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Reviews of Modern Physics

Phase diagram and correlation exponents for interacting fermions in one dimension

Eugene B. Kolomeisky, Joseph P. Straley

Published Jan 1, 1996• 48 citations
Publisher

Abstract

A theory is given for the ground-state properties of short-ranged interacting spin-\textonehalf{} fermions moving in a periodic potential, based on a combination of renormalization-group ideas, the harmonic (Luttinger) liquid approach, and the symmetry properties of the Hubbard model, for a wide range of interactions and arbitrary fillings. For rational filling factors the outcome depends nontrivially on the parity of the order of the $2{k}_{F}$ umklapp scattering: even values fall into a universality class characterized by a separation of charge and spin degrees of freedom in the long-wavelength limit, while the action for the case of odd orders does not have this property, thus leading to distinct phase diagrams. It is argued that even when the long-wavelength action does exhibit a spin-charge separation it is not always an accurate approximation, particularly for the case of strong repulsion between electrons of the same spin. This leads to a classification of the possible ground states, associated correlation exponents, and phase transitions. The translational correlation exponents upon approaching insulator phases (existing at exact commensuration) by changing electron density can be related to the fractional charge of the solitons, using the Landauer rule.

DOI
10.1103/revmodphys.68.175

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A theory is given for the ground-state properties of short-ranged interacting spin-\textonehalf{} fermions moving in a periodic potential, based on a combination of renormalization-group ideas, the harmonic (Luttinger) liquid approach, and the symmetry properties of the Hubbard model, for a wide range of interactions and arbitrary fillings. For rational filling factors the outcome depends nontrivially on the parity of the order of the 2kF2{k}_{F}2kF​ umklapp scattering: even values fall into a universality class characterized by a separation of charge and spin degrees of freedom in the long-wavelength limit, while the action for the case of odd orders does not have this property, thus leading to distinct phase diagrams. It is argued that even when the long-wavelength action does exhibit a spin-charge separation it is not always an accurate approximation, particularly for the case of strong repulsion between electrons of the same spin. This leads to a classification of the possible ground states, associated correlation exponents, and phase transitions. The translational correlation exponents upon approaching insulator phases (existing at exact commensuration) by changing electron density can be related to the fractional charge of the solitons, using the Landauer rule.
Close to the transition between incommensurate conductor and commensurate insulator, charge transport in a system of one-dimensional interacting spin-1/2 fermions moving in a periodic potential is done by solitons (i.e., the few extra particles or holes) having a charge QQQ which is a fraction of the elementary charge eee. We show that QQQ is directly related to the limiting value at commensuration of the exponent gc∗{g}_{c}^{*}gc∗​ of the charge-charge correlation function: gc∗=(Q/e)2{g}_{c}^{*}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(Q/e{)}^{2}gc∗​=(Q/e)2. We give some simple rules for determining both charge and spin correlation exponents and point out their dependence on the parity of the order of the underlying umklapp scattering.
The low--energy excited states of a system of interacting one--dimensional fermions in a conducting state are collective charge and spin density oscillations. The unusual physical properties of such a system (called ``Luttinger liquid'') are characterized by the velocities uρu_ρuρ​ and uσu_σuσ​ of the charge and spin excitations, as well as by a parameter KρK_ρKρ​ that determines the power law behavior of correlation functions. Umklapp scattering occuring at half--filling or other commensurate bandfilling can lead to a transition into an insulating state, characterized in particular by a gap in the charge excitations (the Mott--Hubbard gap). The properties in the vicinity of the transition are shown to depend on both the way the transition is approached (constant bandfilling and varying interaction, or constant interaction and varying bandfilling) and on the ``order'' of the commensurability. In particular, even and odd fractional fillings show quite different behavior. This behavior is illustrated in detail using lattice models like the Hubbard model and its extensions.
The extended Hubbard model in one dimension is studied for half-filled bands using a continuum theory based on the Luttinger model. Interactions going beyond the Luttinger model are treated with a renormalization group. We compute correlation functions and use these to establish the phase diagram of the model. It compares favorably with earlier numerical work. Specifically, we identify additional interaction terms coupling charge and spin degrees of freedom; one of them is shown to mediate the crossover from a continuous to a first-order transition from spin- to charge-density waves at a finite value of U=2V. Our approach suggests viewing the model on this line as two weakly coupled interpenetrating systems of up-spin and down-spin electrons with strong backscattering interactions favoring charge-density waves separately in each of the subsystems. A first-order transition occurring for attractive interactions is interpreted as a characteristic instability of the Luttinger liquid towards phase separation. We finally discuss a possible equivalence to some recent studies of coupled two-dimensional XY models.
We systematically explore the ground state properties of one dimensional fermions with long-range interactions decaying in a power law ∼[Formula: see text] through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if [Formula: see text] is larger than some threshold, otherwise the theory breaks down. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of [Formula: see text] has small interactions and finite charge gaps, but, interestingly, it shows metallic nature at the same time. We obtain approximate phase diagrams for the entire parameter space and for band fillings equal to 1/2 and 1/3. Finally, we compare certain bosonization and field theory formulas with our quasi-exact numerical results, from which disagreements are found.
We determine the ground-state phase diagram for the 1/r1/r1/r-Hubbard model with repulsive nearest-neighbor interaction at half band filling using the density-matrix renormalization group method. Due to the absence of Umklapp scattering, the phase diagram displays finite regions for the three generic phases, namely, a Luttinger liquid metal for weak interactions, a Mott-Hubbard insulator for dominant Hubbard interactions, and a charge-density-wave insulator for dominant nearest-neighbor interactions. Up to moderate interaction strengths, the quantum phase transitions between the metallic and insulating phases are continuous, i.e., the gap opens continuously as a function of the interaction strength. We conclude that generic short-range interactions do not change the nature of the Mott transition qualitatively.
Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant 2kF2{k}_{\mathrm{F}}2kF​ spin correlations to dominant 4kF4{k}_{\mathrm{F}}4kF​ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.