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Optical detection of critical exponents in the Tomonaga–Luttinger liquid
Physica B 249—251 (1998) 185—190
Optical detection of critical exponents in the Tomonaga—Luttinger liquid Tetsuo Ogawa* Department of Physics, Tohoku University, Aoba-ku, Sendai 980—8578, Japan
Abstract Optical responses of the Tomonaga—Luttinger liquid are investigated theoretically in terms of the power-law singularities. Critical exponents in the edge anomalies of valence-band photoemission, core-level photoemission, and one-photon absorption processes reflect different aspects of low-energy critical properties. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Tomonaga—Luttinger liquid; Photoemission; One-photon absorption; Critical exponents; Orthogonality catastrophe
1. Introduction Low-energy characteristics of one-dimensional (1D) electron systems are studied extensively. When such systems remain gapless, the many-body correlation, the quantum fluctuation, and the infrared divergence [1] play crucial roles in their low-energy properties. When interactions among particles are of a short-range type and when the backward, umklapp, and spin-flip scatterings can be neglected, critical properties of quantum 1D electrons are universally described as the Tomonaga—Luttinger (TL) liquid [2,3]. Studies on the TL liquids have been focused to their spectral and transport properties as well as the mathematical structure; there are just a few studies on their optical properties. The * Fax: #81 22 217 6447; e-mail: [email protected]. tohoku.ac.jp.
main aim of this paper is to show that the optical responses of the TL liquid are also useful tools for investigating the critical properties of 1D degenerate electrons. Several optical transition probabilities reflect the final-state interaction due to an optically generated hole potential. This interaction brings about two kinds of intrinsic dynamical effects in degenerate electron systems: an orthogonality catastrophe and a many-body excitonic correlation. The former suppresses the transition but the latter enhances it. This is a characteristic point in contrast to those in semiconductors [4]. The most striking feature in optical responses of the TL liquids is the power-law singularity, which is characterized by some critical exponents. We pay attention to the valence-band photoemission spectrum, the core-level photoemission spectrum, and the one-photon absorption spectrum. These optical transition processes yield
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T. Ogawa / Physica B 249—251 (1998) 185—190
the power-law anomaly in the edge spectra with different critical exponents. In this paper, we obtain analytical expressions of the critical exponents to clarify what kinds of low-energy characteristics are reflected in them.
p # + (g d #g d ) 4, s,s{ 4M s,~s{ ¸ s,s{ ] + [. (p). (!p) 1s 1s{ p;0 #. (!p). (p)], 2s 2s{
2. The Tomonaga—Luttinger liquid The 1D degenerate electron systems near the Fermi level are characterized by two (right- and left-moving) branches of electrons. When the interparticle interaction in 1D is of the short-range type, the Coulomb scattering strengths are represented by constant parameters: g , g , and g [5—10] 2 4 1 with conventional notation, where g and g cor2 4 respond to the strengths of inter- and intra-branch forward scatterings, and g , the backward-scatter1 ing strength. Then the backward scattering is neglected because Dg DKDg DADg D. We further classify 2 4 1 the interaction parameters in terms of the mutual spin orientation; the scattering between parallelspin (E) particles and that between antiparallel-spin (o) ones are labeled separately as g and g , rei, iM spectively, for i"2,4. For convenience, we define the parameters, K and K , for characterizing the o p interaction, i.e., K ,[(v #g !g )/(v #g # l F 4l 2l F 4l g )]1@2, for l"o, p with g ,(g #g )/2, g , 2l io i, iM ip (g !g )/2, and v is the spin-independent Fermi i, iM F velocity. According to usual bosonization, we linearize the conduction-band dispersion near the Fermi points: e(k)K+ e (k) with e (k),$v (kGk ), where jj j F F k is the Fermi wave number and j"1 ( j"2) F corresponds to the right-(left-) moving branch. In this case, the 1D electron systems are described by the following Hamiltonian: H%"H% #H% , where 0 */5 2pv H% " F + + [. (p). (!p) 0 14 1s ¸ s p;0 #. (!p). (p)], 2s 2s
(1)
p H% " + (g d #g d ) */5 ¸ 2, s,s{ 2M s,~s{ s,s{ ] + [. (p). (!p)#. (!p). (p)] 1s 2s{ 1s 2s{ p;0
(2)
where . (p) is the spin-dependent density-fluctujs ation operator for spin s"C, B"$1 in the jth branch and ¸ is the system size. The electronic Hamiltonian, H%, is described also as a sum of two diagonal parts describing the charge- and spin-density fluctuations: H%"+ + vp l/o,p p;0 l ](as a #bs b ). Here a and b (a and b ) lp lp op op pp pp lp lp are boson operators for the charge (spin) fluctuation, which are related to the density-fluctuation [. (p) operators [7,8,11] via as "(p/¸p)1@2+ lp s/B1 1s coshc $. (p)sinhc ], for l"o, p with lp 2s lp cosh(2c )"v (1#g )/v and sinh(2c )"v g /v . lp F 2l l lp F 2l l Both the charge and spin density fluctuations have linear gapless dispersions with constant velocities: v "2(v #g )(K #K~1)~1"[(v #g )2 l F 4l l F 4l l !g2 ]1@2, for l"o, p. 2l Peculiar features of the TL liquid are found in its dynamical properties, which are characterized by anomalous power laws in correlation functions. We shall study the single-electron Green function: G% (x, t)"!iH(t)St (x, t)ts (0, 0)T, where H(t) is js js js the unit-step function and t (x,t) is the Heisenberg js representation of the field operator t (x) for an js electron in the jth branch with spin s. Average is made by the Fermi vacuum at zero temperature. Using the diagonalized Hamiltonian, we have eB*kFx K#i(v tGx) F G% (x,t)" lim js 2p d#i(v tGx) F d?0 1 ] < l/o,p JK#i(vltGx) ]
C
D
clTL K2 , (K#iv t)2#x2 l
(3)
where j"1 (2) corresponds to upper (lower) sign of the right-hand side, K and d are cutoffs, and the exponent is cl "1(K #K~1!2)*0. Eq. (3) TL 8 l l gives a universal behavior of the Green function, which is independent of detailed cutoff forms. For
T. Ogawa / Physica B 249—251 (1998) 185—190
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t"0, the single-electron Green function decays as G% (x, 0)&x~a~1 with the exponent a, js 2(co #cp )*0, which appears in all single-elecTL TL tron properties; e.g., the momentum distribution function n(k)&1!CDk!k Dasign(k!k ) and the 2 F F single-particle density of states N(u)&DuDa.
liquid band and a hole is suddenly generated, i.e., the numbers of both electron and hole change simultaneously by one.
3. Power-law singularities in optical spectra
The valence-band photoemission spectroscopy measures the energy distribution of the photoelectrons ejected from a partially filled band by absorbing photons of a fixed frequency (Fig. 1a). The valence-band photoemission spectrum P (e) PES as a function of the photoelectron energy e (measured from the Fermi energy) reflects directly the density of states N(u). Then P (e)&ebTLH(e). The PES critical exponent b is identical to a: b "a, i.e., TL TL 1 b " (K #K~1#K #K~1!4) p o p TL 4 o
Before the optical transition occurs, we assume that the partially filled band has N electrons forming the TL liquid and there is no positive hole in any levels and bands. Hereafter this initial state is called the N-electron—0-hole state, denoted as the (N, 0) state. In this section, we consider three types of final states after optical transitions, as shown in Fig. 1. (a) From (N, 0) to (N!1, 0): After an electron in the N-electron TL system is excited to be emitted as a photoelectron, the final state is the (N!1, 0) state, where no optical hole is generated and the number of electrons only decreases by one. (b) From (N, 0) to (N, 1): When an electron in a core level is excited by a larger-energy photon to be detected as a photoelectron, the final state is the (N, 1), where the number of electrons in the TL liquid remains unchanged and a hole is abruptly generated. Here the Auger processes are neglected. (c) From (N, 0) to (N#1, 1): When an electron in a core level is excited by a photon to an empty states of the partially-filled band, the final state is the (N#1, 1), where an electron is added to the TL
3.1. The valence-band photoemission: (N, 0)P(N!1, 0)
C A
BD
2 ~1@2 1 g 2l " + 1! !1*0. 2 v #g F 4l l/o,p (4) The valence-band photoemission spectrum can be a direct evidence for the absence of a discontinuous jump at k"k in the momentum distribution of F the TL liquid. Thus nonzero b is a direct evidence TL of the TL liquid. The valence-band photoemission spectrum is the Fourier transform of the time-domain singleelectron Green function G% (t)"!i H(t) js
Fig. 1. Schematic diagrams of three kinds of optical transitions. (a) The valence-band photoemission process describes a transition from (N, 0) to (N!1, 0) state. (b) The core-level photoemission process is a transition from (N, 0) to (N, 1) state. (c) The one-photon absorption process indicates a transition from (N, 0) to (N#1, 1) state.
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]Se*tH%@+t e~*tH%@+ts T&t~bTL~1. Thus the valjs js ence-band photoemission spectroscopy is an effective tool for investigating the TL liquid properties, particularly its single-electron properties.
3.2. The core-level photoemission: (N, 0)P(N, 1) We here mention the orthogonality theorem [12]. When dynamical and local perturbations are applied to degenerate electron systems, low-energy excitations near the Fermi energy come into play and give rise to an infrared divergence. The overlap integral between the total electronic wavefunctions, DW T and DW T, of the Fermi sea without and with i f a local potential becomes zero as the system size goes to infinity, i.e., these two states are orthogonal: DSW DW TD2"n~b0#@2. Here n is the number of (sf i F F wave) electrons and b is the critical exponent for 0# the orthogonality catastrophe. To detect phenomena related to the orthogonality property, sudden appearance of an external local potential is necessary. This can be done in the core-level photoemission spectroscopy [13]. The core-level photoemission spectroscopy measures the energy distribution of the photoelectrons ejected from a core level in a metal by absorbing photons of a fixed frequency (Fig. 1b). If the existence of conduction electrons is neglected, only a sharp line will be observed. When the photon energy is high enough for the photoelectron to leave the metal instantly without being affected by a final-state interaction, the sharp line broadens with a long tail on its lower-energy side. The corelevel photoemission spectrum, P (e), the prob#03% ability of finding an ejected electron at an energy e lower than the main peak, is a direct measure of the probability that the conduction electrons will be excited with an excitation energy e due to the hole potential. When an optical hole with spin !s (an excited electron with spin s) is created, the electron-hole (e—h) Coulomb attraction abruptly appears. Here the hole is assumed to be localized. The Hamiltonian describing the e—h correlation is given by H) (s)"H)#H)(s) with H)"!(g) /2) 2o p o o */5 ](pK /¸)1@2+ Jp(as #a #bs #b ) and o p;0 op op op op
H)(s)"!(g) /2)(pK /¸)1@2s+ Jp(as #a # 2p p p;0 p pp pp bs #b ), where g) "(g) #g) )/2 and g) " 2p 2M 2, 2o pp pp (g) !g) )/2 denote the strength of the e—h for2M 2, ward scattering. Using this e—h interaction, we shall calculate the single-hole Green function of the TL liquid in time domain; G)(t), s !iH(t)e~*E0t@+Se*tH%@+e~*t*H%`H)*/5(s)+@+T&t~b0#, where t) is the field operator of a localized hole with spin s s and E is the energy separation between the core 0 level and the Fermi level. Here the critical exponent b is independent of the hole spin and is given as 0# 2 1 g) 2l b " + (K #K~1)4 0# 64 l l v #g 4l l/o,p F ]M1G[1!4(K #K~1)~2]1@2N l l 2 ~1@2 1 g) g 2l 2l " + 1! 4 v #g v #g 4l F 4l l/o,p F ~3@2 g 2l ] 1# *0, (5) v #g F 4l where K (1 (K '1) corresponds to ! (#) sign l l of Eq. (5). Asymptotic behavior of the core-level photoemission spectrum near the main peak is given by the Fourier transform of G)(t), i.e., P (e)& s #03% eb0#~1H(e), showing the power-law divergence near e&0 because of 0)b (1. Appearance of the 0# power-law singularity instead of the d-function peak means that the quasiparticle picture is invalid in the TL liquid, resulting from simultaneous excitation of many particle-vacancy pairs near the Fermi level.
A
A
B
A
BA B
B
3.3. The one-photon absorption: (N, 0)P(N#1, 1) We shall consider the one-photon absorption process (Fig. 1c). The absorption spectrum ¼ (u) !"4 also exhibits the power-law singularity near the Fermi edge with an exponent b . This is called the FES Fermi-edge singularity (FES) [5,6,14—16]. The critical exponent b is directly related to a powerFES law exponent of the current—current correlation function C (t) at zero temperature: C (t)" s s Se*tH%@+t e~*t*H%`H)*/5(s)+@+tsT"SJs(t)J (0)T, with s s s s t ,t (0)#t (0). Here the current operator is s 1s 2s
T. Ogawa / Physica B 249—251 (1998) 185—190
189
Table 1 Comparison among the critical exponents of power-law singularities in several optical spectra. Corresponding correlation functions are also given. Here t is the field operator of an electron, H% is the electronic Hamiltonian, and Hh means the e—h interaction */5 Optical spectra
Critical exponents
Corresponding correlation functions
Valence-band photoemission P (e) PES (N,0)P(N!1,0) Core-level photoemission P (e) #03% (N,0)P(N,1) One-photon absorption ¼ (u) !"4 (N,0)P(N#1,1)
b TL
Single-electron Green function G%(t)JSe*tH%@+te~*tH%@+tsT&t~bTL~1 Single-hole Green function G)(t)JSe*tH%@+e~*t*H%`H)*/5+@+T&t~b0# Current—current correlation function C(t)JSe*tH%@+te~*t*H%`H)*/5+@+tsT&t~bTL~b0#~b%9~1
b !1 0# b #b #b TL 0# %9
J (t),e*tH%@++ º ts e~*tH%@+, where the unitary s j/1,2 s js operator º is defined through ºsH%º " s s s H%#H) (s). For large t, C (t) decays as a power */5 s law: C (t)&t~bFES~1. Then the one-photon absorps tion spectrum, which is the Fourier transform of C (t), shows the power-law anomaly, i.e., s ¼ (u)&ubFESH(u), where u is a photon energy !"4 measured from the Fermi edge. The FES exponent consists of three parts as b "b #b #b , where b and b are positFES TL 0# %9 TL 0# ive (corresponding to suppression of the transition probability) and given in Eq. (4) and Eq. (5), respectively, but b is negative, indicating the excitonic %9 enhancement of the transition probability, i.e., 1 g) 2l (K #K~1) b "! + %9 l l 4 v #g 4l l/o,p F 2 ~1@2 1 g g) 2l 2l "! + )0. 1! 2 v #g v #g 4l F 4l l/o,p F (6)
C A
BD
Thus the FES exponent in one-photon absorption spectra contains all information on the TL-liquid character (b *0), the orthogonality catastrophe TL (0)b (1), and the many-body excitonic correla0# tions (b )0). Therefore the FES exponent can be %9 either negative or positive depending on the balance between the positive parts b #b and the TL 0# negative part b . The negative (positive) b results %9 FES in the power-law divergence (convergence) of the edge spectra. The e—h attraction (g) '0 and 2, g) '0) is necessary to obtain the divergent edge. 2M Here we note that the FES exponent of the TL liquid is independent of the hole motion [5,6]. Comparison among several exponents is given in Table 1.
4. Concluding remarks and summary We note that there are some problems on optical responses of the TL liquids to be solved theoretically. For example, we should study effects of the e—h backward scattering (denoted as g) ) on the 1 critical exponents. In addition, roles of bound states should be clarified, which are excluded in the TL picture. To summarize, we have investigated three important quantities characterizing the many-body effects in the TL liquid: (a) the single-electron Green function G%(t), (b) the localized-hole Green function G)(t), and (c) the current-current correlation function C(t). G%(t) has an exponent b , which reveals TL the TL properties, e.g., the power-law rounding of the momentum distribution at k . While G)(t) conF tains the orthogonality exponent b . Many-body 0# excitonic correlations are reflected in C(t) as the exponent b . We have stressed that these critical %9 exponents are separately measurable in actual optical experiments [17]; The valence-band photoemission experiments reveal the single-particle density of states, hence b is clarified. The core-level TL photoemission spectroscopy gives us b . The one0# photon absorption spectra exhibit the Fermi-edge singularity with the exponent b #b #b . TL 0# %9 Acknowledgements The author is grateful to K. Ohtaka, A. Furusaki, and N. Nagaosa for discussion. A part of this work is supported by CREST (Core Research for Evolutional Science and Technology) of Japan Science
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and Technology Corporation (JST) and by a Grant-in-Aid for Scientific Research on Priority Areas, “Mutual Quantum Manipulation of Radiation Field and Matter,” from the Ministry of Education, Science, Sports and Culture of Japan. References [1] J. Kondo, A. Yoshimori (Eds.), Fermi Surface Effects, Springer, Berlin, 1988. [2] S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. [3] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. [4] T. Ogawa, Y. Kanemitsu (Eds.), Optical Properties of Low-Dimensional Materials, World Scientific, Singapore, 1995.
[5] T. Ogawa, A. Furusaki, N. Nagaosa, Phys. Rev. Lett. 68 (1992) 3638. [6] T. Ogawa, A. Furusaki, N. Nagaosa, Jpn. J. Appl. Phys. Suppl. 32 (1993) 76. [7] H. Otani, T. Ogawa, Phys. Rev. B 53 (1996) 4684. [8] T. Ogawa, H. Otani, J. Phys. Soc. Japan 64 (1995) 3664. [9] T. Ogawa, H. Otani, Surf. Sci. 361/362 (1996) 476. [10] T. Ogawa, Jpn. J. Appl. Phys. Suppl. 34 (1995) 146. [11] H. Otani, T. Ogawa, Phys. Rev. B 54 (1996) 4540. [12] P.W. Anderson, Phys. Rev. Lett. 18 (1967) 1049. [13] S. Doniach, M. Sunjic, J. Phys. C 3 (1970) 285. [14] K.D. Schotte, U. Schotte, Phys. Rev. 182 (1969) 479. [15] T. Ogawa, Phys. Stat. Sol. (b) 188 (1995) 83. [16] N. Nagaosa, T. Ogawa, Solid State Commun. 88 (1993) 295. [17] A. Fujimori, Y. Tokura (Eds.), Spectroscopy of Mott Insulators and Correlated Metals, Springer, Berlin, 1995.
Optical detection of critical exponents in the Tomonaga—Luttinger liquid Tetsuo Ogawa* Department of Physics, Tohoku University, Aoba-ku, Sendai 980—8578, Japan
Abstract Optical responses of the Tomonaga—Luttinger liquid are investigated theoretically in terms of the power-law singularities. Critical exponents in the edge anomalies of valence-band photoemission, core-level photoemission, and one-photon absorption processes reflect different aspects of low-energy critical properties. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Tomonaga—Luttinger liquid; Photoemission; One-photon absorption; Critical exponents; Orthogonality catastrophe
1. Introduction Low-energy characteristics of one-dimensional (1D) electron systems are studied extensively. When such systems remain gapless, the many-body correlation, the quantum fluctuation, and the infrared divergence [1] play crucial roles in their low-energy properties. When interactions among particles are of a short-range type and when the backward, umklapp, and spin-flip scatterings can be neglected, critical properties of quantum 1D electrons are universally described as the Tomonaga—Luttinger (TL) liquid [2,3]. Studies on the TL liquids have been focused to their spectral and transport properties as well as the mathematical structure; there are just a few studies on their optical properties. The * Fax: #81 22 217 6447; e-mail: [email protected]. tohoku.ac.jp.
main aim of this paper is to show that the optical responses of the TL liquid are also useful tools for investigating the critical properties of 1D degenerate electrons. Several optical transition probabilities reflect the final-state interaction due to an optically generated hole potential. This interaction brings about two kinds of intrinsic dynamical effects in degenerate electron systems: an orthogonality catastrophe and a many-body excitonic correlation. The former suppresses the transition but the latter enhances it. This is a characteristic point in contrast to those in semiconductors [4]. The most striking feature in optical responses of the TL liquids is the power-law singularity, which is characterized by some critical exponents. We pay attention to the valence-band photoemission spectrum, the core-level photoemission spectrum, and the one-photon absorption spectrum. These optical transition processes yield
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T. Ogawa / Physica B 249—251 (1998) 185—190
the power-law anomaly in the edge spectra with different critical exponents. In this paper, we obtain analytical expressions of the critical exponents to clarify what kinds of low-energy characteristics are reflected in them.
p # + (g d #g d ) 4, s,s{ 4M s,~s{ ¸ s,s{ ] + [. (p). (!p) 1s 1s{ p;0 #. (!p). (p)], 2s 2s{
2. The Tomonaga—Luttinger liquid The 1D degenerate electron systems near the Fermi level are characterized by two (right- and left-moving) branches of electrons. When the interparticle interaction in 1D is of the short-range type, the Coulomb scattering strengths are represented by constant parameters: g , g , and g [5—10] 2 4 1 with conventional notation, where g and g cor2 4 respond to the strengths of inter- and intra-branch forward scatterings, and g , the backward-scatter1 ing strength. Then the backward scattering is neglected because Dg DKDg DADg D. We further classify 2 4 1 the interaction parameters in terms of the mutual spin orientation; the scattering between parallelspin (E) particles and that between antiparallel-spin (o) ones are labeled separately as g and g , rei, iM spectively, for i"2,4. For convenience, we define the parameters, K and K , for characterizing the o p interaction, i.e., K ,[(v #g !g )/(v #g # l F 4l 2l F 4l g )]1@2, for l"o, p with g ,(g #g )/2, g , 2l io i, iM ip (g !g )/2, and v is the spin-independent Fermi i, iM F velocity. According to usual bosonization, we linearize the conduction-band dispersion near the Fermi points: e(k)K+ e (k) with e (k),$v (kGk ), where jj j F F k is the Fermi wave number and j"1 ( j"2) F corresponds to the right-(left-) moving branch. In this case, the 1D electron systems are described by the following Hamiltonian: H%"H% #H% , where 0 */5 2pv H% " F + + [. (p). (!p) 0 14 1s ¸ s p;0 #. (!p). (p)], 2s 2s
(1)
p H% " + (g d #g d ) */5 ¸ 2, s,s{ 2M s,~s{ s,s{ ] + [. (p). (!p)#. (!p). (p)] 1s 2s{ 1s 2s{ p;0
(2)
where . (p) is the spin-dependent density-fluctujs ation operator for spin s"C, B"$1 in the jth branch and ¸ is the system size. The electronic Hamiltonian, H%, is described also as a sum of two diagonal parts describing the charge- and spin-density fluctuations: H%"+ + vp l/o,p p;0 l ](as a #bs b ). Here a and b (a and b ) lp lp op op pp pp lp lp are boson operators for the charge (spin) fluctuation, which are related to the density-fluctuation [. (p) operators [7,8,11] via as "(p/¸p)1@2+ lp s/B1 1s coshc $. (p)sinhc ], for l"o, p with lp 2s lp cosh(2c )"v (1#g )/v and sinh(2c )"v g /v . lp F 2l l lp F 2l l Both the charge and spin density fluctuations have linear gapless dispersions with constant velocities: v "2(v #g )(K #K~1)~1"[(v #g )2 l F 4l l F 4l l !g2 ]1@2, for l"o, p. 2l Peculiar features of the TL liquid are found in its dynamical properties, which are characterized by anomalous power laws in correlation functions. We shall study the single-electron Green function: G% (x, t)"!iH(t)St (x, t)ts (0, 0)T, where H(t) is js js js the unit-step function and t (x,t) is the Heisenberg js representation of the field operator t (x) for an js electron in the jth branch with spin s. Average is made by the Fermi vacuum at zero temperature. Using the diagonalized Hamiltonian, we have eB*kFx K#i(v tGx) F G% (x,t)" lim js 2p d#i(v tGx) F d?0 1 ] < l/o,p JK#i(vltGx) ]
C
D
clTL K2 , (K#iv t)2#x2 l
(3)
where j"1 (2) corresponds to upper (lower) sign of the right-hand side, K and d are cutoffs, and the exponent is cl "1(K #K~1!2)*0. Eq. (3) TL 8 l l gives a universal behavior of the Green function, which is independent of detailed cutoff forms. For
T. Ogawa / Physica B 249—251 (1998) 185—190
187
t"0, the single-electron Green function decays as G% (x, 0)&x~a~1 with the exponent a, js 2(co #cp )*0, which appears in all single-elecTL TL tron properties; e.g., the momentum distribution function n(k)&1!CDk!k Dasign(k!k ) and the 2 F F single-particle density of states N(u)&DuDa.
liquid band and a hole is suddenly generated, i.e., the numbers of both electron and hole change simultaneously by one.
3. Power-law singularities in optical spectra
The valence-band photoemission spectroscopy measures the energy distribution of the photoelectrons ejected from a partially filled band by absorbing photons of a fixed frequency (Fig. 1a). The valence-band photoemission spectrum P (e) PES as a function of the photoelectron energy e (measured from the Fermi energy) reflects directly the density of states N(u). Then P (e)&ebTLH(e). The PES critical exponent b is identical to a: b "a, i.e., TL TL 1 b " (K #K~1#K #K~1!4) p o p TL 4 o
Before the optical transition occurs, we assume that the partially filled band has N electrons forming the TL liquid and there is no positive hole in any levels and bands. Hereafter this initial state is called the N-electron—0-hole state, denoted as the (N, 0) state. In this section, we consider three types of final states after optical transitions, as shown in Fig. 1. (a) From (N, 0) to (N!1, 0): After an electron in the N-electron TL system is excited to be emitted as a photoelectron, the final state is the (N!1, 0) state, where no optical hole is generated and the number of electrons only decreases by one. (b) From (N, 0) to (N, 1): When an electron in a core level is excited by a larger-energy photon to be detected as a photoelectron, the final state is the (N, 1), where the number of electrons in the TL liquid remains unchanged and a hole is abruptly generated. Here the Auger processes are neglected. (c) From (N, 0) to (N#1, 1): When an electron in a core level is excited by a photon to an empty states of the partially-filled band, the final state is the (N#1, 1), where an electron is added to the TL
3.1. The valence-band photoemission: (N, 0)P(N!1, 0)
C A
BD
2 ~1@2 1 g 2l " + 1! !1*0. 2 v #g F 4l l/o,p (4) The valence-band photoemission spectrum can be a direct evidence for the absence of a discontinuous jump at k"k in the momentum distribution of F the TL liquid. Thus nonzero b is a direct evidence TL of the TL liquid. The valence-band photoemission spectrum is the Fourier transform of the time-domain singleelectron Green function G% (t)"!i H(t) js
Fig. 1. Schematic diagrams of three kinds of optical transitions. (a) The valence-band photoemission process describes a transition from (N, 0) to (N!1, 0) state. (b) The core-level photoemission process is a transition from (N, 0) to (N, 1) state. (c) The one-photon absorption process indicates a transition from (N, 0) to (N#1, 1) state.
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]Se*tH%@+t e~*tH%@+ts T&t~bTL~1. Thus the valjs js ence-band photoemission spectroscopy is an effective tool for investigating the TL liquid properties, particularly its single-electron properties.
3.2. The core-level photoemission: (N, 0)P(N, 1) We here mention the orthogonality theorem [12]. When dynamical and local perturbations are applied to degenerate electron systems, low-energy excitations near the Fermi energy come into play and give rise to an infrared divergence. The overlap integral between the total electronic wavefunctions, DW T and DW T, of the Fermi sea without and with i f a local potential becomes zero as the system size goes to infinity, i.e., these two states are orthogonal: DSW DW TD2"n~b0#@2. Here n is the number of (sf i F F wave) electrons and b is the critical exponent for 0# the orthogonality catastrophe. To detect phenomena related to the orthogonality property, sudden appearance of an external local potential is necessary. This can be done in the core-level photoemission spectroscopy [13]. The core-level photoemission spectroscopy measures the energy distribution of the photoelectrons ejected from a core level in a metal by absorbing photons of a fixed frequency (Fig. 1b). If the existence of conduction electrons is neglected, only a sharp line will be observed. When the photon energy is high enough for the photoelectron to leave the metal instantly without being affected by a final-state interaction, the sharp line broadens with a long tail on its lower-energy side. The corelevel photoemission spectrum, P (e), the prob#03% ability of finding an ejected electron at an energy e lower than the main peak, is a direct measure of the probability that the conduction electrons will be excited with an excitation energy e due to the hole potential. When an optical hole with spin !s (an excited electron with spin s) is created, the electron-hole (e—h) Coulomb attraction abruptly appears. Here the hole is assumed to be localized. The Hamiltonian describing the e—h correlation is given by H) (s)"H)#H)(s) with H)"!(g) /2) 2o p o o */5 ](pK /¸)1@2+ Jp(as #a #bs #b ) and o p;0 op op op op
H)(s)"!(g) /2)(pK /¸)1@2s+ Jp(as #a # 2p p p;0 p pp pp bs #b ), where g) "(g) #g) )/2 and g) " 2p 2M 2, 2o pp pp (g) !g) )/2 denote the strength of the e—h for2M 2, ward scattering. Using this e—h interaction, we shall calculate the single-hole Green function of the TL liquid in time domain; G)(t), s !iH(t)e~*E0t@+Se*tH%@+e~*t*H%`H)*/5(s)+@+T&t~b0#, where t) is the field operator of a localized hole with spin s s and E is the energy separation between the core 0 level and the Fermi level. Here the critical exponent b is independent of the hole spin and is given as 0# 2 1 g) 2l b " + (K #K~1)4 0# 64 l l v #g 4l l/o,p F ]M1G[1!4(K #K~1)~2]1@2N l l 2 ~1@2 1 g) g 2l 2l " + 1! 4 v #g v #g 4l F 4l l/o,p F ~3@2 g 2l ] 1# *0, (5) v #g F 4l where K (1 (K '1) corresponds to ! (#) sign l l of Eq. (5). Asymptotic behavior of the core-level photoemission spectrum near the main peak is given by the Fourier transform of G)(t), i.e., P (e)& s #03% eb0#~1H(e), showing the power-law divergence near e&0 because of 0)b (1. Appearance of the 0# power-law singularity instead of the d-function peak means that the quasiparticle picture is invalid in the TL liquid, resulting from simultaneous excitation of many particle-vacancy pairs near the Fermi level.
A
A
B
A
BA B
B
3.3. The one-photon absorption: (N, 0)P(N#1, 1) We shall consider the one-photon absorption process (Fig. 1c). The absorption spectrum ¼ (u) !"4 also exhibits the power-law singularity near the Fermi edge with an exponent b . This is called the FES Fermi-edge singularity (FES) [5,6,14—16]. The critical exponent b is directly related to a powerFES law exponent of the current—current correlation function C (t) at zero temperature: C (t)" s s Se*tH%@+t e~*t*H%`H)*/5(s)+@+tsT"SJs(t)J (0)T, with s s s s t ,t (0)#t (0). Here the current operator is s 1s 2s
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Table 1 Comparison among the critical exponents of power-law singularities in several optical spectra. Corresponding correlation functions are also given. Here t is the field operator of an electron, H% is the electronic Hamiltonian, and Hh means the e—h interaction */5 Optical spectra
Critical exponents
Corresponding correlation functions
Valence-band photoemission P (e) PES (N,0)P(N!1,0) Core-level photoemission P (e) #03% (N,0)P(N,1) One-photon absorption ¼ (u) !"4 (N,0)P(N#1,1)
b TL
Single-electron Green function G%(t)JSe*tH%@+te~*tH%@+tsT&t~bTL~1 Single-hole Green function G)(t)JSe*tH%@+e~*t*H%`H)*/5+@+T&t~b0# Current—current correlation function C(t)JSe*tH%@+te~*t*H%`H)*/5+@+tsT&t~bTL~b0#~b%9~1
b !1 0# b #b #b TL 0# %9
J (t),e*tH%@++ º ts e~*tH%@+, where the unitary s j/1,2 s js operator º is defined through ºsH%º " s s s H%#H) (s). For large t, C (t) decays as a power */5 s law: C (t)&t~bFES~1. Then the one-photon absorps tion spectrum, which is the Fourier transform of C (t), shows the power-law anomaly, i.e., s ¼ (u)&ubFESH(u), where u is a photon energy !"4 measured from the Fermi edge. The FES exponent consists of three parts as b "b #b #b , where b and b are positFES TL 0# %9 TL 0# ive (corresponding to suppression of the transition probability) and given in Eq. (4) and Eq. (5), respectively, but b is negative, indicating the excitonic %9 enhancement of the transition probability, i.e., 1 g) 2l (K #K~1) b "! + %9 l l 4 v #g 4l l/o,p F 2 ~1@2 1 g g) 2l 2l "! + )0. 1! 2 v #g v #g 4l F 4l l/o,p F (6)
C A
BD
Thus the FES exponent in one-photon absorption spectra contains all information on the TL-liquid character (b *0), the orthogonality catastrophe TL (0)b (1), and the many-body excitonic correla0# tions (b )0). Therefore the FES exponent can be %9 either negative or positive depending on the balance between the positive parts b #b and the TL 0# negative part b . The negative (positive) b results %9 FES in the power-law divergence (convergence) of the edge spectra. The e—h attraction (g) '0 and 2, g) '0) is necessary to obtain the divergent edge. 2M Here we note that the FES exponent of the TL liquid is independent of the hole motion [5,6]. Comparison among several exponents is given in Table 1.
4. Concluding remarks and summary We note that there are some problems on optical responses of the TL liquids to be solved theoretically. For example, we should study effects of the e—h backward scattering (denoted as g) ) on the 1 critical exponents. In addition, roles of bound states should be clarified, which are excluded in the TL picture. To summarize, we have investigated three important quantities characterizing the many-body effects in the TL liquid: (a) the single-electron Green function G%(t), (b) the localized-hole Green function G)(t), and (c) the current-current correlation function C(t). G%(t) has an exponent b , which reveals TL the TL properties, e.g., the power-law rounding of the momentum distribution at k . While G)(t) conF tains the orthogonality exponent b . Many-body 0# excitonic correlations are reflected in C(t) as the exponent b . We have stressed that these critical %9 exponents are separately measurable in actual optical experiments [17]; The valence-band photoemission experiments reveal the single-particle density of states, hence b is clarified. The core-level TL photoemission spectroscopy gives us b . The one0# photon absorption spectra exhibit the Fermi-edge singularity with the exponent b #b #b . TL 0# %9 Acknowledgements The author is grateful to K. Ohtaka, A. Furusaki, and N. Nagaosa for discussion. A part of this work is supported by CREST (Core Research for Evolutional Science and Technology) of Japan Science
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and Technology Corporation (JST) and by a Grant-in-Aid for Scientific Research on Priority Areas, “Mutual Quantum Manipulation of Radiation Field and Matter,” from the Ministry of Education, Science, Sports and Culture of Japan. References [1] J. Kondo, A. Yoshimori (Eds.), Fermi Surface Effects, Springer, Berlin, 1988. [2] S. Tomonaga, Prog. Theor. Phys. 5 (1950) 544. [3] J.M. Luttinger, J. Math. Phys. 4 (1963) 1154. [4] T. Ogawa, Y. Kanemitsu (Eds.), Optical Properties of Low-Dimensional Materials, World Scientific, Singapore, 1995.
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