Abstract
We study NN spinless fermions in their ground state confined by an external potential in one dimension with long range interactions of the general Calogero-Sutherland type. For some choices of the potential this system maps to standard random matrix ensembles for general values of the Dyson index \betaβ. In the fermion model \betaβ controls the strength of the interaction, \beta=2β=2 corresponding to the noninteracting case. We study the quantum fluctuations of the number of fermions N_DND in a domain DD of macroscopic size in the bulk of the Fermi gas. We predict that for general \betaβ the variance of N_DND grows as A_{\beta} \log N + B_{\beta}AβlogN+Bβ for N \gg 1N≫1 and we obtain a formula for A_\betaAβ and B_\betaBβ. This is based on an explicit calculation for \beta\in\left\{ 1,2,4\right\}β∈{1,2,4} and on a conjecture that we formulate for general \betaβ. This conjecture further allows us to obtain a universal formula for the higher cumulants of N_DND. Our results for the variance in the microscopic regime are found to be consistent with the predictions of the Luttinger liquid theory with parameter K = 2/\betaK=2/β, and allow to go beyond. In addition we present families of interacting fermion models in one dimension which, in their ground states, can be mapped onto random matrix models. We obtain the mean fermion density for these models for general interaction parameter \betaβ. In some cases the fermion density exhibits interesting transitions, for example we obtain a noninteracting fermion formulation of the Gross-Witten-Wadia model.
Highlights
The rationale behind this conjecture is that (i) the expression (38) for C(x, y) on macroscopic scale is valid for arbitrary β, a result which comes naturally from the Coulomb gas calculations [43, 44, 84] (ii) the above calculations for β = 1, 2, 4 show that the β-dependence of the constant part, cβ, is determined from microscopic scales only. We expect that it is independent of the random matrix theory (RMT) ensemble in the Table 1. This constant is given for general β by formula (17), which we will justify in Section 3 based on previous works on the circular β ensemble (CβE)
We calculated the counting statistics for several models of N 1 interacting spinless fermions in their ground state in one dimension confined by an external potential, see Tables 1 and 2
We found that the variance of the number of fermions in a macroscopic interval [a, b] in the bulk of the Fermi gas grows with N as Aβ log N + Bβ + o(1)
Summary
The full counting statistics (FCS), which measures the fluctuations of the number of particles ND inside a domain D has been studied extensively in the context of shot noise [1], quantum transport [2], quantum dots [4,5], non-equilibrium Luttinger liquids [3] as well as in quantum spin chains and fermionic chains [6,7,8,9,10,11]. In the absence of external potential and at zero temperature it is well known that both the variance of ND and the entropy grow as ∼ Rd−1 log R with the typical size R of the domain D in space dimension d [18,19,20,21,22,23] These results have been extended for noninteracting fermions in the presence of a confining potential. They are independent of the size of the interval (within the bulk) and (ii) they are universal, i.e., independent of the precise shape of the potential (assumed to be smooth) This conjecture was used to obtain a prediction for the entanglement entropy for noninteracting fermions in a potential in [52]. In the rest of this section we first present the main models that we will study, we explain the main idea of the method and we present the main results
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