Spectral functions of two-band spinless fermion and single-band spin-$\frac{1}{2}$ fermion models

We examine zero-temperature one-particle spectral functions for the one-dimensional two-band spinless fermions with different velocities and general forward-scattering interactions. By using the bosonization technique and diagonalizing the model to two Tomonaga-Luttinger liquid Hamiltonians, we obtain general expressions for the spectral functions, which are given in terms of the Appell hypergeometric functions. For the case of identical two-band fermions, corresponding to the SU(2) symmetric spin-1/2 fermions with repulsive interactions, the spectral functions can be expressed in terms of the Gauss hypergeometric function and are shown to recover the double-peak structure suggesting the well-known “spin-charge” separation. By tuning the difference in velocities for the two-band fermions, we clarify the crossover in spectral functions from the spin-charge separation to the decoupled fermions. We discuss the relevance of our results to the spin-1/2 Hubbard model under a magnetic field, which can be mapped onto two-band spinless fermions.

Figure 1

Exponents ${\eta }_A$ for possible order parameters as a function of $v_2/v_1$ for $g_a/\left(\pi v_1\right)=0.1$ (a) and $g_a/\left(\pi v_1\right)=0.0$ (b) with fixed $g/\left(\pi v_1\right)=0.3$.

Figure 2

Phase diagram on a plane of $v_2/v_1$ and $g_a/\left(\pi v_1\right)$ with fixed $g/\left(\pi v_1\right)=0.3$. In the “unstable” region, $u_2$ becomes imaginary.

Figure 3

Spectral functions $A_{R,s}(q=0.01,\omega )$ with SU(2) symmetry for several choices of ${\gamma }_{\rho }$ ($=0.2$, $0.4$, $0.6$, and $0.8$ from top to bottom) with fixed $u_{\rho }=2$, $u_{\sigma }=1$, and $q=0.01$. The dots represent the value at $\omega =u_{\rho }q$ given by Eq. (4.24).

Figure 4

$u_2/u_1$ dependence of the exponents ${\nu }_{1,\pm }$, ${\nu }_{1,\pm }^{\prime }$ (top) and ${\nu }_{2,\pm }$, ${\nu }_{2,\pm }^{\prime }$ (bottom) for $K_1=K_2=0.8$ and $g/\left(\pi u_1\right)=0.3$. The two-component TL liquid is unstable for $u_2<u_{2c}\approx 0.0576u_1$.

Figure 5

$u_2/u_1$ dependence of the exponents ${\beta }_{1,\pm }$, ${\beta }_{1,\pm }^{\prime }$ (top) and ${\beta }_{2,\pm }$, ${\beta }_{2,\pm }^{\prime }$ (bottom) for $K_1=K_2=0.8$ and $g/\left(\pi u_1\right)=0.3$.

Figure 6

Values for $u_2/u_1$ satisfying ${\beta }_{2,+}=0$ (solid line) and ${\beta }_{1,-}=0$ (dotted line), with fixed $K(=K_1=K_2)=1$ and $0.8$.

Figure 7

Velocities as a function of the magnetic field for $U/t=3$ and $\rho =1/2$.

Figure 8

Spectral functions $A_{R,1}(q,\omega )$ and $A_{R,2}(q,\omega )$ for $U/t=3$ and $q=0.01/a$, with fixed $h/t=0.2$ (top) and 0.6 (bottom).

Figure 9

Contour plot of the spectral function $A_a(k,\omega )$ for $a=2$ (left) and $a=1$ (right), with fixed $U/t=3$ and $h/t=0.2$ (top) and $h/t=0.6$ (bottom). The dotted lines denote the bare dispersion ${\varepsilon }_a\left(k\right)\simeq u_a(k-k_{F,a})$ where the Fermi momenta are given by $(k_{F,2},k_{F,1})\simeq (0.64,0.93)$ for $h/t=0.2$ and $(0.35,1.22)$ for $h/t=0.6$. In the insets, the bare energy dispersions are shown and the rectangular represent the region where the spectral functions are plotted in the main figures.

Figure 10

Exponents ${\beta }_{a,\pm }$ and ${\beta }_{a,\pm }^{\prime }$ as a function of the magnetic field for $U/t=3$ and $\rho =1/2$.

Figure 11

$U$ dependence of the critical value for $h/t$ satisfying ${\beta }_{2,+}=0$ (solid line) and ${\beta }_{1,-}=0$ (dotted line), for $\rho =1/2$. The dash line denotes the upper bound of $h/t$ for the real $u_2$, i.e., corresponding to $u_{2c}$.