Mathematical Physics
- [1] arXiv:2405.15029 [pdf, ps, html, other]
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Title: Layers of planar hexagonal heterostructure modeled by quantum graphsComments: We demonstrate that if the magnetic flux is constant in the hexagonal network and is a rational multiple of 2π, then there will be values of this flux such that, for certain boundary conditions at the vertices (modeling the hBN), the conical touches in the operator scattering relation will cease to exist and we guarantee the existence of gapsSubjects: Mathematical Physics (math-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Functional Analysis (math.FA); Spectral Theory (math.SP)
The work presents a study on the quantum theory of periodic graphs applied to mono- and bilayer hexagonal materials. Different parameters associated with the atoms present at the vertices of these materials were analyzed, verifying the existence of gaps in the spectral bands and expressing the width of these openings according to the parameters. The study was extended to heterostructures with mixed layers and "sandwiches" of graphene and hexagonal boron nitride. The dispersion relationships obtained in these models were analyzed and it was concluded that the inclusion of hBN layers on graphene layers can induce a gap in the graphene. Furthermore, it was observed that the inclusion of a single layer of graphene between two layers of hBN reduces the width of the spectral gap. The interaction between carbon atoms and nitrogen and boron atoms was pointed out as responsible for these results. Finally, the inclusion of a magnetic field in the hBN layer was considered, demonstrating that specific values of magnetic flux can eliminate conical touches in the dispersion relation of the operator and ensure the existence of gaps.
- [2] arXiv:2405.15048 [pdf, ps, other]
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Title: Two-center problem with harmonic-like interactions: periodic orbits and integrabilityComments: 22 pagesSubjects: Mathematical Physics (math-ph)
We study the classical planar two-center problem of a particle $m$ subjected to harmonic-like interactions with two fixed centers. For convenient values of the dimensionless parameter of this problem we use the averaging theory for showing analytically the existence of periodic orbits bifurcating from two of the three equilibrium points of the Hamiltonian system modeling this problem. Moreover, it is shown that the system is generically non-integrable in the sense of Liouville-Arnold. The analytical results are complemented by numerical computations of the Poincaré sections as well as providing some explicit periodic orbits.
- [3] arXiv:2405.15315 [pdf, ps, html, other]
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Title: 2D discrete Yang-Mills equations on the torusComments: 17 pagesSubjects: Mathematical Physics (math-ph); Numerical Analysis (math.NA)
In this paper, we introduce a discretization scheme for the Yang-Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang-Mills equations are presented in the form of both a system of difference equations and a matrix form.
- [4] arXiv:2405.15639 [pdf, ps, html, other]
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Title: Well Posed Origin Anywhere Consistent Systems in Celestial MechanicsComments: 12 pagesSubjects: Mathematical Physics (math-ph); Earth and Planetary Astrophysics (astro-ph.EP); Dynamical Systems (math.DS); Space Physics (physics.space-ph)
Certain measurements in celestial mechanics necessitate having the origin O of a Cartesian coordinate system (CCS) coincide with a point mass. For the two and three body problems we show mathematical inadequacies in Newton's celestial mechanics equations (NCME) when the origin of a coordinate system coincides with a point mass. A certain system of equations of relative differences implied by NCME is free of these inadequacies and is invariant with respect to any CCS translation. A new constant of motion is derived for the relative system. It shows that the universe of relative differences of the $N$-body problem is ``restless''.
New submissions for Monday, 27 May 2024 (showing 4 of 4 entries )
- [5] arXiv:2405.14924 (cross-list from math.PR) [pdf, ps, html, other]
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Title: Upper tail large deviations of the directed landscapeComments: 62 pages, 2 figuresSubjects: Probability (math.PR); Mathematical Physics (math-ph)
Starting from one-point tail bounds, we establish an upper tail large deviation principle for the directed landscape at the metric level. Metrics of finite rate are in one-to-one correspondence with measures supported on a set of countably many paths, and the rate function is given by a certain Kruzhkov entropy of these measures. As an application of our main result, we prove a large deviation principle for the directed geodesic.
- [6] arXiv:2405.14929 (cross-list from cond-mat.str-el) [pdf, ps, html, other]
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Title: Lieb-Schultz-Mattis theorems and generalizations in long-range interacting systemsComments: 4.5 pages + supplemental materialSubjects: Strongly Correlated Electrons (cond-mat.str-el); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph); Operator Algebras (math.OA); Quantum Physics (quant-ph)
In a unified fashion, we establish Lieb-Schultz-Mattis (LSM) theorems and their generalizations in systems with long-range interactions. We show that, for a quantum spin chain, if the interactions decay fast enough as their ranges increase and the Hamiltonian has an anomalous symmetry, the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, these theorems hold when the interactions decay faster than $1/r^2$, with $r$ the distance between the two interacting spins. Moreover, any pure state with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of the theorems through various examples.
- [7] arXiv:2405.14958 (cross-list from hep-th) [pdf, ps, html, other]
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Title: Dirichlet Scalar Determinants On Two-Dimensional Constant Curvature DisksComments: 36 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We compute exactly the scalar determinants $\det(\Delta+M^{2})$ on the two-dimensional round disks of constant curvature $R=0$, $\mp 2$, for any finite boundary length $\ell$ and mass $M$, with Dirichlet boundary conditions, using the $\zeta$-function prescription. When $M^{2}=\pm q(q+1)$, $q\in\mathbb N$, a simple expression involving only elementary functions and the Euler $\Gamma$ function is found. Applications to two-dimensional Liouville and Jackiw-Teitelboim quantum gravity are presented in a separate paper.
- [8] arXiv:2405.15620 (cross-list from math.NT) [pdf, ps, html, other]
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Title: Measures, modular forms, and summation formulas of Poisson typeSubjects: Number Theory (math.NT); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)
In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call $k$-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct $k$-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn-Gonçalves, Lev-Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional $k$-spherical measures.
- [9] arXiv:2405.15648 (cross-list from cond-mat.str-el) [pdf, ps, html, other]
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Title: A journey on self-$G$-alityComments: 14 pagesSubjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
We explore topological manipulations in one spatial dimension, which are defined for a system with a global symmetry and map the system to another one with a dual symmetry. In particular, we discuss fusion category symmetries enhanced by the invariance of the actions of topological manipulations, i.e., self-$G$-alities for topological manipulations. Based on the self-$G$-ality conditions, we provide LSM-type constraints on the ground states of many-body Hamiltonians. We clarify the relationship between different enhanced symmetries and introduce the notion of $\textit{codimension-two transitions}$. We explore concrete lattice models for such self-$G$-alities and find how the self-$G$-ality structures match the IR critical theories.
Cross submissions for Monday, 27 May 2024 (showing 5 of 5 entries )
- [10] arXiv:1904.01048 (replaced) [pdf, ps, html, other]
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Title: Non-compact quantum spin chains as integrable stochastic particle processesComments: 35 pages, 2 figures, v2: typos fixed and references added, v3: typo fixed, v4: minor correctionSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Probability (math.PR)
In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in [Sasamoto-Wadati], [Barraquand-Corwin] and [Povolotsky] in the context of KPZ universality class. We show that they may be mapped onto an integrable $\mathfrak{sl}(2)$ Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a "dual model" of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of $\mathcal{N}=4$ super Yang-Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the $\mathfrak{sl}(2|1)$ superstring that has been derived directly from $\mathcal{N}=4$ SYM.
- [11] arXiv:2107.01720 (replaced) [pdf, ps, html, other]
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Title: Exact solution of an integrable non-equilibrium particle systemComments: 45 pages, 2 figures, v2: minor improvements, v3: typo fixedSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Probability (math.PR); Exactly Solvable and Integrable Systems (nlin.SI)
We consider the integrable family of symmetric boundary-driven interacting particle systems that arise from the non-compact XXX Heisenberg model in one dimension with open boundaries. In contrast to the well-known symmetric exclusion process, the number of particles at each site is unbounded. We show that a finite chain of $N$ sites connected at its ends to two reservoirs can be solved exactly, i.e. the factorial moments of the non-equilibrium steady-state can be written in closed form for each $N$. The solution relies on probabilistic arguments and techniques inspired by integrable systems. It is obtained in two steps: i) the introduction of a dual absorbing process reducing the problem to a finite number of particles; ii) the solution of the dual dynamics exploiting a symmetry obtained from the Quantum Inverse Scattering Method. Long-range correlations are computed in the finite-volume system. The exact solution allows to prove by a direct computation that, in the thermodynamic limit, the system approaches local equilibrium. A by-product of the solution is the algebraic construction of a direct mapping between the non-equilibrium steady state and the equilibrium reversible measure.
- [12] arXiv:2403.18576 (replaced) [pdf, ps, html, other]
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Title: Logarithmic correlation functions in 2D critical percolationComments: V2: Significantly revised version, several new results addedSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th); Probability (math.PR)
It is believed that the large-scale geometric properties of two-dimensional critical percolation are described by a logarithmic conformal field theory, but it has been challenging to exhibit concrete examples of logarithmic singularities and to find an explanation and a physical interpretation, in terms of lattice observables, for their appearance. We show that certain percolation correlation functions receive independent contributions from a large number of similar connectivity events happening at different scales. Combined with scale invariance, this leads to logarithmic divergences. We study several logarithmic correlation functions for critical percolation in the bulk and in the presence of a boundary, including the four-point function of the density (spin) field. Our analysis confirms previous findings, provides new explicit calculations and explains, in terms of lattice observables, the physical mechanism that leads to the logarithmic singularities we discover. Although we adopt conformal field theory (CFT) terminology to present our results, the core of our analysis relies on probabilistic arguments and recent rigorous results on the scaling limit of critical percolation and does not assume a priori the existence of a percolation CFT. As a consequence, our results provide strong support for the validity of a CFT description of critical percolation and a step in the direction of a mathematically rigorous formulation of a logarithmic CFT of two-dimensional critical percolation.
- [13] arXiv:2404.06606 (replaced) [pdf, ps, html, other]
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Title: Internal Lagrangians and spatial-gauge symmetriesComments: 16 pagesSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Differential Geometry (math.DG)
A direct reformulation of the Hamiltonian formalism in terms of the intrinsic geometry of infinitely prolonged differential equations is obtained. Concepts of spatial equation and spatial-gauge symmetry of a Lagrangian system of equations are introduced. A non-covariant canonical variational principle is proposed and demonstrated using the Maxwell equations as an example. A covariant canonical variational principle is formulated. The results obtained are applicable to any variational equations, including those that do not originate in physics.
- [14] arXiv:2405.10646 (replaced) [pdf, ps, html, other]
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Title: On pressureless Euler equation with external forceComments: reference added 24 pages, 4 figuresSubjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI); Fluid Dynamics (physics.flu-dyn)
Hodograph equations for the n-dimensional Euler equations with the constant pressure and external force linear in velocity are presented. They provide us with solutions of the Euler in implicit form and information on existence or absence of gradient catastrophes. It is shown that in even dimensions the constructed solutions are periodic in time for particular subclasses of external forces. Several particular examples in one, two and three dimensions are considered, including the case of Coriolis external force.
- [15] arXiv:2210.17382 (replaced) [pdf, ps, html, other]
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Title: On D. Peterson's presentation of quantum cohomology of $G/P$Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Representation Theory (math.RT)
We prove in full generality that the $T$-equivariant quantum cohomology of any flag variety $G/P$ is isomorphic to the coordinate ring of a stratum of the Peterson scheme associated to the Langlands dual group scheme $G^{\vee}$. This result was discovered by Dale Peterson but remains unpublished. Our isomorphism is constructed using Yun-Zhu's isomorphism and Peterson-Lam-Shimozono's homomorphism.
- [16] arXiv:2212.09688 (replaced) [pdf, ps, html, other]
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Title: Master Actions and Helicity Decomposition for Spin-4 Models in $3D$Comments: 16 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
The present work introduces a master action that interpolates between four self-dual models, $SD(i)$, for describing massive spin-4 particles in $D=2+1$ dimensions. These models are designated by $i=1,2,3$ and $4$, representing the order in derivatives. Our results show that the four descriptions are quantum equivalent through comparison of their correlation functions, up to contact terms. A geometrical approach is demonstrated to be a useful tool in describing the third and fourth order models. The construction of the master action relies on the introduction of mixing terms, which must be free of particle content. We use the helicity decomposition method to verify the absence of particle content in these terms, ensuring the proper functioning of the master action.
- [17] arXiv:2302.00684 (replaced) [pdf, ps, html, other]
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Title: Infinite Derivatives vs Integral Operators. The Moeller-Zwiebach PuzzleComments: 19 pages, 3 figuresJournal-ref: 2024 J. Phys. A: Math. Theor. 57 235202Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We study the relationship between integral and infinite-derivative operators. In particular, we examine the operator $p^{\frac12\,\partial_t^2}\,$ that appears in the theory of $p$-adic string fields, as well as the Moyal product that arises in non-commutative theories. We also attempt to clarify the apparent paradox presented by Moeller and Zwiebach, which highlights the discrepancy between them.
- [18] arXiv:2307.06803 (replaced) [pdf, ps, html, other]
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Title: Generalized double affine Hecke algebras, their representations, and higher Teichm\"uller theoryComments: 51 pages, 23 figures; final accepted versionSubjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph); Representation Theory (math.RT)
Generalized double affine Hecke algebras (GDAHA) are flat deformations of the group algebras of $2$-dimensional crystallographic groups associated to star-shaped simply laced affine Dynkin diagrams. In this paper, we first construct a functor that sends representations of the $\tilde D_4$-type GDAHA to representations of the $\tilde E_6$-type one for specialised parameters. Then, under no restrictions on the parameters, we construct embeddings of both GDAHAs of type $\tilde D_4$ and $\tilde E_6$ into matrix algebras over quantum cluster $\mathcal{X}$-varieties, thus linking to the theory of higher Teichmüller spaces. For $\tilde E_6$, the two explicit representations we provide over distinct quantum tori are shown to be related by quiver reductions and mutations.
- [19] arXiv:2307.11451 (replaced) [pdf, ps, html, other]
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Title: The five gradients inequality on differentiable manifoldsComments: 32 pages -- accepted in Journal de Mathématiques Pures et AppliquéesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Differential Geometry (math.DG); Metric Geometry (math.MG)
The goal of this paper is to derive the so-called five gradients inequality for optimal transport theory for general cost functions on two class of differentiable manifolds: locally compact Lie groups and compact Riemannian manifolds.
- [20] arXiv:2311.15523 (replaced) [pdf, ps, html, other]
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Title: The $D$-module mirror conjecture for flag varietiesComments: Reformulated main result; simplified exposition; added referencesSubjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph); Representation Theory (math.RT)
Rietsch constructed a candidate $T$-equivariant mirror LG model for any flag variety $G/P$. In this paper, we prove the following mirror symmetry prediction: the small $T\times\mathbb{G}_m$-equivariant quantum cohomology of $G/P$ equipped with quantum connection is isomorphic as $D$-modules to the Brieskorn lattice associated to the LG model equipped with Gauss-Manin connection.
- [21] arXiv:2311.16994 (replaced) [pdf, ps, html, other]
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Title: From Snyder space-times to doubly $\kappa$-dependent Yang quantum phase spaces and their generalizationsComments: 10 pages; published versionJournal-ref: Physics Letters B, 138729 (2024)Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
We propose the doubly $\kappa$-dependent Yang quantum phase space which describes the generalization of $D = 4$ Yang model. We postulate that such model is covariant under the generalized Born map, what permits to derive this new model from the earlier proposed $\kappa$-Snyder model. Our model of $D=4$ relativistic Yang quantum phase space depends on five deformation parameters which form two Born map-related dimensionful pairs: $(M,R)$ specifying the standard Yang model and $(\kappa,\tilde{\kappa})$ characterizing the Born-dual $\kappa$-dependence of quantum space-time and quantum fourmomenta sectors; fifth parameter $\rho$ is dimensionless and Born-selfdual. In the last section, we propose the Kaluza-Klein generalization of $D=4$ Yang model and the new quantum Yang models described algebraically by quantum-deformed $\hat{o}(1,5)$ algebras.
- [22] arXiv:2402.13328 (replaced) [pdf, ps, html, other]
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Title: On energy-aware hybrid modelsSubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph)
This study proposes deterministic and stochastic energy-aware hybrid models that should enable simulations of idealized and primitive-equations Geophysical Fluid Dynamics (GFD) models at low resolutions without compromising on quality compared with high-resolution runs. Such hybrid models bridge the data-driven and physics-driven modelling paradigms by combining regional stability and classical GFD models at low resolution that cannot reproduce high-resolution reference flow features (large-scale flows and small-scale vortices) which are, however, resolved. Hybrid models use an energy-aware correction of advection velocity and extra forcing compensating for the drift of the low-resolution model away from the reference phase space. The main advantages of hybrid models are that they allow for physics-driven flow recombination within the reference energy band, reproduce resolved reference flow features, and produce more accurate ensemble forecasts than their classical GFD counterparts.
Hybrid models offer appealing benefits and flexibility to the modelling and forecasting communities, as they are computationally cheap and can use both numerically-computed flows and observations from different sources. All these suggest that the hybrid approach has the potential to exploit low-resolution models for long-term weather forecasts and climate projections thus offering a new cost effective way of GFD modelling.
The proposed hybrid approach has been tested on a three-layer quasi-geostrophic model for a beta-plane Gulf Stream flow configuration. The results show that the low-resolution hybrid model reproduces the reference flow features that are resolved on the coarse grid and also gives a more accurate ensemble forecast than the physics-driven model. - [23] arXiv:2402.19230 (replaced) [pdf, ps, html, other]
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Title: A Simple and Efficient Joint Measurement Strategy for Estimating Fermionic Observables and HamiltoniansComments: 10 + 8 pages, 7 figures. v2: Sections reordered, improved narration, corrected typos, appendix slightly rewritten for claritySubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
We propose a simple scheme to estimate fermionic observables and Hamiltonians relevant in quantum chemistry and correlated fermionic systems. Our approach is based on implementing a measurement that jointly measures noisy versions of any product of two or four Majorana operators in an $N$ mode fermionic system. To realize our measurement we use: (i) a randomization over a set of unitaries that realize products of Majorana fermion operators; (ii) a unitary, sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries; (iii) a measurement of fermionic occupation numbers; (iv) suitable post-processing. Our scheme can estimate expectation values of all quadratic and quartic Majorana monomials to $\epsilon$ precision using $\mathcal{O}(N \log(N)/\epsilon^2)$ and $\mathcal{O}(N^2 \log(N)/\epsilon^2)$ measurement rounds respectively, matching the performance offered by fermionic shadow tomography. In certain settings, such as a rectangular lattice of qubits which encode an $N$ mode fermionic system via the Jordan-Wigner transformation, our scheme can be implemented in circuit depth $\mathcal{O}(N^{1/2})$ with $\mathcal{O}(N^{3/2})$ two-qubit gates, offering an improvement over fermionic and matchgate classical shadows that require depth $\mathcal{O}(N)$ and $\mathcal{O}(N^2)$ two-qubit gates. By benchmarking our method on exemplary molecular Hamiltonians and observing performances comparable to fermionic classical shadows, we demonstrate a novel, competitive alternative to existing strategies.