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Resonant Touchdown of Fisher Zeros and Super-Kink Dynamical Quantum Phase Transitions via Lagrangian-Engineered Riemann Potentials

A Citizen of the Republic of Korea

Published Sep 21, 2026• 0 citations

Abstract

Dynamical quantum phase transitions (DQPT) provide a non-equilibrium framework for probing quantum criticality, where the non-analytic kinks in the Loschmidt echo rate function λ(t) signal the crossing of Fisher zeros across the real-time axis. In disordered many-body localized (MBL) systems, spatial inhomogeneity typically induces multi-frequency dephasing that melts these non-analytic signatures into diffuse crossovers. Here, we demonstrate that deterministic on-site potential landscapes derived from the unfolded non-trivial zeros of the Riemann zeta function prevent the thermalized melting of DQPT signatures through Gaussian Unitary Ensemble (GUE) level repulsion. While strong unengineered Riemann potentials (W = 2.0J) induce multi-frequency spectral beating resulting in dense multi-kink profiles, we formulate an augmented Lagrangian variational framework that directly targets the trade-off between isochronous periodicity and singular non-analyticity. By minimizing a multi-objective functional encompassing Fisher zero proximity, high-frequency beating penalties, and GUE level-repulsion kernels, the optimization converges to a dual-champion configuration h*. Under quantum quench dynamics of a one-dimensional transverse-field Ising chain (L=10, 1024-dimensional Hilbert space), this engineered landscape drives the Loschmidt echo down to the numerical machine-precision floor (L_min = 2.03e-13), generating an ultra-sharp super-kink with peak amplitude λ_max = 2.9219, nearly three times the baseline transition height. These results reveal a direct route for engineering singular non-equilibrium criticality in digital quantum processors.

DOI: 10.5281/zenodo.22869220

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Dynamical quantum phase transitions (DQPT) provide a non-equilibrium framework for probing quantum criticality, where the non-analytic kinks in the Loschmidt echo rate function λ(t) signal the crossing of Fisher zeros across the real-time axis. In disordered many-body localized (MBL) systems, spatial inhomogeneity typically induces multi-frequency dephasing that melts these non-analytic signatures into diffuse crossovers. Here, we demonstrate that deterministic on-site potential landscapes derived from the unfolded non-trivial zeros of the Riemann zeta function prevent the thermalized melting of DQPT signatures through Gaussian Unitary Ensemble (GUE) level repulsion. While strong unengineered Riemann potentials (W = 2.0J) induce multi-frequency spectral beating resulting in dense multi-kink profiles, we formulate an augmented Lagrangian variational framework that directly targets the trade-off between isochronous periodicity and singular non-analyticity. By minimizing a multi-objective functional encompassing Fisher zero proximity, high-frequency beating penalties, and GUE level-repulsion kernels, the optimization converges to a dual-champion configuration h*. Under quantum quench dynamics of a one-dimensional transverse-field Ising chain (L=10, 1024-dimensional Hilbert space), this engineered landscape drives the Loschmidt echo down to the numerical machine-precision floor (L_min = 2.03e-13), generating an ultra-sharp super-kink with peak amplitude λ_max = 2.9219, nearly three times the baseline transition height. These results reveal a direct route for engineering singular non-equilibrium criticality in digital quantum processors.

We establish a unified framework for dynamical quantum phase transitions (DQPTs) in non-Hermitian systems that encompasses both biorthogonal and self-norm nonbiorthogonal formulations for pure and mixed states under quantum quench protocols. Our framework provides explicit expressions for the Loschmidt amplitude, Loschmidt echo, and rate function, revealing a universal geometric signature of DQPTs in the two-band model: orthogonality of two related vectors in two-dimensional real space. Strikingly, we demonstrate that nonbiorthogonal quenches from non-Hermitian to Hermitian Hamiltonians under chiral symmetry exhibit emergent topological characteristics of DQPTs, unveiling their fundamental features beyond conventional Hermitian regimes. This work establishes fundamental geometric and topological principles governing quantum criticality in open systems, with implications for quantum sensing and many-body physics in dissipative environments.

We investigate dynamical quantum phase transitions (DQPTs) in the extended Agassi model following sudden quenches of same level pairing (

Σ

), monopole-monopole (

χ

) and cross level pairing (

Λ

) strengths. Using exact diagonalization, we characterize the non-equilibrium dynamics through the Loschmidt echo and intensive return-rate function. Quenches across equilibrium phase boundaries produce cusp-like structures in the return rate that shows non-analyticities with increasing system size. Equilibrium finite size to large scaling identifies the

χ

-driven transition as a second order phase transition similar to mean-field, Dicke, and Lipkin-Meshkov-Glick (LMG) models critical behaviour with the exponent of the order parameter

β\simeq 1/2

, while the

Σ

- and

Λ

-driven transitions are first order. We further characterize the associated dynamical singularities through a local scaling exponent

α

, with

α_χ\simeq 2/3

,

α_Λ\simeq 1/3

, and

α_Σ\simeq 3/2

of all three parameters. The distinct scaling behaviours across the parameter channels show that dynamical criticality is not determined solely by the equilibrium order of the transition, revealing multiple dynamical scaling regimes. These non-analyticities are associated with the Fisher zeros of the Loschmidt amplitude, which cross the real time axis at critical times and the time derivatives of the return-rate function to characterize the order of DQPTs within the extended Agassi model.

Dynamical quantum phase transitions (DQPTs) are temporal singularities marked by zeros of the Loschmidt echo, yet their underlying quantum-information structure remains elusive. Here, we introduce a momentum-resolved entanglement entropy as a direct probe of DQPTs in translation-invariant free systems. We analytically establish that every critical momentum mode

k^{*}

associated with a DQPT saturates its entanglement to the maximal value

\ln{2}

, coinciding with the vanishing of the Loschmidt echo. Crucially, we demonstrate that this maximal entanglement universally suppresses information scrambling: a momentum-resolved out-of-time-ordered correlator (OTOC) vanishes identically for all times at

k^{*}

. These three signatures -- Fisher zeros, maximal entanglement, and vanished OTOC -- are proved to be equivalent in both the transverse-field Ising and Su-Schrieffer-Heeger models, despite their distinct bipartitions (momentum-pair vs. sublattice). Our results establish a unified, information-theoretic framework for DQPTs, revealing them a points where quantum correlations saturate and information flow halts. This work elevates entanglement and scrambling to central dynamical order parameters, offering a universal perspective on nonequilibrium quantum critically.

In the context of closed quantum systems, when a system prepared in its ground state undergoes a sudden quench, the resulting Loschmidt echo can exhibit zeros, resembling the Fisher zeros in the theory of classical equilibrium phase transitions. These zeros lead to nonanalytical behavior of the corresponding rate function, which is referred to as the dynamical quantum phase transition (DQPT). In this paper, we investigate DQPTs in the context of open quantum systems that are coupled to both Markovian and non-Markovian dephasing baths via a conserved quantity. The general framework is corroborated by studying the nonequilibrium dynamics of a transverse-field Ising ring. Our paper might shed light on theoretical developments of DQPTs in the strong system-bath coupling regime.

The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Establishing connections between the RH and physical phenomena could offer new perspectives on its physical origin and verification. Here, we establish a direct correspondence between the nontrivial zeros of the zeta function and dynamical quantum phase transitions (DQPTs) in two realizable quantum systems, characterized by the averaged accumulated phase factor and the Loschmidt amplitude, respectively. This precise correspondence reveals that the RH can be viewed as the emergence of DQPTs at a specific temperature. We experimentally demonstrate this correspondence on a five-qubit spin-based system and further propose an universal quantum simulation framework for efficiently realizing both systems with polynomial resources, offering a quantum advantage for numerical verification of the RH. These findings uncover an intrinsic link between nonequilibrium critical dynamics and the RH, positioning quantum computing as a powerful platform for exploring one of mathematics' most enduring conjectures and beyond.

A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that nonanalyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices, we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples---both in the integrable and the nonintegrable cases---where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.