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arXiv:2604.01856 · Physical Review Research

Curvature-induced bound states in quantum wires

Tim Bergmann, Benjamin Schwager, Jamal Berakdar

quant-phSubmitted Apr 2, 2026• Announced Apr 3, 2026• 1 citation
arXivHTMLPDF

Abstract

A classical particle under spatial constraints is strictly confined to live on a specific space manifold or path, but this assumption is incompatible with the zero-point fluctuations of a quantum particle. One way to describe quantum mechanics under constraints is the confinement potential approach (CPA). For a non-relativistic particle, the CPA maps the problem onto the solution of a Schrödinger-type equation in an isometrically embedded Riemannian submanifold of Euclidean space while the motion along orthogonal directions are decoupled and spatially confined. This approach respects quantum uncertainty, and one of its key results is the appearance of geometry- and metric-induced potentials that affect the stationary states and the dynamics of the particle. For particles constrained to different spaces, such as structures hosting sharp bents, vertices, wedges, conical apices, tips, or self-intersections, a formalism beyond the CPA is needed. Here, a step towards a CPA extension for irregular spaces is presented. After classifying the possible geometric irregularities concerning the CPA formalism, the presentation is focused on a sharply bent quantum wire modeled as an embedded curve with singular (but absolute integrable) curvature. For a subclass fulfilling the additional requirement that the geometric potential is a distribution of first order, a solution scheme for the confined Schrödinger equation is presented based on singular Sturm-Liouville theory and operator theoretic methods. The analytical considerations and numerical simulations evidence the existence of curvature-induced bound states with non-differentiable wave functions localized around the singular point, with an extension well beyond the singularity. Furthermore, a multitude of scattering states appear that may affect the transport and optical properties of the system.

Journal reference
Phys. Rev. Research 8, 033228 (2026)
DOI
10.1103/rt1z-yf5j
Comments
21 pages, 5 figures, submitted to Phys. Rev. Research

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A classical particle under spatial constraints is strictly confined to live on a specific space manifold or path, but this assumption is incompatible with the zero-point fluctuations of a quantum particle. One way to describe quantum mechanics under constraints is the confinement potential approach (CPA). For a non-relativistic particle, the CPA maps the problem onto the solution of a Schrödinger-type equation in an isometrically embedded Riemannian submanifold of Euclidean space while the motion along orthogonal directions are decoupled and spatially confined. This approach respects quantum uncertainty, and one of its key results is the appearance of geometry- and metric-induced potentials that affect the stationary states and the dynamics of the particle. For particles constrained to different spaces, such as structures hosting sharp bents, vertices, wedges, conical apices, tips, or self-intersections, a formalism beyond the CPA is needed. Here, a step towards a CPA extension for irregular spaces is presented. After classifying the possible geometric irregularities concerning the CPA formalism, the presentation is focused on a sharply bent quantum wire modeled as an embedded curve with singular (but absolute integrable) curvature. For a subclass fulfilling the additional requirement that the geometric potential is a distribution of first order, a solution scheme for the confined Schrödinger equation is presented based on singular Sturm-Liouville theory and operator theoretic methods. The analytical considerations and numerical simulations evidence the existence of curvature-induced bound states with non-differentiable wave functions localized around the singular point, with an extension well beyond the singularity. Furthermore, a multitude of scattering states appear that may affect the transport and optical properties of the system.
The motion of a particle rigidly bounded to a surface is discussed, considering the Schr\"odinger equation of a free particle constrained to move, by the action of an external potential, in an infinitely thin sheet of the ordinary three-dimensional space. Contrary to what seems to be the general belief expressed in the literature, this limiting process gives a perfectly well-defined result, provided that we take some simple precautions in the definition of the potentials and wave functions. It can then be shown that the wave function splits into two parts: the normal part, which contains the infinite energies required by the uncertainty principle, and a tangent part which contains "surface potentials" depending both on the Gaussian and mean curvatures. An immediate consequence of these results is the existence of different quantum mechanical properties for two isometric surfaces, as can be seen from the bound state which appears along the edge of a folded (but not stretched) plane. The fact that this surface potential is not a bending invariant (cannot be expressed as a function of the components of the metric tensor and their derivatives) is also interesting from the more general point of view of the quantum mechanics in curved spaces, since it can never be obtained from the classical Lagrangian of an a priori constrained particle without substantial modifications in the usual quantization procedures. Similar calculations are also presented for the case of a particle bounded to a curve. The properties of the constraining spatial potential, necessary to a meaningful limiting process, are discussed in some detail, and, as expected, the resulting Schr\"odinger equation contains a "linear potential" which is a function of the curvature.
Investigating a particle, a charged particle in an electromagnetic field and a particle with spin-orbit interaction confined to a space curve in square and circular cases, respectively, we first find that their effective Hamiltonian can be defined in terms of the ground states of their confined dimensions. And we obtain interesting results: the torsion-induced geometric potential appears in the square case, while in the circular case it vanishes; the geometrically induced gauge potential and Zeeman coupling appear in the circular case; and the geometric gauge potential, curvature and torsion simultaneously emerge in the spin-orbit interaction in the circular case. In the example, a helix nanowire, the previous results are all demonstrated. These results provide a way to generate an artificial gauge field and to manipulate spin transport by designing the geometry of curved nanodevice.
Quantum mechanics is sensitive to the geometry of the underlying space. Here we present a framework for quantum scattering of a nonrelativistic particle confined to a two-dimensional space. When the motion manifold hosts localized curvature modulations, scattering occurs from an emergent geometric potential and the metric tensor field. Analytical and full numerical simulations identify the geometric potential as the primary source for low-energy scattering, while the metric tensor field of the curved space governs high-energy diffraction. Compared to flat spaces, important differences in the validity range of perturbation approaches are found and demonstrated by full numerical simulations using combined finite element and boundary element methods. As an illustration, we consider a Gaussian-shaped dent leading to effects known as gravitational lensing. Experimentally, the considered setup is realizable based on geometrically engineered two-dimensional materials.
We study quantum mechanics on the limiting case of a highly curved wire by approximating the physics around the curved region by three boundary condition parameters coming from the two-interval Sturm-Liouville theory. Since the geometric potential in this case is strong and non-integrable, these parameters depend crucially on the regularization of the curve. Hence, unless we know precisely the shape of the curve, any physical prediction is meaningless. In this context, the method presented in this paper becomes not only a useful approximation, but also a necessary scheme to deal with quantum mechanics on highly curved wires and (possibly) corners.
We investigate bound states of a non-relativistic scalar particle in a three-dimensional helically twisted (torsional) geometry, considering both the free case and the presence of external radial interactions. The dynamics is described by the Schrödinger equation on a curved spatial background and, when included, by minimal coupling to a magnetic vector potential incorporating an Aharonov--Bohm flux. After separation of variables, the problem reduces to a one-dimensional radial eigenvalue equation governed by an effective potential that combines torsion-induced Coulomb-like and centrifugal-like structures with magnetic/flux-dependent terms and optional model interactions. Because closed-form analytic solutions are not reliable over the parameter ranges required for systematic scans, we compute spectra and eigenfunctions numerically by formulating the radial equation as a self-adjoint Sturm--Liouville problem and solving it with a finite-difference discretization on a truncated radial domain, with explicit convergence control. We analyze four representative scenarios: (i) no external potential, (ii) Cornell-type confinement, (iii) Kratzer-type interaction, and (iv) the small-oscillation regime around the minimum of a Morse potential. We present systematic trends of the low-lying levels as functions of the torsion parameter, magnetic field, and azimuthal sector, and we show that geometric couplings alone can produce effective confinement even in the absence of an external interaction.
When a quantum particle moves in a curved space, a geometric potential can arise. In spite of a long history of extensive theoretical studies, to experimentally observe the geometric potential remains a challenge. What are the physically observable consequences of such a geometric potential? Solving the Schr\"odinger equation on a truncated conic surface, we uncover a class of quantum scattering states that bear a strong resemblance to the quasiresonant states associated with atomic collapse about a Coulomb impurity, a remarkable quantum phenomenon in which an infinite number of quasiresonant states emerge. A characteristic defining feature of such collapse states is the infinite oscillations of the local density of states (LDOS) about the zero energy point separating the scattering from the bound states. The emergence of such states in the curved (Riemannian) space requires neither a relativistic quantum mechanism nor any Coulomb impurity: they have zero angular momentum and their origin is purely geometrical, hence the term ``geometry-induced wave-function collapse.'' We establish the collapsing nature of these states through a detailed comparative analysis of the behavior of the LDOS for both the zero and finite angular momentum states as well as the corresponding classical picture. Potential experimental schemes to realize the geometry-induced collapse states are articulated. Not only does our paper uncover an intrinsic connection between the geometric potential and atomic collapse, it also provides a method to experimentally observe and characterize geometric potentials arising from different subfields of physics. For example, in nanoscience and nanotechnology, curved geometry has become increasingly common. Our finding suggests that wave-function collapse should be an important factor of consideration in designing and developing nanodevices.