precursor of DQPTs. We apply the method to quenched ground states in the Ising model with tunable interaction range and demonstrate close agreement between the approximate and exact distributions of zeros. We prove that the approximate construction becomes exact for BCS ground-state quenches in two-band models. To describe short-time dynamics, we introduce a minimal Gaussian model with a nearly equidistant spectrum. Slow dephasing continuously deforms the initial zero pattern into the asymptotic two-level structure, explaining anomalous DQPTs as a delayed approach of zeros to the real-time axis. Our results identify the energy envelope as the key ingredient shaping dynamical critical behavior and provide a universal interpretation of the whole zero distribution of the complex-time survival amplitude.