Spin–Charge Separation and Spectral Observables

Spin–charge separation is experimentally meaningful only when it changes a response function. In a spinful Luttinger liquid, injecting one electron excites independent charge and spin sectors with velocities $u_c$ and $u_s$. Their convolution replaces a quasiparticle pole by branch-cut continua with distinct dispersing thresholds.

Required background. Spinful and Multicomponent Luttinger Liquids supplies the separated fields and electron exponent; Lehmann Representations and Spectral Functions fixes the retarded spectral convention. Helpful background. Photoemission and ARPES supplies the matrix-element and resolution conditions for comparing the spectral function with measured intensity.

Factorized electron propagation

For right movers, a spin-$\sigma$ electron factorizes into charge and spin vertices,

$$\psi_{R\sigma}\propto\eta_{R\sigma} \exp\left[-\frac{i}{\sqrt2} (\phi_c-\theta_c+\sigma\phi_s-\sigma\theta_s)\right].$$

At the Gaussian fixed point its Euclidean correlator is a product of charge and spin factors. In a spin-rotation-invariant system, $K_s=1$ and the spin factor propagates at $u_s$; the charge factors propagate at $u_c$ with interaction-dependent powers. Because the two light cones differ, there is no single spacetime trajectory carrying all of the injected electron’s quantum numbers.

For $K_c=1$ and $u_c=u_s=v_F$, the factors recombine into the free pole. Interactions usually make $u_c\ne u_s$ and give an anomalous exponent

$$\alpha_{\mathrm{bulk}}=\frac{K_c+K_c^{-1}-2}{4}.$$

Marginally irrelevant spin backscattering adds logarithmic corrections; curvature broadens and reshapes high-energy edges. The fixed-point factorization therefore predicts the low-energy singularity structure, not a complete line shape.

Spectral normalization and threshold structure

Define

$$G^R(k,t)=-i\Theta(t)\langle\{c_k(t),c_k^\dagger(0)\}\rangle, \qquad A(k,\omega)=-2\operatorname{Im}G^R(k,\omega).$$

For a canonical fermion,

$$\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}A(k,\omega)=1.$$

Near $k=k_F+q$, the linear Luttinger theory places nonanalytic edges near $\omega=u_sq$ and $\omega=u_cq$ (with particle/hole and chirality qualifications). The weight between them comes from sharing momentum and energy between the two sectors. The edge exponents depend on $K_c$ and on which side of the threshold is approached. Fourier transforms of the finite-temperature factorized correlator give the universal scaling form Orgad 2001, §§ 2–4.

The momentum-integrated local spectrum obeys

$$\int\frac{dk}{2\pi}A(k,\omega)\propto |\omega|^{\alpha_{\mathrm{bulk}}}$$

in the bulk scaling window. At an open end the exponent is instead $(K_c^{-1}-1)/2$ for SU(2)-invariant spinful fermions. A suppressed tunneling density of states is therefore consistent with a Luttinger liquid but is not by itself evidence for two distinct velocities.

What different probes see

Angle-resolved photoemission approximately measures occupied spectral weight multiplied by a matrix element and convolved with instrumental resolution. Separate dispersing features can constrain $u_s/u_c$, but phonons, multiple bands, surface states, and nonlinear thresholds are alternatives. Photoemission on the quasi-one-dimensional cuprate SrCuO$_2$ revealed two dispersive structures consistent with spinon and holon branches Kim et al. 1996, pp. 4054–4057; the inference relies on comparison with a microscopic chain model, not on branch counting alone.

The dynamical charge structure factor couples directly to charge density and has a low-energy mode near $\omega=u_c|q|$. The spin structure factor couples to the spin sector and has a corresponding velocity $u_s$. Tunneling probes the local convolution of both sectors. Time-domain wave-packet experiments can spatially separate charge and spin pulses, provided dispersion and boundaries are controlled. Agreement of velocities extracted from several such observables is substantially stronger evidence than a fit to one spectrum.

Nonlinear and finite-temperature limits

Linear bosonization predicts straight thresholds extending indefinitely, which no lattice band possesses. At fixed nonzero $q$ and very high resolution, curvature is a dangerously important correction: mobile-impurity theory replaces fixed-point exponents by momentum-dependent edge exponents Imambekov, Schmidt, and Glazman 2012, §§ II–IV. Temperature rounds singularities on a scale $T$, disorder relaxes momentum, and interchain hopping can eventually produce higher-dimensional coherence. A credible comparison must state a window where $T$, transverse hopping, disorder broadening, and curvature are all smaller than the resolved separation.

Exercises

1. Check the spectral sum rule for a free level $G^R(k,\omega)=(\omega-\varepsilon_k+i0^+)^{-1}$.

Solution

$\operatorname{Im}(x+i0^+)^{-1}=-\pi\delta(x)$, so $A(k,\omega)=2\pi\delta(\omega-\varepsilon_k)$. Its integral with $d\omega/(2\pi)$ equals one.

2. Evaluate the bulk tunneling exponent at $K_c=1/2$.

Solution

$\alpha_{\mathrm{bulk}}=(1/2+2-2)/4=1/8$. The local spectral weight vanishes as $|\omega|^{1/8}$, even though the system remains gapless.

References

• Imambekov, A., T. L. Schmidt, and L. I. Glazman. “One-Dimensional Quantum Liquids: Beyond the Luttinger Liquid Paradigm.” Reviews of Modern Physics 84 (2012): 1253–1306. DOI.
• Kim, C., A. Y. Matsuura, Z.-X. Shen, N. Motoyama, H. Eisaki, S. Uchida, T. Tohyama, and S. Maekawa. “Observation of Spin-Charge Separation in One-Dimensional SrCuO$_2$.” Physical Review Letters 77 (1996): 4054–4057. DOI.
• Orgad, D. “Spectral Functions for the Tomonaga–Luttinger and Luther–Emery Liquids.” Philosophical Magazine B 81 (2001): 377–398. DOI.