Real-Space Winding Number

• Real-space winding number is an integer invariant that classifies nontrivial topological phases by quantifying protected edge modes in disordered systems.
• It employs methods like the Kitaev flow and Bott index to compute mappings between real-space lattices and internal manifolds, ensuring robust quantization.
• Experimental implementations in cold-atom, photonic, and topolectrical systems verify the observable bulk-boundary correspondence across diverse setups.

A real-space winding number is an integer-valued topological invariant defined directly in the position representation, characterizing nontrivial topology in condensed matter systems, field theory, and nonlinear sigma models, without requiring translational symmetry or momentum-space representations. Real-space winding numbers apply to mappings between real-space lattices and internal manifolds or to the algebraic "flow" and commutator expressions involving position and Hamiltonian or unitary operators. They play a central role in classifying topological phases, diagnosing localized states, and ensuring robust quantization in the presence of disorder, interactions, and open boundaries.

1. Fundamental Definitions and Classifications

The real-space winding number is fundamentally associated with integer-valued homotopy classes of mappings between real-space and internal spaces, or equivalently with integer invariants in the algebra of local operators. For one-dimensional (1D) systems, the archetypal setting involves a chiral-symmetric Hamiltonian acting on a lattice, where the winding number counts the number of protected zero-energy edge modes under open boundary conditions. The construction generalizes to higher dimensions, notably in three-dimensional (3D) chiral insulators, nonlinear sigma models, higher-order topological insulators (HOTIs), and non-Hermitian settings such as systems with the skin effect.

The winding number is preserved under smooth deformations that do not close bulk energy gaps or introduce singularities into the relevant mapping, thus serving as a robust classifier of topological phases.

2. Real-Space Flow Approach: Generalization and Computation

The Kitaev "flow" formalism provides a topologically rigorous definition of the winding number in real space for odd spatial dimensions. For 1D systems, given a lattice with sites indexed by integers, a system is partitioned into two regions (e.g., $i \ge 0$ and $i < 0$) using projectors $\Pi$, and the flow is defined via traces and commutators: ${\cal F}_1(U) = \Tr[U^\dagger[\Pi,U]] = \Tr(U^\dagger\Pi U - \Pi)$ where $U$ is a unitary derived from the Hamiltonian (often the unitarized off-diagonal block for chiral systems). For translationally invariant systems, this reduces to the canonical momentum-space winding number. The flow is well-defined and quantized even without translational symmetry, remaining robust against disorder or quasiperiodicity.

In three dimensions, the flow generalizes to a trace involving commutators with half-space projectors or position operators: ${\cal F}_3(U) = -2\pi i \Tr\left(\epsilon^{ABCD} U^\dagger\Pi_A U\Pi_B U^\dagger\Pi_C U\Pi_D\right)$ with appropriate choices of regions indexed by $A,B,C,D$, and with projectors $\{\Pi_A\}$ onto orthants of the cubic lattice. This expression, as shown by combinatorial arguments, matches the conventional Brillouin-zone winding number when translation symmetry is present and remains topologically valid otherwise (Hamano et al., 2024).

Numerically, real-space flow invariants are efficiently computed by introducing truncated projectors to localize the calculation in large finite systems, observing convergence to quantized plateaus that identify the bulk invariant irrespective of boundary conditions.

3. Alternative Real-Space Representations: Bott Index and Projected Position

Independent and equivalently powerful approaches recast the winding number as the index (specifically, a Bott index) quantifying commutation between almost-unitary operators in real space. For 1D chiral-symmetric systems, after singular value decomposition and flattening, the winding number may be written as

$\nu = \frac{1}{2\pi i} \Tr\log[\mathcal{X}_A\mathcal{X}_B^{-1}]$

where $\mathcal{X}_A$, $i < 0$0 are projected position operators on the two sublattices, or, equivalently, as a trace formula involving commutators with position operators (Lin et al., 2021). Quantization is exact as long as the chiral symmetry and spectral gap persist, even in disordered systems.

For higher-order topological insulators (HOTIs), specifically in 2D and 3D chiral settings, a "multipole winding number" is constructed by applying corner-twisted boundary conditions, then evaluating commutators with "multipole" position-like operators in the bulk: $i < 0$1 where $i < 0$2 is the flattened off-diagonal block, and $i < 0$3 encodes the real-space quadrupole moment (Lin et al., 2024).

4. Real-Space Winding in Non-Hermitian and Disordered Systems

For non-Hermitian systems, notably those with point gaps and the non-Hermitian skin effect, real-space winding numbers are extended via polar decompositions: $i < 0$4 where $i < 0$5 emerges from the phase factor in the polar decomposition of $i < 0$6 and $i < 0$7 is the trace per unit volume. This invariant is quantized, self-averaging in the thermodynamic limit, and robust under arbitrary local disorder as long as the point gap at $i < 0$8 does not close, enabling accurate phase diagrams even in strongly disordered, non-Hermitian phases (Claes et al., 2020).

5. Lattice Discretization and Numerical Implementation in Higher Dimensions

For mappings from discrete $i < 0$9-dimensional tori to $\Pi$0 (e.g., $\Pi$1), lattice discretizations of the winding number are constructed to minimize finite-size effects: $\Pi$2 with $\Pi$3 a finite-difference approximation to $\Pi$4 involving four-point stencils for improved accuracy. Applying a lattice gradient flow (with an "over-improved" action) suppresses discretization errors and stabilizes the winding sector under local noise or coarseness. The method is verified to yield near-integer results with errors less than $\Pi$5 even on coarse lattices, and generalizes to $\Pi$6 with corresponding combinatorics (Morikawa et al., 2024).

6. Physical Significance, Bulk-Boundary Correspondence, and Experimental Relevance

Real-space winding numbers underpin bulk-boundary correspondence: a nonzero invariant signals the presence of robust, topologically protected zero modes (e.g., at the edge in 1D, at corners in HOTIs, or at boundaries in 3D insulators). Unlike Berry phase, twisted boundary, or momentum-space invariants, real-space formulas require no integration over auxiliary parameters and remain directly accessible in both numerical and experimental contexts, independent of crystalline order.

In cold-atom, photonic, and topolectrical circuit platforms, real-space invariants can be probed via dynamical or site-resolved measurements; for HOTIs, implementation of twisted boundary conditions and projective measurements of position-like operators enables bulk extraction of higher-order topological invariants (Lin et al., 2024). In non-Hermitian systems, the real-space winding number provides the precise criterion for predicting the emergence of the skin effect, even in regimes where bulk-boundary correspondence in momentum space fails (Claes et al., 2020).

7. Generalizations and Extensions

The notion of real-space winding number has broad extensions:

• In nonlinear field theories, real-space winding numbers classify lump soliton solutions (e.g., maps $\Pi$7 in spinor Bose–Einstein condensates), with explicit surface-integral expressions and homotopy invariance (He et al., 2023).
• The method extends to real (non-integer) winding numbers as analytic continuations in certain soliton/vortex models, allowing for perturbative expansions in the winding parameter and Padé extrapolation to large values (Ohashi, 2015).
• For higher dimensions and topological invariants associated with multipole moments, real-space winding numbers implemented through multipole generators capture boundary-obstructed phenomena not seen by conventional invariants (Lin et al., 2024).

These approaches collectively establish real-space winding numbers as indispensable tools for classifying, computing, and measuring topological phenomena in both theoretical models and experimental systems across dimensions and symmetry classes.