Basic Many-Body Quantum Mechanics

1 Basic Many-Body Quantum Mechanics 1.1 Slater Determinants and Matrix Elements The solutions of eigenvalue equations like the time-independent one-electron Schrödinger equation hwi = ǫi wi form a a complete set of spin-orbitals {wi ≡ ϕi (x)χi } , where ϕi (x) are normalized space orbitals and χi =↑ or ↓ . The set can be taken orthogonal and ordered in ascending energy or in any other arbitrary way. Any one-electron state can be expanded as a linear combination of the wi . Moreover, we can think of a state for N electrons obtained as follows. Choose in any way N spin-orbitals out of the set {wi }, keep them in the original order but call them v1 , v2 . . . vN ; now let |v1 , v2 . . . vN | be the state with one electron in each. Imagine labeling1 the indistinguishable electrons with numbers 1, 2, · · · N. In this many-body state one has an amplitude Ψ (1, 2, . . . N ) ≡ Ψ ((x1 , χ1 ), (x2 , χ2 ), . . . (xN , χN )) of having electron i in the one-particle state (xi , χi ). How to calculate Ψ ? A product like  v1 v2 . . . vN = N k vk is in conflict with the Pauli principle because it fails to be antisymmetric in the exchange of two particles. However, the remedy is easy, because anti-symmetrized products are a basis for the antisymmetric states. To this end, let P : {1, 2, . . . N } → {P1 , P2 , . . . PN } be one of the N ! permutations of N objects. If N = 3, the set of 6 permutations comprises the rotations {(1, 2, 3), (2, 3, 1), (3, 1, 2)} and {(2, 1, 3), (3, 2, 1), (1, 3, 2)}. Anticipating some Group Theory The last three are just transpositions, that is, they are obtained from the fundamental permutation (1,2,3) by one exchange. One can multiply two permutations Q and P; the product QP is the permutation obtained by performing P and then Q and the result is: {Q1 , Q2 , . . . QN }{P1 , P2 , . . . PN } = {QP1 , QP2 , . . . QPN }. (1.1) For instance2 , (3,2,1)(2,1,3)=(2,3,1) . All permutations can be obtained from transpositions by multiplication. The inverse of a permutation is the one that 1 The electrons are identical, but this does not prevent us from labeling them; rather it imposes that the wave function changes sign for each exchange of labels. 2 in words,P sends 1 → 2 and then Q sends 2 → 2, and in the same way 2 → 1 → 3 and 3 → 3 → 1. 2 1 Basic Many-Body Quantum Mechanics upon multiplication restores the standard (ascending) order (1, 2, . . . N ). Any permutation can be shown to be a product of transpositions, usually in more than one way: for example, (2,3,1) is obtained from the standard order by exchanging 1 and 2 and then 1 and 3 but also exchanging 1 and 3 and then 2 and 3. P has a parity or signature (−)P defined such that an exchange is odd, two are even, and so on, so in the above example (1,2,3), (2,1,3)and (3,1,2) are even and the others odd. In Chapter 7 we shall see that such simple observations have far reaching consequences. Determinants The antisymmetrizer operator A= 1  (−)P P N! (1.2) P converts any product into a normalized Slater determinant, so we may write a physically acceptable solution as Ψ (1, 2, . . . N ) = A N  k 1  vk = √ (−)P vQ1 (1)vQ2 (2) . . . vQN (N ), N! Q (1.3) or, equivalently, ⎛ v1 (1) v2 (1) 1 ⎜ v1 (2) v2 (2) Ψ (1, 2, . . . N ) = √ ⎜ ⎝ ... ... N! v1 (N ) v2 (N ) ⎞ . . . vN (1) . . . vN (2) ⎟ ⎟. ... ... ⎠ . . . vN (N ) (1.4) Note that the transposed matrix is equivalent since the determinant is the same. Exchanging two rows, that is, two electrons, one gets a - sign. A permutation of the electrons is equivalent to the inverse permutation of the spinorbitals, and PΨ (1, 2, . . . N ) = Ψ (P1 , P2 , . . . PN ) = (−)P Ψ (1, 2, . . . N ). The set of all determinants is complete provided an arbitrary order is fixed for the one-electron states (otherwise the set is overcomplete). Suppose we solve hwi = ǫi wi for one electron and then consider the same N problem with N electrons and H = i h(i). This problem is no harder than for a single electron, and the N −body Schrödinger equation is solved by (1.4) with energy eigenvalue ǫ1 + ǫ2 + . . . + ǫN . 1.1.1 Many-electron Matrix Elements The matrix element Ψ |F |Φ of operators F (1, 2, . . . N ) between determinantal states, when we expand the determinants, means a sum of (N !)2 terms. This grows disastrously with N ; however there are simple rules to calculate 1.1 Slater Determinants and Matrix Elements 3 such matrix elements. Let Φ(1, . . . N ) be a determinant made by N spinorbitals u1 , u2 . . . uN ; taken out of the orthogonal set {wi } (they may be same as in Ψ , in which case we are dealing with expectation values). One can readily observe these rules by working out a 2 × 2 example, while the proof requires a trick which is explained in Sect. 1.1.2. The simplest case is f = 1, and the rule is: the overlap between determinants is the determinant of the matrix with elements the one-electron overlaps ui |vj : Φ|Ψ = Det [{ui |vj }] . (1.5) This useful result holds even if u and v spinorbitals are taken from different sets w and w′ . The overlap of a determinant with itself is indeed 1, as it should, which verifies the normalization of determinants. The one-electron matrix elements also imply a spin scalar product. N For one-body operators F (1, 2, . . . N ) = i f (i), where f acts on one electron, the rule is simple: determinants gives the same results as simple product wave functions, and antisymmetry has no consequences. The expectation values are given by Ψ |F |Ψ = N  i ui |f |ui . (1.6) For example, if we pick f (i) = δ(x − xi ), F = ρ(x) is the number density 2 and one finds ρ(x) = N i |ui (x)| . Off diagonal elements vanish if Ψ and Φ differ by more than 1 spinorbital; if they differ only by one spinorbital,vk in Ψ and uk in Φ, (1.7) vk = uk ⇒ Φ|F |Ψ = uk |f |vk . pairs N Two-body operators of the form F = i=j fij = 12 i,j,i=j fij , like the Coulomb interaction, have vanishing matrix elements when the two determinants differ by more than 2 spin-orbitals. For the rest, the best way to recall the result is by the interaction vertices in Figure 1.1.1 below (embryos of the Feynman diagrams that we introduce later). One must just note carefully which lines enter at 1 and (left and right ) and which are outgoing. The order of labels is: the one entering at 1, the one entering at 2, the one outgoing at 1. the one outgoing at 2. If vi and vj in Φ replace ui and uj in Ψ, ui = vi , uj = vj , then Φ|F |Ψ = vi (1)vj (2)|f (1, 2)|ui (1)uj (2) − vi (1)vj (2)|f (1, 2)|ui (2)uj (1) (1.8) where the second, exchange term comes from the antisymmetry. These correspond to vertices a) and b) below, respectively. If f does not depend on spin the exchange term vanishes for opposite spins. The two-body operator matrix element describes the collision of a couple of electron, while all the others are spectators. If Φ and Ψ are the same, except 4 1 Basic Many-Body Quantum Mechanics ui ui uj vj vi uj vj vi a) uk u uk i vk c) b) uj ui ui d) vk e) ui uj f) Fig. 1.1. Interaction vertices for ui = vi , we get the vertex c) and its exchange companion d), representing the two terms in the expression Φ|F |Ψ = N  [vk (1)ui (2)|f (1, 2)|uk (1)ui (2) i=k −ui (1)vk (2)|f (1, 2)|uk (1)ui (2) ]. (1.9) This is similar to (1.8), but there is a summation over all the spinorbitals present in both determinants, which act as background particles while the i-th electrons jumps from v(i) to u(i). Finally, for the expectation value we get the vertices e) and f) , that is, Ψ |F |Ψ = N  [ui (1)uj (2)|f (1, 2)|ui (1)uj (2) j =i −uj (1)ui (2)|f (1, 2)|ui (1)uj (2) ]. (1.10) 1.1.2 Derivation of the Rules A direct expansion of  Ψ |F |Φ = 1  (−)P+Q N! P,Q  uP1 (1)uP2 (2) . . . uPN (N )|f |vQ1 (1)vQ2 (2) . . . vQN (N ) (1.11) involves N !2 terms and is formidable unless N is small. However, the proof of the above rules is easily obtained by a trick, that I exemplify in the case of one-body operators. The matrix element uP1 (1)uP2 (2) . . . uPN (N )|f (1, · · · , N )|vQ1 (1)vQ2 (2) . . . vQN (N ) is a multiple integral; we permute the names of dummy variables and get uP1 (P1 )uP2 (P2 ). . . uPN (PN )|f |vQ1 (P1 )vQ2 (P2 ) . . . vQN (PN ) ; f does not change since it must depend onparticles in a symmetric way. The bra is independent of P, since i uPi | = i ui |, and we obtain 1.2 Second Quantization 5 u1 (1)u2 (2) . . . uN (N )|f |vQ1 (P1 )vQ2 (P2 ) . . . vQN (PN ) . Now the Q summation yields back the Ψ determinant with a permutation P of the electrons, that is, Ψ (−)P ; the (−)P factor cancels the one already present in (1.11). Hence, ⎛ ⎞ v1 (1) v2 (1) . . . vN (1) ⎜ v1 (2) v2 (2) . . . vN (2) ⎟ ⎟ . (1.12) Φ|F |ψ = u1 (1)u2 (2) . . . uN (N )|f | ⎜ ⎝ ... ... ... ... ⎠ v1 (N ) v2 (N ) . . . vN (N ) More explicitly,  1 (−)Q vQ1 (1)vQ2 (2) . . . vQN (N ). Φ|f |ψ = √ u1 (1)u2 (2) . . . uN (N )|f | N! Q (1.13) 1.2 Second Quantization 1.2.1 Bosons Since the time-independent Schrödinger equation for the Harmonic Oscillator, − mω 2 x2 h̄2 d2 ψ ψ = Eψ + 2m dx2 2 (1.14) h̄ mω , has a characteristic length x0 = ator one introduces the annihilation oper ix0 p x + . (1.15) x0 h̄ 1 a= √ 2 this is equivalent to x a + a† = √ , x0 2 ix0 p a − a† = √ , h̄ 2 (1.16) with the commutation relation [a, a† ]− = 1; (1.17) the Hamiltonian can be rewritten H= a† a + 1 2  h̄ω. (1.18) If ψ is a solution of (1.14) with eigenvalue E, aψ must be solution with eigenvalue E − h̄ω. The conclusions are: 1) aψ0 = 0 if ψ0 is the ground state 6 1 Basic Many-Body Quantum Mechanics and 2) a† is a creation operator that in fact creates excitations like a destroys them. One then learns that i) this represents a boson field with one degree of freedom (the x) ii) when dealing with real physical fields one never observes the oscillators but only the excitations, e.g. photons for the electromagnetic field. The noninteracting bosons in a field mode can be created in any number, and each adds the same energy to the field. The oscillator does not exist at all, but the unique property of the oscillator potential which has infinitely many states with uniform spacing h̄ω makes it a perfect representation for the field. Example: Coupled Boson Representation of Angular Momentum Schwinger [8] has shown how one can build a representation of the angular momentum operators including components Ji , shift J± and more exotic K± operators that conserve m but raise or lower j. All this was obtained using creation and annihilation operators of a couple of modes, and everything comes from a simple observation. For instance let j = 23 in units of h̄ and 2 in terms of consider the following scheme: For any j, we can write j = n1 +n 2 2 2 n2 j = n1 +n jz = n1 −n 2 2 3 3/2 −3/2 2 3/2 −1/2 1 3/2 1/2 0 3/2 3/2 n1 0 1 2 3 2 integers ≥ 0 in several ways, and each entry corresponds to a choice of jz ; n2 increases from 0 in 2j + 1 steps. So, a J+ should add 1 to n1 and remove 1 from n2 , while K± should add ±1 to both. Two harmonic oscillators can provide the occupation number operators to represent that. So, j= n̂1 + n̂2 ; 2 jz = n̂1 − n̂2 . 2 (1.19) To extend this idea, one can observe that introducing a spinor operator  a1 ψ= (1.20) a2 we may write j= This extends naturally to h̄ † ψ ψ; 2 jz = h̄ † ψ σz ψ. 2 − → h̄ † − σψ j = ψ → 2 (1.21) (1.22) 1.2 Second Quantization 7 which implies trivially jx = h̄ † (a a2 + a†2 a1 ); 2 1 and hence j+ = h̄a†1 a2 ; jy = ih̄ (−a†1 a2 + a†2 a1 ) 2 j− = h̄a†2 a1 . (1.23) (1.24) Indeed, it is a simple matter to verify that [jx , jy ] = ih̄jz , (1.25) j 2 = h̄2 j(j + 1). (1.26) One can change j by K+ = h̄a†1 a†2 ; K− = h̄a1 a2 . (1.27) Using j and m = jz in (1.19) one finds n1 and n2 , hence (a† )j+m (a†2 )j−m |0 . |jm =  1 (j + m)!(j − m)! (1.28) This is an alternative way to derive results like Clebsh-Gordan coefficients and the like. 1.2.2 Field Quantization and Casimir Effect The electromagnetic field fluctuations in vacuo have a macroscopic consequence named Casimir effect. This is of interest for fundamental physics but also for potential applications. Let two square mirrors of side L be put in front of each other at a distance s. Roughly speaking, this causes boundary conditions E = 0 of vanishing parallel electric field component on both surfaces, at frequencies below the plasma frequency ωp of the metal. The field between the mirrors is constrained and has a reduced zero point energy; thus, the radiation pressure is lower than in vacuum and a macroscopic attraction between the mirrors appears. The effect was discovered by Casimir [6] theoretically and then verified experimentally [7],[200]. It is important at mμ distances, so it is longer ranged than Van der Waals forces, which are mainly due to the fluctuating instantaneous dipoles on non-polar systems. To understand this in more detail consider a metallic pillbox,with reflecting walls, a square basis of side L and hight s. How much is the zero-point πb πc energy U (s) in the pillbox? Each wave-vector k = ( πa s , L , L ), (with integer a, b and c) contributes h̄ck and the sum diverges of course, so we impose an exponential ultraviolet cutoff α, removing the short wavelengths such that kα ≫ 1. They are not involved anyhow because at ultraviolet frequencies the mirrors are transparent. 8 1 Basic Many-Body Quantum Mechanics U (s) = h̄c  abc   aπ 2 s + bπ L 2 +  cπ 2 L  2 a b 2 c 2 e−α ( s ) +( L ) +( L ) . (1.29) We can calculate this exactly for large enough L and the result diverges as α → 0. One finds (see Appendix 1)  2 h̄cπ 2 L2 1 d 1 (1.30) U (s) = α 2 dα αes −1 Using the expansion  Bn y n y 1 1 y2 1 y4 = 1 − y + − + ... = ey − 1 2 6 2! 30 4! n! n (the Bn are called Bernoulli numbers), one obtains:  2  s d 1 α2 1 h̄cπ 2 L2 1 U (s) = + − − + · · · 2 dα α2 2α 12s 30 4!s3 (1.31) (1.32) The first two terms lead to the aforementioned divergence: should we try to remove all the radiation from the cavity, including the high frequency modes, that would cost us infinite energy. However, the divergence disappears if we ask: what changes if we shift one side of the cavity by 1 cm? To better answer this question, suppose a cavity of length R is divided in two equal halves by a mirror: evidently the energy of the vacuum is the diverging quantity 2U (R/2). If instead the mirror is at distance s from one end and R − s from the other, the vacuum energy must be U (s) + U (R − s), which also diverges. The finite difference  R } (1.33) ΔE(s) = lim {U (s) + U (R − s) − 2U R→∞ 2 has the physical meaning of an energy that must be supplied to the system in order to shift the mirror to the middle of the cavity. If the cavity is large, this can be identified with the interaction energy at distance s. Eventually one can let α → 0. The zero point energy decrease per unit surface is thus π 2 h̄c , (1.34) 720 s3 and since the radiation pressure is proportional to the energy density one observes an attractive force ΔE = π 2 h̄c . 240 s4 Measuring distances s in μm, one finds F = − (1.35) 0.013 dyne/cm2 . (1.36) s4 This force and its dependence on material and surface properties is actively investigated and could be used to operate nano-machines. F =− 1.2 Second Quantization 9 1.2.3 Fermions The second quantization formalism for Fermions was invented in order to deal with phenomena like neutron decay n → p + e + ν̄ or pair creation in particle physics, but to create an electron-positron pair one needs about a million eV. In condensed matter physics the typical energy scale is much less than that, yet many important phenomena are naturally described in terms of the creation (or annihilation) of fermion quasi-particles. Electronhole pairs can be created very much like electron-positron ones. In scattering processes, when all the particles are conserved, one can proceed with Slater determinants in first quantization; however, second quantization formalism is much easier to work with. The change from bosons to fermions replaces permanents with determinants. In place of a N-times excited oscillator representing N bosons in a given mode, we now consider N -fermion determinants |u1 u2 . . . uN |, where the spinorbitals are chosen from a complete orthonormal set {wi }. The index i can be discrete or continuous but implies a fixed ordering of the complete set. In this way, one can convene e.g. that in |u1 u2 . . . uN | the indices 1 · · · N are in increasing order thereby avoiding multiple counting of the same state. The zero-particles or vacuum state |vac replaces the oscillator ground state. For the determinants, it is generally preferable to use a compact notation like  u (1) u (2) m m which contains the |um un | rather than the explicit √12 Det un (1) um (2) 3 same information. Consider the following correspondence between determinants and states of the Hilbert space with various numbers of electrons: First Quantization No − electrons state(vacuum) 1 − body state uk 2 − body determinant |um un | 3 − body determinant |um un up | ... Second Quantization |vac c†k |vac c†m c†n |vac † † † cm cn cp |vac ... (1.37) Up to now the second-quantization side looks very similar to the compact notation for determinants: the new idea is using the operator c†m , clearly deserving the name of electron creation operator in spin-orbital m, in order to express all other operators. The left column introduces an occupation number representation of the basis of the Hilbert space; second quantization builds such a representation by creation operators c†m . Adding a particle to any state cannot lead to the vacuum state, vac|c†m = 0. (1.38) 3 mathematically, it is an isomorphism; it can be thought of as a change in notation. 10 1 Basic Many-Body Quantum Mechanics Moreover, since a determinant is odd when columns are exchanged, we want an anticommutation rule [c†m , c†n ]+ ≡ c†m c†n + c†n c†m = 0. (1.39) It follows that the square of a creation operator vanishes. By definition, c†m {c†n c†r |vac } = c†m c†n c†r |vac (1.40) The notation suggests that c†m is the Hermitean conjugate of cm ; this is called annihilation operator. Taking the conjugate of (1.40) {vac|cr cn }cm = vac|cr cn cm (1.41) and taking the scalar product with c†m c†n c†r |vac , we deduce that {vac|cr cn }cm | c†m c†n c†r |vac = 1. (1.42) If now we consider cm as acting on the right, we see that it is changing the 3-body state c†m c†n c†r |vac into the 2-body one c†n c†r |vac . Thus, annihilation operator is a well deserved name: an annihilation operator cm for a fermion in the spin-orbital state um removes the leftmost state in the determinant leaving a N − 1 state determinant: c1 |u1 u2 . . . uN | = |u2 . . . uN | (1.43) cm |vac = 0. (1.44) and It obeys the conjugate of the anticommutation rules (1.39), namely, [cm , cn ]+ ≡ cm cn + cn cm = 0, c2m = 0. (1.45) cn c†m c†n c†r |vac , n, m, r all different. (1.46) Next consider Since the creation operators anticommute, we get −cn c†n c†m c†r |vac = c†m c†r |vac since the m state is created at the leftmost place in the determinant but is annihilated at once. This shows that creation and annihilation operators also anticommute, (1.47) [cn , c†m ]+ = 0, n = m. As long as the indices are different c and c† all anticommute, so the pairs cn cm ,cn c†m ,c†n cm and c†n c†m can be carried through any product of creation or annihilation operators where the indices n,m do not occur. Next we note that c†p |vac ≡ |p is a one-body wave function; cp c†p |vac = |vac and c†p cp c†p |vac = c†p |vac . Now one can check that 1.2 Second Quantization np ≡ c†p cp 11 (1.48) is the occupation number operator, having eigenvalue 1 on any determinant where p is occupied and 0 if p is empty. On the other hand, cp c†p having eigenvalue 0 on any determinant where p is occupied and 1 if p is empty. thus in any case cp c†p + c†p cp = 1. Since this holds on all the complete set it is an operator identity and we may complete the rules with [cp , c†q ]+ = δpq . (1.49) Note that n†p = np and n2p = np . 1.2.4 Basis Change in Second Quantization and Field Operators We can readily go from basis set {an } to a new set {bn }; since  |bn >= |ak >< ak |bn > (1.50) k the rule is b†n =  k a†k < ak |bn >, bn =  k ak < bn |ak > . (1.51) It is often useful to go from any set {un } to the coordinate representation introducing the creation and annihilation field operators  † Ψ (x) = n c†n u†n (x) (1.52) Ψ (x) = n cn un (x), (here u†n denotes the conjugate spinor). Note that c†p |vac is a one-electron state and corresponds to the first-quantized spinor up (x); Ψ † (y)|vac is a one-electron state and corresponds to the first-quantized spinor with spatial † wave function n un (y)un (x) = δ(x − y); thus it is a perfectly localized electron. The rules are readily seen to be [Ψ (x), Ψ (y)]+ = 0, [Ψ † (x), Ψ † (y)]+ = 0, (1.53) and [Ψ † (y), Ψ (x)]+ =  p,q [c†p , cq ]+ up †(x)uq (y) =  p,q up †(x)up (y) = δ(x − y) where the δ also imposes the same spin for both spinors. A one-body operator V (x) in second-quantized form becomes   V̂ = dxΨ † (x)V (x)Ψ (x) = Vp,q c†p cq . p,q (1.54) (1.55) 12 1 Basic Many-Body Quantum Mechanics This gives the correct matrix elements between determinantal states, as one can verify. The above expressions imply spin sum along with the space integrals, although this was not shown explicitly; let me write the spin components, for one-body operators:  V̂ = dxΨα† Vα,β (x)Ψβ (1.56) α,β For the spin operators, setting h̄ = 1, and using the Pauli matrices, Sz =  0 1 1 + and the rule (1.55) one finds 2 σz , S = 00     1 dx Ψ↑† (x)Ψ↑ (x) − Ψ↓† (x)Ψ↓ (x) , S + = dxΨ↑† (x)Ψ↓ (x). (1.57) Sz = 2 Often we shall use a discrete basis and notation and we shall write  † S+ = ck↑ ck↓ (1.58) k which is obtained from (1.57) by taking a Fourier transform in discrete notation. A two-body operator U (x, y) becomes    Û = dx dyΨ † (x)Ψ † (y)U (x, y)Ψ (y)Ψ (x) = Uijkl c†i c†j cl ck (1.59) ijkl (please note the order of indices carefully). The Hamiltonian for N interacting electrons in an external potential ϕ(x) is the true many-body Hamiltonian in the non-relativistic limit that we shall often regard as the full many-body problem for which approximations must be sought. It may be written H (r1 , r2 , . . . , rN ) = H0 (r1 , r2 , . . . , rN ) + U (r1 , r2 , . . . , rN ) (1.60) where H0 is the free part H0 = T + Vext =    1 h0 (i) − ∇2i + V (ri ) = 2 i i (1.61) with T the kinetic energy and Vext the external potential energy while U= 1 uC (ri − ri ) 2 (1.62) i=j is the Coulomb interaction. This Hamiltonian may be written in secondquantized form 1.2 Second Quantization 13 H = H0 + U,  H0 = drΨσ† (r)h0 Ψσ (r), σ   1  U = dxdyψα† (x)ψβ† (y)uC (x − y)αγ,βδ Ψδ (y)Ψγ (x). (1.63) 2 α,β,γ,δ Often the spin indices are understood as implicit in the integrations. It should be kept in mind that relativistic corrections are needed in most problems with light elements and the relativistic formulation is needed when heavy elements are involved. Fortunately, the ideas that we shall develop lend themselves to a direct generalization to Dirac’s framework. 1.2.5 Hubbard Model for the Hydrogen Molecule The Hubbard Model is a lattice of atoms or sites that can host one electron per spin; there is a hopping term between nearest neighbors like in a tightbinding model and a repulsion U between two electrons on the same atom. The Hubbard Hamiltonian   † cjσ ciσ + U ni↑ ni↓ , (1.64) H =K +W =t i,j ,σ i where K stands for the kinetic energy while W accounts for the on-site repulsive interaction. The summation on i, j runs over sites i and j which are nearest neighbors in a cubic lattice. This is often called trivial Hubbard Model to distinguish it from its extensions, involving degenerate orbitals and off-site interactions, that have been studied for many purposes.4 To model H2 in the same spirit we represent the 1s orbitals of both atoms by two sites a and b and Ĥ = T̂ + Ŵ with   T̂ = th c†aσ cbσ + c†bσ caσ (1.65) σ the kinetic energy, with th > 0 the hopping integral; Ŵ = U (n̂a↑ n̂a↓ + n̂b↑ n̂b↓ ) . 4 (1.66) Some people blame the Hubbard Model and its extensions as too idealized to be realistic. Indeed nobody would use them to refine well-understood properties of Silicon. However, there are lots of problems involving strong correlations and e.g. transport, spectroscopies, time-dependent perturbations, which are far too hard for an ab-initio description. Hubbard-like models are primarily conceptual tools aimed at a semi-quantitative understanding. We shall see particularly in Chapters 4, 5 and 10 that often they allow to deal with highly excited states of strongly interacting system very successfully. The Bosonic Hubbard Model is also important, e.g. in the rapidly developing subject of Cold Bosonic Atoms in Optical Lattices (see Ref. [15]. 14 1 Basic Many-Body Quantum Mechanics We wish to solve with two electrons of opposite spin (the ms = 0 sector) so we take  N̂ = (n̂aσ + n̂bσ ) = 2. σ This is conserved. If U = 0, one solves the single-electron problem, and finds the orbitals |a ± |b (1.67) ϕ± = √ 2 with energy eigenvalues ε ± = ± th (1.68) and the ground state Ψ = ϕ−↑ ϕ−↓  has energy E = −2th . In the interacting case, we choose a basis |v1 >= |a ↑ a ↓>, |v2 >= |a ↑ b ↓>, |v3 >= |b ↑ a ↓>, |v4 >= |b ↑ b ↓> . ǫ U th Fig. 1.2. Singlet eigenvalues of the Hydrogen molecule model versus U . th There is a single state in the ms = 1 sector, so out of the 4 states in the ms = 0 sector we expect one triplet and 3 singlets. We form the matrices W = U Diag (1, 0, 0, 1) and ⎛ ⎞ 0110 ⎜1 0 0 1⎟ ⎟ T̂ = th ⎜ (1.69) ⎝1 0 0 1⎠. 0110 One the eigenvalues: E = 0 for the triplet, and E0 = U, E± =  finds   1 2 2 for the singlets, with E− the ground state (remark16th + U 2 U ± ably) for any U > 0. Magnetism never obtains in this model. 1.3 Schrieffer-Wolff Canonical Transformation 15 1.3 Schrieffer-Wolff Canonical Transformation One often meets problems with Hamiltonians H = H0 + λV (1.70) such that the interaction λV takes the system to an enlarged Hilbert space, involving extra degrees of freedom not in action in the simple problem described by H0 . Let A denote the restricted space and B the enlargement. Typically,  HA 0 , (1.71) H0 = 0 HB and V = 0 v† v 0  (1.72) is the mixing term. A standard way to solve such problems, that we shall meet several times in this book, is by a canonical transformation H → H̃ = U HU −1 where U is designed such that H̃ is block-diagonal:  H̃A 0 H̃ = 0 H̃B (1.73) (1.74) The transformation must be unitary in order to preserve the norm of states, to this end we want U −1 = U † ; this is granted if U = eS with S = −S † . Thus, expanding the exponentials, 1 H̃ = eS He−S = H + [S, H] + [S, [S, H]] + · · · 2 (1.75) Now we insert (1.70) with S = λS1 +λ2 S2 +· · · and separate orders. Including up to second-order, [S, H]− = λ[S1 , H0 ]− + λ2 ([S1 , V ]− + [S2 , H0 ]− ), (1.76) [S, [S, H]− ]− = λ2 [S1 , [S1 , H0 ]− . (1.77) At order λ, we want to have nothing and we require that S1 be such that V + [S1 , H0 ] = 0, (1.78) that is, 0 v† v 0  +[ 0 −s† s 0  ,  HA 0 ] = 0. 0 HB − (1.79) 16 1 Basic Many-Body Quantum Mechanics where we tried the solution 0 −s† s 0 S1 =  . (1.80) We immediately obtain two conditions, v = −sHA + HB s and v † = −HA s† + s† HB . Picking H0 eigenstates |m in the A subspace and |ν in (B) (A) the B subspace, with eigenvalues Em and Eν , we obtain sνn = vνn (B) Eν − (A) En , (s† )mν = (v † )mν (B) Eν (A) − Em . (1.81) The second-order contribution to (1.75), using (1.76),(1.77) and (1.79), is ] + [S2 , H0 ]. We may set S2 = 0 since  −(s† v + v † s) 0 [S1 , V ] = (1.82) 0 s† v + v † s λ2 2 [S1 , V already gives a Hermitean, block-diagonal result. Thus,to second order, H̃A = HA + Hint (1.83) where λ2 [S1 , V ]. (1.84) 2 The effect of V can be obtained by working within the A subspace with a renormalized Hamiltonian (see (10.48),( 1.82)) with elements   † † vνn vmν vmν vνn 1 + . (1.85) (Hint )mn = (B) (B) (A) (A) 2 En − Eν ν∈B Em − Eν Hint = If the energy separation of A and B is large, the dependence of the energy denominators on m, n is negligible, and we may write Hint = −  ν∈B v † |ν ν|v (B) Eν − E (A) ; (1.86) the denominator is a positive excitation energy. 1.4 Variational Principle The energy of a quantum system is a quadratic functional of the wave function φ. Consider a small variation φ → φ + αη where η ia an arbitrary function of the same variables on which φ depends, while α → 0 is a complex parameter, E stationary ⇐⇒ {δE = 0, η arbitrary}; (1.87) 1.4 Variational Principle 17 subject to the condition that the norm is conserved, namely, δ(E − λN ) = 0. (1.88) The Lagrange multiplier λ is fixed by the condition N =< φ(λ)|φ(λ) >= 1. Applying Lagrange’s method, one finds that the following statements are equivalent: {Hφ = Eφ, < φ|φ >= 1} ⇔ {δ(E−λN ) = 0, λ = E} ⇔ {δ(E) = 0, N = 1}. (1.89) This is an exact refurmulation of Quantum Mechanics Example Given the Hamiltonian ⎛ −3 1 ⎜ 1 0 ⎜ H =⎜ ⎜ 1 0 ⎝ 1 0 1 0 11 00 00 00 00 ⎞ 1 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0 find variationally the eigenfunctions of the form ⎛ ⎞ α ⎜β ⎟ ⎜ ⎟ 2 2 ⎟ ψ=⎜ ⎜ β ⎟ . Normalizzation requires N = ψ|ψ = α + 4β = 1 while E = ⎝β ⎠ β ψ|H|ψ = 8αβ−3α2 , thus we must look for the extrema of f (α, β) = E−λN. One finds ⎧ ∂f ⎨ ∂α = 0 =⇒ 4β = (3 + λ)α ⎩ ∂f ∂β = 0 =⇒ α = λβ The compatibility condition λ(3 + λ) ⎛ = 4⎞yields λ = −4, λ = 1. For λ = −4 −4 ⎜ 1 ⎟ ⎜ ⎟ ⎟ da α = −4βone finds ψ−4 = √120 ⎜ ⎜ 1 ⎟ which is the ground state with ⎝ 1 ⎠ 1 ⎛ ⎞ 1 ⎜1⎟ ⎜ ⎟ ⎟ eigenvalue ǫ = −4. For λ = 1 da α = β one finds ψ1 = √15 ⎜ ⎜ 1 ⎟ which is the ⎝1⎠ 1 exact excited state witheigenvalue ǫ = 1. 18 1 Basic Many-Body Quantum Mechanics 1.5 Variational Approximations One can choose a trial function φ(x, {λ1 , λ2 , · · · λn }) depending on parameters {λ1 , λ2 , · · · λn } and look for the extremum. If φ is not already normalized, the normalization condition can be enforced by Lagrange’s method. If the exact ground state φ belongs to the class of functions, it corresponds to the minimum energy, otherwise the minimum always overestimates the ground state energy. Some variational approximations, like the Hartree-Fock scheme and the Bardeen- Cooper - Schrieffer theory of superconductivity, have been highly successful. The excited states also correspond to extrema of the functional, however there are severe limitations to the method. The trouble is that the true eigenstates are orthogonal, but this cannot be granted in general in a limited class of functions. We need the orthogonality. For example we cannot give any meaning to an excited state which fails to be orthogonal to the ground state. However, the lowest state of any symmetry can always be found variationally, since it is automatically orthogonal to the ground state. A symmetry is an operator X, which is unitary (that is XX † = 1), such that [H, X]− = 0. The eigenstates of an unitary operator X belong to different eigenvalues are orthogonal. Indeed, if Xφ1 = eiα φ1 aand Xφ2 = eiβ φ2 , (φ1 , φ2 ) = (φ1 , X † Xφ2 ) = ei(β−α) (φ1 , φ2 ) and with α = β, this requires (φ1 , φ2 ) = 0. 1.6 Non-degenerate Perturbation Theory The standard perturbation series yields [25] the corrected eigenvalues (0) Em = Em + m|H ′ |m + (0)  ˜ |m|H ′ |n |2 (0) n (0) Em − En + ··· (1.90) where Em are unperturbed eigenvalues and m|H ′ |n are perturbation matrix elements; ˜m excludes the terms with zero denominators. The perturbed wave functions are:   m|H ′ |m ˜ (0) k|H ′ |m (0) (1 − ) ψk ψn = ψn + (0) (0) (0) (0) Em − Ek Em − Ek k   k|H ′ |n n|H ′ |m ˜ + ··· (1.91) + (0) (0) (0) (0) n (Em − Ek )(Em − En ) (0) where ψn are the unperturbed ones. A much more general form of perturbation theory will be developed starting from Chapter 11.