Counting interacting electrons in one dimension

The calculation of the full counting statistics of the charge within a finite interval of an interacting one-dimensional system of electrons is a fundamental, yet as of now, unresolved problem. Even in the noninteracting case, charge counting turns out to be more difficult than anticipated because it necessitates the calculation of a nontrivial determinant and requires regularization. Moreover, interactions in a one-dimensional system are best described using bosonization. However, this technique rests on a long-wavelength approximation and is a priori inapplicable for charge counting due to the sharp boundaries of the counting interval. To mitigate these problems, we investigate the counting statistics using several complementary approaches. To treat interactions, we develop a diagrammatic approach in the fermionic basis, which makes it possible to obtain the cumulant generating function up to arbitrary order in the interaction strength. Importantly, our formalism preserves charge quantization in every perturbative order. We derive an exact expression for the noise and analyze its interaction-dependent logarithmic cutoff. We compare our fermionic formalism with the results obtained by other methods, such as the Wigner crystal approach and numerical calculations using the density-matrix renormalization group. Surprisingly, we show good qualitative agreement with the Wigner crystal for weak interactions, where the latter is in principle not expected to apply.

Figure 1

Dependence of Hartree-Fock correction of the cumulant generating function on $\lambda$ and the occupancy for an interval of length $L=10$. Solid and dashed lines correspond to real and imaginary components. (a) The interaction correction to the generating function for low electron densities. The insets demonstrate the dependence of correction to the first (right) and second (left) cumulants (i.e., noise and charge) on occupancy and show the increase of their absolute values at increase of interval length $L=10,20,30,40,50$. The equidistance of the first cumulant lines shows us an expected linear dependence on $L$. (b) The interaction correction to the generating function for densities close to the first critical density $k_{C,1}/\pi \approx 0.054$ for $L=10$. The inset demonstrates the sign change (or potential discontinuities) around $k_{C,n}$ of the generating function at $\lambda =\pi$.

Figure 2

The noise lowest-order interaction correction and its dependence on occupancy $k_F$ and the length of the measuring interval $L$. The main logarithmic plot demonstrates the dependence on $L$ at $k_F=\pi /20$. Dots correspond to our perturbative approach, solid curve is Wigner crystal result, and dashed lines are interaction corrections obtained from the formulas given in the plot's legend. The offset of the data in the main plot is governed by the cutoff parameter $\kappa$ [see Eqs. (19) and (20)] pictured in the inset, with color scheme labeling corresponding to the legend of the main plot.

Figure 3

Interaction dependence of the offset. Dots are obtained by DMRG simulation, while lines illustrate the formulas in the legend at the occupancy of the $k_F=\pi /2$.

Figure 4

The principle of a requantization of the boson field $\mathrm{\Phi }$. (a) Possible configurations of $\mathrm{\Phi }$ as a function of $x$ for standard Luttinger liquid theory. The black and gray curves represent possible quantum superpositions of two different realizations of $\mathrm{\Phi }$. Only the total difference $\mathrm{\Phi }\left(0\right)-\mathrm{\Phi }\left(L\right)$ is an integer multiple of $2\pi$, but for arbitrary positions $\mathrm{\Phi }\left(a\right)-\mathrm{\Phi }\left(b\right)$ the field may assume arbitrary values. Thus, any local charge operator is not quantized in general. (b) Configuration of $\mathrm{\Phi }$ including a mass term in the nonrelativistic limit $c\to \infty$, again for two different quantum realizations (black and gray). Here, the mass term leads to a steplike behavior, such that $\mathrm{\Phi }\left(a\right)-\mathrm{\Phi }\left(b\right)$ is an integer multiple of $2\pi$ independent of $a$ and $b$. Thus, the local charge is always guaranteed to be quantized.

Figure 5

DMRG computations of the parity expectation values. The color designates the interaction strength. The data points are collected for $k_F$ from $\pi /20$ to $\pi /10$, and the interval length varies in $L\in [1\cdots 199]$ at total number of sites 200.