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

Free fermions on a line: Asymptotics of the entanglement entropy and entanglement spectrum from full counting statistics

Roman Süsstrunk, Dmitri A. Ivanov

We consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length L , the leading asymptotic behavior of this entropy is known to be logarithmic in L . We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers.We consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length L , the leading asymptotic behavior of this entropy is known to be logarithmic in L . We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers.

arXiv:1208.5845Europhysics Letters (EPL)Dec 1, 201240 citations

Entanglement entropy and particle number cumulants of disordered fermions

I. S. Burmistrov, Konstantin S. Tikhonov, I. V. Gornyi +1 more

arXiv:1701.05876Annals of PhysicsJun 2, 201712 citations

Entanglement in fermionic systems

Bei Zeng

Jan 1, 20120 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.