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31:
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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
1243:
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
1679:
Ishii, H; Kataura, H; Shiozawa, H; Yoshioka, H; Otsubo, H; Takayama, Y; Miyahara, T; Suzuki, S; Achiba, Y; Nakatake, M; Narimura, T; Higashiguchi, M; Shimada, K; Namatame, H; Taniguchi, M (4 December 2003). "Direct observation of
Tomonaga–Luttinger-liquid state in carbon nanotubes at low
944:
754:
reformulated the theory in terms of Bloch sound waves and showed that the constraints proposed by
Tomonaga were unnecessary in order to treat the second-order perturbations as bosons. But his solution of the model was incorrect; the correct solution was given by
1255:
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).
917:
1161:
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
846:
1154:" - or charge density waves) propagating at a velocity that is determined by the strength of the interaction and the average density. For a non-interacting system, this wave velocity is equal to the
1110:{\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)}
1195:
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).
1238:
937:
1184:), and most of the work consists in transforming back to obtain the properties of the particles themselves (or treating impurities and other situations where '
1180:
of the Fermi liquid (which carry both spin and charge). The mathematical description becomes very simple in terms of these waves (solving the one-dimensional
750:
in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963,
756:
698:
1569:
Blumenstein, C.; Schäfer, J.; Mietke, S.; Meyer, S.; Dollinger, A.; Lochner, M.; Cui, X. Y.; Patthey, L.; Matzdorf, R.; Claessen, R. (October 2011).
1314:
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853:
1130:
can then be used to predict spin-charge separation. Electron-electron interactions can be treated to calculate correlation functions.
778:
Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron
Hamiltonian:
1805:
1240:. However, in contrast to the Fermi liquid, their decay at large distances is governed by yet another interaction-dependent exponent.
691:
1643:
1379:
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Attempts to demonstrate
Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in
1119:
Expressions for bosons in terms of fermions are used to represent the
Hamiltonian as a product of two boson operators in a
1489:
Mattis, Daniel C.; Lieb, Elliott H. (1965). "Exact
Solution of a Many-Fermion System and Its Associated Boson Field".
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Likewise, the tunneling rate into a
Luttinger liquid is suppressed to zero at low voltages and temperatures, as a
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electrons hopping along one-dimensional chains of molecules (e.g. certain organic molecular crystals)
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is separated into left and right moving electrons and undergoes linearization with the approximation
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1412:(1 June 1950). "Remarks on Bloch's Method of Sound Waves applied to Many-Fermion Problems".
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1324:(the Luttinger liquid model also works for integer spins in a large enough magnetic field)
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Haldane, F.D.M. (1981). "'Luttinger liquid theory' of one-dimensional quantum fluids".
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like a power law in voltage and temperature, with an interaction-dependent exponent).
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Among the physical systems believed to be described by the
Luttinger model are:
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Theoretical model describing interacting fermions in a one-dimensional conductor
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1279:' (one-dimensional strips of electrons) defined by applying gate voltages to a
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1571:"Atomically controlled quantum chains hosting a Tomonaga–Luttinger liquid"
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waves are the elementary excitations of the Luttinger liquid, unlike the
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Exact solution of a many-fermion system and its associated boson field
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Among the hallmark features of a Luttinger liquid are the following:
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912:{\displaystyle \epsilon _{k}\approx \pm v_{\rm {F}}(k-k_{\rm {F}})}
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96:
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1358:. Series on Directions in Condensed Matter Physics. Vol. 20.
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although the latter is often considered a more trivial example.
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1452:(1963). "An Exactly Soluble Model of a Many-Fermion System".
841:{\displaystyle H=\sum _{k}\epsilon _{k}c_{k}^{\dagger }c_{k}}
374:
71:
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1356:
Luttinger Model: The First 50 Years and Some New Directions
81:
1320:
a 1D 'chain' of half-odd-integer spins described by the
746:
The Tomonaga–Luttinger's liquid was first proposed by
1216:
947:
925:
856:
786:
1731:
Chudzinski, P.; Jarlborg, T.; Giamarchi, T. (2012).
1618:
Mattis, Daniel C.; Lieb, Elliot H. (February 1965).
1150:) density to some external perturbation are waves ("
1353:
1232:
1109:
931:
911:
840:
739:). Such a model is necessary as the commonly used
1819:
719:, is a theoretical model describing interacting
1354:Mastropietro, Vieri; Mattis, Daniel C. (2013).
1420:(4). Oxford University Press (OUP): 544–569.
692:
1733:"Luttinger-liquid theory of purple bronze
1617:
1488:
1300:electrons moving along edge states in the
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685:
29:
1770:
1594:
1448:
1408:
1525:
1202:Around impurities, there are the usual
1820:
1317:in quasi-one-dimensional atomic traps
743:model breaks down for one dimension.
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1622:. Vol. 6. pp. 98–106.
1460:(9). AIP Publishing: 1154–1162.
1329:Lithium molybdenum purple bronze
666:
665:
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1808:(Stuttgart University, Germany)
1491:Journal of Mathematical Physics
1454:Journal of Mathematical Physics
1414:Progress of Theoretical Physics
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1672:
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1562:
1497:(2). AIP Publishing: 304–312.
1302:fractional Quantum Hall Effect
906:
885:
1:
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1233:{\displaystyle 2k_{\text{F}}}
1528:J. Phys. C: Solid State Phys
1281:two-dimensional electron gas
1206:in the charge density, at a
1199:on the interaction strength.
7:
1548:10.1088/0022-3719/14/19/010
1335:
1133:
10:
1854:
1781:10.1103/PhysRevB.86.075147
1636:10.1142/9789812812650_0008
242:Spin gapless semiconductor
1121:Bogoliubov transformation
773:
717:Tomonaga–Luttinger liquid
182:Electronic band structure
1838:Condensed matter physics
1266:condensed matter physics
932:{\displaystyle \Lambda }
92:Bose–Einstein condensate
23:Condensed matter physics
1192:for one technique used.
727:) in a one-dimensional
1234:
1188:' is important). See
1164:spin-charge separation
1111:
1004:
933:
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1833:Statistical mechanics
1814:(FreeScience Library)
1755:in the charge regime"
1283:, or by other means (
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237:Topological insulator
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1204:Friedel oscillations
1142:The response of the
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255:Electronic phenomena
102:Fermionic condensate
1828:Theoretical physics
1702:10.1038/nature02074
1694:2003Natur.426..540I
1628:1994boso.book...98M
1587:2011NatPh...7..776B
1540:1981JPhC...14.2585H
1503:1965JMP.....6..304M
1466:1963JMP.....4.1154L
1434:10.1143/ptp/5.4.544
1426:1950PThPh...5..544T
1364:2013SDCMP..20.....M
1306:Quantum Hall Effect
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262:Quantum Hall effect
1806:Short introduction
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748:Sin-Itiro Tomonaga
659:Physics portal
1759:Physical Review B
1688:(6966): 540–544.
1645:978-981-02-1847-8
1596:10.1038/nphys2051
1534:(19): 2585–2609.
1511:10.1063/1.1704281
1474:10.1063/1.1704046
1381:978-981-4520-71-3
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175:Electronic phases
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