---
title: Helical Quantum Two-Body Problem
url: https://www.emergentmind.com/topics/helical-quantum-two-body-problem
type: topic
---

# Helical Quantum Two-Body Problem

to=arxiv_search.search +天天中彩票{"query":"\"helical quantum two-body problem\" OR helix two-body problem helical Majorana", "max_results": 10}
to=arxiv_search.search  天天中彩票公司json{"query":"all:\"helical quantum two-body problem\" OR ti:\"Two-impurity helical Majorana problem\" OR ti:\"Phase diagram of two interacting helical states\" OR ti:\"Quantum few-body bound states of dipolar particles in a helical geometry\"", "max_results": 10}
The helical quantum two-body problem denotes a class of minimal quantum problems in which two interacting degrees of freedom are constrained by helical kinematics or by helical edge structure. In current arXiv usage, the phrase spans at least three technically distinct settings: two charged particles confined to a geometric helix and interacting through the full three-dimensional Coulomb law; two aligned dipoles confined to a helical trap, which produces an oscillatory effective interaction with multiple attractive wells; and interacting helical edge systems in topological matter, including two spin-\(1/2\) impurities coupled to a helical Majorana edge and two coupled helical edge modes whose low-energy description separates into total and relative sectors [2603.09093, 1507.08138, 1412.5317, 1601.03851]. Across these realizations, the common structural feature is that helicity reorganizes the relative coordinate, the effective interaction, or the dissipative environment in a way not available in straight one-dimensional geometries.

## 1. Geometric formulation on a helix

In the geometric realizations, a helix of radius \(R\) and pitch \(h\) is parameterized by an angular variable \(\phi\), with arc-length scale
\[
a=\sqrt{R^2+\left(\frac{h}{2\pi}\right)^2},
\qquad s=a\phi .
\]
For two identical particles constrained to move on the same helix, the center-of-mass coordinate \(S=(s_1+s_2)/2\) separates from the relative coordinate \(s=s_1-s_2\), and the relative Schrödinger equation takes the standard form
\[
-\frac{\hbar^2}{2\mu}\frac{d^2\psi}{ds^2}+V(s)\psi(s)=E\psi(s),
\qquad \mu=\frac{m}{2},
\]
because the helix has constant curvature and torsion [2603.09093].

For Coulomb-interacting particles, the effective interaction is obtained by evaluating the full three-dimensional distance between two points on the helix separated by \(\Delta\phi\):
\[
d(\Delta\phi)=\sqrt{2R^2(1-\cos\Delta\phi)+\left(\frac{h}{2\pi}\Delta\phi\right)^2},
\]
so that
\[
V(\Delta\phi)=\frac{k_e q_1 q_2}{d(\Delta\phi)}.
\]
Introducing
\[
\eta=\frac{h}{2\pi R},
\]
the extrema satisfy
\[
\sin\Delta\phi+\eta^2\Delta\phi=0,
\]
and a local minimum of \(V\) occurs precisely when
\[
\cos\Delta\phi_m+\eta^2<0.
\]
This converts a purely repulsive Coulomb law into a multi-well effective potential superimposed on a repulsive background, with the number of wells controlled by geometry alone [2603.09093].

For dipoles aligned along the helix symmetry axis, the projection of the three-dimensional dipole-dipole interaction onto the helix yields
\[
V(\Delta\phi)=\frac{d^2}{4\pi\epsilon_0}\,
\frac{2R^2[1-\cos\Delta\phi]-2h^2\left(\frac{\Delta\phi}{2\pi}\right)^2}
{\left(2R^2[1-\cos\Delta\phi]+h^2\left(\frac{\Delta\phi}{2\pi}\right)^2\right)^{5/2}},
\]
or equivalently a reduced potential
\[
\tilde V(\phi)=
\frac{1-\cos\phi-\left(\frac{h\phi}{2\pi R}\right)^2}
{\left(2[1-\cos\phi]+\left(\frac{h\phi}{2\pi R}\right)^2\right)^{5/2}} .
\]
In this case the helix generates an oscillatory sequence of attractive wells separated by repulsive barriers, with minima slightly below \(2\pi n\), corresponding to approximately one, two, or more winding separations [1507.08138]. At short distance, the interaction is repulsive only for
\[
\frac{h}{R}<\sqrt{2}\,\pi,
\]
which is the regime explicitly retained in the dipolar analysis [1507.08138].

## 2. Bound-state structure and geometry-controlled spectra

The geometric Coulomb problem is governed, after canonical scaling \(s'=s/a\), by the single dimensionless parameter \(\eta=h/(2\pi R)\). The existence and number of wells depend only on the ratio \(r=h/R\), with bifurcation points listed at
\[
r\approx 2.929,\ 1.899,\ 1.513,\ 1.295,\ 1.151,
\]
equivalently
\[
\eta\approx 0.467,\ 0.303,\ 0.241,\ 0.206,\ 0.183,
\]
corresponding to the appearance of one up to five wells [2603.09093]. As \(\eta\) decreases, more wells emerge, outer wells appear at larger \(|s|\), and anharmonicity increases.

Near a minimum \(s_m=a\Delta\phi_m\), both the Coulomb and dipolar problems admit a local harmonic approximation,
\[
V(s)\approx V(s_m)+\frac{1}{2}V''(s_m)(s-s_m)^2,
\]
with local frequency
\[
\omega_m=\sqrt{\frac{V''(s_m)}{\mu}} .
\]
In the Coulomb case, the isolated-well spectra were computed by cutting the potential to a constant at the neighboring maxima, discretizing with 10th-order finite differences, and diagonalizing. Two representative numerical results summarize the dependence on geometry and well order: for the innermost well with \(h=20\), \(R=20\), \(M=1\), there are \(18\) bound states and the level spacings increase approximately linearly and by \(\approx 100\%\) across the ladder; for the innermost well with \(h=12.2\), \(R=10\), \(M=100\), there are \(32\) bound states and the spacings increase only by \(\approx 20\%\), so the spectrum is much closer to harmonic [2603.09093]. The paper interprets this as evidence that anharmonicity depends crucially on \(\eta\) and on the order \(m\) of the well.

The same control appears in the multi-well examples used for dynamics. For \(h=10\), \(R=10\), \(M=1\), the isolated-well bound-state counts are \(N_{b,m}=5,3,1,1,0,0\) for \(m=1,\ldots,6\); for the heavier-mass case \(h=10\), \(R=10\), \(M=10\), they become \(14,7,2,2,2,0\) [2603.09093]. The systematic decrease with well order reflects shallower barriers and stronger anharmonicity in outer wells.

For dipoles, the dimensionless Hamiltonian is
\[
\tilde H=-\frac{1}{2}\frac{d^2}{d\phi^2}+\beta\,\tilde V(\phi),
\qquad
\beta=\frac{\mu d^2}{2\pi\epsilon_0\hbar^2R^3}(1+\eta^2),
\]
and the harmonic expansion around a minimum \(\phi_0\) gives
\[
\tilde E_n\approx \beta \tilde V(\phi_0)+\sqrt{\beta \tilde V''(\phi_0)}\left(n+\frac{1}{2}\right).
\]
For \(h=R\) and \(\beta=1\), the two-dipole system has three bound states below threshold; the ground state is localized near the first minimum at approximately one winding separation, the first excited state near the second well, and the second excited state spreads across the first three wells with the expected nodal structure [1507.08138]. In the strong-coupling regime, the size scales as \(\langle(\delta\phi)^2\rangle\propto \beta^{-1/2}\), while near threshold \(\langle\phi^2\rangle\propto 1/|E|\) [1507.08138].

## 3. Wave-packet dynamics in the helical Coulomb landscape

The most explicit dynamical realization of the helical quantum two-body problem studies Gaussian relative-coordinate wave packets,
\[
\psi(s,0)=\left(\frac{1}{2\pi\sigma^2}\right)^{1/4}
\exp\!\left[-\frac{(s-s_0)^2}{4\sigma^2}+ik_0(s-s_0)\right],
\]
propagated with the MCTDH method on sine-DVR grids such as \(2301\) points on \(s\in[-150,1000]\), \(3301\) points on \([-150,1500]\), \(4001\) points on \([-500,1500]\), and \(4301\) points on \([-150,2000]\), using open boundaries and a regularization near the Coulomb singularity [2603.09093]. The standard probability current,
\[
j(s,t)=\frac{\hbar}{\mu}\operatorname{Im}\!\left[\psi^*(s,t)\frac{\partial\psi}{\partial s}\right],
\]
organizes the emitted pulses and reflected structures.

For the three-well landscape \(h=5.8\), \(R=4\), \(M=1\), with minima at \(s\approx 13,41,69\), only the innermost well supports a single bound state. A far-out packet prepared at \(s_0=220\), \(\Delta s=4\), \(p_0=0\) develops beats superimposed on the reflected pulse, while the integrated well occupations rise first for the outer wells and later for the inner ones. A packet initially centered in the second well at \(s_0=40.99\), \(\Delta s=6\), \(p_0=0\) evolves into a broad double peak roughly aligned with the second and third wells, and the occupation of the second well decays monotonically. A packet in the innermost well at \(s_0=13.63\), \(\Delta s=4.5\), \(p_0=0\) shows monotonous depletion to about \(10\%\) left at \(t=2000\), with weak leaking pulses to the outer wells. With finite incoming momentum \(p_0=-0.154\), the reflected density forms a multi-beat pulse; for \(p_0=-0.8\), over-the-barrier transmission produces separate transmitted and reflected packets [2603.09093].

For the six-well landscape \(h=10\), \(R=10\), \(M=1\), with minima at \(s\approx 32.5,98,163,229,295,361\), the dynamics is substantially richer because the first four wells support bound states. A far-out packet at \(s_0=350\), \(\Delta s=4\), \(p_0=0\) displays multi-scale oscillations, then reshapes into three large peaks followed by a decaying oscillatory tail, and later into a structured pulse with rising-amplitude peaks followed by beats. A packet in the innermost well at \(s_0=32.5\), \(\Delta s=4\), \(p_0=0\) exhibits a persistent broad peak with transient double-peak modulation and “pulsed emission,” seen in the integrated well occupations as mini-plateaus and sequential rises of the outer wells. For \(p_0=-0.3\), double peaks and beats appear on a nonzero background, reflecting interference within and between wells [2603.09093].

The heavier-mass case \(h=10\), \(R=10\), \(M=10\) amplifies the same mechanisms because each well supports more bound levels. For a packet in the second well at \(s_0=98\), \(\Delta s=1.2\), \(p_0=0\), the density retains a broad trapped peak while emitting a prominent pulse between \(s\approx 200\) and \(700\) at \(t=3000\); the corresponding well occupations show rapid initial decay, slower approach to an asymptote, and superimposed plateaus. The paper identifies this as direct evidence for pulsed emission governed by intrawell beating. More generally, the beat frequencies are
\[
\omega_{\text{beat}}=\frac{|E_{n,m}-E_{n',m}|}{\hbar},
\]
so the number of beat scales is controlled by the number of bound states within each well [2603.09093].

## 4. Two localized spins coupled to a helical Majorana edge

A distinct topological realization of the helical two-body problem consists of two spin-\(1/2\) impurities, realized as quantum dots in the local-moment regime, coupled to the helical Majorana edge of a two-dimensional time-reversal-invariant topological superconductor [1412.5317]. The boundary hosts counterpropagating self-adjoint Majorana fields \(\psi_R,\psi_L\) with Hamiltonian
\[
H_0=-iv\int_{-\infty}^{\infty}dx\,
\left[\psi_R\partial_x\psi_R-\psi_L\partial_x\psi_L\right],
\]
and anticommutators
\[
\{\psi_\nu(x),\psi_{\nu'}(x')\}=\delta_{\nu\nu'}\delta(x-x') .
\]
Because of Majorana algebra, only a single nonvanishing spin-density operator exists,
\[
s(x)=i\psi_R(x)\psi_L(x),
\]
which points along a fixed Ising axis
\[
\hat e_I=(\cos\theta,\sin\theta,0).
\]
Accordingly, only the impurity projections
\[
S_{I,j}=\hat e_I\cdot S_j=S_{x,j}\cos\theta+S_{y,j}\sin\theta
\]
couple to the edge, via
\[
H_J=J\,[s(-R/2)S_{I,1}+s(R/2)S_{I,2}] .
\]

The bulk quasiparticles are gapped below \(\Delta\) but mediate static RKKY interactions, so that
\[
H_D=-K\,S_1\cdot S_2+K'S_{y,1}S_{y,2}-B\cdot(S_1+S_2),
\]
with
\[
K\approx \frac{J_b^2}{16\pi v_b R^3}, \qquad K'\approx \frac{3}{2}K
\]
for \(R\ll \xi=v_b/\Delta\). Integrating out the helical Majorana edge at small \(J\) produces an additional Ising interaction
\[
H_M=K_M S_{I,1}S_{I,2},
\qquad
K_M=\frac{\pi v}{4R}(\rho_0J)^2,
\qquad
\rho_0=\frac{1}{2\pi v},
\]
which is antiferromagnetic, strictly Ising, and long-ranged as \(1/R\), in contrast to the bulk \(R^{-3}\) terms [1412.5317].

Combining the bulk and edge contributions yields
\[
H_{\text{eff}}
=
-\frac{K}{2}S^2
+\frac{K_M\cos^2\theta}{2}S_x^2
+\frac{K'+K_M\sin^2\theta}{2}S_y^2
-B\cdot S ,
\qquad
S=S_1+S_2 .
\]
Using the basis \(|S,M\rangle\) quantized along \(y\), the energies are
\[
E_s=K,
\qquad
E_{t,0}=\frac{K_M}{2}\cos^2\theta,
\]
\[
E_{t,\pm}
=
\frac{2K'+K_M(1+\sin^2\theta)}{4}
\pm
\sqrt{B_y^2+\left(\frac{K_M}{4}\cos^4\theta\right)^2} .
\]
At \(B=0\) and \(\theta=\pi/2\), the ground state is always the entangled triplet \(|1,0\rangle\), minimizing both the ferromagnetic \(-K S_1\cdot S_2\) term and the antiferromagnetic Ising term \(K'+K_M\). At \(B=0\) and \(\theta=0\), the ground state changes at a critical separation \(R_c\) determined by \(K'(R_c)=K_M(R_c)\), which yields a quantum phase transition driven by the competition between the bulk \(y\)-axis anisotropy and the edge Ising axis [1412.5317].

The qualitative distinction from the conventional two-impurity Kondo problem is explicit: the helical Majorana edge couples only through a single Ising component, Kondo screening is absent in zero field, and there are no \(2k_F\) oscillations because particle-hole symmetry implies \(k_F=0\) [1412.5317].

## 5. Dissipative reduction and universal quench dynamics

The same two-impurity Majorana system admits an exact low-energy reduction to an Ohmic dissipative problem. Combining the two Majoranas into a chiral Dirac field,
\[
\Psi(x)=\frac{\psi_R(x)+i\psi_L(-x)}{\sqrt{2}},
\qquad
H_0=-iv\int dx\,\Psi^\dagger\partial_x\Psi,
\]
and expanding at energies below \(v/R\), the only marginal local edge coupling is
\[
H_J^{(1)}=J S_I \Psi^\dagger(0)\Psi(0),
\qquad
S_I=S_{I,1}+S_{I,2},
\]
while the operator proportional to \(S_{I,1}-S_{I,2}\) is irrelevant and flows to zero under one-loop RG. Bosonizing \(\Psi(x)=e^{-i\phi(x)}/\sqrt{2\pi R}\) and integrating out the Gaussian bosonic field yields the exact spin action
\[
S_{\text{spin}}
=
-\frac{1}{2}(\rho_0J)^2
\int d\tau d\tau'
\frac{S_I(\tau)S_I(\tau')}{(\tau-\tau')^2+(R/v)^2}
+
\frac{v}{R}(\rho_0J)^2\int d\tau\,S_I^2(\tau),
\]
whose first term is Ohmic damping and whose second reproduces the edge-mediated RKKY interaction up to \(O(1)\) factors [1412.5317].

This action is equivalent to a dissipative Hamiltonian
\[
H=H_{\text{eff}}+(S_x\cos\theta+S_y\sin\theta)\,\mathcal{E}+H_B[\mathcal{E}],
\]
with bath correlator
\[
L(z)=\langle \mathcal{E}(0)\mathcal{E}(z)\rangle
=
\frac{1}{\pi}\int_0^\infty d\omega\,
J(\omega)\,
\frac{\cosh[\omega(-iz+\beta/2)]}{\sinh(\beta\omega/2)},
\]
and Ohmic spectral density
\[
J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/(v/R)},
\qquad
\alpha=\frac{1}{2}(\rho_0J)^2 .
\]
For \(T=0\),
\[
L(z)=-\frac{2\alpha}{(z-iR/v)^2}.
\]

Choosing \(\theta=\pi/2\), \(B_z=0\), and
\[
\epsilon=-B_y+\frac{K'+K_M}{2},
\]
the regime
\[
\max\{|\epsilon|,|B_{x,z}|,T\}\ll K
\]
permits projection to the two lowest triplet states \(|1,1\rangle\) and \(|1,0\rangle\). The resulting two-level Hamiltonian is the Ohmic spin-boson model
\[
H_{SB}=-\frac{B_x}{\sqrt{2}}\sigma_x+\frac{\epsilon+\mathcal{E}}{2}\sigma_z+H_B .
\]
After a quench from \(\epsilon(t<0)\ll 0\) to \(\epsilon(t>0)=0\) at \(t=0\), with \(B_x\) held fixed, the long-time zero-temperature dynamics is
\[
\langle \sigma_z(t)\rangle \simeq e^{-\Gamma t}\cos(\Omega t),
\qquad
\langle S_y(t)\rangle=\frac{\langle \sigma_z(t)\rangle+1}{2},
\]
with
\[
\Gamma=2T_b\sin^2\!\left[\frac{\pi\alpha}{2(1-\alpha)}\right],
\qquad
Q\equiv \frac{\Omega}{\Gamma}
=
\cot\!\left[\frac{\pi\alpha}{2(1-\alpha)}\right],
\]
and
\[
T_b=c_b\left(\frac{RB_x}{v}\right)^{\alpha/(1-\alpha)}B_x .
\]
The quality factor \(Q\) is therefore universal in the sense that it depends only on \(\alpha=(\rho_0J)^2/2\) and is independent of \(B_x\) [1412.5317].

The proposed experimental signatures combine static spectroscopy and time-domain control. The static signal is the long-range Ising RKKY term \(K_M\propto J^2/R\) without \(2k_F\) oscillations; the dynamical signal is weakly damped oscillation of \(\langle S_y(t)\rangle\) after a magnetic-field quench. For \(v\sim 10^5\) m/s and \(R\sim 10\)–\(50\) nm, the cutoff \(v/R\) lies in the \(1\)–\(10\) THz range, while for \(\alpha\ll 1\) and \(B_x\sim 0.05\)–\(0.5\) K, both \(\Omega\) and \(\Gamma\) lie in the GHz range with nanosecond time scales [1412.5317].

## 6. Coupled helical edge modes, topology, and common structure

Another helical two-body problem arises in two stacked quantum spin Hall insulators with helical edge states of the same helicity [1601.03851]. The clean Hamiltonian contains kinetic energy, inter-edge tunneling \(t_\perp\), spin-orbit coupling \(\alpha_{\rm SO}\), and intra- and inter-edge density interactions \(U_0,U\). After diagonalizing the single-particle sector and linearizing around the four Fermi points, bosonization reorganizes the problem into a gapless total sector \((\varphi_+,\theta_+)\) and a relative sector \((\varphi_-,\theta_-)\):
\[
H=H_++H_-,
\]
\[
H_+=\frac{u_+}{2}\int dx\left[\frac{(\partial_x\varphi_+)^2}{K}+K(\partial_x\theta_+)^2\right],
\]
\[
H_-=\frac{u_-}{2}\int dx\left[(\partial_x\varphi_-)^2+(\partial_x\theta_-)^2\right]
-\frac{g'}{\pi a_0^2}\int dx\left[\cos(\sqrt{8\pi}\theta_-)-\cos(\sqrt{8\pi}\varphi_-+2\Delta k_F x)\right].
\]
Because helicity fixes the bare relative-sector Luttinger parameter to \(K_-=1\), both cosine operators are marginal at tree level. At energies above \(t_\perp\), the theory remains on a self-dual manifold with no one-loop RG flow. At energies below \(t_\perp\), the oscillatory \(\cos(\sqrt{8\pi}\varphi_-+2\Delta k_Fx)\) term averages to zero, leaving a sine-Gordon theory in \(\theta_-\) with BKT-type flow to strong coupling. Consequently, the relative mode becomes gapped only when \(t_\perp\neq 0\), with gap
\[
\Delta_{\rm edge}\sim
t_\perp\exp\!\left[-\frac{\pi}{2|\tilde g|}\right]
\propto
t_\perp
\exp\!\left(-\frac{\pi^2 u_-}{a_0|U-U_0|}\right).
\]

The sign of \(g'=(U_0-U)a_0/2\pi\) selects the gapped phase. For \(U_0>U\), the field \(\theta_-\) pins at \(\sqrt{\pi/2}\,n\), producing a spin-nematic phase with dominant order parameter
\[
\mathcal O_{\rm SN}
=
\frac{2}{\pi a_0}\cos(\sqrt{2\pi}\theta_-)\cos(\sqrt{2\pi}\varphi_+),
\]
which corresponds, in the tilted spin basis, to a spiral pattern of spin currents between the edges. For \(U>U_0\), \(\theta_-\) shifts by half a period and the dominant order is
\[
\mathcal O_{\rm SDW}
=
\frac{2}{\pi a_0}\sin(\sqrt{2\pi}\theta_-)\cos(\sqrt{2\pi}\varphi_+),
\]
a spiral spin-density wave with antiferromagnetic alignment between edges. The phase boundary lies at \(U_0=U\), where the relative mode remains gapless [1601.03851].

The gapped relative mode has direct topological and transport consequences. A nonmagnetic impurity generates a backscattering operator proportional to \(\cos(\sqrt{2\pi}\theta_-)\cos(\sqrt{2\pi}\varphi_+)\). In the spin-nematic phase, \(\langle \cos(\sqrt{2\pi}\theta_-)\rangle\neq 0\), so this operator is relevant for repulsive interactions and localizes the system, driving the conductance to zero as \(T\to 0\). In the spin-density-wave phase, \(\langle \cos(\sqrt{2\pi}\theta_-)\rangle=0\), so single-impurity backscattering averages to zero and the conducting charge mode remains protected. For random disorder, integrating out the massive relative sector maps the problem to the Giamarchi–Schulz model with effective Luttinger parameter \(K'=2K\), implying that disorder is relevant only if \(K<3/4\) [1601.03851].

Strong nonmagnetic impurities that pinch off a finite segment produce boundary terms incompatible with the bulk pinning of \(\theta_-\) in the spin-density-wave phase. The resulting kink has magnitude \(\pm \tfrac{1}{2}\sqrt{\pi/2}\), carries fractional spin
\[
S^z=\pm \frac{1}{4},
\]
and corresponds to localized zero-energy boundary states. Tunneling spectroscopy is predicted to show zero-bias anomalies localized at the endpoints, while away from the endpoints the local density of states develops a hard gap of size \(\Delta_{\rm edge}\) [1601.03851].

Taken together, these realizations suggest a broad organizing principle: in helical systems, the minimal two-body sector is rarely a trivial reduction of a straight one-dimensional problem. In geometric helices, the embedding in \(\mathbb R^3\) turns Coulomb or dipolar interactions into oscillatory effective potentials with geometry-tunable wells and nontrivial bound-state ladders. In helical topological systems, helicity restricts the operator content, so the effective low-energy problem becomes a competition of anisotropic RKKY couplings, a gapped relative mode, or an Ohmic dissipative two-level system. The recurring outcome is that helicity reshapes the relative coordinate into the central dynamical object, whether through multi-well confinement, emergent topological order, or universal quench dynamics [2603.09093, 1507.08138, 1412.5317, 1601.03851].

Source: https://www.emergentmind.com/topics/helical-quantum-two-body-problem