Fate of 1D Spin-Charge Separation Away from Fermi Points

Abstract

We consider the dynamic response functions of interacting one dimensional spin-1/2 fermions at arbitrary momenta. We build a nonperturbative zero-temperature theory of the threshold singularities using mobile impurity Hamiltonians. The interaction induced low-energy spin-charge separation and power-law threshold singularities survive away from Fermi points. We express the threshold exponents in terms of the spinon spectrum.PACS numbers: 71.10.PmThe low-energy excitations of interacting spin-1/2 fermions confined to one dimension (1D) is well represented by two collective bosonic modes. These modes are the quantized waves of spin and charge densities. Their spectra are linear; the corresponding velocities, v s and v c , differ from each other. A microscopic consideration [1,2] of the repulsive interaction between spinful fermions leads to v c > v s .The spectra of the collective modes can be probed in a momentum-resolved tunneling [3,4] or in an ARPES [5] or photoemission [6] experiment. In these methods, a spin-1/2 fermion with a given momentum tunnels into or out of the studied system. The tunneling inevitably perturbs each of the two collective modes. For example, in the case of a low-energy particle (k F ≫ k − k F > 0), the small difference k − k F is shared between the excitations of the two modes. The Luttinger liquid (LL) theory predicts [2] that at given k the tunneling probability is singular at energies v s (k−k F ) and v c (k−k F ), corresponding to the entire momentum k − k F given to the "spinon" or "holon" belonging to the spin and charge mode, respectively. The exponents of the two power-law singularities depend on a single number, the LL parameter K c for the charge mode. The two sharp peaks in the momentumresolved tunneling probability at energies associated with excitation of the two modes, ω = v s,c · (k − k F ), are the hallmark of the spin-charge separation in the LL.The momentum-resolved tunneling rate is proportional to the fermionic spectral functionRecently, considerable progress was achieved in the analytical theory of dynamic responses of a 1D system away from the Fermi points for spinless fermions [7][8][9][10][11]. The developed methods map the 1D dynamic response problem near the edge of support onto the "mobile quantum impurity" effective Hamiltonian [12]. For spinless fermions in the weak-interaction limit, one may consider the generic spectrum of free fermions exactly, while treating their interaction perturbatively [7,8]. For example, at |k| < k F the threshold coincides with the spectrum of a hole, which can be thought of as a mobile quantum impurity. Because of the interactions, it "shakes up" the fermions in the vicinity of Fermi points, leading to the orthogonality catastrophe and to the power-law behavior of A(k, ω) at the threshold [7,8]. The perturbation theory allows one to identify the quantum numbers of the impurity and to match the phenomenological theory of threshold exponents [10] valid at any interaction strength with the weak-interaction limit.In this Letter we bui...

Charge dynamics in half-filled Hubbard chains with finite on-site interaction

“…15 For spin-1/2 models, this means defining spinless fermions associated with holon and spinon bands that have a finite curvature about the Fermi points. 18,19 The idea behind the effective impurity models for edge singularities is the same for all dynamic correlation functions. Essentially, it involves defining high energy subbands within the dispersion of elementary excitations, in addition to the chiral low energy modes.…”

“…[4][5][6][7][8] In addition, ultracold atoms trapped in optical lattices have emerged as a new means to study coherent dynamics of 1D models, including integrable ones which are not realizable in condensed matter systems. 9 At the same time, significant progress has been achieved in developing analytical [10][11][12][13][14][15][16][17][18][19][20][21][22] and numerical [23][24][25] techniques to study dynamical correlation functions in the high energy regime where conventional Luttinger liquid theory 26,27 does not apply. Analytically, it is possible to compute exponents of power-law singularities that develop near thresholds of the spectrum of dynamical correlation functions at arbitrarily high energies.…”

Charge and spin fractionalization beyond the Luttinger-liquid paradigm

“…The LL theory also predicts the fractionalization of injected charge into two chiral modes (leftand right-going) [6][7][8][9][10] , a phenomenon recently confirmed experimentally 11 . Along the experimental advances also theoretical progress was recently achieved pertaining extensions beyond the LL limit by incorporating nonlinearity of the dispersion, leading to qualitative changes in the spectral function [12][13][14][15][16] and relaxation processes of 1D electronic systems 17 .…”

Dynamical spin structure factor of one-dimensional interacting fermions

“…A detailed analysis of matching between the threshold singularities due to finite mass (Ref. 20) with our RG result is outside the scope of this paper. However, one can compare the high-frequency tails due to each of these effects.…”