物质结构的概念的统一性（巴丁）

A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. Ann. Rev. Mater. Sci. 1980. 10: 1-18 Copyright© 1980 by Annual Reviews Inc. All rights reserved UNITY OF CONCEPTS IN THE STRUCTURE OF MATTER John Bardeen Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 1 INTRODUCTION x8641 In this age of increasing specialization it is comforting to realize that basic physical concepts apply to a wide range of seemingly diverse problems. Progress made in understanding one area may often be applied in many other fields. This is true not only for various fields of materials science but for the structure of matter in general. As examples we illustrate how concepts developed to understand magnetism, superfluid helium, and superconductivity have been extended and applied to such diverse fields as nuclear matter, weak and electromagnetic interactions, quark structure of the particles of high energy physics, and phases of liquid crystals. Theoretical methods used in quantum field theory and in many-body problems of condensed-matter theory have much in common. For example, Greens function methods, Feynman diagrams, and renor­ malization group methods introduced for quantum electrodynamics have also been used in many problems in condensed-matter physics. The concept of spontaneously broken symmetry and associated phase transitions derived for condensed matter are now being widely used for problems of high energy physics. The commonality of physical concepts and methods extending over a broad range of problems is one of the main reasons that the National Science Foundation recently established an Institute for Theoretical Physics at the University of California, Santa Barbara. It is hoped that exchange of ideas between theorists in different disciplines will be mutually beneficial. In this article we are not concerned with mathematical methods but with broad physical concepts useful for a wide variety of problems. One concept of great generality is the method of elementary excitations. 1 0084-6600/80/0801-0001$01.00 A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. 2 BARDEEN Although it originated much earlier, the method in its modern form was developed in large part by Landau and co-workers. One tries to under­ stand the ground-state and low-lying excitations that are approximate eigenstates of the system. The interactions between excitations generally are such that at low temperatures one may treat the system as a dilute gas of noninteracting excitations. At higher temperatures, inter­ actions must be taken into account. In homogeneous systems, because of translational symmetry, excitations are waves and may be characterized by a wave vector, k. Examples are quasiparticle excitations and phonons in metals and spin waves in ferromagnets. When applied to particle physics, the ground state is the vacuum and the particles the excitations of the system. When there is a phase transition, the ordered ground state and the low temperature phase may have lower symmetry than the Hamiltonian describing the system. Further, the ground state may be degenerate. A familiar example is the Heisenberg model of ferromagnetism in which at T = 0 K the spins are aligned along any one of several preferred directions, or, in the isotropic case, along any direction in space. The low temperature phase may be characterized by an order parameter that in this case is taken to be the average magnetization, specified by its three components. Elementary excitations are spin waves in which the magnetization precesses about the direction in the ground state. With increasing temperature, the magnetization decreases and goes to zero at the Curie temperature, Te, where there is a phase transition to the paramagnetic state. It is not always easy to recognize the nature of the order that character­ izes the broken symmetry of the low temperature phase. As first suggested on phenomenological grounds by Ginzburg & Landau, superconductors, as well as superfluid helium, are characterized by a complex order parameter with amplitude and phase. In the two-fluid model, the ampli­ tude is proportional to the superfluid density and the gradient of phase to the superfluid velocity. At the nematic-smectic-A phase transition in liquid crystals, a density wave develops with oscillations along the axis parallel to which the molecules are aligned. A complex order parameter may be used to define the amplitude and phase of the wave. Landau (1) suggested that properties in the vicinity of a second-order phase transition may be treated by expanding the free energy in powers of the order parameter. This implies a similarity of properties between different systems that are characterized by similar order parameters. Later we discuss analogies between superconductors and liquid crystals in the smectic-A phase. In phases characterized by order parameters, there are excitations A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. THE STRUCTURE OF MATTER 3 corresponding to space and time variations of the parameters. Small amplitude oscillations again may be described by a wave vector, k. If the forces are of short range, the frequency goes to zero with increasing wavelength, or as k --+ o. Spin waves in ferro magnets are an example. The quanta of such oscillations are known in high energy physics as Goldstone bosons and correspond to particles of zero mass. With long-range forces, such as Coulomb forces between electrons, the frequency may remain large as k --+ O. An example is plasma oscillations from longitudinal density tluctuations of the electron gas in metals. In this case the frequency is very large, corresponding to the order of 10 eV. The quanta, plasmons, are not normally excited. In high energy physics, the corresponding particles are massive and are known as Higgs bosons. With Higgs bosons, spontaneously broken symmetry may be introduced through an order parameter without getting unwanted mass­ less Goldstone bosons. In addition to the small-amplitude oscillations, there may be large­ amplitude nonlinear excitations that maintain their identity through collisions. They are now often called solitons, the term originating from the solitary water wave observed to flow down a canal following a sudden change in level. Examples are vacancies and interstitials as point defects in crystals, quantized vortex lines in supertluids as line defects, and Bloch walls between magnetic domains in ferro magnets as sheet defects. These play analogous roles in quite different systems. In three­ dimensional crystals, point defects may exist in thermal equilibrium. In two-dimensional systems, dislocations and vortex lines become point defects. And in quasi-one-dimensional systems, walls between domains become point defects. The Landau mean field theory does not apply to critical phenomena at temperatures very close to Te. In this region tluctuations are large and important ones have wavelengths large compared with the lattice or interparticle spacing. It turns out that critical phenomena depend mainly on the number of space dimensions, d, and on the number of parameters, n, required to specify the order parameter. For example, in the Heisenberg ferromagnet, d = 3 and n = 3. For a supertluid or the smectic-A phase, d = 3 and n = 2. Fluctuations are greater in systems of lower dimension­ ality. In strictly one-dimensional systems they are so large that they prevent a phase transition above T = O. We discuss later a number of closely related one- and two-dimensional systems. In systems with many identical particles, the statistics of the particles (Einstein-Bose or Fermi-Dirac) plays an essential role in determining both the ground state and elementary excitations. The helium liquids composed of isotopes of mass three and mass four, and mixtures of the A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. 4 BARDEEN two, provide rich systems for studying the striking differences in behavior that follow from the statistics. One might expect that helium would form the simplest of liquids, but at very low temperatures it exhibits remarkably complex behavior. The Bose liquid, 4He, becomes superfluid below the A.-transition (2.2 K), while 3He is a normal Fermi liquid down to about 10 - 3 K where it undergoes a pairing transition analogous to that of electrons in a superconductor. In the following we give some examples illustrating common features between magnetic systems, liquid helium, helium films, superconductors, and liquid crystals. We then indicate how concepts developed to under­ stand these systems have been applied to the structure of nuclei, nuclear matter, and the particles of high energy physics. In these latter applications the concept of spontaneously broken symmetry plays a key role. 2 MAGNETIC SYSTEMS The study of magnetic systems has yielded a great deal of information about phase transitions in general. The Heisenberg model has been studied for lattices in one, two, and three dimensions (d = 1,2, 3) and for spin orientations in one, two, or three dimensions (n = 1, 2,3) with all combinations possible (2, 3). In the Ising model (n = 1), only two spin orientations are possible. In analogous systems, up and down spin may be replaced by presence or absence of an atom (lattice gas model) or presence of an atom of type A or B (order-disorder systems in alloys). The case n = 2, for which the spin orientations are confined to a plane, gives systems analogous to superfluids with a complex order parameter. As mentioned in the introduction, the Heisenberg model in three dimensions (d = 3) provides a good example of the method of elementary excitations. Depending on the sign of the interaction between neighboring spins, the ground state may be ferromagnetic or antiferromagnetic. The elementary excitations are spin waves, specified by a wave vector, k, in which the spins precess about the direction in the ground state. In the ferromagnetic case the energy for small k is proportional to k2, as it would be for a free particle (momentum p = lik). The low temperature specific heat is proportional to T3/2 as in a classical monatomic gas. The interaction between spin waves is small; it gives a leading term in the expansion of specific heat in powers of the temperature that goes as T3. In the paramagnetic phase above the Curie temperature, Te, the spins become increasingly free to orient; as T ..... 00 the magnetic sus­ ceptibility approaches that of a system of free spins. For T> Te, one may start from a system of free spins and treat the spin-spin interaction A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. THE STRUCTURE OF MATTER 5 energy, J, by a perturbation expansion in powers of J jkBT. Just above Te, fluctuations are large and the expansion converges slowly, diverging as T --t Te. The low temperature expansion in terms of spin waves is in half­ integral powers of kBTjJ, or in inverse powers of the coupling constant. Two-dimensional (2D) Heisenberg systems are of particular interest. In 1944 Onsager gave an exact solution of the 2D Ising model (d = 2, n = 1) that played a very important role not only for this and analogous systems but also for testing approximate methods required for more difficult problems. An interesting system that has attracted a great deal of attention and for which the exact solution is not known is the 2DXY model, correspond­ ing to d = 2, with classical spins, in which the spin-spin interactions include only the components in the plane of the lattice (3). There is a transition temperature, Te, at which the susceptibility diverges and there is an essential singularity in the specific heat, but the anomaly in specific heat is so small that it is practically unobservable. Although there is local order, there appears to be no long-range order in the magnetization below Te. This model is analogous to several other models, including the 2D Coulomb plasma. Presumably a soliton-like excitation in which the spins tend to align along concentric circles surrounding a point defect plays an important role. A mathematically similar system with no long­ range order in magnetization is the IDXYZ model (d = 1, n = 3). 3 QUANTUM FLUIDS In this section we briefly review the differences in the ground state and elementary excitations of liquid 3He and 4He that result from the difference in statistics (4). The gross structure of the liquids, as given, for example, by the pair distribution function, is very similar but the low temperature properties are strikingly different. The Bose liquid 4He becomes superfluid in the phase He II at temperatures below the ,1- transition at T). = 2.2 K, while 3He remains a normal Fermi liquid down to temperatures of the order of lO - 3 K. The properties of He II can be accounted for in quantitative detail by a two-fluid model derived from a spectrum of elementary excitations proposed by Landau. Landau also first gave the correct description of normal Fermi liquiQs of interacting particles in terms of quasiparticle excitations and interactions between them. Liquid 3He and 3 He-4He mixtures have served as model systems for testing predictions of the Fermi liquid theory. First proposed on phenomenological grounds, both of Landau's theories have since been derived from microscopic theory. A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. 6 BARDEEN The A-transition is generally attributed to an Einstein-Bose con­ densation of helium atoms in the liquid. In the ground state of a noninteracting Bose gas, the particles are all in the ground state of zero momentum. At finite temperatures, particles are thermally excited to states of higher momenta, but up to the Einstein-Bose transition temperature (3. 13 K for a gas of the density of 4He), a finite fraction remain in the state p = O. If the ground-state wave function of the interacting system is expanded in terms of the momenta, the number, np, in states of momentum p > 0 graduaIly decreases with increasIng p from a maximum at p = 0 to near zero as p --+ CXJ. However, a finite fraction, estimated to be about 10%, remain at the state p = O. Thus in the liquid at rest, the state p = 0 is macroscopically occupied. If the liquid is flowing with a velocity vs' the state of macroscopic occupation is Ps = mvs rather than p = O. The fraction decreases from 10% at T = 0 to zero at TA, above which the liquid is normal. In contrast, in the ground state of a Fermi system of noninteracting particles, each state of momentum p is doubly occupied by particles of opposite spin up to the Fermi momentum, PF, and those above are un­ occupied. Thus np = 2 for P < PF and np = 0 for P > PF' In the interacting system, np decreases from a maximum at p = 0 to zero as p -+ 00, but a discontinuity in occupancy remains at p = PF. In a strongly interacting system such as 3He, the discontinuity is smaIl. In fact, for the same density a plot of the momentum distribution of the particles of 3He is very much like that of 4He, the differences being the discontinuity at PF in 3He and the finite fraction of particles in the state p = 0 in 4He. The spectra of elementary excitations are quite different. In 3He, there are quasiparticle excitations in one-to-one correspondence with those of the noninteracting system, specified by occupancy of a state of momentum P and spin (J above PF or a missing particle or hole below PF' Excitations in 4He may be specified by a momentum p or wave vector k = p/h. As proposed by Landau, the energy is linear in k for small k, goes over a maximum to the roton minimum at k = kr and increases again beyond. For small k, the excitations correspond to the quanta of longitudinal sound waves and are the most important ones at very low temperatures. Rotons with k-values near the minimum of the excitation spectrum at kr are the most important for temperatures above about 1 K. The existence of a state of macroscopic occupation Ps = mvs in the Bose liquid specifies a particular reference frame and breaks Galilean in variance in the same way that crystalline order does. In a normal liquid, the states available to the system are independent of the reference frame in which they are described, but in the superfluid the states depend on the state of macroscopic occupation. Of course Ps may A nn u. R ev . M at er . S ci . 1 98 0. 10 :1 -1 9. D ow nl oa de d fro m w w w .an nu al re vi ew s.o rg by N an jin g U niv ers ity on 05 /05 /11 . F or pe rso na l u se on ly. THE STRUCTURE OF MATTER 7 take on different values in different reference frames, but it must be specified in order to specify the system. The superfluid properties and the two-fluid model may be accounted for in terms of this picture. If the superfluid is flowing with velocity v s in a narrow channel the excitations come into equilibrium with the walls. These thermal excitations decrease the flow from pVs to PsV., where p is the total density and Ps is defined to be the superfluid density, equal to P at T = 0 and decreasing to zero at T = TA• If the walls are moving with velocity Vo> the total flow is PV=Pnvn+PsV., where Pn=P-Ps is the normal component. This is the basis for the two-fluid model. In quantum mechanics, Galilean in variance is related to the fact that the wave functions of a particle have an arbitrary phase factor. One may replace 1./1 by 1./1' = 1./1 exp [ix(r)] if one replaces p by p' = p- h grad x. The local gauge group corresponding to this transformation is the unitary group U(1). Macroscopic occupation of the state p = Ps gives a spon­ taneous breaking of this gauge group. One may describe a vs(r) that varies slowly in space by taking p.(r) = mvs(r) = h grad ¢(r). Then except for a factor, ¢(r) is the velocity potential for potential flow of the superfluid. This implies no vorticity in the flow, or curl v s = O. Vorticity may be introduced in the form of quantized vortex lines. On the axis of a line, the fluid is normal, Ps = O. Superfluid may circulate around the axis, but the circulation is quantized to be an integral multiple of him. This follows from the requirement that ¢(r) change by a multiple of 2n on a circuit of the axis so that the wave function is single valued. Vortex lines play an important role in the flow properties of superfluid helium. For a given total vorticity, the free energy is lowest if it -is divided into an array of vortex lines of unit circulation, him. Superfluidity is also observed in helium films only a few atoms thick. These may be regarded as essentially two dimensional. In two dimensions there is no reason to expect an Einstein-Bose condensation. Nevertheless, there must be an