Drude weight increase by orbital and repulsive interactions in fermionic ladders

In strictly one-dimensional systems, repulsive interactions tend to reduce particle mobility on a lattice. Therefore, the Drude weight, controlling the divergence at zero-frequency of optical conductivities in perfect conductors, is lower than in noninteracting cases. We show that this is not the case when extending to quasi-one-dimensional ladder systems. Relying on bosonization, perturbative and matrix product states (MPS) calculations, we show that nearest-neighbor interactions and magnetic fluxes provide a bias between back- and forward-scattering processes, leading to linear corrections to the Drude weight in the interaction strength. As a consequence, Drude weights counterintuitively increase (decrease) with repulsive (attractive) interactions. Our findings are relevant for the efficient tuning of Drude weights in the framework of ultracold atoms trapped in optical lattices and equally affect topological edge states in condensed matter systems.

Figure 1

(a) The interacting Creutz model, Eqs. (7)–(10). We represent interactions by colored boxes and tight-binding hopping elements as arrows connecting different lattice sites. (b) depicts the usual one-dimensional case. A single pair of distinct left and right movers is identified close to the chemical potential $\mu$ and $V_{\parallel }$ induces $g_{2/4}$ interaction processes that exactly compensate each other on the lattice. (c) illustrates the two-leg ladder case with magnetic flux ($g=0$ for simplicity). A finite interchain hopping of amplitude $m$ gaps out only one pair among the right or left movers of each singular chain (dashed lines), leading to a suppression of backscattering $g_{2,\parallel }$ processes compared to forward scattering $g_{4,\parallel }$ processes caused by $V_{\parallel }$. As a consequence, the Drude weight increases. We also sketch how $V_{\perp }$ increases backscattering, opposite to the action of $V_{\parallel }$.

Figure 2

Spin components of the bands for different parameters $g,\chi$ as displayed at the top of the figures; we fix $m=t=1$ and $\chi =\pi /2$ for simplicity. Black lines mark the band dispersion and the color represents the band polarization along the canonical axis ($\uparrow$ in blue, $\downarrow$ in red). White implies a perfect superposition of $\uparrow$ and $\downarrow$ species. The green inset circle represents the winding of $\vec{h}\left(k\right)$ and dashed axes mark the origin. The gray region marks chemical potentials to fix a central charge $c=1$ with a single left and right mover at $\pm k_{\mathrm{F}}$, the starting point of the bosonization approach exploited in Sec. 5.

Figure 3

Renormalization of the Drude weight by interaction (different colors and symbols). Continuous lines represent the thermodynamic limit of the perturbative calculation given in Eqs. (16), (18), (19) and symbols represent results obtained by MPS simulations. Numerical values are converged up to the third decimal digit. The system size is $N/L=20/32$. Each set of data is obtained by the interplay of the kinetic term with individual interactions $\mathcal{H}={\mathcal{H}}_{\mathrm{kin}}+{\mathcal{H}}_i$ with weak interaction amplitudes $V_{\parallel }=0.1t$, $V_{\perp }=0.1t$, $V_{\parallel }=V_{\perp }=0.1t$, and $U=0.125t$. The two figures demonstrate the robustness of the strong Drude weight renormalization, namely its increase for $V_{\parallel }>0$ and decrease for $V_{\perp }>0$ around the fine-tuned point of a single Dirac cone $m/t=g/t=\chi /\pi =1$. Hereby, we devote our focus to the influence of the individual interactions by subtracting the corresponding noninteracting value ${\mathcal{D}}_0$ from the total susceptibility, i.e., ${\mathcal{D}}_i=\mathcal{D}-{\mathcal{D}}_0$. In (a) we display the interaction-induced Drude weight shift ${\mathcal{D}}_i$ versus transverse flux $\chi$, in which the curvature of the two $SU\left(2\right)$ symmetric interactions denoted by red (down-triangle) and green (right triangle) is not visible in the chosen scaling. To quantify, the plotted values of $D_U/U$ are all inside the interval $[0,5]\times {10}^{-3}$ and ${\mathcal{D}}_{\perp }/V_{\perp }+{\mathcal{D}}_{\parallel }/V_{\parallel }$ resides in $[-4.2,0]\times {10}^{-3}$. (b) shows ${\mathcal{D}}_i$ as a function of the nearest neighbor interspecies hopping amplitude $g/t$.

Figure 4

Change of the Drude weight obtained by MPS simulations for different interaction amplitudes signalled by different colors and markers. The displayed values are converged up to the third decimal digit. The density of the system is $n=\frac{N}{L}=\frac{20}{32}$, $m/t=\chi /\pi =1$, and $g$ is relaxed to values around the topological transition at $g/t=1$. (a) Drude increase induced by ${\mathcal{H}}_{\parallel }$ and (b) Drude decrease by ${\mathcal{H}}_{\perp }$. (c) Drude renormalization by ${\mathcal{H}}_U$ and (d) by ${\mathcal{H}}_{\parallel }+{\mathcal{H}}_{\perp }$ for the case $V_{\parallel }=V_{\perp }$. The effect is observed for any simulated amplitude of the interactions and irrespective of the underlying topology. Notice that the renormalization of the Drude weight due to ${\mathcal{H}}_{\perp }$ is stronger and very asymmetrical compared to ${\mathcal{H}}_{\parallel }$.

Figure 5

Change of the Drude weight by interaction according to first-order perturbation theory in the thermodynamic limit for a system at incommensurate total density $n=20/32$ and $m/t=1$. (a) ${\mathcal{D}}_{\parallel }/V_{\parallel }$, (b) ${\mathcal{D}}_{\perp }/V_{\perp }$, (c) ${\mathcal{D}}_{\parallel }/V_{\parallel }+{\mathcal{D}}_{\perp }/V_{\perp }$, (d) ${\mathcal{D}}_U/U$. Solid lines mark the Lifshitz phase transitions between central charge $c=1$ and $c=2$. Figure 3 displays cuts of the density plots at $c=1$ close to $g/t=\chi /\pi =1$ with markers representing MPS results at $V_{\perp }/t=0.1t$, $V_{\parallel }/t=0.1t$, and $U/t=0.125t$.

Figure 6

Coupled-wire approach with (a) $N=2$ and (b) $N=20$. We place the Fermi energy in the center of the gap, marked as a red line. In (c), we plot the amplitude ${\left|u_N/u_1\left(k_{\mathrm{F}}\right)\right|}^2$ for the case $m=0.1t$, $g=0$, and $\chi =0.025\pi$ versus number of chains $N$, evidently following an exponential decay. Following Eq. (21) this implies a strong suppression of $g_2$ processes compared to $g_4$ in the $c=1$ gap.

Figure 7

(a)–(b) First two derivatives of $\left|u^{\left(n\right)}/u\left(k_{\mathrm{F}}\right)\right|$, (c)–(d) first two derivatives of $\left|v^{\left(n\right)}/v\left(k_{\mathrm{F}}\right)\right|$ versus $g$ and flux $\chi$ at density $n=2/3$ and $m=1$. Discrepancies between bosonization and perturbation theory arise due to a strong violation of Eq. (D2).

Figure 8

(a) Deformation of a generic ring system to simulate periodic boundary conditions with open boundary MPS algorithms. (b) Generic representative of numerical simulations with different bond dimension $M$ of the ground-state energy dependence $E\left(\mathrm{\Phi }\right)$ for a system at fixed density $n=N/L=20/32$ at parameters $m/t=\chi /\pi =1$, $g/t=1.2$, $U/t=4$, and $V_{\parallel }=V_{\perp }=0$. Crosses represent MPS results and continuous lines represent a quadratic fit. At the presented decimal precision, data and fits for $M=128$ and $M=256$ reside exactly on top of each other. (c) Dependence of the Drude weight on the truncated probability $\mathrm{\Delta }\rho$ of the MPS. In case of $\mathrm{\Delta }\rho <{10}^{-7}$, the data is sufficiently converged up to the necessary precision presented in the main text and the convergence error may be disregarded.