The full counting statistics is studied for a one-dimensional system of noninteracting fermions with and without disorder. For two unbiased $L$ site lattices connected at time $t=0$ , the charge variance increases as the natural logarithm of $t$ , following the universal expression $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$. Since the static charge variance for a length $l$ region is given by $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}l$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent $z=1$ of the disordered free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z\ensuremath{\ne}1$ and a model for entanglement entropy based upon dynamical scaling at the infinite disorder fixed point (IDFP). The latter scaling, which predicts $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\propto}\mathrm{ln}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, appears to better describe the charge variance of disordered one-dimensional fermions. When a bias voltage is introduced, the behavior changes dramatically, and the charge and variance become proportional to ${(\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t)}^{1/\ensuremath{\psi}}$ and $\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$ , respectively. The exponent $\ensuremath{\psi}$ may be related to the critical exponent, characterizing spatial/energy fluctuations at the IDFP.The full counting statistics is studied for a one-dimensional system of noninteracting fermions with and without disorder. For two unbiased $L$ site lattices connected at time $t=0$, the charge variance increases as the natural logarithm of $t$, following the universal expression $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$. Since the static charge variance for a length $l$ region is given by $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}l$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent $z=1$ of the disordered free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z\ensuremath{\ne}1$ and a model for entanglement entropy based upon dynamical scaling at the infinite disorder fixed point (IDFP). The latter scaling, which predicts $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\propto}\mathrm{ln}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, appears to better describe the charge variance of disordered one-dimensional fermions. When a bias voltage is introduced, the behavior changes dramatically, and the charge and variance become proportional to ${(\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t)}^{1/\ensuremath{\psi}}$ and $\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, respectively. The exponent $\ensuremath{\psi}$ may be related to the critical exponent, characterizing spatial/energy fluctuations at the IDFP.

arXiv:1201.3933Physical Review BNov 21, 201228 citations

Entanglement and Subsystem Particle Numbers in Free Fermion Systems

Yanbai Zhang, Huichao Li, Long-Hui Sheng +2 more

Nov 3, 20110 citations

Quantum Monte Carlo Study of a Disordered Fermionic System

J. W. Lee, Shailesh Chandrasekharan A, Harold U. Baranger

APSMar 1, 20040 citations

Counting statistics for noninteracting fermions in a d -dimensional potential

Naftali R. Smith, Pierre Le Doussal, Satya N. Majumdar +1 more

We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.We develop a first-principles approach to compute the counting statistics in the ground state of N noninteracting spinless fermions in a general potential in arbitrary dimensions d (central for d>1). In a confining potential, the Fermi gas is supported over a bounded domain. In d=1, for specific potentials, this system is related to standard random matrix ensembles. We study the quantum fluctuations of the number of fermions N_{D} in a domain D of macroscopic size in the bulk of the support. We show that the variance of N_{D} grows as N^{(d-1)/d}(A_{d}logN+B_{d}) for large N, and obtain the explicit dependence of A_{d},B_{d} on the potential and on the size of D (for a spherical domain in d>1). This generalizes the free-fermion results for microscopic domains, given in d=1 by the Dyson-Mehta asymptotics from random matrix theory. This leads us to conjecture similar asymptotics for the entanglement entropy of the subsystem D, in any dimension, supported by exact results for d=1.

arXiv:2008.01045Physical review. EMar 22, 202135 citations

The full counting statistics is studied for a one-dimensional system of noninteracting fermions with and without disorder. For two unbiased

L

site lattices connected at time

t=0

, the charge variance increases as the natural logarithm of

t

, following the universal expression $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$. Since the static charge variance for a length

l

region is given by $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\approx}\frac{1}{{\ensuremath{\pi}}^{2}}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}l$, this result reflects the underlying relativistic or conformal invariance and dynamical exponent

z=1

of the disordered free lattice. With disorder and strongly localized fermions, we have compared our results to a model with a dynamical exponent $z\ensuremath{\ne}1$ and a model for entanglement entropy based upon dynamical scaling at the infinite disorder fixed point (IDFP). The latter scaling, which predicts $\ensuremath{\langle}\ensuremath{\delta}{N}^{2}\ensuremath{\rangle}\ensuremath{\propto}\mathrm{ln}\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t$, appears to better describe the charge variance of disordered one-dimensional fermions. When a bias voltage is introduced, the behavior changes dramatically, and the charge and variance become proportional to ${(\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t)}^{1/\ensuremath{\psi}}$ and

\mathrm{ln}\phantom{\rule{0.16em}{0ex}}t

, respectively. The exponent $\ensuremath{\psi}$ may be related to the critical exponent, characterizing spatial/energy fluctuations at the IDFP.