Generic short-range interactions in two-leg ladders

We derive a Hamiltonian for a two-leg ladder which includes an arbitrary number of charge and spin interactions which must be weak but are otherwise of arbitrary strength. To illustrate this Hamiltonian we consider two examples and use a renormalization-group technique to evaluate the ground-state phases. The first example is a two-leg ladder with zigzagged legs. We find that increasing the number of interactions in such a two-leg ladder may result in a richer phase diagram, particularly at half-filling where a few exotic phases are possible when the number of interactions are large and the angle of the zigzag is small. In the second example we determine under which conditions a two-leg ladder at quarter-filling is able to support a Tomanaga-Luttinger liquid phase. We show that this is only possible when the spin interactions across the rungs are ferromagnetic. In both examples we focus on lithium purple bronze, a two-leg ladder with zigzagged legs which is thought to support a Tomanaga-Luttinger liquid phase.

Figure 1

(a) A standard two-leg ladder with hopping strengths $t$ and $t_{\perp }$ and several electron-electron interactions defined by $U$ and $X$ where $X=V,J$. (b) A zigzag two-leg ladder where the legs are bent to make a constant angle $\varphi$. Electron-electron interactions can be defined similarly to the standard case.

Figure 2

Phase diagrams at half-filling with $t=t_{\perp }=1$, $U=2$ and (a) $N=2$, $J=0$ (b) $N=2$, $V=0$ (c) $N=10$, $J=0$ (d) $N=10$, $V=0$. Dashed lines indicate where $\varphi$ is small enough so that some interactions not considered are larger than some which are considered.

Figure 3

Phase diagram at half-filling with $N=10$, $t=t_{\perp }=1$, $U=2$ and $\varphi =\pi /4$.

Figure 4

(Color online) Phase diagrams at quarter-filling with $t=1$, $t_{\perp }=0.01$, $U=2$ and (a) $N=2$, $J=0$ (b) $N=2$, $V=0$ (c) $N=10$, $J=0$, and (d) $N=10$, $V=0$. Dashed lines indicate where $\varphi$ is small enough so that some interactions not considered are larger than some which are considered.

Figure 5

Phase diagrams at quarter-filling with $t=1$, $t_{\perp }=0.01$, $U=2$ and (a) $V_{\parallel }=V_d=J_{\parallel }=0$, (b) $V_{\perp }=V_d=J_{\parallel }=0$, (c) $V_{\perp }=V_{\parallel }=J_{\parallel }=0$, and (d) $V_{\perp }=V_{\parallel }=V_d=0$.

Figure 6

(Color online) The critical exponent $\alpha$ at quarter-filling with $t=1$, $t_{\perp }=0.01$, $U=2$, $V_{\parallel }=V_{\perp }$ and $J_{\parallel }=\left(1+2j\right)$ with $j=0,1,2$. While the arrow indicates increasing $J_{\parallel }$ for the curves shown here, it does not indicate a general trend. The lowest solid line and the lowest dashed line diverge so are not TLL, while the other four lines approach constant values and imply a TLL state.

Figure 7

(Color online) The critical exponent $\alpha$ at quarter-filling with $t=1$, $t_{\perp }=0.01$, $U=2$ and (a) $J_{\parallel }=V_{\perp }=V_{\parallel }=V_d=0$ resulting in a TLL, (b) $J_{\parallel }=-J_{\perp }$, $V_{\perp }=V_{\parallel }=V_d=0$ not resulting in a TLL.

Figure 8

(Color online) The renormalized coupling constants (rescaled by $4\pi v_q$) at quarter-filling with $t=1$, $t_{\perp }=0.01$, $U=2$, $J_{\parallel }=-J_{\perp }=0.5$, and $V_{\parallel ,\perp ,d}=0$.