- Email: [email protected]
Conference summary: Theory
PHYSICA
Physica B 186-188 (1993) 1080-1082 North-Holland
Conference summary: Theory Sadamichi Maekawa Department of Applied Physics, Nagoya University, Japan
1. Introductory remarks At the conference, to my knowledge, 85 theoretical papers and 278 experimental papers have been presented. The theoretical papers cover heavy fermions, high-temperature superconductors and their related problems. The experimental papers cover the following three categories: heavy fermions and related problems (241 papers), high-temperature superconductors and related problems (29 papers), and organics, fullerenes and others (8 papers). Considering the vast area discussed during the past five days, to summarize the conference is an overwhelming task. In the following, therefore, I would lik~ to report a rather broad view of the present status of our understanding in the strongly correlated electron systems.
2. Heavy fermions Let me start with heavy fermions. At the conference, special attention has been placed on the Kondo insulators and the related low-carrier compounds. The Kondo insulators are the compounds with Kondo or intermediate-valence ions such as SmB6, YbB12 and CeNiSn. Some of them are not new ones, but they were left without extensive theoretical discussion until recently, although a longstanding effort had been done by some groups. In Kondo insulators a narrow energy gap is observed. One of the most important problems is, therefore, what the origin of the energy gap is. Several mechanisms for the gap have been proposed. The Kondo exchange interaction between a conduction electron and a localized spin with S = 1/2 causes the spin singlet state. A numerical study has revealed that when the number of the conduction electrons is the same as that of the localized spins, the singlet state occurs at each site and the excited state is the spin
Correspondence to: S. Maekawa, Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan.
triplet state with finite energy, resulting in the insulating ground state. Although the calculation was done in the one-dimensional system, the results are considered to hold in the higher-dimensional systems. On the other hand, it was argued that in low-carrier compounds, the Wigner crystalization occurs due to the long ranged Coulomb interaction. It was shown that a self-consistent band calculation with the spin-orbit interaction can derive the gap in the compounds given above. This suggests that the hybridization also plays a role in the formation of the gap. In connection with Kondo insulators, some theoretical problems were raised. One of them is how the energy gap affects the formation of the singlet state when a Kondo impurity is introduced into an insulator. This problem came from the fact that the instability of the Fermi surface brings about a Kondo singlet in a metal. Since the gap is mostly due to the strong correlation in Kondo insulators, the next step will be to treat the gap and the singlet state selfconsistently. The nature of the electronic state near the Fermi level in heavy fermion compounds has been another topic discussed extensively. In metals with usual Kondo impurities, the local Fermi liquid state is obtained. In the case of U 4+ impurities, however, the situation is different. A U 4÷ ion couples to two channels of the conduction electrons. Since the two channels of the conduction electrons screen the pseudo spin of the impurity, the overscreening occurs and results in a local non-Fermi liquid state with unusual physical properties. For instance, the 3' term of the specific heat increases logarithmically at low temperatures. This problem is called the two-channel or multichannel Kondo problem. A similar local non-Fermi liquid state has also been proposed in the system with two Kondo impurities which interact by the RKKY interaction with each other. Both multi-channel and two-impurity Kondo problems propose new quantum liquid states in strongly correlated electron systems and a more complete study will be made before long. Our understanding of the Kondo lattice is much
0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
S. Maekawa / Conference summary: Theory
behind the Kondo impurity case. For examining the electronic properties, many theoretical methods have been developed: the exact-diagonalization method of finite-size clusters treats the local electron correlation exactly and gives much physical insight to the manybody states. Various expansion methods such as 1/d expansion, d being the spatial dimension, and 1 / N expansion, N being the degree of internal degeneracy, were applied to the Kondo lattice. Several operator expansion methods such as slave bosons and Majorana fermions were also introduced. These expansion methods describe the coherent heavy fermion state, although some do not include the local constraint properly and others are a little far from the real world. The variety of theoretical methods seems to be sufficient to guarantee future progress. To examine the transport properties and the superconductivity, the dispersion, the Fermi surface geometry and the pairing interaction in the heavy fermion state must be clarified. Concerning the Fermi surface geometry, the band calculation has been done for many heavy fermion compounds and good agreement between theory and experiment was demonstrated. Heavy fermion superconductivity has been one of the most exciting subjects at the conference. In UPt3, URu2Si 2 and the new members UNi2AI 3 and UPd2AI3, the superconducting state coexists with antiferromagnetic order. The compound Ul_xThxBe~3 is also an example. In CeCu2Si2, on the other hand, it seems that there is a subtle interplay between superconductivity and antiferromagnetism. This is different from the high-temperature superconductivity in the cuprates which will be discussed in the next section. It has been recognized at the conference that the superconducting state is not a conventional one. Is this due to the antiferromagnetic nature? This issue remains open. The heavy fermion superconductivity is sometimes marginal. It is easily destroyed by changing the stoichiometry. It has been shown that in CeCu2Ge 2 the superconductivity is stabilized only under pressure by suppressing the antiferromagnetic order. This fact indicates that the electronic state is strongly dependent on the lattice constants. These experiments show that heavy fermions are sensitive to low-energy excitations of spin, charge and phonons. Such sensitivity often causes confusion in the theoretical discussion. A very unique situation of the heavy fermion superconductivity is that there exist multi-superconducting transitions. Among them, the tetra-critical point in the B - T diagram in UPt 3 received particular attention. It was argued that the interplay between superconductivity and antiferromagnetism caused multi-transitions. In this argument, however, weak spin-orbit coupling
1081
had to be assumed. This assumption is not always justified and will be subject of future study. As mentioned above, the multi-channels of the conduction electrons couple to the U ions. Therefore, in U compounds such as UPt 3 and UBe13 the multichannels should contribute to the electronic states near the Fermi level and the superconductivity. The problem remains to be examined.
3. High-temperature superconductors The parent compounds of high-temperature superconductors (HTSC) are charge transfer insulators. When hole or electron carriers are introduced into the insulators, a many-body singlet state emerges near the Fermi level, which is the split-off state from the ligand oxygen 2p, band or which is the upper Hubbard Cu 3d (x 2 _ y2) band. The singlet state is called the ingap (or simply gap) state and carries the superconductivity. The ingap state is formed in the charge transfer gap upon doping of carriers in a self-consistent way with the Fermi level so that it might have a relation with the low-carrier state in the Kondo insulators. Actually, much discussion about the relation was given in the lobby in the conference. It seems that the problem will be one of the subjects at the next conference. To clarify the origin of superconductivity in the cuprates, we must ask whether the ingap state is a Fermi liquid or non-Fermi liquid. It is convenient to adopt the one-band Hubbard model for this problem, noting that the low-energy excitations are carried by the ingap band and the upper Hubbard band. These bands may be mapped onto the one-band Hubbard model. Since the parent compounds are antiferromagnetic insulators, the charge excitation can further be projected out. Then, we have the so-called t - J model. This model describes the motion of charged carriers in the spin background with antiferromagnetic interaction. At the conference, the electronic state and superconductivity in the cuprates have been examined extensively by using the one-band Hubbard and t - J models. In the one-dimensional Hubbard model, spin and charge degrees of freedom behave separately. As a result, the electronic state is the Luttinger liquid. In the two-dimensional model which is more interesting in connection with the CuO 2 plane in the cuprates, much discussion has been done to examine which liquid, i.e. Fermi or Luttinger, is consistent with experiments. If spin and charge degrees of freedom are separate in the CuO 2 plane, the optical conductivity perpendicular to the plane should be suppressed in the normal state, whereas the superconducting current
1082
S. Maekawa / Conference summary: Theory
may flow between the planes, indicating the absence of the sum rule in the optical conductivity in this direction. Detailed experiments in La 2 xSrxCuO4 have been presented. Although it was argued that they suggested the existence of the sum rule, the energy dependence of the conductivity is so complex that the problem is still controversial. One of the crucial properties in the cuprates is the existence of a (pseudo) spin gap even in the normal state, which has been observed in neutron inelastic scattering experiments. It was shown that the results were consistent with nuclear magnetic resonance experiments: the spin-lattice relaxation time (T1) becomes longer below a certain temperature at which the spin gap appears in the neutron experiments. Based on the resonating valence bond (RVB) picture, various magnetic properties and the phase diagram in the cuprates have been calculated in the t - J model. The results propose that the spin gap in the normal state is due to the formation of a singlet state and that the material dependence of the magnetic properties reflects Fermi surfaces observed by angular-resolved photoemission experiments. As seen in the magnetic as well as transport properties, the normal state is already not normal in HTSC. Thus it is expected that an understanding of the normal state will give us a solution for the mechanism of superconductivity. To my knowledge, this conference was the first to pay much attention to the other transition metal oxides on an equal footing with the cuprates and heavy fermion compounds. These are Ti-oxides and Voxides. In these oxides which are M o t t - H u b b a r d compounds, the fine tuning of the electron correlation, i.e. the so called Hubbard U parameter, is possible. It is promising that this field grows as one of the strongly correlated electron systems in connection with HTSC and heavy fermions. The superconductivity of HTSC is more controversial. In contrast to the heavy fermion case, superconductivity does not coexist with magnetic order, although it seems that there is a general consensus that
the spin-fluctuation plays a role in the electronic state. If the antiferromagnetic spin fluctuation is the main mechanism of the superconductivity, d-wave pairing is more favorable. However, some experiments such as tunneling seem to suggest s-wave pairing. It has been demonstrated experimentally that a phonon mode changes dramatically at the superconducting transition temperature. However, it is not clear whether this change indicates that the phonons induce superconductivity or it is just caused by superconductivity. Considering the anomalous temperature dependence of 1 / T j , possibilities of odd-frequency and odd-parity pairing were also examined.
4. Concluding remarks Discovery of heavy fermion superconductors and HTSC have stimulated the active theoretical investigation on strongly correlated electron systems. In the systems, the electron-electron interaction is not a weak perturbation on the electronic state given by the kinetic energy and the Fermi statistics but must be taken into account from the beginning. A variety of theoretical methods and concepts was developed in the course of the study. Although the problems imply difficult many-body effects, many theoretical issues have been clarified at the conference. It is expected that the theoretical concepts newly presented here will also be confronted with experiment at the next conference. These five days have been full of scientific discussion with colleagues from all over the world. On behalf of all the participants, I would like to thank Prof. T. Kasuya (Chairman) and the organizing committee members for providing all of us with this enjoyable and stimulating conference. I hope we will get together again in the next conference in San Diego and exchange what we will get in a year from now.
Physica B 186-188 (1993) 1080-1082 North-Holland
Conference summary: Theory Sadamichi Maekawa Department of Applied Physics, Nagoya University, Japan
1. Introductory remarks At the conference, to my knowledge, 85 theoretical papers and 278 experimental papers have been presented. The theoretical papers cover heavy fermions, high-temperature superconductors and their related problems. The experimental papers cover the following three categories: heavy fermions and related problems (241 papers), high-temperature superconductors and related problems (29 papers), and organics, fullerenes and others (8 papers). Considering the vast area discussed during the past five days, to summarize the conference is an overwhelming task. In the following, therefore, I would lik~ to report a rather broad view of the present status of our understanding in the strongly correlated electron systems.
2. Heavy fermions Let me start with heavy fermions. At the conference, special attention has been placed on the Kondo insulators and the related low-carrier compounds. The Kondo insulators are the compounds with Kondo or intermediate-valence ions such as SmB6, YbB12 and CeNiSn. Some of them are not new ones, but they were left without extensive theoretical discussion until recently, although a longstanding effort had been done by some groups. In Kondo insulators a narrow energy gap is observed. One of the most important problems is, therefore, what the origin of the energy gap is. Several mechanisms for the gap have been proposed. The Kondo exchange interaction between a conduction electron and a localized spin with S = 1/2 causes the spin singlet state. A numerical study has revealed that when the number of the conduction electrons is the same as that of the localized spins, the singlet state occurs at each site and the excited state is the spin
Correspondence to: S. Maekawa, Department of Applied Physics, Nagoya University, Nagoya 464-01, Japan.
triplet state with finite energy, resulting in the insulating ground state. Although the calculation was done in the one-dimensional system, the results are considered to hold in the higher-dimensional systems. On the other hand, it was argued that in low-carrier compounds, the Wigner crystalization occurs due to the long ranged Coulomb interaction. It was shown that a self-consistent band calculation with the spin-orbit interaction can derive the gap in the compounds given above. This suggests that the hybridization also plays a role in the formation of the gap. In connection with Kondo insulators, some theoretical problems were raised. One of them is how the energy gap affects the formation of the singlet state when a Kondo impurity is introduced into an insulator. This problem came from the fact that the instability of the Fermi surface brings about a Kondo singlet in a metal. Since the gap is mostly due to the strong correlation in Kondo insulators, the next step will be to treat the gap and the singlet state selfconsistently. The nature of the electronic state near the Fermi level in heavy fermion compounds has been another topic discussed extensively. In metals with usual Kondo impurities, the local Fermi liquid state is obtained. In the case of U 4+ impurities, however, the situation is different. A U 4÷ ion couples to two channels of the conduction electrons. Since the two channels of the conduction electrons screen the pseudo spin of the impurity, the overscreening occurs and results in a local non-Fermi liquid state with unusual physical properties. For instance, the 3' term of the specific heat increases logarithmically at low temperatures. This problem is called the two-channel or multichannel Kondo problem. A similar local non-Fermi liquid state has also been proposed in the system with two Kondo impurities which interact by the RKKY interaction with each other. Both multi-channel and two-impurity Kondo problems propose new quantum liquid states in strongly correlated electron systems and a more complete study will be made before long. Our understanding of the Kondo lattice is much
0921-4526/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
S. Maekawa / Conference summary: Theory
behind the Kondo impurity case. For examining the electronic properties, many theoretical methods have been developed: the exact-diagonalization method of finite-size clusters treats the local electron correlation exactly and gives much physical insight to the manybody states. Various expansion methods such as 1/d expansion, d being the spatial dimension, and 1 / N expansion, N being the degree of internal degeneracy, were applied to the Kondo lattice. Several operator expansion methods such as slave bosons and Majorana fermions were also introduced. These expansion methods describe the coherent heavy fermion state, although some do not include the local constraint properly and others are a little far from the real world. The variety of theoretical methods seems to be sufficient to guarantee future progress. To examine the transport properties and the superconductivity, the dispersion, the Fermi surface geometry and the pairing interaction in the heavy fermion state must be clarified. Concerning the Fermi surface geometry, the band calculation has been done for many heavy fermion compounds and good agreement between theory and experiment was demonstrated. Heavy fermion superconductivity has been one of the most exciting subjects at the conference. In UPt3, URu2Si 2 and the new members UNi2AI 3 and UPd2AI3, the superconducting state coexists with antiferromagnetic order. The compound Ul_xThxBe~3 is also an example. In CeCu2Si2, on the other hand, it seems that there is a subtle interplay between superconductivity and antiferromagnetism. This is different from the high-temperature superconductivity in the cuprates which will be discussed in the next section. It has been recognized at the conference that the superconducting state is not a conventional one. Is this due to the antiferromagnetic nature? This issue remains open. The heavy fermion superconductivity is sometimes marginal. It is easily destroyed by changing the stoichiometry. It has been shown that in CeCu2Ge 2 the superconductivity is stabilized only under pressure by suppressing the antiferromagnetic order. This fact indicates that the electronic state is strongly dependent on the lattice constants. These experiments show that heavy fermions are sensitive to low-energy excitations of spin, charge and phonons. Such sensitivity often causes confusion in the theoretical discussion. A very unique situation of the heavy fermion superconductivity is that there exist multi-superconducting transitions. Among them, the tetra-critical point in the B - T diagram in UPt 3 received particular attention. It was argued that the interplay between superconductivity and antiferromagnetism caused multi-transitions. In this argument, however, weak spin-orbit coupling
1081
had to be assumed. This assumption is not always justified and will be subject of future study. As mentioned above, the multi-channels of the conduction electrons couple to the U ions. Therefore, in U compounds such as UPt 3 and UBe13 the multichannels should contribute to the electronic states near the Fermi level and the superconductivity. The problem remains to be examined.
3. High-temperature superconductors The parent compounds of high-temperature superconductors (HTSC) are charge transfer insulators. When hole or electron carriers are introduced into the insulators, a many-body singlet state emerges near the Fermi level, which is the split-off state from the ligand oxygen 2p, band or which is the upper Hubbard Cu 3d (x 2 _ y2) band. The singlet state is called the ingap (or simply gap) state and carries the superconductivity. The ingap state is formed in the charge transfer gap upon doping of carriers in a self-consistent way with the Fermi level so that it might have a relation with the low-carrier state in the Kondo insulators. Actually, much discussion about the relation was given in the lobby in the conference. It seems that the problem will be one of the subjects at the next conference. To clarify the origin of superconductivity in the cuprates, we must ask whether the ingap state is a Fermi liquid or non-Fermi liquid. It is convenient to adopt the one-band Hubbard model for this problem, noting that the low-energy excitations are carried by the ingap band and the upper Hubbard band. These bands may be mapped onto the one-band Hubbard model. Since the parent compounds are antiferromagnetic insulators, the charge excitation can further be projected out. Then, we have the so-called t - J model. This model describes the motion of charged carriers in the spin background with antiferromagnetic interaction. At the conference, the electronic state and superconductivity in the cuprates have been examined extensively by using the one-band Hubbard and t - J models. In the one-dimensional Hubbard model, spin and charge degrees of freedom behave separately. As a result, the electronic state is the Luttinger liquid. In the two-dimensional model which is more interesting in connection with the CuO 2 plane in the cuprates, much discussion has been done to examine which liquid, i.e. Fermi or Luttinger, is consistent with experiments. If spin and charge degrees of freedom are separate in the CuO 2 plane, the optical conductivity perpendicular to the plane should be suppressed in the normal state, whereas the superconducting current
1082
S. Maekawa / Conference summary: Theory
may flow between the planes, indicating the absence of the sum rule in the optical conductivity in this direction. Detailed experiments in La 2 xSrxCuO4 have been presented. Although it was argued that they suggested the existence of the sum rule, the energy dependence of the conductivity is so complex that the problem is still controversial. One of the crucial properties in the cuprates is the existence of a (pseudo) spin gap even in the normal state, which has been observed in neutron inelastic scattering experiments. It was shown that the results were consistent with nuclear magnetic resonance experiments: the spin-lattice relaxation time (T1) becomes longer below a certain temperature at which the spin gap appears in the neutron experiments. Based on the resonating valence bond (RVB) picture, various magnetic properties and the phase diagram in the cuprates have been calculated in the t - J model. The results propose that the spin gap in the normal state is due to the formation of a singlet state and that the material dependence of the magnetic properties reflects Fermi surfaces observed by angular-resolved photoemission experiments. As seen in the magnetic as well as transport properties, the normal state is already not normal in HTSC. Thus it is expected that an understanding of the normal state will give us a solution for the mechanism of superconductivity. To my knowledge, this conference was the first to pay much attention to the other transition metal oxides on an equal footing with the cuprates and heavy fermion compounds. These are Ti-oxides and Voxides. In these oxides which are M o t t - H u b b a r d compounds, the fine tuning of the electron correlation, i.e. the so called Hubbard U parameter, is possible. It is promising that this field grows as one of the strongly correlated electron systems in connection with HTSC and heavy fermions. The superconductivity of HTSC is more controversial. In contrast to the heavy fermion case, superconductivity does not coexist with magnetic order, although it seems that there is a general consensus that
the spin-fluctuation plays a role in the electronic state. If the antiferromagnetic spin fluctuation is the main mechanism of the superconductivity, d-wave pairing is more favorable. However, some experiments such as tunneling seem to suggest s-wave pairing. It has been demonstrated experimentally that a phonon mode changes dramatically at the superconducting transition temperature. However, it is not clear whether this change indicates that the phonons induce superconductivity or it is just caused by superconductivity. Considering the anomalous temperature dependence of 1 / T j , possibilities of odd-frequency and odd-parity pairing were also examined.
4. Concluding remarks Discovery of heavy fermion superconductors and HTSC have stimulated the active theoretical investigation on strongly correlated electron systems. In the systems, the electron-electron interaction is not a weak perturbation on the electronic state given by the kinetic energy and the Fermi statistics but must be taken into account from the beginning. A variety of theoretical methods and concepts was developed in the course of the study. Although the problems imply difficult many-body effects, many theoretical issues have been clarified at the conference. It is expected that the theoretical concepts newly presented here will also be confronted with experiment at the next conference. These five days have been full of scientific discussion with colleagues from all over the world. On behalf of all the participants, I would like to thank Prof. T. Kasuya (Chairman) and the organizing committee members for providing all of us with this enjoyable and stimulating conference. I hope we will get together again in the next conference in San Diego and exchange what we will get in a year from now.