Diamagnetism of doped two-leg ladders and probing the nature of their commensurate phases

We study the magnetic orbital effect of a doped two-leg ladder in the presence of a magnetic field component perpendicular to the ladder plane. Combining both low-energy approach (bosonization) and numerical simulations (density-matrix renormalization group) on the strong coupling limit ($t\text{−}J$ model), a rich phase diagram is established as a function of hole doping and magnetic flux. Above a critical flux, the spin gap is destroyed and a Luttinger liquid phase is stabilized. Above a second critical flux, a reentrance of the spin gap at high magnetic flux is found. Interestingly, the phase transitions are associated with a change of sign of the orbital susceptibility. Focusing on the small magnetic field regime, the spin-gapped superconducting phase is robust, but immediately acquires algebraic transverse (i.e., along rungs) current correlations which are commensurate with the $4k_F$ density correlations. In addition, we have computed the zero-field orbital susceptibility for a large range of doping and interaction ratio $J∕t$: we found strong anomalies at low $J∕t$ only in the vicinity of the commensurate fillings corresponding to $\delta =1∕4$ and $1∕2$. Furthermore, the behavior of the orbital susceptibility reveals that the nature of these insulating phases is different: while for $\delta =1∕4$ a $4k_F$ charge density wave is confirmed, the $\delta =1∕2$ phase is shown to be a bond order wave.

Figure 1

(Color online) The isotropic $t\text{−}J$ ladder under a magnetic field. If the magnetic field has a component perpendicular to the ladder plane, a flux $\varphi$ passes through each plaquette. Below is the gauge of the Peierls substitution with opposite phases $\pm \varphi ∕2$ along the legs.

Figure 2

The four typical shapes of the bands depending on the flux and on the ratio $t_{\perp }∕t_{\Vert }$. Critical fields ${\varphi }_0$ $\left({\varphi }_c\right)$ signal the appearance of a double well (band gap). Note that the D phase always has only two Fermi points whatever the filling, and that the C phase when $\varphi =\pi$ always has four Fermi points.

Figure 3

Phase diagram in the weak-coupling limit for an isotropic ladder restricted to fillings $0⩽n⩽1$. Phases with four (two) Fermi points fall generically into the C1S0 (C1S1) class. At half-filling, interactions will drive the system into a Mott insulating phase for $\varphi <{\varphi }_0$, while a band insulating phase occurs when $\varphi >{\varphi }_c$. Other phases can be found if the ratio of the Fermi velocity is large (C2S1 and C2S2) and if the $d$-band Fermi wave vector is $\pi ∕2$ (C0S1 and C1S2).

Figure 4

(Color online) Orbital susceptibility, charge, and spin gaps for $J∕t=0.5$ and on the $\delta =0.063$ line of Fig. 5. The zeros of the susceptibility precisely probe the different two phase transitions.

Figure 5

Phase diagram of the $t\text{−}J$ model for $J∕t=0.5$, for which the system has dominant superconducting fluctuations at zero flux, determined from the zeros of the susceptibility on a system with $L=32$. Results are very similar to Fig. 3, with a transition to a C1S1 phase at intermediate flux and a reentrance to a C1S0 phase at large flux.

Figure 6

(Color online) Spin correlations $S\left(x\right)$ for fluxes (a) in the C1S1 phase and (b) in the C1S0 phases at low and high fluxes in the phase diagram of Fig. 5 on the $\delta =0.25$ line. Inset: The behavior of the spin gap computed on the same system using Eq. (8). These independent observables confirm the reentrance of the C1S0 phase.

Figure 7

(Color online) Superconducting correlations $P\left(x\right)$ for three values of the flux corresponding to each phase encountered on the $\delta =0.063$ line of the phase diagram of Fig. 5. The decrease exponent is much higher in the C1S1 and high-flux C1S0 phases, and the overall magnitude is strongly reduced, suggesting a metallic state. Note that correlations display oscillations associated with a $4k_F$ contribution.

Figure 8

(Color online) Local hole density and local currents in the ground state of the C1S0 phase (at low flux). The area of the circles is proportional to $h\left(x\right)-\delta$, with $h\left(x\right)$ the local density of holes. An excess of hole is represented by empty circles, while a lack of holes is represented by full circles. The line thickness is proportional to the current strength, and the arrow gives the direction. Note that the transverse current has been rescaled [by a factor $(\times 5)$]; the main screening currents are on the chains. Interestingly, the orbits have a length scale ${\delta }^{-1}$ determined by the average hole density.

Figure 9

(a) Transverse current correlations become algebraic in the C1S0 phase once the magnetic field is turned on (same parameters as in Fig. 8). (b) Demodulation of the signal enables to extract clearly the wave vector $2\pi \delta$ of the correlations.

Figure 10

(Color online) Zero-field susceptibility ${\chi }_0$ as a function of doping $\delta$ and interaction parameters $J∕t$ for a ladder with $L=48$. A strong reduction at low $J∕t$ and commensurabilities $\delta =1∕4$ and $1∕2$ are clearly visible. The computed susceptibility is nearly zero at $\delta =1∕4$, but subject to finite size effects (see Appendix ). On the contrary, the $\delta =1∕2$ phase has a finite susceptibility. Inset: The derivative ${\phantom{\mid }d{\chi }_0∕d\delta \mid }_{\delta =0}$ as a function of $J∕t$.

Figure 11

(Color online) Two-particle charge gap as a function of doping showing the emergence of insulating phases at commensurabilities $\delta =1∕2$ and $\delta =1∕4$ at low $J∕t$. The smallest flux we have access to is enough to destroy the $\delta =1∕4$ CDW phase (see text for details), while the $\delta =0.5$ BOW phase is more stable. Data are computed with an extra $J=+0.3$ at the two extremal rungs.

Figure 12

(Color online) Local expectation values with the conventions of Fig. 8 on the $n=\delta =1∕2$ line (quarter-filling) in the bond-density-wave phase at low flux.

Figure 13

(Color online) Local kinetic bonds $t_{\nu }\left(x\right)$ in the $\delta =1∕2$ BOW phase computed at zero magnetic field. The parallel bond orders are strongly oscillating at wave vector $\pi$, while the transverse bond and the transverse kinetic bond and electronic densities are uniform (see Fig. 12).

Figure 14

Depending on the filling, the number of Fermi points can be either 2 or 4 (sketched on the bands of the B phase of Fig. 2). Note that at low and high fillings, the two Fermi velocities have opposite signs, while at intermediate filling, they have the same sign.

Figure 15

Susceptibility as a probe to the number of Fermi points for $0.5<1-\delta <1.0$ and $t_{\perp }=t_{\Vert }$. ${\varphi }^{2\leftrightarrow 4}$ indicates the transitions from four to two and from two to four Fermi points.

Figure 16

Zero-field orbital susceptibility of the noninteracting system for $t_{\perp }=t_{\Vert }$ from Eqs. (A16, A17). The singularity at $n=0.5$ marks the transition from two to four Fermi points. When $n<0.5$, the susceptibility changes sign for $n\simeq 0.3975\dots$, while the system always has two Fermi points.

Figure 17

(Color online) Finite size effects on the calculation of the zero-field susceptibility at low $J∕t$. (a) energies as a function of flux. (b) ${\chi }_0$ as a function of the doping $\delta$ from Eq. (B2) for two different $d\varphi$.