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Abstract

We generalize our recent results for the hard-wall boundary and interface charges in one-dimensional single-channel continuum [S. Miles et al., Phys. Rev. B 104, 155409 (2021)] and multichannel tight-binding [N. Müller et al., Phys. Rev. B 104, 125447 (2021)] models to the realm of the multichannel continuum systems. Using the technique of boundary Green's functions, we give a rigorous proof that the change in boundary charge upon the shift of the system towards the boundary by the distance x_φ∈[0,L] (where L is a potential periodicity) is given by a perfectly linear function of x_φ plus an integer-valued topological invariant I, the so-called boundary invariant. We provide two equivalent representations for I(x_φ): the winding-number representation and the bound-state representation. The winding-number representation allows one to write I as a winding index of a particular functional of bulk Green's function. The corresponding integration contour is chosen in the complex frequency plane to encircle the occupied part of the spectrum residing on the real axis. In turn, in the bound-state representation, I is expressed through the sum of the winding number of the boundary Green's function and the number of bound states supported by the cavity of size x_φ below the chemical potential. We observe that during a single cycle in the variation of xφ, the boundary invariant exhibits exactly ν downward jumps, each by a unit of electron charge, whenever ν energy bands are completely filled leading to the value I(L)=−ν. Additionally, for translationally invariant models interrupted by a localized impurity we derive the winding-number expression for the excess charge accumulated on the said impurity. We observe that the charge accumulated on a single repulsive impurity is restricted to the values −N_c,⋯,0, where N_c is the number of channels (spin or orbital components) in the system. For systems with weak potential amplitudes, we additionally develop Green's-function-based low-energy theory, allowing one to analytically access the physics of multichannel continuum systems in the low-energy approximation.