Luttinger liquid

Condensed matter physics
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A Luttinger liquid, or Tomonaga–Luttinger liquid, is a theoretical model describing interacting

The Tomonaga–Luttinger's liquid was first proposed by

Theory

Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron Hamiltonian:

$${\displaystyle H=\sum _{k}\epsilon _{k}c_{k}^{\dagger }c_{k}}$$

is separated into left and right moving electrons and undergoes linearization with the approximation ${\displaystyle \epsilon _{k}\approx \pm v_{\rm {F}}(k-k_{\rm {F}})}$ over the range ${\displaystyle \Lambda }$:

$${\displaystyle H=\sum _{k=k_{\rm {F}}-\Lambda }^{k_{\rm {F}}+\Lambda }v_{\rm {F}}k\left(c_{k}^{\mathrm {R} \dagger }c_{k}^{\mathrm {R} }-c_{k}^{\mathrm {L} \dagger }c_{k}^{\mathrm {L} }\right)}$$

Expressions for bosons in terms of fermions are used to represent the Hamiltonian as a product of two boson operators in a Bogoliubov transformation.

The completed bosonization can then be used to predict spin-charge separation. Electron-electron interactions can be treated to calculate correlation functions.

Features

Among the hallmark features of a Luttinger liquid are the following:

• The response of the
  Fermi velocity
  , while it is higher (lower) for repulsive (attractive) interactions among the fermions.
• Likewise, there are spin density waves (whose velocity, to lowest approximation, is equal to the unperturbed Fermi velocity). These propagate independently from the charge density waves. This fact is known as
  spin-charge separation
  .
• backscattering' is important). See bosonization
  for one technique used.
• Even at zero temperature, the particles' momentum distribution function does not display a sharp jump, in contrast to the Fermi liquid (where this jump indicates the Fermi surface).
• There is no 'quasiparticle peak' in the momentum-dependent spectral function (i.e. no peak whose width becomes much smaller than the excitation energy above the Fermi level, as is the case for the Fermi liquid). Instead, there is a power-law singularity, with a 'non-universal' exponent that depends on the interaction strength.
• Around impurities, there are the usual
  wavevector
  of ${\displaystyle 2k_{\text{F}}}$. However, in contrast to the Fermi liquid, their decay at large distances is governed by yet another interaction-dependent exponent.
• At small temperatures, the scattering of these Friedel oscillations becomes so efficient that the effective strength of the impurity is renormalized to infinity, 'pinching off' the quantum wire. More precisely, the conductance becomes zero as temperature and transport voltage go to zero (and rises like a power law in voltage and temperature, with an interaction-dependent exponent).
• Likewise, the tunneling rate into a Luttinger liquid is suppressed to zero at low voltages and temperatures, as a power law.

The Luttinger model is thought to describe the universal low-frequency/long-wavelength behaviour of any one-dimensional system of interacting fermions (that has not undergone a phase transition into some other state).

Physical systems

Attempts to demonstrate Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in condensed matter physics.

Among the physical systems believed to be described by the Luttinger model are:

• artificial '
  AFM
  , etc.)
• electrons in carbon nanotubes^[3]
• electrons moving along edge states in the
  Quantum Hall Effect
  although the latter is often considered a more trivial example.
• electrons hopping along one-dimensional chains of molecules (e.g. certain organic molecular crystals)
• fermionic atoms in quasi-one-dimensional atomic traps
• a 1D 'chain' of half-odd-integer spins described by the
  Heisenberg model
  (the Luttinger liquid model also works for integer spins in a large enough magnetic field)
• electrons in Lithium molybdenum purple bronze.^[4]

See also

• Fermi liquid

Bibliography

• Mastropietro, Vieri; Mattis, Daniel C. (2013). Luttinger Model: The First 50 Years and Some New Directions. Series on Directions in Condensed Matter Physics. Vol. 20.
  ISBN 978-981-4520-71-3. {{cite book}}: |journal= ignored (help
  )
• ISSN 0033-068X
  .
• ISSN 0022-2488
  .
• Mattis, Daniel C.; Lieb, Elliott H. (1965). "Exact Solution of a Many-Fermion System and Its Associated Boson Field". Journal of Mathematical Physics. 6 (2). AIP Publishing: 304–312.
  ISSN 0022-2488
  .
• Haldane, F.D.M. (1981). "'Luttinger liquid theory' of one-dimensional quantum fluids". J. Phys. C: Solid State Phys. 14 (19): 2585–2609.
  doi:10.1088/0022-3719/14/19/010
  .