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arXiv:1211.0982 · Physical Review B

Class of variational Ansätze for the spin-incoherent ground state of a Luttinger liquid coupled to a spin bath

Mohammad Soltanieh-ha, Adrian E. Feiguin

Published Nov 19, 2012• 5 citations
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Abstract

Interacting one-dimensional electron systems are generally referred to as ``Luttinger liquids,'' after the effective low-energy theory in which spin and charge behave as separate degrees of freedom with independent energy scales. The ``spin-incoherent Luttinger liquid'' describes a finite-temperature regime that is realized when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale. Similar physics can take place in the ground state, when a Luttinger liquid is coupled to a spin bath, which effectively introduces a ``spin temperature'' through its entanglement with the spin degree of freedom. We show that the spin-incoherent state can be written as a factorized wave function, with a spin wave function that can be described within a valence bond formalism. This enables us to calculate exact expressions for the momentum distribution function and the entanglement entropy. This picture holds not only for two antiferromagnetically coupled $t\ensuremath{-}J$ chains, but also for the $t\ensuremath{-}J$-Kondo chain with strongly interacting conduction electrons. We argue that this theory is quite universal and may describe a family of problems that could be dubbed ``spin incoherent.''

DOI
10.1103/physrevb.86.205120

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Interacting one-dimensional electron systems are generally referred to as Luttinger liquids,'' after the effective low-energy theory in which spin and charge behave as separate degrees of freedom with independent energy scales. The spin-incoherent Luttinger liquid'' describes a finite-temperature regime that is realized when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale. Similar physics can take place in the ground state, when a Luttinger liquid is coupled to a spin bath, which effectively introduces a spin temperature'' through its entanglement with the spin degree of freedom. We show that the spin-incoherent state can be written as a factorized wave function, with a spin wave function that can be described within a valence bond formalism. This enables us to calculate exact expressions for the momentum distribution function and the entanglement entropy. This picture holds not only for two antiferromagnetically coupled $t\ensuremath{-}J$ chains, but also for the $t\ensuremath{-}J$-Kondo chain with strongly interacting conduction electrons. We argue that this theory is quite universal and may describe a family of problems that could be dubbed spin incoherent.''
Interacting one-dimensional electron systems are generally referred to as "Luttinger liquids", after the effective low-energy theory in which spin and charge behave as separate degrees of freedom with independent energy scales. The "spin-incoherent Luttinger liquid" describes a finite-temperature regime that is realized when the temperature is very small relative to the Fermi energy, but larger than the characteristic spin energy scale, and it is realized for instance in the strongly interacting Hubbard chain (with U → ∞). Similar physics can take place in the ground-state, when a Luttinger Liquid is coupled to a spin bath, which effectively introduces a "spin temperature" through its entanglement with the spin degree of freedom. We show that the spin-incoherent state can be exactly written as a factorized wave-function, with a spin wave-function that can be described within a valence bond formalism. This enables us to calculate exact expressions for the momentum distribution function and the entanglement entropy. This picture holds not only for two antiferromagnetically coupled t-J chains, but also for the t-J-Kondo chain with strongly interacting conduction electrons. In chapter 3 we argue that this theory is quite universal and may describe a family of problems that could be dubbed "spin-incoherent".
In contrast to the well-known Fermi-liquid theory of three dimensions, interacting one-dimensional and quasi-one-dimensional systems of fermions are described at low energy by an effective theory known as Luttinger liquid theory. This theory is expressed in terms of collective many-body excitations that show exotic behavior such as spin-charge separation. Luttinger liquid theory is commonly applied on the premise that ``low energy'' describes both the spin and charge sectors. However, when the interactions in the system are very strong, as they typically are at low particle densities, the ratio of spin to charge energy may become exponentially small. It is then possible at very low temperatures for the single-spin excitation energy to be low compared to the characteristic single excitation charge energy, but still high compared to the characteristic spin energy. This energy window of near ground-state charge degrees of freedom but highly thermally excited spin degrees of freedom is called a spin-incoherent Luttinger liquid. The spin-incoherent Luttinger liquid exhibits a higher degree of universality than the Luttinger liquid and its properties are qualitatively distinct. In this Colloquium some recent theoretical developments in the field are detailed and experimental indications of such a regime in gated semiconductor quantum wires are described.
It is commonly believed that strongly interacting one-dimensional Fermi systems with gapless excitations are effectively described by Luttinger liquid theory. However, when the temperature of the system is high compared to the spin energy, but small compared to the charge energy, the system becomes "spin incoherent." We present numerical evidence showing that the one-dimensional "t-J-Kondo" lattice, consisting of a t-J chain interacting with localized spins, displays all the characteristic signatures of spin-incoherent physics, but in the ground state. We argue that similar physics may be present in a wide range of strongly interacting systems.
We study a lattice realization of a Luttinger liquid interacting with quantum spins in terms of an antiferromagnetic S=1∕2S=1∕2S=1∕2 Heisenberg chain, where each spin is also coupled to a $\ensuremath{\sigma}=1∕2$ Kondo spin degree of freedom. This model describes the low-energy spin dynamics in quasi-one-dimensional materials, where the electronic spins of the magnetic ions interact with those of impurities, nuclei, and possibly other spin species present in their environment. For large ferromagnetic and antiferromagnetic Kondo interaction ${J}^{\ensuremath{'}}$, there are two phases corresponding to an effective spin-1 Heisenberg chain and a dimerized spin-1∕21∕21∕2 ladder, respectively. For weak Kondo interaction, we establish that the Kondo interaction drives the system to a strong coupling regime. This suggests that ${J}^{\ensuremath{'}}=0$ is the only critical point in the system.
We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification g=2gcg=2{g}_{c}g=2gc​, where ggg is the interaction parameter in a spinless Luttinger liquid and gc{g}_{c}gc​ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.