Field theory for a fermionic ladder with generic intrachain interactions

An effective low-energy field theory is developed for a system of two chains. The main novelty of the approach is that it allows one to treat generic intrachain repulsive interactions of arbitrary strength. The chains are coupled by direct tunneling and four-fermion interactions. At low energies, the individual chains are described as Luttinger liquids with an arbitrary ratio of spin $v_s$ and charge $v_c$ velocities. A judicious choice of the basis for the decoupled chains greatly simplifies the description and allows one to separate high- and low-energy degrees of freedom. In a direct analogy to the bulk cuprates, the resulting effective field theory distinguishes between three qualitatively different regimes: (i) small doping ($v_c\ll v_s$), (ii) optimal doping ($v_s\approx v_c$), and (iii) large doping ($v_s\ll v_c$). I discuss the excitation spectrum and derive expressions for the electron spectral function, which turns out to be highly incoherent. The degree of incoherence increases when one considers an array of ladders (stripe phase).

Figure 1

The succession of transformations leading from the original formulation (1) to Eq. (30). The framed fields are the ones taking part in the final effective theory.

Figure 2

The spectral function $A(\omega ,\pm Q+k_F)=A(-\omega ,\pm Q+k_F)$ (61) as a function of $\omega /M_B>0$ (blue). The sharp maximum at $\omega =\epsilon (k)$ is replaced by a power-law singularity. The magenta graph shows the same function convoluted with the Gaussian resolution function $(3/\sqrt{\pi }{M}_B)\exp [-9{\omega }^2/M_B^2]$.

Figure 3

The schematic phase diagram of a doped two-leg ladder.

Figure 4

The spectral function $A(\omega ,Q)$ (82) as a function of $\omega /M_B$ (blue figure). The threshold singularity present for a single ladder is removed by the emission of soft phase fluctuations (compare with Fig. 2). The magenta figure represents the spectral function convoluted with the Gaussian resolution function $(3/\sqrt{\pi }{M}_B)\exp [-9{\omega }^2/M_B^2]$.