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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
1233:
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
1669:
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
934:
744:
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
1245:
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).
907:
1151:
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
836:
1144:" - 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
1100:{\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)}
1185:
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).
1228:
688:
927:
1174:), and most of the work consists in transforming back to obtain the properties of the particles themselves (or treating impurities and other situations where '
1170:
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
740:
in 1950. The model showed that under certain constraints, second-order interactions between electrons could be modelled as bosonic interactions. In 1963,
1559:
Blumenstein, C.; Schäfer, J.; Mietke, S.; Meyer, S.; Dollinger, A.; Lochner, M.; Cui, X. Y.; Patthey, L.; Matzdorf, R.; Claessen, R. (October 2011).
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can then be used to predict spin-charge separation. Electron-electron interactions can be treated to calculate correlation functions.
768:
Luttinger liquid theory describes low energy excitations in a 1D electron gas as bosons. Starting with the free electron
Hamiltonian:
1795:
1230:. However, in contrast to the Fermi liquid, their decay at large distances is governed by yet another interaction-dependent exponent.
1827:
1137:
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1822:
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Attempts to demonstrate
Luttinger-liquid-like behaviour in those systems are the subject of ongoing experimental research in
436:
1109:
Expressions for bosons in terms of fermions are used to represent the
Hamiltonian as a product of two boson operators in a
91:
1817:
1270:
611:
1479:
Mattis, Daniel C.; Lieb, Elliott H. (1965). "Exact
Solution of a Many-Fermion System and Its Associated Boson Field".
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1311:
616:
241:
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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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1402:(1 June 1950). "Remarks on Bloch's Method of Sound Waves applied to Many-Fermion Problems".
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1314:(the Luttinger liquid model also works for integer spins in a large enough magnetic field)
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1683:
1617:
1576:
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1148:, while it is higher (lower) for repulsive (attractive) interactions among the fermions.
1516:
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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1269:' (one-dimensional strips of electrons) defined by applying gate voltages to a
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1699:
1561:"Atomically controlled quantum chains hosting a Tomonaga–Luttinger liquid"
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waves are the elementary excitations of the Luttinger liquid, unlike the
710:
96:
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1141:
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Exact solution of a many-fermion system and its associated boson field
1585:
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1128:
Among the hallmark features of a Luttinger liquid are the following:
369:
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902:{\displaystyle \epsilon _{k}\approx \pm v_{\rm {F}}(k-k_{\rm {F}})}
1761:
1348:. Series on Directions in Condensed Matter Physics. Vol. 20.
374:
71:
1361:
1298:
although the latter is often considered a more trivial example.
81:
1442:(1963). "An Exactly Soluble Model of a Many-Fermion System".
831:{\displaystyle H=\sum _{k}\epsilon _{k}c_{k}^{\dagger }c_{k}}
1346:
Luttinger Model: The First 50 Years and Some New Directions
1310:
a 1D 'chain' of half-odd-integer spins described by the
736:
The Tomonaga–Luttinger's liquid was first proposed by
1206:
937:
915:
846:
776:
1721:
Chudzinski, P.; Jarlborg, T.; Giamarchi, T. (2012).
1608:
Mattis, Daniel C.; Lieb, Elliot H. (February 1965).
1140:) density to some external perturbation are waves ("
1222:
1099:
921:
901:
830:
729:). Such a model is necessary as the commonly used
709:, is a theoretical model describing interacting
1344:Mastropietro, Vieri; Mattis, Daniel C. (2013).
1410:(4). Oxford University Press (OUP): 544–569.
682:
8:
1723:"Luttinger-liquid theory of purple bronze
1290:electrons moving along edge states in the
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675:
29:
18:
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797:
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1192:Around impurities, there are the usual
21:
1651:
1641:
1387:
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1307:in quasi-one-dimensional atomic traps
733:model breaks down for one dimension.
7:
1086:
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1046:
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990:
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961:
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14:
1612:. Vol. 6. pp. 98–106.
1450:(9). AIP Publishing: 1154–1162.
1319:Lithium molybdenum purple bronze
656:
655:
642:
1798:(Stuttgart University, Germany)
1481:Journal of Mathematical Physics
1444:Journal of Mathematical Physics
1404:Progress of Theoretical Physics
1487:(2). AIP Publishing: 304–312.
1292:fractional Quantum Hall Effect
896:
875:
1:
1223:{\displaystyle 2k_{\text{F}}}
1518:J. Phys. C: Solid State Phys
1271:two-dimensional electron gas
1196:in the charge density, at a
1189:on the interaction strength.
1538:10.1088/0022-3719/14/19/010
1844:
1771:10.1103/PhysRevB.86.075147
1626:10.1142/9789812812650_0008
242:Spin gapless semiconductor
1111:Bogoliubov transformation
707:Tomonaga–Luttinger liquid
182:Electronic band structure
1828:Condensed matter physics
1256:condensed matter physics
922:{\displaystyle \Lambda }
92:Bose–Einstein condensate
23:Condensed matter physics
1182:for one technique used.
717:) in a one-dimensional
1224:
1178:' is important). See
1154:spin-charge separation
1101:
994:
923:
903:
832:
1823:Statistical mechanics
1804:(FreeScience Library)
1745:in the charge regime"
1273:, or by other means (
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1102:
944:
924:
904:
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237:Topological insulator
1204:
1194:Friedel oscillations
1132:The response of the
935:
913:
844:
774:
255:Electronic phenomena
102:Fermionic condensate
1818:Theoretical physics
1692:10.1038/nature02074
1684:2003Natur.426..540I
1618:1994boso.book...98M
1577:2011NatPh...7..776B
1530:1981JPhC...14.2585H
1493:1965JMP.....6..304M
1456:1963JMP.....4.1154L
1424:10.1143/ptp/5.4.544
1416:1950PThPh...5..544T
1354:2013SDCMP..20.....M
1296:Quantum Hall Effect
1091:
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262:Quantum Hall effect
1796:Short introduction
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919:
899:
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738:Sin-Itiro Tomonaga
649:Physics portal
1749:Physical Review B
1678:(6966): 540–544.
1635:978-981-02-1847-8
1586:10.1038/nphys2051
1524:(19): 2585–2609.
1501:10.1063/1.1704281
1464:10.1063/1.1704046
1371:978-981-4520-71-3
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175:Electronic phases
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1440:Luttinger, J. M.
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1312:Heisenberg model
1286:carbon nanotubes
1250:Physical systems
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727:carbon nanotubes
703:Luttinger liquid
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267:Spin Hall effect
157:Phase transition
127:Luttinger liquid
64:States of matter
47:Phase transition
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1571:(10): 776–780.
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1812:Categories
1546:References
1198:wavevector
713:(or other
532:Louis NĂ©el
522:Schrieffer
430:Scientists
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