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Physical review. B, Condensed matter

Charge fluctuations and fractional charge of fermions in 1 + 1 dimensions

Yitzhak Frishman, B. Horovitz

Published Feb 15, 1983• 20 citations
Publisher

Abstract

Charge fluctuations of solitons with arbitrary fractional fermion number in 1 + 1 dimensions are calculated, generalizing the result of Kivelson and Schrieffer for solitons with fermion number \textonehalf{}. The soliton charge is measured by a sampling function $f(x)$ such that $f(x)\ensuremath{\simeq}1$ over a region of width $L$ around the soliton and then falls to zero in a distance $l$. It is shown that vacuum fluctuations vanish as ${l}^{\ensuremath{-}1}$ for large $l$ while the additional fluctuations due to the presence of a soliton vanish as either $\mathrm{exp}(\ensuremath{-}\frac{L}{\ensuremath{\xi}})$ or $\mathrm{exp}(\ensuremath{-}2{\ensuremath{\Delta}}_{0}L)$; $\ensuremath{\xi}$ is the soliton width and ${\ensuremath{\Delta}}_{0}$ is the mass gap of a ground state. This result establishes that the fractional fermion charge is a well-defined observable.

DOI
10.1103/physrevb.27.2565

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Charge fluctuations of solitons with arbitrary fractional fermion number in 1 + 1 dimensions are calculated, generalizing the result of Kivelson and Schrieffer for solitons with fermion number \textonehalf{}. The soliton charge is measured by a sampling function f(x)f(x)f(x) such that f(x)\ensuremath≃1f(x)\ensuremath{\simeq}1f(x)\ensuremath≃1 over a region of width LLL around the soliton and then falls to zero in a distance lll. It is shown that vacuum fluctuations vanish as l\ensuremath−1{l}^{\ensuremath{-}1}l\ensuremath−1 for large lll while the additional fluctuations due to the presence of a soliton vanish as either exp(\ensuremath−L\ensuremathξ)\mathrm{exp}(\ensuremath{-}\frac{L}{\ensuremath{\xi}})exp(\ensuremath−\ensuremathξL​) or exp(\ensuremath−2\ensuremathΔ0L)\mathrm{exp}(\ensuremath{-}2{\ensuremath{\Delta}}_{0}L)exp(\ensuremath−2\ensuremathΔ0​L); \ensuremathξ\ensuremath{\xi}\ensuremathξ is the soliton width and \ensuremathΔ0{\ensuremath{\Delta}}_{0}\ensuremathΔ0​ is the mass gap of a ground state. This result establishes that the fractional fermion charge is a well-defined observable.
The magnitude of quantum fluctuations of the charge of a fractionally charged soliton is calculated. The soliton charge operator is defined as ${\stackrel{^}{Q}}_{s}={\stackrel{^}{Q}}_{f}\ensuremath{-}〈0|{\stackrel{^}{Q}}_{f}|0〉$, where ${\stackrel{^}{Q}}_{f}$ is the integral of the charge-density operator sampled by a function fff peaked at the position of the soliton, and falling smoothly to zero on a scale LLL. ∣0〉|0〉∣0〉 is the ground state of the system in the absence of solitons. It is shown that the mean-square fluctuation of ${\stackrel{^}{Q}}_{s}$ taken about its fractional average value Qs{Q}_{s}Qs​ vanishes as $O(\frac{{\ensuremath{\xi}}_{0}}{L})$ for $Lgg{\ensuremath{\xi}}_{0}$, where ${\ensuremath{\xi}}_{0}$ is the width of the soliton. Thus, as $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the soliton is an eigenfunction of the charge operator with fractional eigenvalue. We also show that the portion of the charge fluctuations that are due to the soliton falls as $\mathrm{exp}(\frac{\ensuremath{-}L}{{\ensuremath{\xi}}_{0}})$ as $L\ensuremath{\rightarrow}\ensuremath{\infty}$. Nonetheless, the charge of the entire system, including all solitons, is integral.
Fermion number fractionization in quantum field theory on a finite interval is studied for a (1 + 1) dimensional fermion-soliton system with explicit charge conjugation symmetry breaking. The effects of boundary conditions on the fractional fermion number and the connection with the corresponding open space problem are investigated. It is argued that the open space fractional charges can be correctly reproduced from the finite interval results only through a careful definition of what is meant by the soliton charge. This definition of the charge distinguishes between the fermionic and boundary induced charges in the system, and isolates the soliton from possibly other charged topological objects in the system. It therefore gives a true measure of the localized fractional fermion number induced on the soliton of interest. It is then rigorously proven that the corresponding charge fluctuations vanish, and hence that the induced fractional charge on the soliton is a quantum observable.
We study the soliton modes carrying fractional quantum numbers in one-dimensional superfluids. In the sss-wave pairing superfluid with the phase of the order parameter twisted by opposite angles ±φ/2\pm φ/2±φ/2 at the two ends there is an emergent complex Z2Z_2Z2​ soliton mode carrying fractional spin number φ/(2π)φ/(2π)φ/(2π) if there is only one pairing branch. We demonstrate that in finite systems of length LLL, the spin density for one pairing branch in the presence of a single soliton mode consists of two terms, a localized spin density profile carrying fractional quantum number φ/(2π)φ/(2π)φ/(2π), and a uniform background −φ/(2πL)-φ/(2πL)−φ/(2πL). The latter one vanishes in the thermodynamic limit leaving a single soliton mode carrying fractional excitation, however it is essential to keep the total quantum number conserved in finite systems. This analysis is also applicable to other systems with fractional quantum numbers, thus provides a mechanism to understand the compatibility of the emergence of fractional charges with the integral quantization of charges in a finite system. For the ppp-wave pairing superfluid with the chemical potential interpolating between the strong and weak pairing phases, the Z2Z_2Z2​ soliton is associated with a Majorana zero mode. By introducing the dimension density, we argue that the Majorana zero mode may be understood as an object with 1/2 dimension of the single particle Hilbert space. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
We study the soliton modes carrying fractional quantum numbers in one-dimensional fermionic systems. In particular, we consider the solitons in the one-dimensional fermionic superfluids. For the s -wave order parameter with phase twisted by an angle φ , the complex Z 2 soliton may occur carrying a localized fractional quantum number φ /(2 π ). If the system is finite with length L , we show the existence of a uniform background − φ /(2 πL ) which, though vanishing in the thermodynamic limit, is essential to maintain the conservation of the total integer quantum number. This analysis is also applicable to other systems with fractional quantum numbers, thus providing a mechanism to understand the compatibility of the emergence of fractional quantum number in the thermodynamic limit of a finite system with only integer quantum numbers. For the p -wave pairing case, the Majorana zero mode may occur associated with a real Z 2 soliton, and the fractionalized quantum number is the dimension of the single particle Hilbert space, which turns out to be 1/2. Again for a finite system with length L , there is an accompanying uniform dimension density −1/(2 L ) to maintain the dimension of the Hilbert space invariant. We conjecture a connection of the dimension density of one-dimensional solitons with the quantum dimension of topological excitations.
We reconsider the problem of “binding” of a pair of fractionally charged solitons in “1100” charge order backgrounds, using the variational Monte Carlo method based on the matrix-product state. In the “1100” charge order ground state of the one-dimensional extended Peierls–Hubbard model, the 2kF bond order wave (BOW) coexists with the 4kF BOW. We have found that the fractionally charged soliton with −e/2, −e being an electronic charge, is accompanied with not only a kink in the 2kF BOW but also a kink in the 4kF BOW, which was not allowed to exist in the dimer model used in the previous study. Calculating the total energy of a pair of fractionally charged solitons with the same charge as a function of the soliton–soliton distance, we have found that the soliton–soliton interaction is repulsive. Therefore the “binding” of the solitons does not occur and the elementary charge excitation from the “1100” charge order ground state is a fractionally charged soliton.