Quantum Theory of Condensed Matter, Lecture notes of Physics

Download Quantum Theory of Condensed Matter and more Lecture notes Physics in PDF only on Docsity! Quantum Theory of Condensed Matter John Chalker Physics Department, Oxford University 2013 I aim to discuss a reasonably wide range of quantum-mechanical phenomena from condensed matter physics, with an emphasis mainly on physical ideas rather than mathematical formalism. The most important prerequisite is some understanding of second quantisation for fermions and bosons. There will be two problems classes in addition to the lectures. Michaelmas Term 2013: Lectures in the Fisher Room, Dennis Wilkinson Building, Physics Department, on Wednesdays at 10:00 and Fridays at 11:00. OUTLINE • Overview • Spin waves in magnetic insulators • One-dimensional quantum magnets • Superfluidity in a weakly interacting Bose gas • Landau’s theory of Fermi liquids • BCS theory of superconductivity • The Mott transition and the Hubbard model • The Kondo effect • Disordered conductors and Anderson localisation • Anderson insulators • The integer and fractional quantum Hall effects 1 w it h U (1 ) sy m m et ry W ea k c o rr el at io n s O v er al l sc h em e S tr o n g c o rr el at io n s W ea k c o rr el at io n s B o so n s F er m io n s Id ea l B o se g as Id ea l F er m i g as B C S s u p er co n d u ct o r L an d au fe rm i li q u id H u b b ar d m o d el + M o tt i n su la to r K o n d o ef fe ct A n d er so n i n su la to r Q H E O rd er ed m ag n et S u p er fl u id S tr o n g fl u ct u at io n s in 1 D w ea k in te ra ct io n s w ea k at tr ac ti o n w ea k re p u ls io n ra n d o m i m p u ri ty p o te n ti al re p u ls io n + p er io d ic l at ti ce m ag n et ic f ie ld in 2 D R ep u ls io n a t o n e la tt ic e si te E q u iv al en ce 2 Many-Particle Quantum Systems 1 Identical particles in quantum mechanics Many-particle quantum systems are always made up of many identical particles, possibly of several different kinds. Symmetry under exchange of identical particles has very important consequences in quantum mechanics, and the formalism of many-particle quantum mechanics is designed to build these consequences properly into the theory. We start by reviewing these ideas. Consider a system ofN identical particles with coordinates r1, . . . rN described by a wavefunctionψ(r1 . . . rN ). For illustration, suppose that the Hamiltonian has the form H = − ~2 2m N∑ i=1 ∇2 i + N∑ i=1 V (ri) + ∑ i<j U(ri − rj) . Here there are three contributions to the energy: the kinetic energy of each particle (∇2 i operates on the coordinates ri); the one-body potential energy V (r); and the two-particle interaction potentialU(ri−rj). To discuss symmetry under exchange of particles, we define the exchange operator Pij via its action on wavefunctions: Pijψ(. . . ri . . . rj . . .) = ψ(. . . rj . . . ri . . .) . Since [H,Pij ] = 0, we can find states that are simultaneous eigenstates of H and Pij . Moreover, a system that is initially in an eigenstate of Pij will remain in one under time evolution with H. For these reasons we examine the eigenvalues of Pij . Since (Pij)2 = 1, these are +1 and −1. Now, it is an observational fact (explained in relativistic quantum field theory by the spin-statistics theorem) that particles come in two kinds and that particles of a given kind are always associated with the same eigenvalue of the exchange operator: +1 for bosons and −1 for fermions. 1.1 Many particle basis states In a discussion of many-particle quantum systems we should restrict ourselves to wavefunctions with the appropri- ate symmetry under particle exchange. We can do this by using a set of basis states that has the required symmetry. As a starting point, suppose that we have a complete, orthonormal set of single-particle states φ1(r), φ2(r) . . .. Next we would like to write down a wavefunction representing an N -particle system with one particle in state l1, one in state l2 and so on. The choice φl1(r)φl2(r) . . . φlN (r) is unsatisfactory because for general l1, l2 . . . it has no particular exchange symmetry. Instead we take ψ(r1 . . . rN ) = N ∑ distinct perms. (±1)Pφk1(r1) . . . φkN (rN ) . (1) Several aspects of the notation in Eq. (1) require comment. The sign inside the brackets in (±1)P is +1 for bosons and −1 for fermions. The set of labels {k1 . . . kN} is a permutation of the set {l1 . . . lN}. The permutation is called even if it can be produced by an even number of exchanges of adjacent pairs of labels, and is odd otherwise; the integer P is even or odd accordingly. The sum is over all distinct permutations of the labels. This means that if two or more of the labels ln are the same, then permutations amongst equal labels do not appear as multiple contributions to the sum. Finally, N is a normalisation, which we determine next. To normalise the wavefunction, we must evaluate∫ ddr1 . . . ∫ ddrN ψ ∗(r1 . . . rN )ψ (r1 . . . rN ) . Substituting from Eq. (1), we obtain a double sum (over permutations k1 . . . kN and h1 . . . hN ) of terms of the form ∫ ddr1 φ ∗ k1(r)φh1 (r1) . . . ∫ ddrN φ ∗ kN (r)φhN (r1) . 5 These terms are zero unless k1 = h1, k2 = h2, and . . . kN = hN , in which case they are unity. Therefore only the diagonal terms in the double sum contribute, and we have∫ . . . ∫ |ψ|2 = |N |2 ∑ dist. perms. (±1)2P = |N |2 N ! n1!n2! . . . where the n1, n2 . . . are the numbers of times that each distinct orbital appears in the set {l1 . . . lN}, and the ratio of factorials is simply the number of distinct permutations. Hence we normalise the wavefunction to unity by taking N = ( n1! n2! . . . N ! )1/2 . 1.2 Slater determinants For fermion wavefunctions we can get the correct signs by thinking of Eq. (1) as a determinant ψ(r1 . . . rN ) = 1√ N ! ∣∣∣∣∣∣ φl1(r1) . . . φl1(rN) . . . φlN (r1) . . . φlN (rN) ∣∣∣∣∣∣ . (2) Note that this determinant is zero either if two orbitals are the same (li = lj) or if two coordinates coincide (ri = rj), so the Pauli exclusion principle is correctly built in. Note also that, since the sign of the determinant is changed if we exchange two adjacent rows, it is necessary to keep in mind a definite ordering convention for the single particle orbitals φl(r) to fix the phase of the wavefunction. For bosons, we should use an object similar to a determinant, but having all terms combined with a positive sign: this is known as a permanent. 1.3 Occupation numbers We can specify the basis states we have constructed by giving the number of particles nl in each orbital l. Clearly, for fermions nl = 0 or 1, while for bosons nl = 0, 1, . . .. These occupation numbers are used within Dirac notation as labels for a state: |n1, n2, . . .〉. 1.4 Fock space Combining states |n1, n2, . . .〉 with all possible values of the occupation numbers, we have basis vectors for states with any number of particles. This vector space is known as Fock space. Using it, we can discuss processes in which particles are created or annihilated, as well as ones with fixed particle number, described by wavefunctions of the form ψ(r1 . . . rN ). 1.5 The vacuum state It is worth noting that one of the states in Fock space is the vacuum: the wavefunction for the quantum system when it contains no particles, written as |0〉. Clearly, in recognising this as a quantum state we have come some way from the notation of single-body and few-body quantum mechanics, with wavefunctions written as functions of particle coordinates. Of course, |0〉 is different from 0, and in particular 〈0|0〉 = 1. 1.6 Creation and annihilation operators Many of the calculations we will want to do are carried out most efficiently by introducing creation operators, which add particles when they act to the right on states from Fock space. Their Hermitian conjugates are annihilation operators, which remove particles. Their definition rests on the set of single particle orbitals from which we built Fock space: c†l adds particles to the orbital φl(r). More formally, we define c†l1c † l2 . . . c†lN |0〉 (3) to be the state with coordinate wavefunction ψ(r1, . . . rN ) = 1√ N ! ∑ all perms (±1)Pφk1(r1) . . . φkN (rN ) = (n1!n2! . . .)1/2|n1, n2 . . .〉 . (4) 6 A detail to note is that the sum in Eq. (4) is over all permutations, while that in Eq. (1) included only distinct permutations. The difference (which is significant only for bosons, since it is only for bosons that we can have nl > 1), is the reason for the factor (n1!n2! . . .)1/2 appearing on the right of Eq. (4). This choice anticipates what is necessary in order for boson creation and annihilation operators to have convenient commutation relations. Annihilation operators appear when we take the Hermitian conjugate of Eq. (3), obtaining 〈0| clN . . . cl2cl1 . Let’s examine the effect of creation and annihilation operators when they act on various states. Since c†l |0〉 is the state with coordinate wavefunction φl(r), we know that 〈0|cl c † l |0〉 = 1, but for any choice of the state |φ〉 other than the vacuum, c†l |φ〉 contains more than one particle and hence 〈0|cl c † l |φ〉 = 0. From this we can conclude that cl c † l |0〉 = |0〉 , demonstrating that the effect of cl is to remove a particle from the state |nl=1〉 ≡ c†l |0〉. We also have for any |φ〉 the inner products 〈0|c†l |φ〉 = 〈φ|cl |0〉 = 0, and so we can conclude that cl |0〉 = 〈0|c†l = 0 . 1.7 Commutation and anticommutation relations Recalling the factor of (±1)P in Eq. (4), we have for any |φ〉 c†l c † m|φ〉 = ±c†mc † l |φ〉 , where the upper sign is for bosons and the lower one for fermions. From this we conclude that boson creation operators commute, and fermion creation operators anticommute: that is, for bosons [c†l , c † m] = 0 and for fermions {c†l , c † m} = 0 , where we use the standard notation for an anticommutator of two operatorsA andB: {A,B} = AB+BA. Taking Hermitian conjugates of these two equations, we have for bosons [cl, cm] = 0 and for fermions {cl, cm} = 0 . Note for fermions we can conclude that (cl ) 2=(c†l ) 2=0, which illustrates again how the Pauli exclusion principle is built into our approach. Finally, one can check that to reproduce the values of inner products of states appearing in Eq. (4), we require for bosons [cl , c † m] = δlm and for fermions {cl , c † m} = δlm . To illustrate the correctness of these relations, consider for a single boson orbital the value of |[(c†)n|0〉]|2. From Eq. (4) we have |[(c†)n|0〉]|2 = n!. Let’s recover the same result by manipulating commutators: we have 〈0|(c)n(c†)n|0〉 = 〈0|(c)n−1([c, c†] + c†c)(c†)n−1|0〉 = m〈0|(c)n−1(c†)n−1|0〉+ 〈0|c†(c)n−mc†c(c)m(c†)n−1|0〉 = n〈0|(c)n−1(c†)n−1|0〉+ 〈0|c†(c)n−1(c†)n−1|0〉 = n(n− 1) . . . (n− l)〈0|(c†)n−l(c)n−l|0〉 = n! 〈0|0〉 . Of course, manipulations like these are familiar from the theory of raising and lowering operators for the harmonic oscillator. 7 Using the proposed second-quantised form for Â, we have 〈A〉 = ∑ lmpq Almpq〈0|cycxc † l c † mcpcqc † ac † b|0〉 . We can simplify the vacuum expectation value of products of creation and annihilation operators such as the one appearing here by using the appropriate commutation or anticommutation relation to move annihilation operators to the right, or creation operators to the left, whereupon acting on the vacuum they give zero. In particular cpcqc † ac † b|0〉 = (δaqδbp ± δapδbq)|0〉 and 〈0|cycxc † l c † m = 〈0|(δymδxl ± δylδxm) . Combining these, we recover Eq. (9). 2 Diagonalisation of quadratic Hamiltonians If a Hamiltonian is quadratic (or, more precisely, bilinear) in creation and annihilation operators we can diagonalise it, meaning we can reduce it to a form involving only number operators. This is an approach that applies directly to Hamiltonians for non-interacting systems, and also to Hamiltonians for interacting systems when interactions are treated within a mean field approximation. 2.1 Number-conserving quadratic Hamiltonians Such Hamiltonians have the form H = ∑ ij Hija † iaj . Note that in order for the operator H to be Hermitian, we require the matrix H to be Hermitian. Since the matrix H is Hermitian, it can be diagonalised by unitary transformation. Denote this unitary matrix by U and let the eigenvalues of H be εn. The same transformation applied to the creation and annihilation operators will diagonalise H. The details of this procedure are as follows. Let α†l = ∑ i a†iUil . Inverting this, we have ∑ α†l (U †)lj = a†j and taking a Hermitian conjugate ∑ l Ujlαl = aj . Substituting for a†’s and a’s in terms of α†’s and α’s, we find H = ∑ lm α†l (U †HU)lmαm = ∑ n εnα † nαn ≡ ∑ n εnn̂n . Thus the eigenstates of H are the occupation number eigenstates in the basis generated by the creation operators α†n. 2.2 Mixing creation and annihilation operators: Bogoliubov transformations There are a number of physically important systems which, when treated approximately, have bilinear Hamiltoni- ans that include terms with two creation operators, and others with two annihilation operators. Examples include superconductors, superfluids and antiferromagnets. These Hamiltonians can be diagonalised by what are known as Bogoliubov transformations, which mix creation and annihilation operators, but, as always, preserve commutation relations. We now illustrate these transformations, discussing fermions and bosons separately. 10 2.2.1 Fermions Consider for fermion operators the Hamiltonian H = ε(c†1c1 + c†2c2) + λ(c†1c † 2 + c2c1) , which arises in the BCS theory of superconductivity. Note that λ must be real for H to be Hermitian (more generally, with complex λ the second term of H would read λc†1c † 2 + λ∗c2c1). Note as well the opposite ordering of labels in the terms c†1c † 2 and c2c1, which is also a requirement of Hermiticity. The fermionic Bogoliubov transformation is c†1 = ud†1 + vd2 c†2 = ud†2 − vd1 , (10) where u and v are c-numbers, which we can in fact take to be real, because we have restricted ourselves to real λ. The transformation is useful only if fermionic anticommutation relations apply to both sets of operators. Let us suppose they apply to the operators d and d†, and check the properties of the operators c and c†. The coefficients of the transformation have been chosen to ensure that {c†1, c † 2} = 0, while {c†1, c1} = u2{d†1, d1}+ v2{d†2, d2} and so we must require u2 + v2 = 1, suggesting the parameterisation u = cos θ, v = sin θ. The remaining step is to substitute inH for c† and c in terms of d† and d, and pick θ so that terms in d†1d † 2+d2d1 have vanishing coefficient. The calculation is clearest when it is set out using matrix notation. First, we can write H as H = 1 2 ( c†1 c2 c†2 c1 ) ε λ 0 0 λ −ε 0 0 0 0 ε −λ 0 0 −λ −ε   c1 c†2 c2 c†1 + ε where we have used the anticommutator to make substitutions of the type c†c = 1− c c†. For conciseness, consider just the upper block ( c†1 c2 )( ε λ λ −ε )( c1 c†2 ) and write the Bogoliubov transformation also in matrix form as( c1 c†2 )( cos θ sin θ − sin θ cos θ )( d1 d†2 ) . We pick θ so that ( cos θ − sin θ sin θ cos θ )( ε λ λ −ε )( cos θ sin θ − sin θ cos θ ) = ( ε̃ 0 0 −ε̃ ) , where ε̃ = √ ε2 + λ2. Including the other 2× 2 block ofH, we conclude that H = ε̃(d†1d1 + d†2d2) + ε− ε̃ . 2.2.2 Bosons The Bogoliubov transformation for a bosonic system is similar in principle to what we have just set out, but different in detail. We are concerned with a Hamiltonian of the same form, but now written using boson creation and annihilation operators: H = ε(c†1c1 + c†2c2) + λ(c†1c † 2 + c2c1) . We use a transformation of the form c†1 = ud†1 + vd2 c†2 = ud†2 + vd1 . 11 Note that one sign has been chosen differently from its counterpart in Eq. (10) in order to ensure that bosonic commutation relations for the operators d and d† imply the result [c†1, c † 2] = 0. We also require [c1, c † 1] = u2[d1, d † 1]− v2[d2, d † 2] = 1 and hence u2 − v2 = 1. The bosonic Bogoliubov transformation may therefore be parameterised as u = cosh θ, v = sinh θ. We can introduce matrix notation much as before (but note some crucial sign differences), with H = 1 2 ( c†1 c2 c†2 c1 ) ε λ 0 0 λ ε 0 0 0 0 ε λ 0 0 λ ε   c1 c†2 c2 c†1 − ε , where for bosons we have used the commutator to write c†c = c c† − 1. Again, we focus on one 2× 2 block ( c†1 c2 )( ε λ λ ε )( c1 c†2 ) and write the Bogoliubov transformation also in matrix form as( c1 c†2 )( u v v u )( d1 d†2 ) . Substituting for c and c† in terms of d and d†, this block of the Hamiltonian becomes ( d†1 d2 )( u v v u )( ε λ λ ε )( u v v u )( d1 d†2 ) . In the fermionic case the matrix transformation was simply an orthogonal rotation. Here it is not, and so we should examine it in more detail. We have( u v v u )( ε λ λ ε )( u v v u ) = ( ε[u2 + v2] + 2λuv 2εuv + λ[u2 + v2] 2εuv + λ[u2 + v2] ε[u2 + v2] + 2λuv ) . It is useful to recall the double angle formulae u2 + v2 = cosh 2θ and 2uv = sinh 2θ. Then, setting tanh 2θ = −λ/ε we arrive at H = ε̃(d†1d1 + d†2d2)− ε+ ε̃ . with ε̃ = √ ε2 − λ2. (11) Note that in the bosonic case the transformation requires ε > λ: if this is not the case, H is not a Hamiltonian for normal mode oscillations about a stable equilibrium, but instead represents a system at an unstable equilibrium point. 2.3 Fourier transform conventions We will use Fourier transforms extensively, because much of the time we will be considering systems that are translation-invariant, and the plane waves used in these transforms are eigenfunctions of translation operators. For convenience, we collect here some definitions. Although we are generally interested in the thermodynamic limit (the limit of infinite system size), it is usually clearest and cleanest to write transforms in the first instance for a finite system. In order to preserve translation invariance, we take this finite system to have periodic boundary conditions. Since some details differ, we consider lattice and continuum problems separately. 12 at every site in the lattice. This state is in fact an exact eigenstate of the Hamiltonian, Eq. (19) and is a ground state. Other, symmetry-related ground states are obtained by acting on this one with the total spin lowering operator, or with global spin rotation operators. Next we would like to understand excitations from this ground state. Wavefunctions for the lowest branch of excitations can also be written down exactly, but to understand states with many excitations present we need to make approximations, and the Holstein-Primakoff transformation provides a convenient way to do so. Using this transformation and omitting the higher order terms, the Hamiltonian may be rewritten approximately as H = −J ∑ 〈rr′〉 S2 − JS ∑ 〈rr′〉 [ b†rbr′ + b†r′br − b † rbr − b † r′br′ ] . (22) Applying the approach of Section 2.1, we can diagonalise Eq. (22) by a unitary transformation of the creation and annihilation operators. In a translationally invariant system this is simply a Fourier transformation. Suppose the sites form a simple cubic lattice with unit spacing. Take the system to be a cube with side L and apply periodic boundary conditions. The number of lattice sites is then N = L3 and allowed wavevectors are k = 2π L (l,m, n) with l,m, n integer and 1 ≤ l,m, n ≤ L . Boson operators in real space and reciprocal space are related by br = 1√ N ∑ k e−ik·r bk and b†r = 1√ N ∑ k eik·r b†k . We use these transformations, and introduce the notation d for vectors from a site to its nearest neighbours, and z for the coordination number of the lattice (the number of neighbours to a site: six for the simple cubic lattice), to obtain H = −JS2N z 2 − JS ∑ rd ∑ kq 1 N eir·(k−q)[eid·q − 1]b†kbq = −JS2N z 2 + ∑ q ε(q)b†qbq , where ε(q) = 2JS(3− cos qx − cos qy − cos qz) . In this way we have approximated the original Heisenberg Hamiltonian, involving spin operators, by one that is quadratic in boson creation and annihilation operators. By diagonalising this we obtain an approximate description of the low-lying excitations of the system as independent bosons. The most important feature of the result is the form of the dispersion a small wavevectors. For q  1 we have ε(q) = JSq2 +O(q4), illustrating that excitations are gapless. This is expected because these excitations are Goldstone modes: they arise because the choice of ground state breaks the continuous symmetry of the Hamiltonian under spin rotations. The fact that dispersion is quadratic, and not linear as it is, for example for phonons, reflects broken time-reversal symmetry in the ground state of the ferromagnet. 4.3 Heisenberg antiferromagnet We start again from the Heisenberg Hamiltonian, but now with antiferromagnetic interactions. H = J ∑ 〈rr′〉 Sr · Sr′ = J ∑ 〈rr′〉 [ SzrS z r′ + 1 2 ( S+ r S − r′ + S−r S + r′ )] . (23) We will only consider bipartite lattices: those for which the sites can be divided into two sets, in such a way that sites in one set have as their nearest neighbours only sites from the other set. The square lattice and the simple cubic lattice are examples, and we will treat the model in d dimensions on a hypercubic lattice. Approximating the quantum spins in the first instance as classical vectors, the exchange energy of a nearest neighbour pair is minimised when the two spins are antiparallel. For the lattice as a whole, the classical ground states are ones in which all spins on one sublattice have the same orientation, which is opposite to that of spins on the other 15 sublattice. This is a classical Néel state. The corresponding quantum state, taking the axis of orientation to be the z-axis, is defined by the property Szr |0〉 = ±S|0〉 with the sign positive on one sublattice, and negative on the other. In contrast to the fully polarised ferromagnetic state, this is not an exact eigenstate of the Hamiltonian. We can see this by considering the action of the term S+ r S − r′ . If the site r is on the up sublattice and r′ on the down sublattice, the operator simply annihilates |0〉. But if the sublattice assignments for the sites are the other way around, we generate a component in the resulting wavefunction that is different from |0〉: in this component the spin at site r has Sz = S − 1 and that at r′ has Sz = −(S − 1). To find out what the quantum ground state is, and to study excitations, we will again use the Holstein Primakoff transformation. Before we can do so, however, we need to adapt our spin coordinates to suit the classical Néel state. That is, we rotate axes in spin space for sites on the down sublattice, so that local z-axis is aligned with the spin direction in the classical Néel state. The required transformation is Sz → −Sz Sx → −Sx Sy → Sy . As is necessary, this preserves the commutation relations, which inversion (S → −S) would not do. After the transformation the Hamiltonian reads H = −J ∑ 〈rr′〉 [ SzrS z r′ + 1 2 ( S+ r S + r′ + S−r S − r′ )] . (24) We use the Holstein Primakoff transformation denoting the boson annhiliation operator on sites from the up sub- lattice by ar and those from the down sublattice by br. (Note that Néel order means the magnetic unit cell has twice the volume of the chemical one.) Up to terms of quadratic order, we have H = −J ∑ 〈rr′〉 S2 + JS ∑ 〈rr′〉 [ a†rar + b†r′br′ + arbr′ + b†r′a † r ] . (25) Fourier transforming, this becomes H = −JS2N z 2 + JSd ∑ k [ a†kak + b†−kb−k + γ(k) ( akb−k + b†−ka † k )] (26) where we have introduced the quantity γ(k) = 1 d d∑ α=1 cos(kα) , which lies in the range −1 ≤ γ(k) ≤ 1, and has the small k expansion γ(k) ≈ 1 − k2/2d. To diagonalise the quadratic Hamiltonian of Eq. (26), we need to use the bosonic Bogoliubov transformation, as introduced in Section 2.2.2. We find H = −JS(S + 1)N z 2 + ∑ k ε(k) ( α†kαk + β†−kβ−k + 1 ) (27) with ak = ukαk − vkβ†−k and b−k = ukβ−k − vkα†k where uk = cosh(θk) , vk = sinh(θk) , and sinh(2θk) = γ(k)√ 1− γ(k)2 . The spinwave energy is ε(k) = JSd(1− γ2(k))1/2 . For small k the antiferromagnetic spinwave energy varies as ε(k) ∝ k: a linear dependence on wavevector, in contrast to the quadratic variation for a ferromagnet, because the Néel state does not break time-reversal symmetry in a macroscopic sense (a symmetry of the state is time reversal, implying spin inversion, combined with exchange of sublattices). 16 4.4 Fluctuations and the order parameter We can describe the fact that the ground states we have considered break spin rotation symmetry by using an order parameter. For the ferromagnet this is simply the magnetisation, and for the antiferromagnet it is the sublattice magnetisation. It is interesting to ask how the value of the order parameter is affected by fluctuations. Because the classical ferromagnetic state is also an exact quantum eigenstate of the Heisenberg Hamiltionian, there are no zero-point fluctuations in the ferromagnet, and in that case we will be interested in thermal fluctuations. On the other hand, we have seen that the classical Néel state is not an exact eigenstate, and so in this case quantum fluctuations are important as well. 4.4.1 Thermal fluctuations in a ferromagnet The magnetisation (per site) is M = 1 N ∑ r 〈Szr 〉 . Using the Holstein Primakoff transformation at leading order we have M = S − 1 N ∑ k 〈b†kbk〉 ≡ S −∆S . (28) Now, since the excitations are bosons with (like photons) no fixed number, the thermal average 〈b†kbk〉 is given by the Planck distribution. Using results from Section 2.3 to turn the sum on k into an integral, we have ∆S = 1 Ω ∫ BZ ddk 1 eβε(k) − 1 , The most interesting aspects of this result are the generic ones, which emerge at low temperature. In that regime only low energy spinwaves are excited, and for these we can take the small wavevector form for their energy, finding ∆S ∼ kBT J ∫ √kBT/J 0 kd−1dk 1 k2 . The integral is divergent for T > 0 in d = 1 and d = 2, showing that low-range order is not possible in the Heisenberg ferromagnet in low dimensions (an illustration of the Mermin-Wagner theorem, which says that a continuous symmetry cannot be broken spontaneously at finite temperature for d ≤ 2). In d = 3 we have ∆S ∝ T 3/2. The calculation of the spinwave contribution to the heat capacity is also interesting, but left as an exercise. 4.4.2 Quantum fluctuations in an antiferromagnet The sublattice magnetisation on the ‘up’ sublattice is S −∆S = S − 1 N ∑ r 〈a†kak〉 , but now we need to relate ak via the Bogoliubov transformation to the bosons that diagonalise the Hamiltonian. We find 〈a†kak〉 = u2k〈α † kαk〉+ v2k[〈β†−kβ−k〉+ 1] . At zero temperature the boson occupation numbers are zero, and ∆S = 1 N ∑ k v2k = 1 2Ω ∫ BZ ddk [(1− γ2(k))−1/2 − 1] . The most interesting question is to examine whether this integral converges. If it does, then for sufficiently large S the sublattice magnetisation is non-zero. But if it diverges, then our whole theoretical approach will collapse, because we started from the idea of a ground state with Néel order. A divergence can come only from points near the Brillouin zone center, where γ(k) approaches 1. Expanding around this point, we have ∆S ∼ ∫ kd−1dk 1 k . This integral is logarithmically divergent in one dimension, but convergent (since the Brillouin zone boundary sets an upper limit) in higher dimensions. We will see in the next section that one-dimensional antiferromagnets are particularly interesting, precisely because they have large quantum fluctuations. 17 (k) k ε (k) k ε (k) k ε Figure 1: Ground state and excited states of spin one-half XY chain, in Jordan-Wigner fermion description 5.1.4 Excitations Just as the ground state in this one-dimensional model is different from the ordered state we have discussed for higher-dimensional systems, so are excitations in one dimension are different from the spin waves we treated in higher dimensions. Note that the total number of Jordan-Wigner fermions, measured relative to a half-filled lattice, is proportional to the z-component of total spin. Excitations that do not change Sztot therefore involve a rearrangement of fermions in orbitals, without change in total fermion number. The simplest such excitation involves creating a particle-hole pair on top of the ground state. It can be characterised by its total momentum q, and low-lying states are of two types, with either |q|  1 or q ≈ π, as illustrated in Fig. 1. More generally, a given total excitation momentum q can be distributed between the particle and hole in a range of ways, so that a range of total energy is possible for a given momentum. This means that, instead of the sharp dispersion relation we found for spin waves, we have in this one-dimensional model a continuum of excitation energies – see Fig. 2. 0 excitation energy k π 2π Figure 2: Range of possible energies for particle-hole excitations, as a function of total momentum. 5.2 Integer spin chains Historically, many aspects of the behaviour of the spin-half chain were understood before higher spin versions (Bethe’s exact solution of the spin-half Heisenberg model was published in 1931, although it took over 30 years before its physical interpretation was complete). It was Haldane’s work in 1983 that showed there is qualitatively different behaviour in integer spin chains. We were able to discuss behaviour for spin one-half in a relatively simple fashion by treating the XY model, rather than the Heisenberg case. For spin one there is similarly a simplification, developed by AKLT (Affleck, Kennedy, Lieb and Tasaki). As even this simplified version is considerably more complicated than the free fermion problem arising from the spin-half XY model, we will discuss it only in a pictorial way. The essential idea is to view the spin at each site in a spin-one chain as being a composite of two spin-half objects, taken in a symmetric combination. A wavefunction for the chain can be constructed by forming singlets across each bond in such a way that at each site, one spin-half is paired with the site to the left, and the other with the site to the right, as sketched in Fig. 3. Such a state is an exact ground state for a special Hamiltonian that has both Heisenberg (Sn · Sn+1) and biquadratic ([Sn · Sn+1]2) exchange with suitably chosen relative strengths. 20 To see this, consider – for the wavefunction we have described – possible values of the total spin Stot n,n+1 of two neighbouring sites, n and n + 1. This total spin is built from four spin-half objects, two of which are in a singlet state. It may therefore take the values 0 or 1, but cannot take the value 2. Such a state is annihilated by the projection operator P2(Sn + Sn+1) onto Stot n,n+1 = 2. It is therefore a zero-energy eigenstate of the Hamiltonian H = J ∑ n P2(Sn + Sn+1) (36) and is a ground state for antiferromagnetic exchange (J > 0), sinceH in this case is a sum of non-negative terms. The projection operator can be written explicitly as P(L) = |Sn + Sn+1|2(|Sn + Sn+1|2 − 2), and expansion of this expression yields Heisenberg and biquadratic terms as discussed. Figure 3: Schematic representation of the AKLT wavefunction. Boxes represent sites of the spin chain, and small circles represent spin one-half objects that together form spin one degrees of freedom. Dashed lines indicate that spin one-half objects from adjacent sites are in singlet states. It is plausible and true (though the proof takes some work) that this wavefunction has only short range spin correlations. Note that if we wished to construct a similar state for spin one-half, we would be forced to break translation symmetry, because with just a single spin-half object at each site, we can form singlets only across alternate bonds. 6 Weakly interacting Bose gas As a final example of a system of bosons, we treat excitations in a Bose gas with repulsive interactions between particles, using an approximation that is accurate if interactions are weak. There is good reason for wanting to understand this problem in connection with the phenomenon of superfluidity: the flow of Bose liquids without viscosity below a transition temperature, as first observed below 2.1 K in liquid 4He. Indeed, an argument due to Landau connects the existence of superfluidity with the form of the excitation spectrum, and we summarise this argument next. 6.1 Critical superfluid velocity: Landau argument Consider superfluid of mass M flowing with velocity v, and examine whether friction can arise by generation of excitations, characterised by a wavevector k and an energy ε(k). Suppose production of one such excitation reduces the bulk velocity to v −∆v. From conservation of momentum Mv = Mv −M∆v + ~k and from conservation of energy 1 2 Mv2 = 1 2 M |v −∆v|2 + ε(k) . From these conditions we find at large M that k, v and ε(k) should satisfy ~k · v = ε(k). The left hand side of this equation can be made arbitrarily close to zero by choosing k to be almost perpendicular to k, but it has a maximum for a given k, obtained by taking k parallel to v. If ~kv < ε(k) for all k then the equality cannot be satisfied and frictional processes of this type are forbidden. This suggests that there should be a critical velocity vc for superfluid flow, given by vc = mink[ε(k)/k]. For vc to be non-zero, we require a real, interacting Bose liquid to behave quite differently from the non-interacting gas, since without interactions the excitation energies are just those of individual particles, giving ε(k) = ~2k2/2m for bosons of mass m, and hence vc = 0. Reassuringly, we will find from the following calculation that interactions have the required effect. For completeness, we should note also that while a critical velocity of the magnitude these arguments suggest is observed in appropriate experiments, in others there can be additional sources of friction that lead to much lower values of vc. 21 6.2 Model for weakly interacting bosons There are two contributions to the Hamiltonian of an interacting Bose gas: the single particle kinetic energy HKE and the interparticle potential energyHint. We introduce boson creation and annihilation operators for plane wave states in a box with side L, as in Section 2.3. Then HKE = ∑ k ~2k2 2m c†kck . Short range repulsive interactions of strength parameterised by u are represented in first-quantised form by Hint = u 2 ∑ i 6=j δ(ri − rj) . Using Eq. (8) this can be written as Hint = u 2L3 ∑ kpq c†kc † pcqck+p−q . With this, our model is complete, with a HamiltonianH = HKE +Hint. 6.3 Approximate diagonalisation of Hamiltonian In order to apply the techniques set out in Section 2.1 we should approximate H by a quadratic Hamiltonian. The approach to take is suggested by recalling the ground state of the non-interacting Bose gas, in which all particles occupy the k = 0 state. It is natural to suppose that the occupation of this orbital remains macroscopic for small u, so that the ground state expectation value 〈c†0c0〉 takes a value N0 which is of the same order as N , the total number of particles. In this case we can approximate the operators c†0 and c0 by the c-number √ N0 and expandH in decreasing powers of N0. We find Hint = uN2 0 2L3 + uN0 2L3 ∑ k6=0 [ 2c†kck + 2c†−kc−k + c†kc † −k + ckc−k ] +O([N0]0) . At this stage N0 is unknown, but we can write an operator expression for it, as N0 = N − ∑ k 6=0 c†kck . It is also useful to introduce notation for the average number density ρ = N/L3. Substituting for N0 we obtain Hint = uρ 2 N + uρ 2 ∑ k6=0 [ c†kck + c†−kc−k + c†kc † −k + ckc−k ] +O([N0]0) and hence H = uρ 2 N + 1 2 ∑ k 6=0 [ E(k) ( c†kck + c†−kc−k ) + uρ ( c†kc † −k + ckc−k )] + . . . (37) with E(k) = ~2k2 2m + uρ . At this order we have a quadratic Hamiltonian, which we can diagonalise using the Bogoliubov transformation for bosons set out in Section 2.2.2. From Eq. (11), we find that the dispersion relation for excitations in the Bose gas is ε(k) = [( ~2k2 2m + uρ )2 − (uρ)2 ]1/2 . At large k (~2k2/2m  uρ), this reduces to the dispersion relation for free particles, but in the opposite limit it has the form ε(k) ' ~vk with v = √ uρ m . In this way we obtain a critical velocity for superfluid flow, which is proportional to the interaction strength u, illustrating how interactions can lead to behaviour quite different from that in a non-interacting system. 22 using the first terms in a Taylor series, as εkσ − µ = ~2kF m∗ (k − kF) + ∑ qσ′ f(kσ,qσ′)δnqσ′ . (39) Here the zeroth order term in δn is assumed linear in the radial deviation k − kF from the Fermi surface, and its magnitude is characterised by an effective mass m∗, while the first order terms involve the Landau f -parameters. Note that for physically important excited states, both the relevant values of k − kF and the fraction of non-zero δnqσ′ are small, so that the two terms retained in Eq. (39) are comparable in magnitude, and parametrically larger that the neglected higher order terms. At this stage, the approach seems unpromising, because the expansion coef- ficients involve not simply a few fitting parameters but instead an unknown function f(kσ,qσ′). We make things manageable by separating δnqσ′ into spherical harmonics, and recognising that only the lowest two harmonics are generated in situations of physical interest. In turn, and assuming a spherically symmetric Fermi surface, only the zeroth and first harmonics of f(kσ,qσ′) are important, and symmetrising also in spin labels we are left with just three significant Landau parameters. Together with the effective mass they characterise interaction effects. In more detail, we expect f(kσ,qσ′) to depend (for k and q close to the Fermi surface) only on the angle θ between these wavevectors, and so (suppressing spin labels) we write f(kσ,qσ′) = ∑ l flPl(cos θ) with Pl(cos θ) the Legendre polynomials. Similarly, we write f(k ↑,q ↑) = f(k ↓,q ↓) = f skq + fakq and f(k ↑,q ↓) = f(k ↓,q ↑) = f skq − fakq , and finally we use the density of states at the Fermi surface ν(EF) to form dimensionless combinations F = ν(EF)f . The Fermi liquid is then parameterised by F s 0 F a 0 F s 1 and m∗ and of these only three are independent, because m∗ and F s 1 are related. 7.4 Measuring Landau parameters To understand the physical significance of these parameters, we should consider the situations in which each of them becomes important, by examining different ways of exciting the Fermi liquid. 7.4.1 Heat capacity Finite temperature generates a distribution of excitations in which there are equal numbers of quasiparticles and quasiholes, so that the density integrated over the radial component of wavevector vanishes:∫ dk δnkσ = 0 . For this reason interactions affect the heat capacity CV only via the value of effective mass, and CV = π2 3 k2Bν(EF)T = kFk 2 B 3~2 m∗T . 7.4.2 Compressibility An increase in density can be represented as an isotropic, spin-independent δnkσ . Let δn = ∑ kσ δnkσ . The resulting change in the total energy of the system is δE = ~2kF m∗ ∑ kσ (k − kF)δnkσ + 1 2 ∑ kσ,qσ′ f(kσ,qσ′) δnkσ δnqσ′ = Eold F δn+ 1 2 (Enew F − Eold F )δn+ (f s0 + fa0 ) ( δn 2 )2 + (f s0 − fa0 ) ( δn 2 )2 . (40) 25 Now, we also have the relation δn = (Enew F − Eold F )ν(EF) and so the change in energy as a result of a volume change is δE = Eold F δn+ 1 2ν(EF) [1 + F s 0 ](δn)2 . From the energy change we can obtain the compressibility κ, since this quantity, the pressure p, the volume V and the energy E of a system are related by p = −∂E ∂V and κ−1 = −V ∂p ∂V giving κ = ν(EF) V · 1 1 + F s 0 . In this result, the first factor is the contribution from the degeneracy pressure of free fermions (note that we have chosen to define ν(EF) for the system as a whole rather then per unit volume, and so it is proportional to V ), while the second factor represents the influence of interactions between quasiparticles, which reduce the compressibility if they are repulsive (F s 0 > 0), as one would expect. 7.4.3 Susceptibility We can probe the Landau parameter F a 0 by considering a measurement of the Pauli susceptibility, since a Zee- man field generates a spherically symmetric distribution of quasiparticles with opposite signs of δnkσ for spins orientated parallel or antiparallel to the Zeeman field of strength H . Let δn↑ ≡ ∑ k δnk↑ = −δn↓ ≡ ∑ k δnk↓ . Then the magnetisation of the system (writing g for the g-factor of the quasiparticles) is M = 1 2 gµB(δn↑ − δn↓) = gµBδn↑ and the change in total energy, consisting of Zeeman, kinetic and interaction terms, is δE = −gµ0µBHδn↑ + ~2kF m∗ ∑ kσ (k − kF)δnkσ + 1 2 ∑ kσ,qσ′ f(kσ,qσ′) δnkσδnqσ′ = −gµ0µBHδn↑ + 2 ν(EF) (δn↑) 2 + 2F a 0 ν(EF) (δn↑) 2 . Minimising with respect to δn↑ yields the equilibrium value of the magnetisation and the susceptibility χ = ∂M ∂H = µ2 Bµ0ν(EF) 1 + F a 0 . In this expression the numerator is the free fermion result modified by replacing bare mass with effective mass, while the denominator includes the influence of interactions between quasiparticles. Note that an attractive inter- action between quasiparticles with the same spin leads to a negative value for F a 0 and an enhancement of χ. In the limitF a 0 → −1 this produces an instability towards ferromagnetic order. 7.4.4 Galilean invariance The requirement of Galilean invariance leads to the relation m∗ m = 1 + F s 1 3 and the derivation of this result is set as Question 1 on Problem Sheet 2. 26 7.4.5 Fermi liquid parameters for 3He It is interesting to see the measured values of the Landau parameters for 3He, shown in the table below. Note that the effective mass is greatly enhanced compared to the bare mass, and that this enhancement increases with increasing density. Note also that the liquid is quite close to a ferromagnetic instability. m∗/m F s 1 F s 0 F a 0 Low pressure (0.3 atmospheres) 3.1 6.3 10.8 -0.67 High pressure (27 atmospheres) 5.8 14.4 75.6 -0.72 Table 1: Fermi liquid parameters for liquid 3He (from Pines and Nozieres, The Theory of Quantum Liquids). 8 BCS theory of superconductivity We have seen that the Fermi liquid is stable to weak repulsive interactions, in the sense that excitations retain their character though their energy is modified. Attractive interactions by contrast lead to a qualitative change in the ground state and low-temperature properties, no matter how weak they are. The central idea of BCS theory is that electron-phonon interactions lead to the formation of bound pairs of electrons, known as Cooper pairs, which in a sense Bose condense. However, the characteristic size of Cooper pairs – the coherence length – is much larger than their separation, so binding and condensation must be treated together in the theory. The same fact also leads to a simplification: since each Cooper pair interacts with many others, mean field theory is a good approximation. From a historical perspective, it is striking how long the interval was between the experimental discovery of superconductivity, by Onnes in 1911, and the theoretical understanding due to Bardeen, Cooper and Schrieffer in 1957: this serves to underline what a revolutionary advance their treatment of a cooperative quantum phenomenon represents. 8.1 Electron-phonon interactions Experiments on the isotope effect showed that phonons are central to superconductivity. In the ideal case, for different isotopes of the same superconductor the energy scales represented by the critical temperature Tc and the critical field Hc vary with isotope mass like phonon frequencies, as (ionic mass)−1/2. For a pair of electrons that are close in energy, phonon exchange generates an attractive interaction that beats the obvious screened Coulomb repulsion. To derive this effective interaction we start from the Hamiltonian H = H0 +H1 written in terms of electron operators c†k and ck, and phonon operators a†q and aq as H0 = ∑ k ε(k)c†kck + ~ω ∑ q a†qaq and H1 = ∑ kq ( Mc†k+qckaq + h.c. ) . Here ε(k) is the electron dispersion relation and the phonons are represented as Einstein oscillators, all with frequency ω; the electron-phonon coupling is represented by the matrix element M ; and we have omitted spin labels, though they will be crucial later. We wish to focus on the electron system. To this end we eliminate the electron-phonon coupling by means of a canonical transformation, which we determine perturbatively. We write H̃ = e−SHeS = H+ [H, S] + 1 2 [[H, S], S] + . . . and at leading order we fix S simply by setting H1 + [H0, S] = 0 , (41) yielding H̃ = H0 + 1 2 [H1, S] ≡ H0 +Hint . (42) 27 Dropping the fluctuation term and setting U ∑′ k bk ≡ ∆, this is H− µN = ∑ kσ ξk(c†k↑ck↑ + c†−k↓c−k↓)− ′∑ k ( ∆ c−k↓ck↑ + ∆∗c†k↑c † −k↓ ) + |∆|2 U . (47) This Hamiltonian is diagonalised by the fermionic Bogoliubov transformation, as described in Section 2.2.1. Set- ting c†k↑ = ukγ † k0 + v∗kγk1 and c†−k↓ = ukγ † k1 − v ∗ kγk0 we require |uk|2 + |vk|2 = 1 to preserve anticommutation relations, and 2ξkukvk +∆v2k−∆∗u2k = 0 to eliminate number-changing terms from the transformed Hamiltonian. Writing ∆ = eiφ|∆| these conditions are met by uk = eiφ/2 cos θk, vk = e−iφ/2 sin θk and cot 2θk = ξk |∆| . Note that for ξk  |∆|, θk → 0 and so c†k↑ ∼ γ † k0 and c†−k↓ ∼ γ † k1. Conversely, for ξk  −|∆|, θk → π/2 and so c†k↑ ∼ γk1 and c†−k↓ ∼ γk0. The γ-particles thus interpolate between electrons and holes, and at the Fermi energy, where ξk = 0 and θk = π/4, have equal electron and hole content. After Bogoliubov transformation, the Hamiltonian is H− µN = ∑ kσ √ ξ2k + |∆|2 ( γ†k0γk0 + γ†k1γk1 ) + ∑ k ( ξk − √ ξ2k + |∆|2 ) + |∆|2 U . (48) Crucially, the value of |∆| must be determined self-consistently. We have |∆| = U ′∑ k |〈c†k↑c † −k↓〉| = U ′∑ k ukv ∗ k ( 1− 〈γ†k0γk0〉 − 〈γ † k1γk1〉 ) . (49) At zero temperature 〈γ†k0γk0〉 = 〈γ†k1γk1〉 = 0. Then with a constant density of states ρ we have 1 = ρU 2 ∫ −~ωD −~ωD dξ√ ξ2 + |∆|2 . For ~ωD  |∆| this gives 1 ≈ ρU ln(~ωD/|∆|) and we find |∆| = 2~ωDe−1/ρU as we did for the pair binding energy in the Cooper problem. At finite temperature 〈γ†k0γk0〉 and 〈γ†k1γk1〉 are determined from the quasiparticle energies using the Fermi distribution, and the gap |∆| decreases as temperature decreases. Above a critical temperature Tc, the only solution to the self-consistency condition is ∆ = 0. Moreover, since at weak coupling ρU is the only parameter, the energy scales set by the zero-temperature gap and the thermal energy at the critical point have a universal relationship that serves as a test of the theory: 2|∆(T=0)| kBTc = 3.53 . The form of the quasiparticle density of states, probed by tunneling spectroscopy, provides another experimental test of these ideas. 9 The Mott transition and the Hubbard model We now consider the combined consequences for electrons in a solid of electron-electron interactions and the background ionic lattice. Our most important conclusion will be that a new type of insulator is possible, in addition to the band insulator familiar from single-particle theory. Specifically, while in a solid without electron-electron interactions we have a insulator (in the absence of band overlap) when the number of electrons per unit cell is even, we shall see that strong correlations in a half-filled band with an odd number of electrons per unit cell can generate a new state known as a Mott insulator. 30 9.1 The Hubbard model In the standard theoretical description of this phenomenon we use a tight-binding model H0 = −t ∑ 〈rr′〉σ c†r σcr′σ and add only intra-site Coulomb interactions, of the form HI = U ∑ r nr↑nr↓ (50) where nrσ = c†rσcrσ . The Hubbard model Hamiltonian is then H = H0 +HI . (51) We would like to understand its phase diagram as a function of U/t, band-filling and temperature. At small U and above one dimension we expect Fermi liquid behaviour, from the same arguments that led to Landau theory, but at large U with one electron per site we find an insulator, by the following argument. In this regime there is a set of low energy states in which every site is singly occupied. Charge motion requires generation of empty and doubly occupied sites, and so is energetically prohibited. 9.1.1 Relation to the Heisenberg model At large U and half filling, the set of low-lying states has a degeneracy of 2N in a system of N lattice sites, arising from the choices of spin orientation at each site, in the limit t/U → 0. Virtual hopping lifts this degeneracy and gives rise to Heisenberg exchange, as follows. Consider a pair of sites and treat the effects of H0 using perturbation theory. We can label eigenstates of the unperturbed Hamiltonian HI as | ↑, ↑〉, | ↑, ↓〉 and so on. The leading contributions to their energies are at second order, and are δE↑,↑ = 0 and δE↑,↓ = −2t2 U from |〈↑, ↓ |H0| ↑↓, 0〉|2 E↑,↓ − E↑↓,0 and similar. Compare these with the energies of eigenstates of the spin HamiltonianH = Js1 ·s2, which are Esinglet = −3J/4 and Etriplet = J/4, and hence split by J . Noting that | ↑, ↑〉 is a triplet state, while | ↑, ↓〉 = 2−1/2[|singlet〉 + |triplet〉], we recognise that the Hubbard model with a half-filled band and large U reduces at low energy to the antiferromagnetic Heisenberg model. We read off the exchange strength, as J = 4t2 U . 9.2 Mott transition 2dt















































































































































































E Density of states U µ µ E Density of states Figure 6: Spectral function expected for Hubbard model at half filling, showing upper and lower Hubbard bands: (left) in the insulating phase at large U/t; (right) in the metallic phase at small U/t. It is an important and heavily-studied problem to understand behviour at half-filling as U/t is reduced. We will restrict ourselves to some cartoons. At large U added electrons each cost energy U as they necessarily hop 31 between doubly occupied sites with an associated kinetic energy scale t. Holes are also mobile with kinetic energy t, but their creation does not involve the energy penalty U . We therefore expect a spectral density at large U as shown in Fig. 6: at half-filling a fully occupied hole band and an empty electron band, both of widthO(t), are split by U and sit symmetrically in energy either side of the chemical potential. Reducing the Hubbard repulsion, these bands overlap at a critical value Uc ∼ 2dt and one expects a phase transition to metallic behaviour. As we increase U starting from the metallic side, we expect the Migdal discontinuity Z (see Fig. 5) to decrease, reaching zero at the transition. Indeed, deep in the Mott insulator, it is easy to see that there is no sign of the Fermi surface, since the ground state occupation is simply 〈c†kck〉 = 1 N ∑ rr′ 〈c†rcr′〉e i(r′−r)·k = 1 2 for all k, independent of wavevector. 10 The Kondo effect We now consider the consequences of large Hubbard repulsion acting just at one site of a lattice, to represent an impurity atom at that point. It is a surprise to find that this situation, apparently so close to a free-particle one, should generate a subtle many-body problem, involving what is known as the Kondo effect. Experimentally, the resistance of many metals and alloys decreases with decreasing temperature as inelastic scattering processes are suppressed. Alloys containing dilute magnetic moments, such as a low concentration of Mn or Fe in Cu, are an exception: they show a resistivity minimum as a function of temperature and below it an increase in resistivity with decreasing temperature. An explanation of the minimum was given by Kondo in 1964, but it took a decade before the nature of the ground state was properly understood, and it was only in 1980 that a model for the phenomenon was solved exactly. 10.1 Model The Kondo Hamiltonian describes a conduction band of independent electrons interacting via exchange with a single local moment on the impurity site, chosen to lie at the origin. Then H = ∑ kσ εkc † kσckσ +H1 with H1 = J S · s(0) . (52) Here S represents the local moment and s(0) is the spin density of the conduction electrons at the impurity site. We can write the exchange interaction as J S · s(0) = J [ Szsz(0) + 1 2 ( S+s−(0) + S−s+(0) )] = J V ∑ kq [ Sz ( c†k↑cq↑ − c † k↓cq↓ ) + ( S+c†k↓cq↑ + S−c†k↑cq↓ )] (53) 10.2 Scattering amplitude The impurity spin mediates interactions between the conduction electrons, since the state of the impurity spin at a given time depends on previous scattering events. To see the consequences, we will calculate the scattering amplitude for an electron, taking as our example an initial state ki ↑ and a final state kf ↑, and working to second order in J , which will give the scattering rate to O(J3). At first order, the amplitude (left in the form of an operator on the impurity spin) is 〈kf ↑ |H1|ki ↑〉 = JSz V . From this we can find the scattering rate atO(J2), which is independent of temperature in the absence of a Zeeman field. At second order we require ∑ v 〈kf ↑ |H1|v〉 1 ε− εv 〈v|H1|ki ↑〉 32 Mobility edge E σ ξ ρ Figure 7: Behaviour in a disordered system of the density of states ρ, localisation length ξ and conductivty σ as a function of energy E. States are localised below a critical energy – the mobility edge – and extended above it. other characteristics. That is to say, for example, we would get the same result for the final cube by combining very small, highly disordered initial cubes or larger, weakly disordered initial cubes, provided the initial value of G were the same in both cases. We express this idea by writing ∂ ln g ∂ lnL = β(g) . The beta function β(g) encapsulates the nature of the transition. Of course, we know only a little about it. We can, however, be confident that it is a continuous, smooth function, since it represents behaviour in a finite sample. We can also pin down its asymptotics quite easily, as follows. Consider first the weak disorder limit. Here we expect metallic behaviour characterised by a fixed value for the conductivity σ of the material, so that G(L) = σLd−2. Hence β(g) = d − 2 at large g. Next consider strong disorder and assume states are localised. In this regime we anticipate G ∝ exp(−L/ξ) which leads to β(g) = ln(g) + constant. Connecting the two limits with a smooth monotonic curve yields the result shown in Fig. 8. (g) d=2 d=1 d=3 ln(g) β Figure 8: Dependence of the beta-function β(g) for Anderson localisation on dimensionless conductance g, shown as a function of ln(g) in dimensions d = 1, 2 and 3. For d = 3 the position ln(gc) of the critical point at which the beta function has a zero is marked by the vertical dashed line. We should now examine the consequences the follow from the assumptions we have made. It is useful to separate various cases. 35 d = 1 and d = 2. In this case we always have β(g) < 0. As a result, g(L) decreases with increasing L, eventually exponentially. That is to say, states are localised in d = 1 and d = 2 by arbitrarily weak disorder. d > 2 and g < gc. In this regime we again have β(g) < 0. With increasing L we find that β(g) becomes increasingly negative, and states are localised. d > 2 and g > gc. In this case g increases with L and at large L reaches values for which β(g) = d− 2, so that we have an Ohmic metal with g(L) ∝ Ld−2 d > 2 and g = gc. In the case of a system with critical disorder the conductance is independent of system size – an example of scale-independence at a critical point. 12 The integer and fractional quantum Hall effects We now switch to a discussion of the consequences of disorder and of electron-electron interactions for a two- dimensional electron gas in a strong magnetic field. This is a situation in which departures from the behaviour of an ideal, non-interacting Fermi gas without impurity scattering are essentially guaranteed, because the single particle eigenstates of the ideal problem are macroscopically degenerate. Our interest is in the ways this degeneracy may be lifted. The experimental systems are electrons (or holes) in quasi two-dimensional semiconductor structures - either metal-oxide-semiconductor field effect transistors fabricated on silicon, or heterostructures or quantum wells fab- ricated from GaAs and AlxGa1−xAs. Carrier motion in the third dimension is frozen out quantum mechanically when temperature and electron density are both small enough that only the lowest sub-band is occupied. The quantum mechanics of electron motion in a magnetic field with flux density B is characterised by two scales. One – the cyclotron frequency ωc = eB/m∗ – is classical and material-dependent through the value of the effective mass. The other – the magnetic length lB = (~/eB)1/2 – is quantum-mechanical and material- independent. The energy spectrum for the single-particle problem without disorder consists of a sequence of Landau levels, with energies E = ( n+ 1 2 ) ~ωc ± 1 2 g∗µBB . For electrons in free space the orbital splitting ~ωc is almost degenerate with the spin splitting gµBB, but for electrons in a semiconductor ~ωc is typically much larger than g∗µBB, since effective masses are typically smaller than the bare mass, and the effective g-factor g∗ is also reduced because of spin-orbit interactions. The number of states within each spin-split Landau level is given simply by the number Nφ of magnetic flux quanta passing through the area of the system. Comparing this degeneracy with the number Ne of electrons we obtain the most important parameter characterising the system: the Landau level filling factor ν = Ne/Nφ. For an electron gas with number density n we have the relation n = ν eB h . As a prelude to a discussion of experimental observations, it is useful to recall the Hall effect in an ideal system, as shown in Fig. 9. Taking the electrons to have a drift velocity vDrift, the current is I = evDriftnw = e2 h νBvDriftw while the Hall voltage is VH = BvDriftw. Combining these two expressions, the Hall conductivity is σxy ≡ I/VH = e2ν/h. Experimentally, studying the resistivity tensor ρ as a function of magnetic field strength or electron density, around certain filling factors accurately quantised plateaus are observed in ρxy accompanied by vanishing ρxx. Under these conditions the forms of the resistivity tensor ρ and the conductivity tensor σ are ρ = h νe2 ( 0 1 −1 0 ) and σ ≡ ρ−1 = νe2 h ( 0 −1 1 0 ) . It is striking at first sight that both ρxx and σxx should vanish: the important point is simply that the current density is perpendicular to the field, and so dissipation vanishes. This observed behaviour is quite different from that expected for a clean, single-particle system. Instead of a Hall conductance proportional to filling factor and vanishing dissipative conductance, plateaus are seen in Hall conductance as a function of filling factor and the dissipative conductance, though small in quantum Hall plateaus, has peaks at the transtions between plateaus. 36 H B I w V Figure 9: Schematic view of the Hall effect in a two-dimesional system: current flow I in the presence of a magnetic field B generates a Hall voltage VH in a sample of width w. 12.1 Integer quantum Hall effect The existence of plateaus around integer values of ν can be understood if we suppose that disorder affects the states in a Landau level as illustrated in Fig. 10, with Anderson localised states in both the low and high energy tails of the disorder-broadened Landau level, and a divergence in the localisation length near the Landau level centre. Then, if the Fermi energy lies between the centres of two Landau levels, changes in filling factor do not alter the number of extended, current-carrying states that are occupied, and so the Hall conductance is unchanged. In this situation the absence of dissipation can also be understood, since dissipative processes require excitation of an electron initially in a current-carrying state to an empty final state. As the occupied, current carrying states are buried a finite energy below the Fermi energy, such processes are suppressed at low temperature by an activation factor. extended



























































































































































































































































































































































































































































































































































localised E ρ,ξ ξ ρ Figure 10: Disorder-broadened Landau level, showing localisation physics necessary to explain the integer quan- tum Hall effect. The dependence of the density of states ρ and the localisation length ξ is given as a function of energy E. Almost all states in the Landau level are localised, but the localisation length diverges at a critical point near the centre of the Landau level. 12.1.1 Exactness of quantisation While we can understand the existence of Hall plateaus if we accept this picture for the influence of disorder on electron eigenstates, the precise quantisation of Hall conductance immediately becomes a surprise, since one might have expected the reduction in the number of extended states to be accompanied by a reduced value for the Hall conductance. Clearly, the remaining extended states must carry an extra current in a way that exactly compensates for their reduced number. One way to understand the exactness of quantisation is from the thought-experiment illustrated in Fig 11. We consider a quantum Hall sample in the form of an annulus. In addition to the magnetic field responsible for the quantum Hall effect, which pierces the surface of the annulus, we introduce a second magnetic flux Φ, threading 37 12.2.1 Single particle in a magnetic field As a first step, it is useful to summarise some results for the quantum mechanics of a particle in a magnetic field. We take charge −e and magnetic field (0, 0,−B), with e and B positive. The Hamiltonian is H = 1 2m∗ (π2 x + π2 y) with ~π = −i~~∇+ e ~A and [πx, πy] = −i~e(∂xAy − ∂yAx) = i~2 l2B . We define raising and lowering operators a† = lB√ 2~ (πx − iπy) and a = lB√ 2~ (πx + iπy) with [a, a†] = 1 so that H = ~ωc(a †a+ 1/2) . Now, it is convenient to combine coordinates into a complex number, by writing z = x + iy and z = x − iy. We also use the notation ∂z = 1 2 (∂x − i∂y) and ∂z = 1 2 (∂x + i∂y) , which is set up so that, for example, ∂zz = ∂zz = 1 and ∂zz = ∂zz = 0 . Choosing units in which lB = 1 and taking the gauge ~A = (B/2)(y,−x, 0) we then have a = − i√ 2 (2∂z + 1 2 z) . Ground state wavefunctions ψm satisfy aψm = 0, and a complete set is ψm ∝ zme−|z| 2/4 withm = 0, 1, 2 . . .. The proabability density of themth state is a ring with (after restoring units) radius (2m)1/2lB and width lB. 12.2.2 Two particle problem Unsymmetrised two-particle basis states from the lowest Landau level have the form ψ(z1, z2) ∝ zl1zm2 e−(|z1| 2+|z2|2)/4 with l,m non-negative integers. We will consider combinations of these that are eigenfunctions of relative and centre-of-mass angular momentum. They have the form ψ(z1, z2) ∝ (z1 − z2)l(z1 + z2)me−(|z1| 2+|z2|2)/4 . The value of l completely determines the typical particle separation in the state and fixes the energy given a form for the interaction potential. For this reason, and in contrast to a two-particle problem without restriction to a single Landau level, the pair is incompressible. By this we mean that the two-particle system cannot respond smoothly to an external potential that for example squeezes the particles together. Instead, with increasing external potential we expect a sequence of level crossings at which the ground state value of l changes: at each crossing there is a jump in the separation of the particles, but between jumps their separation is independent of the external potential. Note that l is required by the statistics of the particles to be odd for fermions, or even for bosons. 40 12.2.3 Laughlin wavefunction With this discussion of the two-particle problem as orientation, we can consider the many-particle wavefunction proposed by Laughlin ψ(z1 . . . zN ) = ∏ i<j (zi − zj)qe− 1 4 ∑ k |zk| 2 . As motivation for this form, one can begin with the intention of writing a variational state containing so-called Jastrow factors of the form f(zi − zj). Trial wavefunctions of this form have been used successfully, for example, in the theory of superfluid 4He, where an appropriate choice for the function making up the Jastrow factor enables one to build two-particle correlations into the wavefunction that minimise the energy of the system. Restriction to the lowest Landau dramatically constrains this choice of function, to the discrete set (zi − zj)q , with q a positive integer, which must be odd for fermions or even for bosons. The value q = 1 describes a full Landau level. Higher values ensure that the probability density falls faster to zero as a pair of particles approach each other (as |zi − zj |2q) than in a generic antisymmetric state, and for this reason are good variational choices when the interparticle interaction is repulsive. In fact, this wavefunction is an exact ground state for a certain interparticle potential (a generalised delta function), and is known from numerical studies on systems containing a small number of particles to have very high overlap with exact ground states for systems with Coulomb interactions. In many-particle quantum mechanics, knowledge of the wavefunction is not all, since extraction of physical information from this function of many coordinates constitutes a problem as hard as the one in classical statistical mechanics of calculating physical averages from a Boltzmann factor that is a function of the coordinates of all particles in a system. The simplest question we might ask is what filling factor is represented by the Laughlin wavefunction. Ex- panding, we see that the largest power of each coordinate is zq(N−1)i . Recalling that the single-particle wavefuction z e−|z| 2/4 has its probability density concentrated on a ring of radius ∝ m1/2, we see that in the Laughlin state N particles fill the area that would be occupied by qN + 1 − q particles if the Landau level were completely filled. Hence we see for large N that ν = 1 q . As a next and much harder step we would like to understand correlations in the Laughlin state. A good approach to this problem is to use what is called the plasma analogy: we think of the probability density arising from this wavefunction as if it were the Boltzmann weight for a problem in classical statistical mechanics. We can read off the Hamiltonian for this classical problem, which turns out to be a Coulomb gas, or plasma. If we have good physical intuition for the statistical mechanics of plasmas, we can apply this to draw conclusions about the Laughlin state. Some details are as follows. We define a fictitious inverse temperature β and classical HamiltonianHcl via |ψ(z1 . . . zn)|2 = e−βHcl . This yields Hcl = 1 2β ∑ k |zk|2 − 2q β ∑ i<j ln |zi − zj | . To interpret this form we should recall electrostatics in two dimensions: a point charge Q at the origin gives rise at radius r to an electric field E(r) = Q 2πε0r and a potential V (r) = − Q 2πε0 ln r . It is convenient to set the (arbitrary) inverse temperature to β = 4πε0/q. Then the two-particle term inHcl becomes − q2 2πε0 ∑ i<j ln |zi − zj | , which represents the electrostatic interaction of particles with charge q. The single particle term is q 8πε0 ∑ k |zk|2 . 41 It would arise for particles of charge q moving in an electrostatic potential |z|2/(8πε0). We can view this potential as arising from a background charge distribution, and find the density of this charges using Poisson’s equation. We obtain a density −ε0∇2 1 8πε0 |z|2 = − 1 2π . As a first conclusion from this picture, we can recover the value of the Landau level filling factor ν. The plasma consisting of N particles each with charge q will arrange itself so as to cancel on average the background charge, by adopting a mean number density of 1/(2πq). Since this is the number density of electrons in the Laughlin state (remember that we set lB to unity), and since its value is 1/q of the value for the filled Landau level, we obtain again ν = q−1. Going futher, we can use the plasma analogy to discuss electron-electron correlations in the Laughlin wavefunction. In fact, the plasma is known to be fluid for q . 70 and crystalline for q & 70, and so we see that the Laughlin wavefunction represents a liquid state for the important values q = 3, 5 . . .. One can also use Laughlin’s approach to write down a wavefunction including a hole, and use the plasma analogy to examine its properties. We have ψhole(z1 . . . zN ) = ∏ l (zl − ξ)× ψLaughlin(z1 . . . zN ) . The extra factor has the effect of excluding electrons from the vicinity of the point ξ where the hole is located. Within the plasma analogy we find Hcl → Hcl − q 2πε0 ∑ l ln |zl − ξ| . Thus the hole translates to a particle of unit charge in the plasma, located at ξ and interacting with other particles, of charge q at zl. Since plasmas screen, this will induce a compensatiing reduction in the plasma density around it, but since the plasma particles have charge q, a deficit of only 1/q plasma particles is sufficient for exact compensation. Back in the language of electrons, a deficit of 1/q of an electron means that the hole has charge 1/q. This is a spectacular instance of fractionalisation, and fractional charge has been observed reasonably directly in shot noise measurements on current carried by fractional quantum Hall quasiparticles. 42