$ H = \int_0^L dx \left[ \partial_x \Psi^{\dagger}(x) \partial_x \Psi(x) + c \Psi^{\dagger} (x) \Psi^{\dagger}(x) \Psi(x) \Psi(x) \right] $
\[ \begin{align} \Psi_N(\{ x \} | \{ \lambda \}) &= \prod_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) sgn (\lambda_{j_1} - \lambda_{j_2}) \times \\ &\times \sum_{P \in \pi_N} (-1)^{[P]} e^{i \sum_{j=1}^N \lambda_{P_j} x_j + \frac{i}{2} \sum_{N \geq j_1 > j_2 \geq 1} sgn(x_{j_1} - x_{j_2}) \phi (\lambda_{P_{j_1}} - \lambda_{P_{j_2}})} \end{align} \]
in which $\phi(\lambda) \equiv 2\mbox{atan}(\lambda/c)$
\[ {\cal S}^{a \bar{a}} (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k(j-j')} \int_{-\infty}^\infty dt e^{i\omega t} \langle \frac{1}{2} \left[ {\cal O}^a_j (t), ({\cal O}^a_{j'} (0))^\dagger \right] \rangle \]
M. Panfil and J-SC, PRA 89 (2014)
We will consider a sudden switch of the interaction:
$ c (t) = \left\{ \begin{array}{ll} c_i & t < 0, \\ c_f & t > 0 \end{array} \right. $
starting in ground/(any eigen)state of $H(c_i)$ at $t < 0$
$ S^Q = L\int_0^\infty d\lambda \rho(\lambda) \log\left(\frac{\lambda^2}{c^2} \left( \frac{1}{4} + \frac{ \lambda^2}{c^2} \right) \right) + S^{YY} $
Saddle-point equation:
$ \ln \eta(\lambda) = g(\lambda) - h - \int_{-\infty}^\infty \frac{d\lambda'}{2\pi} K (\lambda - \lambda') \ln \left[ 1 + \eta^{-1}(\lambda') \right], $
with driving term $ g(\lambda) = \ln \left[ \frac{\lambda^2}{c^2} \left( \frac{\lambda^2}{c^2} + \frac{1}{4} \right) \right] $
Analytic solution for steady state:
$ \begin{align} \rho_{sp} (\lambda) &= -\frac{\gamma}{4\pi} \frac{1}{1 + a(\lambda)} \frac{\partial a(\lambda)}{\partial \gamma},\\ a(\lambda) &\equiv \frac{2\pi}{\frac{\lambda}{n} \sinh \frac{2\pi \lambda}{c}} I_{1 - 2i\frac{\lambda}{c}} \left(\frac{4}{\sqrt{\gamma}}\right) I_{1 + 2i\frac{\lambda}{c}} \left(\frac{4}{\sqrt{\gamma}}\right) \end{align} $
Steady state distribution: non-thermal
J. De Nardis, B. Wouters, M. Brockmann & J-SC, PRA 89, 2014
After initial dephasings:
'hydrodynamic' evolution
described by local GGE
$\partial_t \rho (\lambda) + \partial_x (v^{\tiny \mbox{eff}}(\lambda) \rho (\lambda)) = 0$
with effective velocities determined by the dressing equation
Effective dynamics:
"flea gas" of scattering quasiparticles
JSC, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6, 070 (2019)
JSC, B. Doyon, J. Dubail, R. Konik and T. Yoshimura, SciPost Phys. 6, 070 (2019)
Phys. Rev. Lett. 123, 130602 (2019)
Phys. Rev. B 103, 165121 (2021)
Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo,
Jean-Sébastien Caux, Manuele Landini, and Hanns-Christoph Nägerl, arXiv:2505.10550
Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo,
Jean-Sébastien Caux, Manuele Landini, and Hanns-Christoph Nägerl, arXiv:2505.10550
A. Scheie, P. Laurell, B. Lake, S. E. Nagler, M. B. Stone, JSC, D. A. Tennant
Nature Communications 13, 5796 (2022)
A. Scheie, P. Laurell, B. Lake, S. E. Nagler, M. B. Stone, JSC, D. A. Tennant
Nature Communications 13, 5796 (2022)
Selectively (patch-wise) Fourier transforming: $k=\pi/2$ (mod $2\pi$) states are responsible for the quantum wake dynamics
Beyond the reach of any low-energy method