Dynamically enhanced unconventional superconducting correlations in a Hubbard ladder

We propose theoretically how unconventional superconducting pairing in a repulsively interacting Hubbard ladder can be enhanced via the application of a Floquet driving. Initially the Hubbard ladder is prepared in its charge-density-wave dominated ground state. A periodic Floquet drive is applied that modulates the energy offset of the two legs and effectively reduces the tunneling along the rungs. This modulation of the energy offsets might be caused by the excitation of a suitable phononic mode in solids or a superlattice modulation in cold atomic gases. We use state-of-the-art matrix product state methods to monitor the resulting real-time dynamics of the system. We find an enormous enhancement of the unconventional superconducting pair correlations by approximately one order of magnitude.

Figure 1

Ground state phase diagram of the repulsively interacting Hubbard ladder for $U=8J_{\parallel }$ at filling $n=0.9375$. For intermediate hopping values ($0.9⪅J_{\perp }/J_{\parallel }⪅1.65$) the exponent $K_d>1$ shows that the unconventional pair correlation is dominant, whereas at low and large values of the ratio of the tunneling the charge density wave is dominant since $K_d<1$. This causes a reentrance behavior wrt. the ratio of the tunnelings. The crossover takes place at $K_d\approx 1$ (marked by the dashed horizontal line). $K_d$ is extracted from the decay of the unconventional pair correlations calculated via iDMRG. The unit cell in the iDMRG calculation has size $32\times 2$. The arrow sketches the nonequilibrium situation in the effective Floquet model $H^{\text{eff}}$. Starting with the $\mathrm{CDW}{}^{4k_F}$ with $J_{\perp }/J_{\parallel }=3.9598$, the frequency of the periodic driving is chosen such that the final effective system is an unconventional superconductor $\left(\mathrm{SC}{}^d\right)$ with $J_{\perp }^{\text{eff}}/J_{\parallel }=1.4$.

Figure 2

(a) The unconventional pair correlations and density-density correlations for a repulsive Hubbard ladder with $U=8J_{\parallel },J_{\perp }=1.4J_{\parallel }$ at filling $n=0.9375$ calculated with DMRG for a ladder of size $384\times 2$ using a bond dimension of up to $M=4000$ and a truncation error $E={10}^{-8}$ and with iDMRG for unit cell of size $32\times 2$ using a bond dimension of up to $M=5000$ and a truncation error $E={10}^{-12}$. The correlations show an algebraic decay with additional oscillations and are fitted (dashed lines) by $l^{-\nu }\left(a+bcos\left(2\pi n(l-l_0)\right)\right)$ in order to extract the exponents where $\nu ,a,b$ and $l_0$ are the fitting parameters. The extracted exponents are $K_d=1.43$ (DMRG), $K_d=1.42$ (iDMRG), and $K_n=1.48$ (DMRG), which identify the dominating $d$-wave superconductivity. As a guide to the eye, the fitted algebraic decay without oscillations is shown with solid lines. (b) The exponential decay of the spin correlations.

Figure 3

The dynamics of (a) unconventional pair correlations and (b) density-density correlations at different distances $l=1$ to $l=7$ (from top to bottom at $t=T_{\text{ramp}}$) for two different ramp procedures. Solid lines show the results of the driven Floquet system and dashed lines the results of the slow parameter change of $J_{\perp }^{\text{eff}}$. The increase of the unconventional pair correlations during the ramp time is larger than the increase of the density-density correlations. The curves for two ramp procedures are hardly distinguishable beside additional oscillations in the density-density correlations of the Floquet driven system. The system is $32\times 2$ ladder with filling $n=0.9375$. The parameters are $J_{\perp }=3.9598J_{\parallel },U=8J_{\parallel },A_f=88.8J_{\parallel }$. The frequency of periodic driving is $\hslash \omega =100J_{\parallel }$ with $T=\frac{2\pi }{\omega }$. Symbols mark the corresponding effective ground states.

Figure 4

The unconventional pair correlations and density-density correlations at initial time $t=0$ and at the end of the ramp $t=T_{\text{ramp}}$ for three different ramp times $T_{\text{ramp}}=50T,100T,150T$. The correlations in the ground state of the effective Hamiltonian with $J_{\perp }^{\text{eff}}=1.4J_{\parallel }$ are plotted for the comparison. Parameters are the same as in Fig. 3.