Hierarchical Winding Number Method

• The hierarchical winding number method is a systematic framework that classifies stationary states in ring-lattice BECs based on quantized phase windings.
• It distinguishes between superfluid and Josephson regimes by correlating energy and angular momentum dependencies with barrier height and lattice discretization.
• The approach offers practical insights into vortex dynamics and current stability, aiding experimental design in atomtronics and quantum simulation.

The hierarchical winding number method in the context of Bose-Einstein condensates (BECs) confined within ring-shaped lattices refers to the systematic classification and analysis of stationary states according to their quantized vorticity (winding number) and the influence of system parameters such as barrier height, particle number, and lattice discretization. This framework captures the topological constraints, energy landscapes, and angular momentum quantization that govern the behavior of supercurrents and vortices in BEC ring lattices, elucidating both the “superfluid” and “Josephson” (lattice) regimes.

1. Winding Number as a Topological Invariant

In a toroidal condensate partitioned into $N_c$ sites by radial barriers, the winding number $n$ denotes the number of $2\pi$ phase windings around the ring: $\psi_n(r, \theta) = \psi_0(r) e^{in\theta}$ where $\theta$ is the angular coordinate, $\psi_0(r)$ is the ground-state amplitude, and $n$ is an integer. This quantized topological charge characterizes persistent currents (superflows), determines the angular momentum per particle, and sets the foundation for classifying stationary states. The winding number is robust as a topological invariant and essential for describing vortex structures in multiply connected geometries.

2. Energy and Angular Momentum Dependencies

The energy and angular momentum of stationary states display sharply contrasting dependencies on $n$ in the two principal regimes of the ring lattice:

Low-Barrier (Superfluid) Regime:

• Energy per particle is quadratic: $E(n) = K n^2 + E_0$, with $K$ determined by the condensate geometry and density profile.
• Angular momentum per particle is linear: $L_z = \hbar n$.
• The maximum metastable winding number $\nu$ is a function of both barrier height $V_b$ and particle number $N$.

High-Barrier (Tight-Binding/Josephson) Regime:

• The ring behaves as an $N_c$-site Josephson junction array (discrete symmetry).
• Energy and angular momentum become periodic functions:

$E(n) \propto \cos\left(\frac{2\pi n}{N_c}\right)$

$L_z(n) \propto \sin\left(\frac{2\pi n}{N_c}\right)$

• Both are periodic in $n$ with period $N_c$.
• The maximum possible winding number for stable states is set only by $N_c$: $|n| < N_c/2$ (for even $N_c$), with “inversion” and “erasure” phenomena occurring for higher $n$.

3. Regime Classification and Physical Mechanisms

The phase diagram for the ring lattice distinguishes clear dynamical and energetic regimes:

• Low-barrier regime: The BEC is effectively contiguous and supports a hierarchy of vortex states up to $n \leq \nu$. When $n$ exceeds $\nu$, the system reduces its topological charge via the dynamic passage of vortex-antivortex pairs or vortex escape, resulting in energy and angular momentum signatures diagnostic of the maximum sustainable current.
• High-barrier regime: The system fragments into $N_c$ weakly coupled condensates, and stable winding states are indexed modulo $N_c$. Charge “inversion” ($n \mapsto n-N_c$ for $n > N_c/2$) and charge “erasure” (at $n = N_c/2$ for even $N_c$) are emergent phenomena stemming from the group-theoretic discrete symmetry. Here, angular momentum vanishes at $n = N_c/2$ even as the energy is maximal, a direct consequence of the sinusoidal winding dependence.

This structure instantiates a natural hierarchy: at low barriers, the hierarchy is set by $N$ and $V_b$; at high barriers, it is set purely by $N_c$, reflecting the discretization of the ring.

4. Detailed Parameter Dependence

The ability to support a given winding number depends acutely on $N$, $V_b$, and $N_c$:

• For a given $N$, increasing $V_b$ rapidly lowers the maximum sustainable $\nu$ (e.g., at $N=10^3$, $\nu=7$ for $V_b=0$; for $N=10^5$, $\nu=9$).
• As the chemical potential $\mu$ crosses below the minimum barrier height, the system undergoes a crossover to the Josephson regime, beyond which $N$ becomes irrelevant in determining the winding hierarchy, and $N_c$ dominates.
• The threshold at which this transition occurs depends on particle number, with larger $N$ pushing the transition to higher $V_b$.

5. Connections to Discrete Symmetry and Hierarchical Group Theory

Prior studies by Pérez-García et al. and Ferrando established a group-theoretical perspective in systems with discrete rotational symmetry, predicting charge erasure and inversion phenomena, and prescribing a modulo $N_c$ hierarchy for winding states. The current analysis directly corroborates and extends these results in the ring lattice context, mapping the full hierarchy of stationary states and transition points as barrier heights and system sizes are varied.

This framework underpins the use of discrete-symmetry-based indices to classify stationary states, providing a robust bridge to lattice models of superfluidity (e.g., Bose-Hubbard models on a ring), atomtronic circuits, and applications in quantum computation involving persistent currents defined by the winding index.

6. Summary Table: Regime Characteristics

Regime	Energy vs $n$	Angular Momentum vs $n$	Maximum Winding $\nu$	Dominant Parameters
Low Barrier (contiguous)	$E \propto n^2$	$L_z \propto n$	Depends on $N$, $V_b$	$N$, $V_b$
High Barrier (lattice)	$E \propto \cos(2\pi n/N_c)$	$L_z \propto \sin(2\pi n/N_c)$	$|n| < N_c/2$ (even $N_c$)	$N_c$

7. Implications and Applications

The hierarchical winding number framework offers direct theoretical and practical benefits:

• It rigorously predicts current-carrying and vortex-carrying metastable states in ring-lattice BECs as functions of experimental control parameters.
• The method reveals the failure mechanisms—such as charge inversion and erasure—that limit superfluid current sustainability, critical for designing atomtronic circuits with quantized persistent currents.
• It provides clear criteria for the construction of effective models (e.g., Bose-Hubbard rings with specified occupancy and site number), supporting developments in quantum simulation, macroscopic quantum coherence, and the realization of quantum bits using persistent current states.
• The hierarchical organization, grounded in both topology and discrete symmetry, forms the foundation for further exploration in systems with more complex geometries, multi-component condensates, and engineered Hamiltonians with controllable symmetry properties.