Tuning between singlet, triplet, and mixed pairing states in an extended Hubbard chain

We study spin-half fermions in a one-dimensional extended Hubbard chain at low filling. We identify three triplet and one singlet pairing channels in the system, which are independently tunable as a function of nearest-neighbor charge and spin interactions. In a large-size system with translational invariance, we derive gap equations for the corresponding pairing gaps and obtain a Bogoliubov–de Gennes Hamiltonian with its nontrivial topology determined by the interplay of these gaps. In an open-end system with a fixed number of particles, we compute the exact many-body ground state and identify the dominant pairing revealed by the pair density matrix. Both cases show competition between the four pairing states, resulting in broad regions for each of them and relatively narrow regions for mixed-pairing states in the parameter space. Our results enable the possibility of tuning a nanowire between singlet and triplet pairing states without breaking time-reversal or $\text{SU}\left(2\right)$ symmetry, accompanied by a change in the system's topology.

Figure 1

Mean-field phase diagram characterizing singlet, triplet, and mixed pairing states under time-reversal and $\text{SU}\left(2\right)$ symmetries (when the three triplet couplings are equal, $g_{\uparrow }=g_{\downarrow }=g_{+}\equiv g_{\gamma }$). The diagram is obtained by numerically solving the gap equations, which determine the equilibrium state of the system. Singlet ($|{\Delta }_{-}|$) and triplet ($|{\Delta }_t|$) pairing strengths are illustrated by vector arrows $\left(\right|{\Delta }_{-}|,|{\Delta }_t\left|\right)$ as a function of attractive singlet and triplet pair couplings, $g_{-}$ and $g_{\gamma }$, respectively (notice that both couplings are negative). Each arrow has length proportional to $\sqrt{|{\Delta }_{-}{|}^2+{\left|{\Delta }_t\right|}^2}$ and slope equal to $|{\Delta }_{-}/{\Delta }_t|$. Purely triplet and purely singlet regions (filled with horizontal and vertical arrows, respectively) sandwich a relatively narrow mixed-pairing region (shadowed, filled with finite-slope arrows), with boundaries marked by gray solid lines. The red dashed line in the mixed region indicates the boundary between topologically trivial (below) and nontrivial (above) regions as the chemical potential $\mu =-1.7t$. There is no pairing beyond the left and bottom axes of this diagram.

Figure 2

(a) Intraspin pair fraction $P_{\updownarrow }$ (solid curve, axis on the left of graph) and the single-particle density of states (DoS) at the Fermi level (dashed curve, axis on the right of graph) vs the second-nearest-neighbor tunneling $t^{\prime }$. (b) Intraspin pair fraction $P_{\updownarrow }$ vs filling $n$ at attractive ($g_{\updownarrow }=-0.1t$, solid curve) and repulsive ($0.1t$, dashed) pairing interactions while the other two couplings are set zero, $g_{\pm }=0$. (c),(d) Interspin triplet and singlet pair fractions $P_{\pm }$ vs $n$ in attractive ($g_{\pm }=-0.2t$, respectively, solid curves) and repulsive ($0.2t$, dashed) cases, with the other two couplings set to zero as denoted in the plots. The relative pair fraction is measured from that of a free system, so negative values mean the disfavor of pairing.

Figure 3

Four cases shows relative pair fractions $P_{\alpha }$ (bottom panels), entanglement entropy $\delta S$ (middle) and pair susceptibility $\chi$ (top) tuned with the pairing couplings $g$. The intraspin, interspin triplet, and singlet pair fractions ($P_{\alpha =\updownarrow ,+,-}$) are represented by red triangles, blue squares, and green circles, respectively. The filled (empty) symbols denote states that show (do not show) the three signatures for a stable pairing state discussed in text. (a) Tuning between singlet and interspin triplet pairings as the singlet coupling $g_{-}$ varies, while the top panel shows pair susceptibilities ${\chi }_{--}$ (green solid curve) and ${\chi }_{+-}$ (blue dashed). (b) Tuning between interspin triplet and singlet pairings as the interspin triplet coupling $g_{+}$ varies, presented together with ${\chi }_{++}$ (blue solid) and ${\chi }_{-+}$ (green dashed). (c) Tuning between intraspin and interspin triplet pairings as the intraspin coupling $g_{\updownarrow }$ varies, presented together with ${\chi }_{\updownarrow \updownarrow }$ (red solid) and ${\chi }_{+\updownarrow }$ (blue dashed). (d) Tuning between intraspin and singlet pairings as the intraspin coupling $g_{\updownarrow }$ varies, presented together with ${\chi }_{\updownarrow \updownarrow }$ (red solid) and ${\chi }_{-\updownarrow }$ (green dashed).

Figure 4

(a)–(f) State diagrams showing stable pairing regions in $g_{+}$-$g_{\updownarrow }$ plane at $g_{-}/t=0$, $-0.04$, $-0.08$, $-0.1$, $-0.2$, and $-0.4$, respectively. Red triangles, blue squares, and green circles represent intraspin, interspin triplet, and singlet pairing, respectively, while the magenta diamonds and cyan stars represent a mix of intraspin and interspin pairings as well as that of interspin triplet and singlet pairings, respectively. The dashed lines $g_{+}=g_{\updownarrow }$ indicates $\text{SU}\left(2\right)$ symmetry of the system.

Figure 5

The relative pair fraction $P_N$ vs normalized coupling constant $g$ at half filling with three different particle numbers of $N=6$, 8, and 10 (blue dotted, green dashed, and red solid curves, respectively). In the region of $g/{\epsilon }_1<-1$, larger $N$ means larger $P_N$, so the system is regarded as a superconducting state. The inset shows that the phase-transition point $P_N-P_{N-2}=0$ approaches the exact solution $g_c=-{\epsilon }_1$ as $N$ increases. The red circles, which are for an infinitely large-size system predicted by the mean-field (MF) treatment, show that $P$ rises from zero exactly at $g_c$.

Figure 6

The relative pair fraction $P$ vs on-site interaction $U$ in a Hubbard chain with various numbers of particles $N$ and sites $L$. The red solid, green dashed, blue dotted, and purple dot-dashed curves represent the cases of $(N,L)=(8,20)$, $(8,18)$, $(8,16)$, and $(4,10)$, respectively. In the attractive interaction region ($U<0$), increasing size at fixed $N$ (as the system dilutes) and at fixed $N/L$ (extends) coincides with positive and increasing $P$, so the system is regarded as a stable pairing state or a superconducting state in the thermodynamic limit. Such trends do not hold in the repulsive interaction region ($U>0$). Therefore, we obtain the transition point at $U=0$, consistent with the solution from the BCS gap equation (see text).