Random singlet-like phase of disordered Hubbard chains

Local moment formation is ubiquitous in disordered semiconductors such as Si:P, where it is observed both in the metallic and the insulating regimes. Here, we focus on local moment behavior in disordered insulators, which arises from short-ranged, repulsive electron-electron interactions. Using density matrix renormalization group and strong-disorder renormalization group methods, we study paradigmatic models of interacting insulators: one-dimensional Hubbard chains with quenched randomness. In chains with either random fermion hoppings or random chemical potentials, both at and away from half-filling, we find exponential decay of disorder-averaged charge and fermion two-point correlations, but power-law decay of disorder-averaged spin correlations that are indicative of the random singlet phase. The numerical results can be understood qualitatively by appealing to the large-interaction limit of the Hubbard chain, in which a remarkably simple picture emerges.

Figure 1

Density-density, fermion two-point and pair-field correlation functions on $L=144$ Hubbard chains at electron filling $n=11/12$ with random potential, random hopping as well as for the disorder-free systems. Additionally, results for the noninteracting ($U=0$) but site-disordered system are shown. All data are shown for $r\le L/2$. The random-potential system has parameters $U=6\overline{t}$, $W_{\mu }/2=3\overline{t}$, $W_t=0$; the random-hopping system has parameters $U=12\overline{t}$, $W_{\mu }=0$, $W_t/2=0.65\overline{t}$; the clean system has parameters $U=6\overline{t}$, $W_{\mu }=W_t=0$; and the Anderson system has $U=0$, $W_{\mu }/2=3\overline{t}$, $W_t=0$. (a) Disorder-averaged density-density correlations $\overline{C_n\left(r\right)}$ [Eq. (2)] decay exponentially with distance $r$ for the disordered systems, as opposed to decaying with a power law as in the clean system. (b) Disorder-averaged electron two-point functions decay exponentially in all disordered systems. (c) Disorder-averaged superconducting pair-field correlations decay exponentially in the disordered systems, in direct contrast to the power-law decay in the nondisordered Luttinger liquid. In the disordered systems, error bars (omitted) are on the order of statistical fluctuations and are mostly not visible.

Figure 2

Disorder-averaged spin-spin correlation $\overline{C_{\sigma }\left(r\right)}$ [Eq. (3)] for $L=144$-site chains. Data are shown for distances $r\le L/3$, as fluctuations increase with distance. Black dashed lines indicate $r^{-2}$ decay. (a) $\overline{C_{\sigma }\left(r\right)}$ in random Heisenberg chain (yellow) and random-hopping Hubbard chain at half-filling (purple) (see main text), averaged over at least 500 realizations. In both chains, $\overline{C_{\sigma }\left(r\right)}$ exhibits long-distance $r^{-2}$ behavior for $r>10$. Error bars (omitted) are not visible for $r>10$. (b) $\overline{C_{\sigma }\left(r\right)}$ in random-hopping and random-potential systems at $n=11/12$ electron filling, averaged over at least 1000 realizations, also exhibit decay close to $r^{-2}$. Error bars (omitted) are on the order of statistical fluctuations.

Figure 3

Spin correlation statistics in a $L=144$ site chain with random potential $W_{\mu }/2=3\overline{t}$. Disorder-averaged spin correlations $\overline{C_{\sigma }\left(r\right)}$ and the root-mean-square spin correlations $\sqrt{\overline{{\left[C_{\sigma }\left(r\right)\right]}^2}}$ shown for $r\le L/3$. In the absence of disorder, the two quantities should be equivalent. Their difference here indicates the importance of rare regions at a strong disorder fixed point. A dotted line shows a $1/r^2$ decay and a dashed line shows a $1/r$ decay.

Figure 4

Results of numerically implementing bond decimation, assuming charge localization, for various system sizes at electron density $n=11/12$, averaged over 2000 realizations. (a) Log-log plot showing $r^{-2}$ scaling of probability $P_s\left(r\right)$ of forming a singlet pair over $r$ lattice sites. (b) Probability distribution of the logarithmic energy scale $\zeta =ln(J_0/J)$, when 50 free spins remain, where $J_0$ is the largest bond energy. Solid lines are fits for $\mathrm{\Gamma }$ according to $P(\zeta ,\mathrm{\Gamma })=e^{-\zeta /\mathrm{\Gamma }}/\mathrm{\Gamma }$. (c) Extracted best-fit values for energy scale $\mathrm{\Gamma }$ as a function of system size $L$ follows a trend $\mathrm{\Gamma }\sim \sqrt{L}$, a clear signature of infinite randomness.

Figure 5

Density-density correlation functions in half-filled Hubbard chains of length $L=144$ with (a) weak and (b) strong site and bond disorder. In both cases, density correlations in the chain with random hopping remain dominated by interactions and exhibit little change as the disorder is increased. In contrast, the chain with random potential undergoes a transition in behavior as the disorder increases. The density fluctuation correlations in the chain with weak random potential appear to have a short-range, interaction dominated region joined to a disorder-dominated region at large distances. At strong disorder, the same system becomes dominated by disorder.

Figure 6

Typical spin correlations $\overline{ln|C_{\sigma }\left(r\right)|}$ in a $L=144$ random potential chain at $n=11/12$ electron filling, with $W_{\mu }/2=3\overline{t}$.