Pair correlations in doped Hubbard ladders

Hubbard ladders are an important stepping stone to the physics of the two-dimensional Hubbard model. While many of their properties are accessible to numerical and analytical techniques, the question of whether weakly hole-doped Hubbard ladders are dominated by superconducting or charge-density-wave correlations has so far eluded a definitive answer. In particular, previous numerical simulations of Hubbard ladders have seen a much faster decay of superconducting correlations than expected based on analytical arguments. We revisit this question using a state-of-the-art implementation of the density matrix renormalization group algorithm that allows us to simulate larger system sizes with higher accuracy than before. Performing careful extrapolations of the results, we obtain improved estimates for the Luttinger liquid parameter and the correlation functions at long distances. Our results confirm that, as suggested by analytical considerations, superconducting correlations become dominant in the limit of very small doping.

Figure 1

Spatial decay of the pair-correlation function (blue markers) for a $2\times 32$ Hubbard model with $U/t=8$ and average filling $n=0.875$, similar to the results obtained in Ref. [14]. The solid, dotted, and dashed lines are reference power-law decays with exponents $\mu =-1/2,-1,-2$, respectively. In this paper we estimate an exponent $\mu \approx -1$ (see exponents in Table 1).

Figure 2

Spatial decay of the pair-correlation function $D\left(r\right)$ on ladders with $L=128$ and $n=0.875$ as a function of distance $r=|i-j|$ for several choices of $i$: fixing $i=18$ (orange line) and $i=20$ (green line), and averaging 11 pairs $(i,j)$ at distance $r$ around the middle according to Eq. (7) (blue line).

Figure 3

Local density profile for ladders with $L=128$ and $n=0.875$. for several bond dimensions $M$, the inset showing details in the center of the system. The uncertainty due to systematic errors in the extrapolated curve $M=\infty$ is shown as a shaded region around the estimate; its size is often smaller than the symbols.

Figure 4

Spatial decay of the pair-correlation function $D\left(r\right)$ for ladders with $n=0.875$ and several bond dimensions $M$ and system sizes $L$. Top panel: Results for $L=128$. Bottom panel: Results are extrapolated to $M=\infty$. The shaded region around the curve shows the confidence range of the systematic error.

Figure 5

Extrapolation of the ground state energy density as a function of the inverse bond dimension $1/M$ (left panel), the truncated weight $\varepsilon$ (right panel, solid circles), and as a function of the energy variance $\mathrm{Var}\left[\widehat{H}\right]$ (right panel, open circles). Results are for $L=128$ and $n=0.875$.

Figure 6

Extrapolation of observables as a function of the inverse bond dimension $1/M$ (left panels), the truncated weight $\varepsilon$ (right panels, solid symbols) and as a function of the energy variance $\mathrm{Var}\left[\widehat{H}\right]$ (right panels, open symbols). Shown are representative examples of (a) the local rung density $n_i$, (b) the density correlation function $N\left(r\right)$, and (c) the pair-correlation function $D\left(r\right)$. Results are for $L=128$ and $n=0.875$.

Figure 7

(a) Fit of the density profile (solid line) for a system of length $L=128$ with filling $n=0.875$ compared to the raw density (markers). The red dotted line is the density offset $n_0$ obtained from the least-square fit, as a reference, and the dashed green line shows the average filling $n$. The fit is restricted to the shaded region to avoid the divergent boundaries. (b) Same as (a) for a system with an additional rung but the same number of holes, i.e., with $N_{\uparrow }=N_{\downarrow }=113$ (two more particles and the same number of holes). (c) Finite-size scaling of the oscillation amplitude $\delta n\left(L\right)$ as a function of the system size $L$ in a double-logarithmic plot for several filling parameters.

Figure 8

Decay of the density-density correlation function $N\left(r\right)$ for several system sizes. The dashed line is a power-law decay obtained with an exponent $-K_{\rho }$ and $K_{\rho }=1.54$ from the $\mathrm{Var}\left[\widehat{H}\right]$ and $\varepsilon$ extrapolations in Table 1; the vertical offset is chosen to show the agreement with the long-distance behavior of the correlation function. (a) and (b) refer to two different average fillings $n=0.875$ and $n=0.9375$, respectively.

Figure 9

Decay of the pair-correlation function $D\left(r\right)$ for several system sizes. The dashed line is a power-law decay obtained with an exponent $-1/K_{\rho }$ and $K_{\rho }=1.54$ from the $\mathrm{Var}\left[\widehat{H}\right]$ and $\varepsilon$ extrapolations in Table 1; the vertical offset is chosen to show the agreement with the long-distance behavior of the correlation function. (a) and (b) refer to two different average fillings $n=0.875$ and $n=0.9375$, respectively.

Figure 10

Finite-size scaling of the oscillation amptitude $\delta n\left(L\right)$ as a function of the system size $L$ in a double-logarithmic plot for an average filling $n=0.96875$.

Figure 11

Decay of the density-density correlation function $N\left(r\right)$ for several system sizes with average filling $n=0.96875$. The dashed line is a power-law decay obtained with an exponent $\mu =-K_{\rho }$ and $K_{\rho }$ from the $\mathrm{Var}\left[\widehat{H}\right]$ and $\varepsilon$ extrapolations in Table 1; the vertical offset is chosen to show the agreement with the long-distance behavior of the correlation function.

Figure 12

Decay of the pair-correlation function $D\left(r\right)$ for several system sizes with average filling $n=0.96875$. The dashed line is a power-law decay obtained with an exponent $\nu =-1/K_{\rho }$ and $K_{\rho }$ from the $\mathrm{Var}\left[\widehat{H}\right]$ and $\varepsilon$ extrapolations in Table 1; the vertical offset is chosen to show the agreement with the long-distance behavior of the correlation function.