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Chapter 1 LUTTINGER LIQUIDS: the BASIC CONCEPTS

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Chapter 1 LUTTINGER LIQUIDS: the BASIC CONCEPTS Chapter 1 LUTTINGER LIQUIDS: THE BASIC CONCEPTS K.Schonhammer¨ Institut fur¨ Theoretische Physik, Universitat¨ Gottingen,¨ Bunsenstr. 9, D-37073 Gottingen,¨ Germany Abstract This chapter reviews the theoretical description of interacting fermions in one dimension. The Luttinger liquid concept is elucidated using the Tomonaga- Luttinger model as well as integrable lattice models. Weakly coupled chains and the attempts to experimentally verify the theoretical predictions are dis- cussed. Keywords Luttinger liquids, Tomonaga model, bosonization, anomalous power laws, break- down of Fermi liquid theory, spin-charge separation, spectral functions, coupled chains, quasi-one-dimensional conductors 1. Introduction In this chapter we attempt a simple selfcontained introduction to the main ideas and important computational tools for the description of interacting fermi- ons in one spatial dimension. The reader is expected to have some knowledge of the method of second quantization. As in section 3 we describe a constructive approach to the important concept of bosonization, no quantum field-theoretical background is required. After mainly focusing on the Tomonaga-Luttinger 1 2 model in sections 2 and 3 we present results for integrable lattice models in section 4. In order to make contact to more realistic systems the coupling of strictly ¢¡ systems as well as to the surrounding is addressed in section 5. The attempts to experimentally verify typical Luttinger liquid features like anoma- lous power laws in various correlation functions are only shortly discussed as this is treated in other chapters of this book. 2. Luttinger liquids - a short history of the ideas As an introduction the basic steps towards the general concept of Luttinger liquids are presented in historical order. In this exposition the ideas are discussed without presenting all technical details. This is done in section 3 by disregarding the historical aspects aiming at a simple presentation of the important practical concepts like the “ bosonization of field operators”. 2.1 Bloch’s method of “sound waves” (1934) In a paper on incoherent x-ray diffraction Bloch [1] realized and used the ¡¥¤¦ ¨§ fact that one-dimensional £ noninteracting fermions have the same type of low energy excitations as a harmonic chain. The following discussion of this connection is very different from Bloch’s own presentation. The low energy excitations determine e.g. the low temperature specific heat. Debye’s famous © -law for the lattice contribution of three dimensional solids ¡¥¤ reads in © ¤ " ! (1.1) ¢#%$'& where ¢# is the sound velocity. At low temperatures the electronic contribution to the specific heat in the “Fermi gas” approximation of Pauli is also linear in © and involves the density of states of the noninteraction electrons at the Fermi energy. This yields for spinless fermions in ¡¥¤¦ © ¤ )(*,+-/. "1032 ! (1.2) $4& 052 082 where is the Fermi velocity. With the replacement #76 the results are identical. This suggests that apart from a scale factor the (low energy) excitation energies and the degeneracies in the two types of systems are identical. For the harmonic chain the excited states are classified by the numbers 9: of phonons in 9=: the modes ;<: whith taking integer values from zero to infinity. The excitation " §HG ¤KJ ! >?£A@B9C:)DFE >FI ;<:DL9=:)D D energy is given by : . For small wave numbers 3M 3M #8P P DON the dispersion is linear ;: . Therefore the excitations energies " § ! # are multiples of £RQ8CSHT for periodic boundary conditions and multiples of U " WV ! # CSHT for fixed boundary conditions. The calculation of the partition function is standard textbook material. This is also true for noninteracting2 9 electrons but there the calculation involves fermionic occupation numbers : 3 which take values zero and one. The two textbook calculations yield Eqs. (1.1) and (1.2), but through the “clever” use of the grand canonical ensemble in order 082 to simplify the fermionic calculation the identity (apart from #X6 ) remains mysterious. A deeper understanding involves two steps: "[Z ¤ § Z ! S\£RQH] 1) Linearization of the kinetic energy Y3: of the free fermions 2 around the Fermi point for fixed boundary conditions or both Fermi points 2 ^_ for periodic boundary conditions. As the argument is simplest for fixed `Ma¤ boundary conditions [2] which lead to ]4CSHT we discuss this case for the U 2 2 2 "c0 V G ! CSHT Y moment. Then the energies YH:)b are integer multiples of where 032 is the Fermi velocity. 2) Classification of any state of the Fermi system by the number 9Cd of upward U 2 shifts by e units of with respect to the ground state. As the fermions are indistinguishable the construction of the @B9=d8E shown in Fig. 1.1, where the highest occupied level in the excited state is connected with the highest occupied level in the ground state and so forth for the second, third f¢f¢f highest levels, completely specifies the excited state. As the 9=d can run from zero to infinity like bosonic quantum numbers and the excitation energy is given by U 2 § J £ge 91d d the canonical partition function for the noninteracting fermions has the same form as the canonical partition function for the harmonic chain U U 2 apart from 6 if one assumes the Fermi sea to be infinitely deep [3]. 2 YH: As we have linearized ;: for small as well as around this equivalence " 2 [ § ! £ ©ah ;Ci ©ahlY only holds for the low temperature specific heats *kj . & If we denote the creation (annihilation) operator of a fermion with cmn¤ "1032 ¤ 5m mc§ ! m m ¢o -/./p £ Y 9=CSHT by and assume a strictly linear dispersion for all 3mrqns a more technical formulation of the discussed equivalence can be given by the exact operator identity u "c032 ¤ t 3m m ! m o © m3vxw z`~ ~ z "[082 ! u z t ¤ ¨§ o £ (1.3) & vxw[{}| | T y Qc z where the operators with are defined as | {< z z u V t MX M o (1.4) Mvxw | { z m V J m3vxw m o and t is the fermionic particle number operator. The proof of z z the “Kronig identity” (1.3) is simple (see Ref. [4]) . The operators obey | ¤as z commutation relations and for & {{ | | z zg1z z u ¤ M M o o f (1.5) & MXvxw | | {{ 4 7 4 4 nF 3 ∆ 1 F{ 1 0 0 groundstate excited state 20 ∆ E0 E0+20 F Figure 1.1. Example for the classification scheme for the excited states in terms of the numbers H xR of upward shifts by units of . In the example shown the nonzero are ¡ ¡ [¤¥ R ¢<£ and . ¬ VK® § o m8vxw P P/§x¨ª©¥« ¯ ¬ ­ ­ ¦ °±£,²¦ For all -particle states b Vac in which the lowest z´z z one-particle levels are all occupied one obtains z ¤O³ o P/§=¨ª©¥« P/§=¨g©¥« ­ ­ ¬ ¬ (1.6) & | | z z ¤ o zz ·¶ for ° , i.e. these operators obey boson commutation relations & & { {}µ | | ³ in this subspace of all possible ¦ -particle states. ~ ~ V G¦¹ G»º ­ © © © I I¼¸ Later it turns out to be useful to work with ¸ , where U U 2 2 2 2 2 ¶ ¹ º ¤ ¤ § ¨§ ­ I I © £9 9 £9 S8Q S8Q is the ground-state energy, 2 ~ V G 9 is the chemical potential of the noninteracting fermions and ¸ . ¨§ Z © £ ¸ Then ¸ is of the form as the rhs of Eq. (1.3) with replaced by . 2.2 Tomonaga (1950): Bloch’s method of sound waves applied to interacting fermions When a two-body interaction between the fermions is switched on, the ground state is no longer the filled Fermi sea but it has admixtures of (multiple) particle- hole pair excitations. In order to simplify the problem Tomonaga studied the high density limit where the range of the interaction is much larger than the interparticle distance, using periodic boundary conditions [5]. Then the Fourier 5 0 [½ [§ P P transform £ of the two-body interaction is nonzero only for values ¸ µ 2 ½ \½ where the cut-off is much smaller than the Fermi momentum h . This implies that for not too strong interaction the ground state and low energy excited states have negligible admixtures of holes deep in the Fermi sea and 2À¿ G¾ 5½ P particles with momenta P . In the two intermediate regions around 2 ^_ the two Fermi points , with particle-hole pairs present, the dispersion Y : is linearized in order to apply Bloch’s “sound wave method” 2ÀÁ 2 032 2 ^_ ^ ¤ ৠN Y £ f Y¨: (1.7) Tomonaga realized that the Fourier components of the operator of the density ¶ ¶ \Æ `Æ Z Z ¯ ¯ m ¤ Å §,È ¡ ¤KÅ § §,È ¡ :)b¨É :bHÉ o Ä Ä £Ç Ç £Ç £Ç Ç `Æ \Æ Z Z¥Ê Ê u ¤ m m o m (1.8) & m mc§ cmˤ m o where £ creates (annihilates) a fermion in the state with momentum Z,Ì 9 , plays a central role in the interaction term, as well as the kinetic energy. Apart from an additional term linear in the particle number operator [4], which is usually neglected, the two-body interaction is given by ¶ ~ ¶ ¶ u Z 0 0 Í m m m ¤ § s § Ä Ä £ £ (1.9) ¸ ¸ m[Îv Q8T Q8TF I ¶ 2 m 5m Ä P P Tomonaga’s important step was to split for h into two parts, one containing operators of “right movers” i.e. involving fermions near the right 2 052 Fermi point with velocity and “left movers” involving fermions near 2 2 0 GF G ~ ~ with velocity ¶ ¶ ¶ u Ï u Ð m¥¤ V mÑ m Ñ m m m m o o m m Ä Ä Ä (1.10) m m I I z ¶ P P where the details of the splitting for small 9 are irrelevant. Apart from the m Ñ Ò square root factor the Ä are similar to the defined in Eq.
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