The Luttinger liquid and integrable models

Abstract

Many fundamental one-dimensional lattice models such as the Heisenberg or the Hubbard model are integrable. For these microscopic models, parameters in the Luttinger liquid theory can often be fixed and parameter-free results at low energies for many physical quantities such as dynamical correlation functions obtained where exact results are still out of reach. Quantum integrable models thus provide an important testing ground for low-energy Luttinger liquid physics. They are, furthermore, also very interesting in their own right and show, for example, peculiar transport and thermalization properties. The consequences of the conservation laws leading to integrability for the structure of the low-energy effective theory have, however, not fully been explored yet. I will discuss the connection between integrability and Luttinger liquid theory here, using the anisotropic Heisenberg model as an example. In particular, I will review the methods which allow to fix free parameters in the Luttinger model with the help of the Bethe ansatz solution. As applications, parameter-free results for the susceptibility in the presence of non-magnetic impurities, for spin transport, and for the spin-lattice relaxation rate are discussed.

23 pages, 5 figures, short review to appear in Int. J. Mod. Phys. B

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• Form factors of the XXZ Heisenberg spin-1/2 finite chain
• Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field
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• Universal theory of nonlinear Luttinger liquids
• Diffusion and ballistic transport in one-dimensional quantum systems
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• Conservation laws, integrability and transport in one-dimensional quantum systems
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• Low energy effective Hamiltonian for the XXZ spin chain
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• Lower bounds for the conductivities of correlated quantum systems
• An Accurate Determination of the Exchange Constant in Sr_2CuO_3 from Recent Theoretical Results
• Boundary susceptibility in the open XXZ-chain
• NMR Response in quasi one-dimensional Spin-1/2 Antiferromagnets
• Possible Localized Modes in the Uniform Quantum Heisenberg Chains of Sr2CuO3

Cited by in corpus (24)

• Families of quasi-local conservation laws and quantum spin transport
• Matrix product solutions of boundary driven quantum chains
• Quantum quench within the gapless phase of the spin-1/2 Heisenberg XXZ spin-chain
• Topological Wigner Crystal of Half-Solitons in a Spinor BEC
• Front dynamics and entanglement in the XXZ chain with a gradient
• Exact real-time longitudinal correlation functions of the massive XXZ chain
• Fast preparation of critical ground states using superluminal fronts
• Exterior integrability: Yang-Baxter form of nonequilibrium steady state density operator
• Inhomogeneous Gaussian Free Field inside the interacting arctic curve
• Closed and Open System Dynamics in a Fermionic Chain with a Microscopically Specified Bath: Relaxation and Thermalization
• Exact results for nonlinear Drude weights in the spin-1/2 XXZ chain
• Conducting fixed points for inhomogeneous quantum wires: a conformally invariant boundary theory
• Density-matrix renormalization group method for the conductance of one-dimensional correlated systems using the Kubo formula
• Thermodynamics of a spin-1/2 XYZ Heisenberg chain with a Dzyaloshinskii-Moriya interaction
• Bloch oscillations in the spin-1/2 XXZ chain
• Electrical and thermal transport in the quasi-atomic limit of coupled Luttinger liquids
• Interacting in-plane molecular dipoles in a zig-zag chain
• Density-matrix renormalization group study of the linear conductance in quantum wires coupled to interacting leads or phonons
• Non-linear Transport by Bethe Bound States
• Kubo's response theory and bosonization with a background gauge field and irrelevant perturbations
• Dressed excitations, thermodynamics and relaxation in the 1D XXZ Heisenberg model
• Drude weights in one-dimensional systems with a single defect
• Momentum-space modulated symmetries in the Luttinger liquid
• Non-integrable fermionic chains near criticality