Spin-charge separation in one-dimensional fermion systems beyond Luttinger liquid theory

We develop a nonperturbative zero-temperature theory for the dynamic response functions of interacting one-dimensional spin-1/2 fermions. In contrast to the conventional Luttinger liquid theory, we take into account the nonlinearity of the fermion dispersion exactly. We calculate the power-law singularities of the spectral function and the charge- and spin-density structure factors for arbitrary momenta and interaction strengths. The exponents characterizing the singularities are functions of momenta and differ significantly from the predictions of the linear Luttinger liquid theory. We generalize the notion of the spin-charge separation to the nonlinear spectrum. This generalization leads to phenomenological relations between threshold exponents and the threshold energy.

Figure 1

(Color online) Spinon and holon spectra, ${ϵ}_s\left(k\right)$ and ${ϵ}_c\left(k\right)$, and spectral function $A\left(k,\omega \right)$ for momenta $k\ge k_F$. For repulsive interactions, a nonlinear fermion spectrum reduces ${ϵ}_s\left(k\right)$ and increases ${ϵ}_c\left(k\right)$ compared to the linear Luttinger liquid spectra. The spectral function has a power-law singularity at the spinon mass shell $\omega \approx {ϵ}_s\left(k\right)$ with exponent ${\mu }^s$ and ${ϵ}_s\left(k\right)$ is the edge of support of $A\left(k,\omega \right)$. The singularity at the holon mass shell $\omega \approx {ϵ}_c\left(k\right)$ is smeared out (dotted line) away from $k_F$ for nonintegrable systems due to holon decay.

Figure 2

(Color online) The spectral function $A\left(q,\omega \right)$ for a linear Luttinger liquid with repulsive interactions along a cut for fixed $q=k-k_F<0$. $A\left(q,\omega \right)$ is characterized by sharp power-law singularities at the holon and spinon mass shells $\omega =v_{c,s}q$ and at the inverted holon mass shell $\omega =-v_{c}q$.

Figure 3

(Color online) The injection of a hole with momentum $k$ and energy $\omega \approx {ϵ}_c\left(k\right)$ leads to the formation of a holon on the mass shell and low-energy excitations around the holon Fermi edge. For clarity, the created low-energy spinons are not displayed. For this particular configuration, energy and momentum conservation lead to $v_c\Delta k=\left|\omega -{ϵ}_c\left(k+\Delta k\right)\right|$. The impurity band (around $k$) and the low-energy band (around $k_F$) can be separated only if $\Delta k⪡k_F-k$. This is the case for $\left|\omega -{ϵ}_c\left(k\right)\right|⪡{\left(k-k_F\right)}^2/\left(2m^{\ast }\right)$.

Figure 4

Holon spectrum ${\omega }_c\left(k_c\right)$ and spinon spectrum ${\omega }_s\left(k_s\right)$ in the noninteracting case. The holon spectrum has Fermi momentum $2k_F$. The spinon spectrum is defined for $-k_F<k_s<k_F$ and the Fermi point is placed at $k_s=0$. The spinon spectrum is particle-hole symmetric, so the dashed line indicates energies of spinon holes.

Figure 5

(Color online) Spinon interaction process generated by the projection of the spin flip interaction in Eq. (30): a spinon particle-hole pair near the left Fermi point scatters into a spinon hole at $k_s<0$ and a spinon near $-k_s>0$. The corresponding interaction operator is similar to the sine-Gordon term (7) and, analogously, its amplitude flows to zero upon bandwidth reduction.

Figure 6

(Color online) Mobile-impurity band structure for the calculation of $A\left(k,\omega \right)$ for $3k_F<k5k_F$ (i.e., $n=2$) and $\omega \approx |{ϵ}_s(k-4k_F)|$. The final state contains a spinon with momentum $k_{d,2}=k-5k_F<0$, a holon at its Fermi momentum as well as two particle-hole pairs in the holon sector which absorb the extra momentum $4k_F$. In this figure, we assume $m_{-}=2$, so the spinon sector contains two additional particle-hole pairs. The Hamiltonian is projected onto narrow bands around these momenta. Within each band, the spectrum can be linearized.

Figure 7

(Color online) Spectral function $A\left(k,\omega \right)$ in the $\left(k,\omega \right)$ plane (a) and along a cross section for fixed $0<k<k_F$ (b). (a) $A\left(k,\omega \right)$ is nonzero in the shaded areas. For repulsive interactions, the edge of support is at the spinon mass shell $\omega =\pm \left|{ϵ}_s\left(k\right)\right|$. At the edge for $\omega \gtrless 0$ in the momentum range $\left(2n-1\right)k_F<k<\left(2n+1\right)k_F$, $A\left(k,\omega \right)$ has a power-law singularity characterized by the exponent ${\mu }_{n,\pm }^s$. A sharp power law at the holon mass shell ${ϵ}_c\left(k\right)$ (dotted lines) exists only for $k\approx \left(2n\pm 1\right)k_F$. The corresponding exponents ${\mu }_{n,-}^c$ are continuous at $\omega =0$. (b) Compared to the Luttinger liquid case, the edge exponents is modified. The singularity at the holon mass shell ${ϵ}_c\left(k\right)$ is smeared out because the band curvature leads in general to a finite holon lifetime.

Figure 8

(Color online) Density structure factor $S\left(k,\omega \right)$. For noninteracting systems $S\left(k,\omega \right)$ is constant for ${\omega }_{-}\left(k\right)<\omega <{\omega }_{+}\left(k\right)$ and vanishes otherwise. For nonzero interactions, a power-law singularity with exponent ${\mu }_n^{\text{DSF}}$ forms at the lower threshold ${\omega }_{-}\left(k\right)$. In the inset: particle-hole excitation with momentum $k$ giving rise to the singularity at ${\omega }_{-}\left(k\right)$.

Figure 9

(Color online) Position of power-law singularities in the spectral function $A\left(k,\omega \right)$ for the strongly interacting Hubbard model. At the main holon branch ${ϵ}_c\left(k\right)$, power-law singularities have an exponent ${\mu }_{0,-}^c$ regardless of the sign of $\omega$. Weaker power laws with exponents ${\mu }_{n,-}^c$ can be found at the shifted holon mass shell ${ϵ}_c\left(k+2n{k}_F\right)$ for $n∊\mathbb{Z}$.

Figure 10

Feynman diagram for the imaginary part of the second order self-energy. The straight lines denote fermions with momentum and spin quantum numbers. The wiggly line depicts the interaction with momentum exchange $q$.

Figure 11

(Color online) Singular lines in the spectral function $A\left(k,\omega \right)$ for a weakly interacting systems with spectrum $ϵ\left(k\right)=\left(k^2-k_F^2\right)/\left(2m\right)$. Up to second order in the interaction strength, singularities appear at the mass shell $\omega \approx ϵ\left(k\right)$ for all $k$ as well as at $\omega \approx -ϵ\left(k\mp 2k_F\right)$ for $k\gtrless \pm k_F$. In the configurations giving rise to the singularities, the energy is carried by a spinon along the red (solid) line or by a holon along the green (dashed) line.