Dynamical correlations and nonequilibrium sum rules in photodoped Hubbard ladders

TL;DR

This work investigates the nonequilibrium dynamics of a half-filled Hubbard ladder under ultrafast optical pumping using matrix product state methods, focusing on the interplay between spin and charge. A central result is a nonequilibrium sum rule, 4W_{S^z}(t)+W_n(t)=2G\rho, which constrains spectral-weight flow and predicts a net transfer from spin to charge upon photodoping. The leg-direction pump more strongly disrupts magnetic correlations than the rung-direction pump, with a concomitant growth of charge correlations and the emergence of a low-energy intraband charge mode, signaling a transition to a correlated metallic state. The findings underscore the importance of treating spin and charge on equal footing in nonequilibrium strongly correlated systems and provide benchmarks for nonequilibrium dynamical structure factors using MPS techniques.

Abstract

Using matrix product state techniques we study the nonequilibrium dynamical response of the half-filled Hubbard ladder when subject to an optical pump. Optical pumping offers a way of producing and manipulating new strongly correlated phenomena by suppressing existing magnetic correlations. The ladder allows the effects of pump directionality to be investigated, and compared to a single chain it has strong spin-charge coupling and a fully gapped excitation spectrum, promising different nonequilibrium physics. We compute time-dependent correlations, including the nonequilibrium dynamical structure factors for spin and charge. By deriving a combined spin-charge sum rule that applies both in and out-of-equilibrium, we show that spectral weight is pumped directly from the antiferromagnetic spin response into a low energy $ω\sim 0$ charge response below the Mott gap. The transfer of weight is pump direction dependent: pumping directed along the legs disrupts magnetic correlations more than pumping in the rung direction, even if the post pump energy density is similar. The charge correlation length is dramatically enhanced by the pump, whilst the spin correlations are most strongly suppressed at nearest and next-nearest neighbour spacings. After the pump the system is in a nonthermal correlated metallic state, with gapless charge excitations and approximately equal spin and charge correlation lengths, emphasising the importance of treating these degrees of freedom on an equal footing in nonequilibrium systems.

Figures (19)

• Figure 1: Energy density of the Hubbard ladder, relative to its ground state, after pumps of the type described in Eq. \ref{['eq:pumpprotocol']} are applied along either the leg $(\parallel)$ or rung $(\perp)$ directions. Data are shown for infinitely long ($L=\infty$) and finite ladders with $L=8$ and periodic boundary conditions (pbc) and $L=6$ and open boundary conditions (obc). The pump parameters are $\Omega = 6.0$ and $\sigma_{p} = 1.0$. Lines are a guide to the eye.
• Figure 2: Upper panel: magnetization (left axis) and doublon number (right axis) as a function of time for the infinite Hubbard ladder when pumped along its legs $(\parallel)$ or along its rungs $(\perp)$ from $t=0$ to $t=\pi$. The pump parameters are $\Omega = 6.0$ and $\sigma_{p}=1.0$. Lower panel: The change in doublon number relative to the initial state $\Delta d=\sum_l(\langle d_{i,l}(t)-d_{i,l}(0)\rangle)/2$, scaled by $z_\alpha A_0^2$ with $z_\parallel=2$ and $z_\perp=1$, showing collapse onto a single curve at short times. Dashed lines show the values of $\Delta E/ (N U z_\alpha A_0^2)$ for the various pumps. For convenience when comparing with Fig. \ref{['fig:energy-density']}, the squared values of $A_0$ are, $(0.45^2)\approx0.2$, $(0.4)^2\approx 0.16$ and $(0.35)^2\approx0.12$.
• Figure 3: Staggered spin-spin correlations $(-1)^x \langle S^z_{i,l} S^z_{i+x,l}\rangle$ along a leg of an infinite ladder as a function of separation $x$ and time $t$. The white dashed line indicates the time at which the pump is turned off. Upper panel: leg-direction pump with $A_0=1$. Middle panel: leg-direction pump with $A_0=0.45$. Lower panel: rung-direction pump with $A_0=1$.
• Figure 4: Connected charge (density-density) correlations $\vert\langle n_{i,l} n_{i+x,l}\rangle_c\vert$ along a leg of an infinite ladder as a function of separation $x$ and time $t$. The white dashed line indicates the time at which the pump is turned off. Upper panel: leg-direction pump with $A_0=1$. Middle panel: leg-direction pump with $A_0=0.45$. Lower panel: rung-direction pump with $A_0=1$.
• Figure 5: Upper panel: staggered spin correlations in the unperturbed, equilibrium system and at $t=10$ for the three different pump protocols discussed in the text. Lower panel: as above, but the connected charge correlations.