Chapter 1 A Quantum Mechanic's Toolbox for Magnetism

Chapter 1A Quantum Mechanic’sToolbox for MagnetismBOULDER SUMMER SCHOOL 2003 LECTURE NOTESDANIEL AROVAS, UCSDIn these lectures I will introduce/review two powerful and commonlyinvokedtechniques in the quantum theory of magnetism: (i) the spin pathintegral, and (ii) the large-N method. Several excellent books with distinctlymodern slants on magnetism have recently appeared, and I commend the interestedstudent to the following monographs:• Eduardo Fradkin, Field Theories of Condensed Matter Systems (Addison-Wesley, 1991)• Assa Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, 1994)• Subir Sachdev, Quantum Phase Transitions (Cambridge University Press,1999)If you find errors or have questions about these lecture notes, please do nothesitate to contact me by email at darovas@ucsd.edu.1

Chapter 2Path Integral for QuantumSpin2.1 Feynman Path IntegralThe path integral formulation of quantum mechanics is both beautiful andpowerful. It is useful in elucidating the quantum-classical correspondenceand the semiclassical approximation, in accounting for interference effects,in treatments of tunneling problems via the method of instantons, etc. Ourgoal is to derive and to apply a path integral method for quantum spin. Webegin by briefly reviewing the derivation of the usual Feynman path integral.Consider the propagator K(x i , x f, T ), which is the probability amplitudethat a particle located at x = x i at time t = 0 will be located at x = x fattime t = T . We may writeK(x i , x f, T ) = 〈 ∣x ∣ 〉∣e −iHT/¯h f xi (2.1)= 〈 ∣x ∣e −iɛH/¯h N1 e −iɛH/¯h 1 · · · 1 e −iɛH/¯h∣ 〉∣ x0(2.2)where ɛ = T/N, and where we have defined x 0 ≡ x i and x N≡ x f. We are2

interested in the limit N → ∞. Inserting (N − 1) resolutions of the identityof the form∫ ∞∣ 〉〈 ∣1 = dx ∣xj j xj , (2.3)−∞we find that we must evaluate matrix elements of the form〈 ∣ ∣ ∫∞〉 〈 ∣ 〉 〈 ∣xj+1∣e −iHɛ/¯h xj ≈ dp j xj+1∣pj pj∣e −iT ɛ/¯h e −iV ɛ/¯h∣ 〉∣ xj=−∞∫ ∞dp j e ip j(x j+1 −x j ) e −iɛp2 j /2m¯h e −iɛV (x j)/¯h . (2.4)−∞The propagator may now be written as〈 ∣ ∣ ∫∞〉xN∣e −iHT/¯h x0 ≈=≡N−1∏dx j−∞j=1( 2π¯hm∫iɛ∫∞−∞k=0) ∞ N ∫ N−1Dx(t) expx(0)=x ix(T )=x f{N−1∏dp k exp i∏dx j exp−∞j=1{ ∫ Ti¯h0dtN−1∑k=0{iɛ¯h[p k (x k+1 − x k ) −N−1∑k=1ɛ2m¯h p2 k − ɛ¯h ] }V (x k)[ (1m xj+1 − x) ] }2 j2 − V (xj )ɛ[12 mẋ2 − V (x)] } , (2.5)where we absorb the prefactor into the measure Dx(t). Note the boundaryconditions on the path integral at t = 0 and t = T . In the semiclassicalapproximation, we assume that the path integral is dominated by trajectoriesx(t) which extremize the argument of the exponential in the last term above.This quantity is (somewhat incorrectly) identified as the classical action,S, and the action-extremizing equations are of course the Euler-Lagrangeequations. Setting δS = 0 yields Newton’s second law, mẍ = −∂V/∂x,which is to be solved subject to the two boundary conditions.The ‘imaginary time’ version, which yields the ‘thermal propagator’, is3

obtained by writing T = −i¯hβ and t = −iτ, in which case〈xf∣ ∣e −βH ∣ ∣ xi〉=∫Dx(τ) expx(0)=x ix(¯hβ)=x f{− 1¯h‘Euclidean action S{ }}E{∫¯hβ [1dτ2 mẋ2 + V (x)] } . (2.6)0The partition function is the trace of the thermal propagator, viz.Z = Tr e −βH =∫ ∞dx 〈 x ∣ ∣ e−βH ∣ ∣ x〉=∫Dx(τ) exp ( − S E[x(τ)]/¯h ) (2.7)−∞x(0)=x(¯hβ)The equations of motion derived from S Eare mẍ = +∂V/∂x, correspondingto motion in the ‘inverted potential’.2.2 Coherent State Path Integral for the ‘Heisenberg-Weyl’ GroupWe now turn to the method of coherent state path integration. In orderto discuss this, we must first introduce the notion of coherent states. Thisis most simply done by appealing to the one-dimensional simple harmonicoscillator,where a and a † are ladder operators,H = p22m + 1 2 mω2 0x 2= ¯hω 0 (a † a + 1 2 ) (2.8)a = l ∂ x + x 2l, a † = −l ∂ x + x 2l(2.9)with l ≡ √¯h/2mω 0 . Exercise: Check that [a, a † ] = 1.4

inserting resolutions of the identity at N − 1 points, as before. We nextevaluate the matrix element〈 ∣ ∣ {〉 〈 ∣ 〉zj∣e −ɛH/¯h zj−1 = zj∣zj−1 · 1 − ɛ¯h〈 ∣ ∣ 〉 }zj∣H∣zj−1 〈 ∣ 〉zj + . . .z j−1≃ 〈 {∣ 〉z ∣zj−1 j exp − ɛ¯h }H(¯z j|z j−1 ) (2.17)whereH(¯z|w) ≡〈z∣ ∣H∣ ∣w〉〈z∣ ∣ w 〉 = e−¯zw 〈 0 ∣ ∣e¯za H(a, a † ) e wa† ∣ ∣ 0 〉 . (2.18)This last equation is extremely handy. It says that if H(a, a † ) is normal orderedsuch that all creation operators a † appear to the right of all destructionoperators a, then H(¯z|w) is obtained from H(a, a † ) simply by sending a † → ¯zand a → w. Note that the function H(¯z|w) is holomorphic in w and in ¯z,but is completely independent of their complex conjugates ¯w and z.The overlap between coherent states at consecutive time slices may bewritten{〈 ∣ 〉 [zj∣zj−1 = exp − 1 ¯z2 j (z j − z j−1 ) − z j−1 (¯z j − ¯z j−1 )] } , (2.19)hence〈zN∣∣zN−1〉· · ·〈z1∣ ∣z0〉= exp{× exp12N−1∑j=1[] }z j (¯z j+1 − ¯z j ) − ¯z j (z j − z j−1 ){12 z 0(¯z 1 − ¯z 0 ) − 1 2 ¯z N (z N − z N−1 ) }(2.20)which allows us to write down the path integral expression for the propagator,〈 ∣ ∣ ∫〉 N−1 ∏zf∣e −βH zi =j=1S E[{z j , ¯z j }]/¯h =N−1∑j=1d 2 z()j2πi exp − S E[{z j , ¯z j }]/¯h(2.21)[1 ¯z 2 j(z j − z j−1 ) − 1 z 2 j(¯z j+1 − ¯z j ) + ɛ¯h ]H(¯z j|z j−1 )1 ¯z ( )2 f zf − z N−1 −1z )2 i(¯z1 − ¯z i6(2.22)

In the limit N → ∞, we identify the continuum Euclidean actionS E[{z(τ), ¯z(τ)}]/¯h =∫¯hβ0{ (1dτ ¯z ∂z2 ∂τ − z ∂¯z ) }+ 1¯h ∂τH(¯z|z)+ 1 ¯z 2 f[zf − z(¯hβ) ] − 1 z ]2 i[¯z(0) − ¯zi(2.23)and write the continuum expression for the path integral,〈zf∣ ∣e −βH ∣ ∣ zi〉=∫D[z(τ), ¯z(τ)] e −S E [{z(τ),¯z(τ)}]/¯h (2.24)The continuum limit is in a sense justified by examining the discrete equationsof motion,1 ∂S E= ¯z k − ¯z k+1 + ɛ¯h∂H(¯z k+1 |z k )(2.25)¯h ∂z k ∂z k1 ∂S E= z k − z k−1 + ɛ¯h∂H(¯z k |z k−1 ), (2.26)¯h ∂¯z k ∂¯z kwhich have the sensible continuum limit¯h ∂¯z∂τ = ∂H(¯z|z)∂z, ¯h ∂z∂τ = −∂H(¯z|z)∂¯z(2.27)with boundary conditions ¯z(¯hβ) = ¯z fand z(0) = z i . Note that there areonly two boundary conditions – one on z(0) and the other on ¯z(¯hβ). Thefunction z(τ) (or its discrete version z j ) is evolved forward from initial dataz i , while ¯z(τ) (or ¯z j ) is evolved backward from final data ¯z f. This is theproper number of boundary conditions to place on two first order differential(or finite difference) equations. It is noteworthy that the action of (2.22) or(2.23) imposes only a finite penalty on discontinuous paths. 1 Nevertheless,the paths which extremize the action are continuous throughout the intervalτ ∈ (0, ¯hβ). As z(τ) is integrated forward from z i , its final value z(¯hβ) will ingeneral be different from z f. Similarly, ¯z(τ) integrated backward from ¯z fwill1 In the Feynman path integral, discontinuous paths contribute an infinite amount tothe action, and are therefore suppressed.7

in general yield an endpoint value ¯z(0) which differs from ¯z i . The differencesz(¯hβ) − z fand ¯z(0) − ¯z i are often identified as path discontinuities, butthe fact is that the equations of motion know nothing about either z for ¯z i .These difference terms do enter in a careful accounting of the action formulae(2.22,2.23), however.The importance of the boundary terms is nicely illustrated in a computationof the semiclassical imaginary time propagator for the harmonic oscillator.With H = ¯hω 0 a † a (dropping the constant term for convenience), wehave〈zf∣ ∣ exp(−β¯hω0 a † a) ∣ ∣ zi〉= e− 1 2 |z f |2 − 1 2 |z i |2 ∞∑m,n=0¯z fm zin 〈 ∣√ m exp(−β¯hω0 a † a) ∣ 〉nm!n!{}= exp − 1|z 2 f| 2 − 1|z 2 i| 2 + ¯z f z i e −β¯hω 0(2.28)The Euclidean action is L E = 1 2¯h(¯zż − z ˙¯z) + ¯hω 0 ¯zz, so the equations ofmotion are¯h ˙¯z = ∂H∂z = ¯hω 0 ¯z , ¯hż = − ∂H∂¯z = −¯hω 0 z (2.29)subject to boundary conditions z(0) = z i , ¯z(¯hβ) = ¯z f. The solution isz(τ) = z i e −ω 0τ, ¯z(τ) = ¯z fe ω 0(τ−¯hβ) . (2.30)Along the ‘classical path’ the Euclidean Lagrangian vanishes: L E = 0. Theentire contribution to the action therefore comes from the boundary terms:S clE /¯h = 0 + 1 2 ¯z f (z f − z i e−β¯hω 0) − 1 2 z i(¯z f e −β¯hω 0− ¯z i )= 1 2 |z f |2 + 1 2 |z i |2 − ¯z fz i e −β¯hω 0, (2.31)What remains is to compute the fluctuation determinant. We writez j = z clj+ η j¯z j = ¯z clj + ¯η j (2.32)8

and expand the action asS E[{z j , ¯z j }] = S E[{z clj , ¯z clj }] + ∂2 S E≡S clE + ¯h 2For general H, we obtain¯η i η j + 1 ∂ 2 S Eη i η j + 1 ∂ 2 S E¯η i¯η j (2.33) + . . .∂¯z i ∂z j 2 ∂z i ∂z j 2 ∂¯z i ∂¯z j) ( )zj+ . . . (2.34)¯z j(¯zi z i) ( A ij B ijC ij A t ijA ij = δ ij − δ i,j+1 + ɛ¯h∂ 2 H(¯z i |z j )δ i,j+1∂¯z i ∂z j(2.35)B ij = ɛ¯h∂ 2 H(¯z i |z i−1 )δ∂¯z i2 i,j (2.36)C ij = ɛ¯h∂ 2 H(¯z i+1 |z i )δ∂zi2 i,j (2.37)(2.38)with i and j running from 1 to N − 1. The contribution of the fluctuationdeterminant to the matrix element is then∫ N−1 ∏j=1d 2 η i2πi exp {− 1 2( ) ( ) ( ) ( ) ( ) }1 1 Akl B Re ηk Im η kl 1 i Re ηlk−i i C kl A lk 1 −i Im η l( ) A B= det −1/2 C A t(2.39)In the case of the harmonic oscillator discussed above, we have B ij = C ij = 0,and since A ij has no elements above its diagonal and A ii = 1 for all i, wesimply have that the determinant contribution is unity.2.3 Coherent States for SpinFor the pros: A. Perelomov, Generalized Coherent States and their Applications(Springer-Verlag, NY, 1986).9

A spin-coherent state ∣ 〉ˆΩ〉 is simply a rotation of the ‘highest weight’ state∣ m = +S , such that the spin is maximally polarized along ˆΩ, i.e.∣ˆΩ · S∣ ˆΩ〉 = +S ˆΩ〉 . (2.40)Note that ∣ ∣ m = +S〉is itself a coherent state with ˆΩ = ẑ. We can effect thisrotation by means of an element R of the group SU(2):R ≡ exp(iχS z ) exp(iθS y ) exp(iφS z ) (2.41)∣ ˆΩ〉 ∣ = R† 〉 ẑ . (2.42)To define and manipulate the spin coherent states, it is useful to introducethe Schwinger representation of quantum spin. You are probably alreadyfamiliar with the Holstein-Primakoff transformation,S + = h † (2S − h † h) 1/2 S z = h † h − S (2.43)S − = (2S − h † h) 1/2 h 0 ≤ h † h ≤ 2S , (2.44)by which a quantum spin can be represented by a single boson. Note thatthe eigenvalues of the boson number operator n h= h † h range over the nonnegativeintegers. There are thus an infinite number of allowed states, butonly a finite number (2S + 1) of states in the Hilbert space for spin. Butthe factor √ 〉 2S − h † h in S + annihilates the state of maximal polarization,∣ m = +S , and thus for any Hamiltonian which can be written in terms ofthe spin algebra operators, the infinite-dimensional boson Hilbert space is effectivelydivided into two parts. The ‘physical’ states all have 0 ≤ n h≤ 2S,and there are no matrix elements connecting this subspace to the ‘unphysical’one where n h> 2S.The square roots are unwieldy, however, and in practice one expands themin powers of (n h/2S), viz.(2S − h † h) 1/2 = √ 2S ·{1 − 1 2( h † h)+ 1 2S 8( h † h) }2+ . . . . (2.45)2SThis expansion forms the basis of spin wave theory. Hence, within spin wavetheory, unphysical states are allowed. For example, an interaction like S + i S − j10

etween spins on sites i and j takes the form 2S h † i h j within the spin waveexpansion. But such a term knows nothing of the border lying at n h= 2Sseparating physical from unphysical states.In the Schwinger representation, two bosons are used, and the constraintis a holonomic one (i.e. one which can be written as an equality):S + = a † b S z = 1 2 (a† a − b † b) (2.46)S − = a b † 2S = a † a + b † b . (2.47)Exercise: Verify that [S + , S − ] = 2S z and [S z , S ± ] = ±S ± for both theHolstein-Primakoff and Schwinger representations. The constraint simplysays n a + n b= 2S, i.e. there are a total of 2S bosons present. Note that theoperators S ± change the number of a and b bosons, but preserve the totaln a + n b, hence they commute with the constraint.A shorthand way of rendering the spin operators in the Schwinger representationis((S = 1 2 a†b †) aσ . (2.48)b)We now investigate the action of the SU(2) rotation R on the Schwingerbosons. We wish to evaluate the expression( )( )a aR † R = e −iφSz e −iθSy e −iχSz e iχSz e iθSy e iφSz . (2.49)bbLet’s work this out:• Rotation about the ẑ-axis:e −iχSz ( ab)e iχSz = e −i χ 2 a†a e i χ 2 b† b/=( ab)e −i χ 2 b†b e i χ 2 a† a( ) e+i χ 2 aa −i χ . (2.50)2 b11

• Rotation about the ŷ axis:.e −iθSy ( ab)e iθSy = e θ 2 (ab† −a † b)=( ab)e θ 2 (ab† −a † b)( )cos(θ/2) a + sin(θ/2) b− sin(θ/2) a + cos(θ/2) b(2.51)The final result isR † ( abwhere u and v are spinor coordinates,) ( ( )ū ¯v aR =−v u)b(2.52)u = exp ( − i 2 χ) exp ( − i 2 φ) cos ( 12 θ) (2.53)v = exp ( − i 2 χ) exp ( + i 2 φ) sin ( 12 θ) . (2.54)Equivalently,u = e iα cos( 1 2 θ) , v = eiα sin( 1 2 θ) eiφ , (2.55)with α = − 1 2 ( χ + φ). We can, without loss of generality, set α = 0, whichremoves a global U(1) phase which has no physical content.Now that we know how the Schwinger bosons themselves transform underSU(2), we investigate the transformation properties of the spin operators S α ,which are bilinear in the Schwinger bosons. We findR † S z R = ( ( ) ( ) ( ( )a † b †) u −¯v 1 0 ū ¯v av ū 0 −1 −v u)b( ( ) (a†b †) cos θ sin θ e −iφ ab)= 1 2hence R † S z R = ˆΩ · S, andsin θ e iφ− cos θ= sin θ cos φ S x + sin θ sin φ S z + cos θ S z , (2.56)S ∣ ∣ ˆΩ〉 = R † S z ∣ ∣ ẑ 〉 = (R † S z R) R †∣ ∣ ẑ 〉 = ˆΩ · S∣ ∣ ˆΩ〉 . (2.57)12

Explicitly, then,∣ ˆΩ〉 = (2S !) −1/2 (ua † + vb † ) 2S∣ ∣ 0〉=2S∑k=0( 2Sk) 1/2u k v 2S−k ∣ ∣ k − S〉(2.58)(2.59)Example: S = 1 2 , θ = 1 2 π, φ = 1 2 π gives ∣ ∣ ˆΩ〉 =1 √2∣ ∣ ↑〉+i √2∣ ∣ ↓〉=∣ ∣ŷ 〉 .A useful property of the coherent states: if〉∣ ψ = f(a † , b † ) ∣ 〉2S∑0 ≡ f k (a † ) k (b † ) ∣ 〉 2S−k 0 , (2.60)k=0then 〈 ˆΩ∣∣ ψ〉= (2S !) −1/2 f(ū, ¯v) , (2.61)i.e. replace a † → ū and b † → ¯v as arguments of f. The overlap of the coherentstates is〈 ˆΩ∣∣ 〉 ˆΩ′ = (ūu ′ + ¯vv ′ ) 2S (2.62)[] S1= (1 + ˆΩ2 · ˆΩ′ ) eiSγ( ˆΩ, ˆΩ ′ )(2.63)whereγ( ˆΩ, ˆΩ′ ) ≡ 2 arg (ūu ′ + ¯vv ′ ) (2.64)[= 2 tan −1 sin 1θ sin 1 2 2 θ′ sin(φ ′ ]− φ)cos 1θ cos 1 2 2 θ′ + sin 1θ sin 1 2 2 θ′ cos(φ ′ − φ)Perhaps the most important result, for our purposes, is the resolution of theidentity:1 = 2S + 1 ∫dΩ ∣ 4πˆΩ〉〈 ˆΩ ∣ . (2.65)As with the case of coherent states for the harmonic oscillator, the spin coherentstates permit a simple resolution of the identity despite their nonorthogonality.The last step, before we tackle the derivation of the spin path integral, isto compute matrix elements in the coherent state basis. We assume that the13

Hamiltonian commutes with the constraint, i.e. it preserves total spin. Themost general such Hamiltonian may be writtenH = ∑ m,n,jC mnj (a † ) m (b † ) n (a) m+j (b) n−j , (2.66)and its matrix elements may be evaluated usingpreserves total S〈 ∣ { }} {ˆΩ1 (a † ) m (b † ) n (a) m+j (b) ∣ 〉n−j ˆΩ2(2.67)=(2S !)(2S − m − n)! (ū 1u 2 + ¯v 1 v 2 ) 2S−m−n ū m 1 ¯v n 1 u m+j2 v n−j2 .Note that the above operator product must be normal-ordered, with annihilationoperators a, b appearing to the right of creation operators a † , b † .)2SExercise: Verify (2.67) by finding the O(¯z 1 z22S term of the matrix elementin the (unnormalized) generalized coherent state∣ z, ˆΩ〉 ≡ ezua † e zvb† ∣ ∣0〉, (2.68)where z is a complex number. Show that a ∣ ∣ z, ˆΩ〉 = zu∣ ∣z, ˆΩ〉 , b∣ ∣z, ˆΩ〉 =zv ∣ ∣ z, ˆΩ〉 , and 〈z, ˆΩ∣∣ z ′ , ˆΩ′ 〉 = exp [¯zz ′ (ūu ′ + ¯vv ′ ) ] . (2.69)Use these results to verify (2.67).As with the case of the coherent state path integral for the Heisenberg-Weylgroup, only diagonal matrix elements are needed. In this case the expression(2.67) simplifies to〈 ˆΩ∣∣ (a † ) m (b † ) n (a) m+j (b) n−j∣ ∣ ˆΩ2〉=(2S !)(2S − m − n)! ūm ¯v n u m+j v n−j . (2.70)We conclude this section with two examples of matrix element computation.14

• O = S + = a † b. Here, (m, n, j) = (1, 0, −1), so• O = (S x ) 2 . First we normal order:S 2 x == 1 4= 1 4( a † b + ab †〈 ˆΩ∣∣ a † b ∣ ∣ ˆΩ〉 = 2S ūv = S sin θ e iφ . (2.71)) 2(2)a † b a † b + a b † a b † + a † b b † a + a b † a † b( { (2,0,−2)}} {a † a † b b +(0,2,2){ }} {b † b † a a +(1,1,0){ }} {(1,0,0){}}{2 a † b † a b + a † a +(0,1,0){}}{b † b−→1(2S)(2S − 4 1)(ū2 v 2 + ¯v 2 u 2 + 2ū¯vuv) + 1 (2S)(ūu + ¯vv)4= S(S − 1)(sin θ cos 2 φ)2 + 1S . (2.72)2)2.4 Valence Bond StatesThe operator A † ij ≡ a † i b† j − b† i a† j creates a singlet ‘valence bond’ betweensites i and j. Exercise: Show that A † ij transforms as an SU(2) singlet, i.e.R † A † ij R = A† ij .Now consider the valence bond solid (VBS) state〉 ∏∣ Ψ(L, m) ≡ (a † i b† j − ∣ 〉 b† i a† j )m 0 , (2.73)〈ij〉∈Lwhere ∣ ∣ 0〉is the Schwinger boson vacuum. Here, the product is over all links〈ij〉 of some regular lattice L. The state ∣ ∣ Ψ(L, m)〉possesses the followingproperties:• ∣ ∣ Ψ(L, m)〉is a singlet, i.e. it has total spin zero.15

where each symbol 1 stands for an insertion of the resolution of the identity,(2.65). We next compute〈 ∣ ∣ {〉 〈 ∣ 〉ˆΩj ∣e −iɛH/¯h ˆΩj−1 = ˆΩj ˆΩj−1 · 1 − iɛ〈 ∣ ∣ 〉 }ˆΩj ∣H ˆΩj−1〈 ∣ 〉¯h ˆΩj + O(ɛ 2 )ˆΩj−1≃ 〈 ∣ 〉 ( )ˆΩj ˆΩj−1 exp − iɛH( ˆΩj | ˆΩj−1 )/¯h , (2.82)where the Hamiltonian is replaced by its coherent state matrix element,〈 ∣ˆΩj H ∣ 〉ˆΩj−1H( ˆΩj | ˆΩj−1 ) =〈ˆΩj∣ ∣ ˆΩj−1〉 . (2.83)Exercise: Show that H( ˆΩj | ˆΩj−1 ) = H(ū j , ¯v j | u j−1 , v j−1 ) is a holomorphicfunction of its arguments.We therefore have〈ˆΩN∣ ∣e −iHT/¯h ∣ ∣ ˆΩ0〉=( 2S + 14π) N−1∫∫dΩ 1 · · ·dΩ N−1e iA[{ ˆΩ j }] , (2.84)where A ≡ S/¯h is given byA = −iN∑j=1ln 〈 ˆΩj∣ ∣ ˆΩj−1〉−ɛ¯hN∑H( ˆΩj | ˆΩj−1 ) . (2.85)Expanding in the difference between ˆΩj and ˆΩj−1 , we may writeln 〈 ∣ 〉 }ˆΩj ˆΩj−1 = 2S ln{1 − ū j (u j − u j−1 ) − ¯v j (v j − v j−1 ){ ( uj − u) (j−1 vj − v) }j−1= −2S ɛ ū j − ¯v j + . . .ɛɛThe continuum limit isj=1≃ −2S ɛ (ū j ˙u j + ¯v j ˙v j ) + O ( ( ˆΩj − ˆΩj−1 ) 2) . (2.86)∫TA[ ˆΩ(t)] = dt{2iS(ū ˙u + ¯v ˙v) − 1¯h }H( ˆΩ) , (2.87)018

where H( ˆΩ) ≡ H( ˆΩ | ˆΩ). Substituting u = cos(θ/2) and v = sin(θ/2) exp(iφ),we obtainū ˙u + ¯v ˙v = i sin 2 (θ/2) ˙φ = i 2 (1 − cos θ) ˙φ = i ˙ω (2.88)2where dω = (1 − cos θ) dφ is the differential element of solid angle. We maynow, finally, write the spin path integral as〈 ∣ ∣ ∫{〉ˆΩf ∣e −iHT/¯h ˆΩi = D ˆΩ(t) exp −iw(0)=w i¯w(T )= ¯w f∫ T0dt[S dωdt + 1¯h H( ˆΩ)] }·〈 ˆΩf∣ ∣ ˆΩ(T )〉 〈 ˆΩ(0)∣ ∣ ˆΩi〉(2.89)where w ≡ v/u = tan(θ/2) exp(iφ) is the stereographic projection of thespinor coordinates (u, v) onto the complex plane.The inclusion of the overlap terms inside the path integral is necessary ifwe are to allow for the possibility of so-called discontinuous paths. Withinthe semiclassical approximation, u(t) and v(t) are integrated forward frominitial data u i and v i while ū(t) and ¯v(t) are integrated backward from finaldata ū f and ¯v f . 4 We encountered an analogous situation with the coherentstate path integral for the Heisenberg-Weyl group, where z(t) was integratedforward from initial data z i and ¯z(t) integrated backward from final data ¯z f .In fact, these paths are perfectly continuous; there simply is no reason whyz(T ) should have any resemblance to z f , or ¯z(0) to ¯z i , since the equations ofmotion know nothing about either z f or ¯z i .The thermal, or imaginary time, propagator in the coherent state representationis〈ˆΩf∣ ∣e −βH ∣ ∣ ˆΩi〉=∫{D ˆΩ(τ) expw(0)=w i¯w(T )= ¯w f∫¯hβ−0dτ[iS dωdτ + 1¯h H( ˆΩ)] }·〈 ˆΩf∣ ∣ ˆΩ(¯hβ)〉 〈 ˆΩ(0)∣ ∣ ˆΩi〉.(2.90)4 The equations of motion may also be written in terms of the stereographic coordinatew = v/u, in which case w(t) is integrated forward from initial data w iand ¯w(t) is integratedbackward from final data ¯w f.19

2.5.1 Gauge Field and Geometric PhaseThe solid angle functional ω[ ˆΩ(t)] may be written∫Td ˆΩω[ ˆΩ(t)] = dt A( ˆΩ) ·dtfor any A( ˆΩ) which satisfies ∇ × A = ˆΩ, i.e.0(2.91)Ω a ∂= ɛ abc∂Ω b Ac ( ˆΩ) (2.92)(To see this, use Stokes’ theorem.) We now derive a useful result:∫ { ∂AbdΩδω[ ˆΩ(t)] b= dt∂Ω a dt δΩa + A a d }dt δΩa ∫ ( ) ∂Ab= dt∂Ω − ∂Aa dΩba ∂Ω b dt δΩa=∫∫dt δΩ a ɛ abc ˙Ω b Ω c =and hence the functional derivative isdt δ ˆΩ · ∂ ˆΩ∂t × ˆΩ , (2.93)δω[ ˆΩ]δ ˆΩ(t)= ∂ ˆΩ∂t × ˆΩ . (2.94)2.5.2 Semiclassical Dynamics∫TÃ[ ˆΩ(t), λ(t)] ≡ A[ ˆΩ(t)] + dt λ(t) ( ˆΩ2 (t) − 1 ) (2.95)0δÃ= −S ∂ ˆΩδ ˆΩ(t) ∂t × 1 ∂HˆΩ − + 2 λ ˆΩ (2.96)¯h ∂ ˆΩδÃδ ˆΩ(t)= ˆΩ2 (t) − 1 . (2.97)20

Setting these variations to zero, we solve for λ(t) by taking the dot product ofthe first equation with ˆΩ(t) and then substituting ˆΩ2 (t) = 1. In this manner,we findλ = 1 ∂H · ˆΩ , (2.98)2¯h ∂ ˆΩThe effect of this is to render all terms on the RHS of (2.96) orthogonal toˆΩ, thereby effectively projecting ∂H/∂ ˆΩ onto this orthogonal subspace. Itis then easy to obtain the equations of motion¯hS ∂ ˆΩ∂t = ∂H∂ ˆΩ× ˆΩ . (2.99)If we write the equations of motion in terms of the spinor coordinates{u, v, ū, ¯v} themselves, it is important to recognize that they must satisfy theconstraint uū + v¯v = 1. A Lagrange multiplier field λ is invoked to imposethis constraint at every value of the time t. This results in the equations ofmotion2i¯hS ˙u = ∂H∂ū + λu∂H2i¯hS ˙v = + λv (2.100)∂¯v−2i¯hS ˙ū = ∂H∂u + λū −2i¯hS ˙¯v = ∂H + λ¯v . (2.101)∂vVarying the action with respect to the Lagrange multiplier field of courseyields the constraint equation. We are then left with five equations in thefive unknowns {u, v, ū, ¯v, λ}, along with the four boundary conditions,u(0) = u i v(0) = v i (2.102)ū(T ) = ū f¯v(T ) = ¯v f. (2.103)Implementing the constraint, one obtains an expression for λ,λ = 2iS(ū ˙u + ¯v ˙v) − ū ∂H∂ū− ¯v∂H∂¯v(2.104)= −2iS(u ˙ū + v ˙¯v) − u ∂H∂u − v ∂H∂v . (2.105)Note that for real θ and φ that (2.100) and (2.101) are related by complexconjugation.21

2.6 Other Useful Representations of the SpinPath Integral2.6.1 Stereographic RepresentationIn the stereographic representation, we writew ≡ v u = tan(θ/2) eiφ . (2.106)One then findsFrom the differential¯wẇ1 + ¯ww = ū ˙u + ¯v ˙v − d ln u . (2.107)dtwe obtaindw = 1 2 sec2 ( θ/2) e iφ dθ + i tan(θ/2) e iφ dφ , (2.108)dw ∧ d ¯w(1 + ¯ww) 2 = 1 2isin θ dθ ∧ dφ . (2.109)The Hamiltonian matrix elements may be recast in terms of w and ¯w. Forexample,S + = a † b −→ 2S ūv = 2S w(2.110)1 + ¯wwS z = 1 2 (a† a − b † b) −→ S (ūu − ¯vv) = S 1 − ¯ww1 + ¯ww . (2.111)〈 ∣ ∣ ∫{〉ˆΩf ∣e −iHT/¯h ˆΩi = D[ ¯w(t), w(t)] exp −i∫ T0dt[¯wẇ2S1 + ¯ww + 1¯h ] }H( ¯w, w) ,(2.112)where the metric D[ ¯w, w] includes the (1 + ¯ww) −2 factor at each time step,and where we’ve omitted the overlap factors at t = 0 and t = T .22

2.6.2 Recovery of Spin Wave TheoryTo recover spin wave theory and the Holstein-Primakoff transformation, definez ≡ u¯v/|v| = cos(θ/2) exp(−iφ). Then |z| 2 = |u| 2 anddz = − 1 2 sin(θ/2) e−iφ dθ − i cos(θ/2) e −iφ dφ . (2.113)We then obtaindz ∧ d¯z = 1 2isin θ dθ ∧ dφ . (2.114)The geometrical phase, which is responsible for the ω[ ˆΩ(t)] term in the actionfunctional, is obtained using¯zż = i 2 (1 − cos θ) dφ − i dφ + d cos2 (θ/2) (2.115)which, after dropping the total time derivatives, yields (i/2) dω. As for theHamiltonian, we haveS + = a † b −→ 2S ūv = 2S ¯z √ 1 − ¯zz (2.116)S z = 1 2 (a† a − b † b) −→ S (ūu − ¯vv) = 2S (¯zz − 1 2 ) . (2.117)This is equivalent to Holstein-Primakoff, with h ≡ √ 2S z as the HP boson.We therefore obtain〈 ∣ˆΩf e −iHT/¯h∣ ∫{〉∣ ˆΩi = D[¯h(t), h(t)] exp −i∫ T0dt[¯hḣ+ 1¯h H(¯h, h)] } , (2.118)where the functional integration is over a disk of area 2πS for each time t.23

Chapter 3Applications of the Spin PathIntegral3.1 Quantum Tunneling of SpinThe theory of quantum spin tunneling has been developed largely by E.Chudnovsky, A. Garg, D. Loss, and others. Consider the following modelHamiltonian,H = K 1 S 2 z + K 2 S 2 y − γHS z , (3.1)where K 1 > K 2 > 0. This describes a spin-S particle with an easy axisalong ˆx and a hard axis along ẑ. To treat this problem by the coherent statepath integral, we need to compute the diagonal matrix element of H in thecoherent state basis. One finds,E(θ, φ) = 〈 ∣ ˆΩ ∣ H ˆΩ〉 = k1 cos 2 θ + k 2 sin 2 (θ) sin 2 (φ) − h cos(θ) , (3.2)where k i = S(S − 1 2 )K i (i = 1, 2), h = γSH, and where we have dropped anunimportant constant. In weak fields h, the energy function E(θ, φ) has thefollowing features:• E(θ, φ) has two degenerate minima with θ 0 = cos −1 (h/2k 1 ) at φ = 024

and at φ = π. The minimum energy is E cl0 = −h 2 /4k 1 .• There is a global maximum with E max = k 1 + h located (assuming h >0) at the South Pole (θ = π), and a local maximum with E ′ max = k 1 − hlocated at the North Pole (θ = 0).• There are two saddle points, located at θ x = cos −1 (h/2 ( k 1 − k 2 ) ) , withφ = ± 1 2 π. The energy of the saddle points is E saddle = k 2 − 1 4 h2 /(k 1 −k 2 ).We therefore expect to find two low-lying states which are linear combinationsof the coherent states ∣ θ = θ0 , φ = 0 〉 and ∣ θ = θ0 , φ = π 〉 . Let usabbreviate these two states ∣ 〉 ∣ 〉 0 and ∣π , respectively. The eigenstates of thesystem should be symmetric and antisymmetric combinations of these states:〉∣ ± = 2−1/2{ ∣∣0 〉 ∣ 〉 }± ∣1 . The tunnel splitting ∆ = E 0 may be obtained byexamining the matrix elements,〈+∣ ∣ e −βH∣ ∣ + 〉 = e −βE 0= 〈 0 ∣ ∣ 〉 〈 ∣e−βH ∣ 〉0 + 0∣e −βH π〈 ∣ ∣ 〉− ∣e −βH − = e−β(E 0 +∆)(3.3)= 〈 0 ∣ ∣e −βH∣ ∣0 〉 − 〈 0 ∣ ∣e −βH∣ ∣π 〉 , (3.4)where E 0 differs from E0 cl due to ‘zero-point energy’, i.e. quantum fluctuations.In reality, there is no reason why the states ∣ 〉 ± should necessarilybe eigenstates of H. What is important, though, is that the antisymmetriccombination projects out all of the ground state. By taking the β → ∞limit, the contribution from admixtures of higher-lying eigenstates to ∣ 〉±can be suppressed. What this means is that we can calculate the exact tunnelsplitting by the formula,{〈 ∣1∣ 〉 〈 ∣0∣e −βH∆ = limβ→∞ β ln ∣ 〉}0 + 0∣e −βH π〈 ∣ ∣ 〉 〈 ∣0∣e −βH ∣ 〉0 − 0∣e −βH . (3.5)πAnother way, of course, to compute the tunnel splitting is to simply numericallydiagonalize the rank-(2J + 1) Hamiltonian matrix. This workswithout fail, but it is not particularly instructive in elucidating the physics25

of spin tunneling. Moreover, it may be that an instanton calculation, whichwe shall presently describe, yields certain analytic results which are usefuland in general impossible to obtain numerically.3.1.1 Instantons and Tunnel SplittingsThe essence of the instanton approach to quantum tunneling is described in abeautiful article by Sidney Coleman, entitled “The Uses of Instantons”. Wewrite the imaginary time matrix element 〈 P f∣ ∣ exp(−βH)∣ ∣Pi〉between pointsP 1 and P 2 as a path integral. In our case, each P labels a spin orientation ˆΩ,and each state ∣ ∣ P〉is a spin coherent state. We extremize the action, applyingthe method of stationary phase. This involves solving the classical equationsof motion, subject to boundary conditions which we shall not fully specify,save to say that the most naïve boundary conditions are simply ˆΩ(0) = ˆΩian ˆΩ(¯hβ) = ˆΩf . 1There may be several instanton paths connecting P 1 and P 2 . Associatedwith each such instanton α is a characteristic time τ α , a classical actionY α + iφ α , written in units of ¯h and separating real and imaginary parts, andalso a ‘fluctuation determinant’ prefactor D α arising from integrating overGaussian fluctuations about the classical instanton trajectory. If we writeξ α = D α e iφα e −Yα , (3.6)then the diagonal matrix element can be written in the ‘dilute instanton gas’approximation as〈P1∣ ∣e −βH ∣ ∣ P1〉=∞∑∑∫¯hβn=0 {α k ,ᾱ k } 0(= cosh ¯hβ∣ ∑ αdτ 1 · · ·∫¯hβdτ 2n ξ α1 ξᾱ1 · · · ξᾱnτ 2n−1∣)∣∣ξ α . (3.7)1 In fact, the proper boundary conditions are u(0) = u i, v(0) = v i, ū(¯hβ) = ū f, and¯v(¯hβ) = ¯v f, as derived above.26

Here we denote the return instantons from P 2 to P 1 with the index ᾱ. Sincethe return path is a time-reversed one, we have ξᾱ = ξ α , i.e. the return pathshave opposite phase.The off-diagonal matrix element, in which paths must begin at P 1 and endat P 2 requires an odd number of instanton events, and is given by∫¯hβ〈 ∣ ∣ 〉 ∑ ∞∑P2∣e −βH P1 = dτ 1 · · · dτ 2n+1 ξ α1 ξᾱ1 · · · ξᾱn ξ αn+1n=0 {α k ,ᾱ k } 0 τ∑2nα=ξ (α∣ ∑ α ξ · sinh ¯hβ∣ ∑ ∣)∣∣ξ α . (3.8)α∣αIf ∑ α ξ α is real, then we can read off the tunnel splitting:∫¯hβ∆ = 2¯h ∑ αD α e iφα e −Yα . (3.9)3.1.2 Garg’s Calculation (1993)Starting from the Euclidean Lagrangian,L E= iS(1 − cos θ) ˙φ +1¯hE(θ, φ) , (3.10)one derives the Euler-Lagrange equations of motion,Note that∂L E∂θ − d ∂L Edt ∂ ˙θ∂L E∂φ − d ∂L Edt ∂ ˙φdEdt= 0 = 1¯h∂E∂θ + iS sin(θ) ˙φ (3.11)= 0 = 1¯h∂E∂φ − iS sin(θ) ˙θ . (3.12)= ˙θ∂E∂θ+ ˙φ∂E∂φ = 0 , (3.13)which says that the energy E(θ, φ) is conserved along the classical trajectories.One can use this result to finesse the instanton calculation and solve27

directly for θ as a function of φ. Energy conservation provides a quadraticequation in cos(θ),E 0 = − h24k 1= k 1 cos 2 (θ) + k 2 sin 2 (θ) sin 2 φ − h cos(θ) , (3.14)the solution of which is written (Garg, 1993),u(φ) = u 0 + i √ λ sin φ √ 1 − u 2 0 − λ sin 2 φ1 − λ sin 2 φ(3.15)where u ≡ cos(θ), u 0 = h/2k 1 ≡ h/h c , and λ = k 2 /k 1 . Note that u =cos θ is complex along the instanton path. Nevertheless, the path obeys theboundary condition that u = u 0 at φ = 0 and φ = π. The dimensionlessinstanton action iswhence∫±πA = Y + iφ = βE 0 + iS dφ { 1 − u(φ) } , (3.16)φ = Im A = ±S∫ π0= ±πSdφ0{1 −{1 −u 0}1 − λ sin 2 φ}. (3.17)u 0√1 − λ2Thus, there are two instantons connecting (θ, φ) = (θ 0 , 0) and (θ, φ) =(θ 0 , π) which wind around the sphere in opposite directions. The tunnelsplitting, according to (3.9), is∆ = 4D e −Y( [cos πS 1 −h2 √ k 2 1 − k 2 2]). (3.18)The tunnel splitting therefore vanishes at a set of dimensionless field strengthsh m , where√ {h m = 2 k1 2 − k22 1 − m + 1 }2. (3.19)S28

Note that for h = 0 the splitting vanishes whenever S = m + 1 , which is to2say whenever the ground state is a Kramers doublet.In Fe 8 clusters, where S = 10, this predicts ten values of h > 0 where∆ vanishes. In fact, experiments by Wernsdorfer and Sessoli see only foursuch vanishings. The reason for this is that the effective Hamiltonian for theexperimental molecule includes a term proportional to J 4 + + J 4 − which is notincluded in the Hamiltonian of (3.1). This new term allows for two additionalinstanton solutions. Moreover, the new solutions exhibit discontinuities inˆΩ(τ) at the boundaries τ = 0 and τ = ¯hβ. This very interesting result wasobtained by Keçecioǧlu and Garg (2002) and is discussed in Garg’s lecturesnotes for this School.3.2 Haldane’s Mapping to the Nonlinear SigmaModelThe many-spin dimensionless action isA = −S ∑ iω[ ˆΩi (t)] − 1¯h∫ Tdt H ( { ˆΩi (t)} ) , (3.20)where the Hamiltonian is that of a Heisenberg antiferromagnet, with diagonalcoherent state matrix elements given by0H ( { ˆΩi (t)} ) = 1 2 S2 ∑ i,jJ ij ˆΩi · ˆΩj . (3.21)( v0)L 2 i vˆΩ i = η i ˆn i√1 − + 0¯hS ¯hS L i , (3.22)where ˆn i · L i = 0. Here, ˆn i is the local Néel field, which varies slowly oncethe sublattice modulation η i extracted from the spin field ˆΩi , L i describes29

ferromagnetic fluctuations about the local Néel order; v 0 is the unit cellvolume. Note that¯hS ∑ ∑∫ˆΩ i = v 0 L i = d d x L(x) , (3.23)iiwhere the RHS is obtained after taking the continuum limit.3.2.1 HamiltonianWe now expand the Heisenberg interaction ˆΩi · ˆΩj in the slowly varyingquantities ˆn i − ˆn j and L i . Since ˆn i is a unit vector, we may writeˆn i · ˆn j = 1 − 1 2 (ˆn i − ˆn j ) · (ˆn i − ˆn j ) . (3.24)We then have}ˆΩ i · ˆΩj = η i η j{1 { − 1 2 (ˆn i − ˆn j ) · (ˆn i − ˆn j ) 1 − 1 ( v0) }(L22 ¯hS i + Lj) 2 + . . .+ v 0¯hS η i ˆn i · (L j − L i ) + v 0¯hS η j ˆn j · (L i − L j )(+ 1 v0) 2 }{L 2 2i + L 2 j − (L i − L j ) 2 . (3.25)¯hSLattice differences may now be expanded in derivatives, asf(R j ) − f(R i ) = (R µ j − Rµ i ) ∂f(R i)∂R µ i+ 1 2 (Rµ j − Rµ i ) (Rν j − R ν i ) ∂2 f(R i )∂R µ i ∂Rν i+ . . .(3.26)Expanding to Gaussian order in the fields ˆn and L and their gradients, wefind}ˆΩ i · ˆΩj = η i η j{1 − 1 2 (Rµ j − Rµ i ) (Rν j − Ri ν ) (∂ µ n a i ) (∂ ν n a i ) + . . .+ 1 2( v0) 2 {}(1 − η¯hSi η j )(L 2 i + L 2 j) − (R µ j − Rµ i ) (Rν j − Ri ν ) (∂ µ L a i ) (∂ ν L a i ) + . . .+ v 0¯hS η i na i (R µ j − Rµ i ) (∂ µL a i ) − v 0¯hS η j na j (R µ j − Rµ i ) (∂ µL a j ) + . . . . (3.27)30

where Q ≡ (π/a, π/a, . . . .π/a) is the zone corner wavevector. The dimensionsof ρ s and χ are:[ρ s ] = E · L 2−d ; [χ] = E · T 2 · L −d . (3.35)3.2.2 Geometric PhaseThe geometric phase contribution to the dimensionless action is writtenA B = −S ∑ iω[ ˆΩi (t)] = −S ∑ i[v]η i ω ˆn i (t) + η 0i¯hS L i(t) . (3.36)We now expand in the notionally small quantity linear in L i , using the resultof (2.94):A B = −S ∑ i= −S ∑ i∫ω[ˆn i ] − S dt ∑ i∫ω[ˆn i ] − 1¯h( v0)L i · ∂ˆn¯hS ∂t × ˆn∫d d x dt ∂ˆn × ˆn · L . (3.37)∂t3.2.3 Emergence of the Nonlinear Sigma ModelA = − 1¯h∫ ∫ {}d d 1x dt ρ 2 s( ∇ˆn) ⃗ 2 + L22χ + L · ∂ˆn∂t × ˆn + gµ H · L B− S ∑ ω[ˆn i ] .i(3.38)Now let us integrate out L. In order to do so, we must introduce a Lagrangemultiplier field λ(x, t) which enforces the local constraint ˆn · L = 0. We then32

must evaluate the functional integral∫ ∫ {I ≡ Dλ(t) DL(t) exp − ī ∫h{ ∫iχ= I 0∫Dλ(t) exp{ ∫i= Ĩ0 exp¯h2¯h∫d d x dt∫ [ Ld d 2 (x dt2χ + L · λˆn + ∂ˆn) ]}∂t × ˆn + gµ H B∫}(d d x dt λˆn + ∂ˆn) 2∂t × ˆn + gµ H B(3.39)[1 χ ( ) ] }∂ˆn 2 ∂ˆn2 + gµB χ H ·∂t∂t × ˆn + 1 χ (gµ2 B) 2 (H × ˆn) 2where I 0 and Ĩ0 are independent of H and ˆn. The complete action functional,including the geometric phase term, is then∫ ∫ [A = d d 1x dt χ ( )∂ˆn 22 −12∂tρ s( ∇ˆn) ⃗ 2 ∂ˆn+ gµ Bχ H ·∂t × ˆn +]+ 1 χ (gµ2 B) 2 (H × ˆn) 2 − S ∑ η i ω[ˆn i ] . (3.40)iDimensional analysis reveals the spin wave velocity c = √ ρ s /χ. Definingx 0 = ct, we find that the quantum field theoretic action, excluding the geometricphase term, isAˆn = ρ s2¯hc∫d d+1 x{12 (∂ µn a )(∂ µ n a )+ 1 c gµ B H · ∂ˆn∂x 0 ׈n+ 12c 2 (gµ B) 2 (H ׈n) 2 }(3.41)where we adopt a Minkowski (+, −, . . . , −) metric. The Euclidean version isA Ḙ n = ρ ∫ {sd d+1 1x22¯hc(∂ µn a ) 2 + i c gµ H · ∂ˆnB∂x × 0 ˆn − 1}2c (gµ B) 2 2 (H × ˆn) 2 .(3.42)Notice the factor of i in the coefficient of the second term. The third term,which is quadratic in H, may be dropped in the case of weak external fields.33

3.2.4 Continuum Limit of the Geometric Phase: d = 1In one space dimension, we have∑(−1) j ω[ˆn j ] = ω[ˆn 0 ] − ω[ˆn 1 ] + ω[ˆn 2 ] − . . . = 1 2jWe now invoke (2.94), which saysδω =to obtain the beautiful result,∫ T0∫ L0dx ∂ω∂x . (3.43)dt ɛ abc ṅ b n c δn a , (3.44)A B = −S ∑ (−1) j ω[ˆn j ]j∫ ∫= 1 S dx dt ˆn · ∂ˆn2∂t × ∂ˆn∂x ≡ 2πS Q tx , (3.45)where Q tx is an integer topological invariant, known as the Pontrjagin indexof the field ˆn(x, t):Q tx = 1 ∫d 2 x ɛ µν ɛ8πabcn a ∂ µ n b ∂ ν n c , (3.46)where x 0 = ct as before. Q tx measures the winding of the field ˆn(x, t) over theunit sphere. To see it is an integer, change variables from local coordinates(n b , n c ) to (ξ 0 , ξ 1 ) in the vicinity of ˆn. The differential surface area elementprojected along n a isdΣ a = 1 2 ɛµν ɛ abc∂n b∂ξ µ ∂n c∂ξ ν , (3.47)and changing variables from (x 0 , x 1 ) to (ξ 0 , ξ 1 ), we obtainQ tx = 1 ∫d 2 ξ , (3.48)4πS 2 internal34

which is manifestly an integer.Put another way, think of ˆn(x, t) as a rubber band draped over the surfaceof a sphere. As time evolves from 0 to T , the configuration of the rubberband changes, but if the configuration itself is periodic, i.e. ˆn(x, 0) − ˆn(x, T ),The Pontrjagin index measures the number of times the rubber band windsaround the sphere. Configurations of ˆn(x, t) which yield a nonzero value ofQ tx are known as skyrmions. An example of a skyrmion configuration on thetwo-dimensional (x, y) (or (x, t)) plane is obtained by identifying the vectorˆn(x, y) with the (inverse) stereographically projected position (x, y). Putanother way, we setvu = tan(θ/2) eiφ ≡ (x + iy)/a , (3.49)where a is an arbitrary length scale. This (exercise!) is equivalent ton x =2axa 2 + x 2 + y 2 , n y =This skyrmion has Pontrjagin index Q xy = 1.2aya 2 + x 2 + y 2 , n z = a2 − x 2 − y 2a 2 + x 2 + y 2 . (3.50)Thermodynamic properties are derived from the Euclidean action,A E d=1 = 2πiS Q tx + ρ ∫sd 2 x ( ∇ˆn)2¯hc⃗ 2 . (3.51)The effect of the geometric phase term, then, is quite simple and in factdiscrete:{e 2πiS +1 if S ∈ ZQtx=(−1) tx it S ∈ Z + 1 . (3.52)2Thus, for integer S, the geometric phase term always contributes a factorof unity, and the full quantum field theoretic action is that of the twodimensionalO(3) model, also called the nonlinear sigma model. For halfoddinteger S, space-time configurations with even and odd Pontrjagin indexdestructively interfere with each other.What have we learned? First of all, we conclude that antiferromagneticHeisenberg chains generically fall into two classes: those with integer spin and35

those with half-odd integer spin. The field theory for the first class is simplythat of the classical O(3) model in two dimensions. The Hohenberg-Mermin-Wagner theorem precludes any spontaneous breaking of the continuous O(3)symmetry in d = 2 at any finite value of ρ s . The system has a gap, andcorrelation functions decay exponentially, up to power law corrections, viz.〈Ψ0∣∣S0 · S j∣ ∣Ψ0〉≃ (−1) j |j| −1/2 exp(−|j|/ξ) , (3.53)where the correlation length ξ, in units of the lattice spacing a, is a functionof the dimensionless quantity ρ s /¯hc.For the second class – the half-odd integer antiferromagnetic chains – thefield theory includes the so-called ‘θ-term’,A θ= θ ∫ ∫dx dt ˆn · ∂ˆn4π∂t × ∂ˆn∂x , (3.54)with θ = 2πS = π mod 2π. While no exact solution to the field theory withthe θ-term is yet known, we nonetheless conclude that all half-odd integerantiferromagnetic chains behave equivalently, since they all map onto thesame model. Since the S = 1 Heisenberg antiferromagnetic chain is known,2from Bethe’s Ansatz, to possess a disordered ground state with gapless excitationsand power law correlations, 〈S 0 · S j 〉 ∼ (−1) j /|j| (up to logarithmiccorrections), we conclude that the same is true for the S = 3, 5 , etc. spin2 2chains.3.2.5 The Geometric Phase in Higher DimensionsSo long as the Néel field ˆn(x, t) is a smooth function of space and time, thereare no interesting topological terms in the field theory in more than onespace dimension. The reason is trivial. Consider a d-dimensional system asa network of parallel one-dimensional chains. Call the longitudinal (chain)coordinate x. For each set of transverse coordinates R ⊥, one can define theinteger Pontrjagin index Q tx (R ⊥). The geometric phase term in the actionis then given byA B= S ∑ iη i ω[ˆn i ] = S ∑ R ⊥η R⊥Q tx (R ⊥) = 0 , (3.55)36

where the last equality follows from the assumed smoothness of ˆn(x, t), whichrequires that Q tx (R ⊥) be independent of R ⊥, since a smooth integer-valuedfunction must be a constant!When the smoothness constraint is relaxed, however, the geometric phaseterm can play an important role. For a two-dimensional antiferromagnet,there exist topology-changing instanton for which ∆Q xy = ±1. Such fieldconfigurations are called ‘hedgehogs’, because the direction of the field ˆn(t, x, y)points radially outward from the center of the hedgehog. For quantumdisorderedtwo-dimensional antiferromagnets (i.e. small ρ s ), Haldane arguedthat geometrical phase considerations associated with the presence of hedgehogswould distinguish not only between integer and half-odd integer S onthe square lattice, but between even and odd integer S as well.37

Chapter 4Large-N TechniquesThe basic idea behind large-N approaches is to extend the global symmetrygroup of some physical model from e.g. O(3), SU(2), etc. to a larger group,such as O(N), SU(N), or Sp(N). If the extension is done in a certain way, theresultant model can be solved exactly in the N → ∞ limit. N plays the roleof 1/¯h, so N → ∞ is a classical limit of sorts, with no quantum fluctuations.Furthermore, one can derive a systematic diagrammatic expansion in powersof 1/N, which can be used to investigate properties at finite N.We shall barely scratch the surface of this subject. My aim here is to guideyou through a large-N calculation for the nonlinear sigma model.4.1 1/N Expansion for an IntegralTo begin, consider the one-dimensional integral,I =∫ ∞dx e −Nf(x) , (4.1)−∞38

where f(x) is some function and N is large. Clearly the integral is dominatedby values of x near the minimum of f(x). Suppose a unique global minimumexists at x = x c . We can then write∫∞I = e −Nf(xc) du e − 1 2 Nf′′ (x c) u 2 e − 1 6 Nf′′′ (x c) u 3 e − 124 Nf′′′′ (x c) u 4 · · ·−∞∫∞= e −Nf(xc) du e − 1 { }2 Nf′′ (x c) u 2 1 − 1 6 Nf′′′ (x c ) u 3 − 124 Nf′′′′ (x c ) u 4 + . . .=−∞( ) 1/2 2π{}e −Nf(xc) 1 − 1Nf ′′ 24(x c )Nf′′′′ (x c )〈u 4 〉 + . . . . (4.2)Thus, we have derived a 1/N expansion for the integral:GaussianfluctuationsO(1/N)correctionsleadingterm { }} { { }} {{ }} { (1 Nf ′′− ln I = Nf(x c ) + ln (x c ))1+22π 8Nf ′′′′ (x c )[f′′(x c ) ] 2 +O(N −2 ) . (4.3)4.2 Large-N Theory of the Nonlinear SigmaModelRecall the Euclidean action for the O(3) nonlinear sigma model,A E= ρ ∫s2¯hc∫¯hβd d x0dx 0 (∂ µ n a ) 2 , (4.4)where ˆn = (n x , n y , n z ) is a three-component unit vector. In the case of Haldane’sderivation of the sigma model action for quantum antiferromagnets,ˆn(x) is physically the Néel field, which varies slowly from site to site eventhough the local magnetization itself oscillates from one sublattice to thenext. 1 It should be emphasized, though, that the D-dimensional nonlin-1 The notation I adopt here is that (d+1)-dimensional vectors are denoted as x ≡ (x 0 , x).39

ear sigma model also describes the finite temperature phase transition of anisotropic D-dimensional ferromagnet.Quantum mechanics is irrelevant at finite temperature, since the imaginarytime variable is bounded: 0 ≤ τ ≤ ¯hβ. At a critical point, the spatial correlationlength diverges as ξ(T ) ∼ |T − T c | −ν , and the temporal correlationlength (or correlation time) diverges along with ξ, as ξ τ ∼ ξ z . Here, ν is thecorrelation length exponent and z the dynamic critical exponent. With ¯hβfinite, however, sufficiently close to T c the correlation time exceeds the thickness¯hβ of the temporal ‘slab’, hence the degrees of freedom at a particularlocation in space are ‘locked’ as a function of imaginary time. Finite T secondorder transitions of a d-dimensional quantum system are therefore describedby a d-dimensional action. 2 At zero temperature, though, the temporal slabis infinitely thick, and one cannot ignore temporal fluctuations. The actionis then for a (d + 1)-dimensional system.It is perhaps worth emphasizing that the continuum effective action for theHeisenberg ferromagnet is given by∫A FM=∫¯hβd d x0dτ{iS v −10 A(ˆn) · ∂ˆn∂τ + 1 2 ρ s ( ⃗ ∇ˆn) 2 }(4.5)where v 0 is the unit cell volume, andρ s =S2 ∑4dv 0RJ(R) R 2 . (4.6)Note the difference between this and the effective action of the antiferromagnet,in which space and time appear symmetrically. The effective (lowenergy)theory for the antiferromagnet possesses a ‘Lorentz invariance’ wherethe speed of light is replaced by the spin wave velocity c = √ ρ s /χ.2 Note that this does not say that quantum mechanics has no effect whatsoever at finitetemperature. Indeed, the partition function for the quantum and classical Heisenbergmodels will be different. What is true is that the critical properties at a finite temperaturesecond order transition are not affected by quantum mechanics.40

Returning to the nonlinear sigma model, the partition function is given bythe functional integral∫Z = e −F/kBT = Dˆn(x) e −A E [ˆn] . (4.7)ˆn 2 =1The extension of the O(3) model to one with an O(N) symmetry is trivial.Simply replace the 3-component unit vector (n x , n y , n z ) by an N-componentone, n = (n 1 , n 2 , . . . , n d). How do we generalize the unit length constraint togeneral N? Let us write the constraint as n 2 (x) = qN, where the parameterq is as yet undetermined. We can envisage two natural extensions to generalN:• Maintain n 2 = 1, i.e. take q = N −1 .• Fix q and let N vary. The length of n then increases with N.It turns out that it is the second of these schemes which generates a proper1/N expansion, as we shall soon see.To enforce the length constraint, we insert into the functional integral aδ-function δ(ˆn 2 − qN) at every space-time point. We write the δ-function asδ(y) =∫i∞dλ e −λy , (4.8)−i∞where the integration contour runs along the imaginary axis, from −i∞ to+i∞. The partition function is then expressed as a double functional integralover the fields ˆn(x) and λ(x),∫Z = D[n(x), λ(x)] e −ÃE [n,λ] , (4.9)where∫à E={ }1d d+1 x2g (∂ µn a ) 2 + λ (n 2 − qN) . (4.10)41

For convenience we have defined the couplingg ≡ ¯hcρ s=The dimensions of g are [g] = L d−1 .¯h √ χ ρ s. (4.11)We now integrate out the n a (x) fields, which are quadratic in ÃE . Writing∫ ∫∫à E= 1 d d+1 x d d+1 x ′ n a (x) K(x, x ′ ) n a (x ′ ) − qN d d+1 x λ(x) (4.12)2withK(x, x ′ ) = − ρ s ∂¯hc ∂x δ(x − ∂µ x′ )∂x ′ µ + 2λ δ(x − x′ ) , (4.13)the partition function can be written in terms of an effective free energywhich is a function of the field λ(x) alone:∫Z = Dλ(x) e −NF eff [λ]/k BT(4.14)wheree −NF eff [λ]/k BT=∫−ÃE [n(x),λ(x)]Dn(x) e= (det K) −N exp{ ∫qN}d d+1 x λ(x) . (4.15)Thus, the effective free energy is∫F eff[λ]/k B T = ln det K − qd d+1 x λ(x) , (4.16)where the determinant of the integral operator K is, as always, defined bythe product of its eigenvalues,ln det K = ∏ nζ n . (4.17)The eigenvalue equation is∫d d+1 x ′ K(x, x ′ ) ψ n (x ′ ) = ζ n ψ n (x) . (4.18)42

We can now see why keeping q finite as N → ∞ generates a true 1/Nexpansion. Had we instead taken n 2 = 1, we would have q = 1/N and theeffective free energy F eff[λ] would not be independent of N.Solution of the N → ∞ TheoryWhen N → ∞ the functional integral is dominated by the saddle point inthe action. Before we solve for this saddle point, let us slightly extend ourmodel to include a coupling to a magnetic field. The augmented action isthen∫ { 1à E= d d+1 x2g (∂ µn a ) 2 + λ (n a n a − qN) − √ }N h a n a . (4.19)The √ N factor preceding h · n ensures that the action will be proportionalto N when h is of O(N 0 ). In the case of the antiferromagnet, where ˆn isthe Néel field, h corresponds to the q = π/a (zone corner) component ofthe physical magnetic field, i.e.a sublattice-staggered magnetic field. This isof course quite unphysical, however our purpose in introducing h is not toinvestigate the effects of an external field per se, but rather as an artifice bywhich we can couple to any condensate, as we shall see presently.To find the saddle point of F eff[λ], we should set its functional variationwith respect to λ(x) to zero. We will assume that the saddle point occursfor real, constant λ. We will justify this by presenting such a solution to theequation δF eff= 0. Note that the saddle point lies off the integration contourfor λ(x), which runs along the imaginary axis.When λ is constant, the model may be solved by Fourier transform. Wewriten a (x) = √ 1 ∑ˆn a (k) e ik·x (4.20)L0 Vwith( 2πj0L 0 = β¯hc , V = L 1 · · · L d , k =43kL 0, 2πj 1L 1, . . . , 2πj )d. (4.21)L d

Expressed in terms of the Fourier modes, the Euclidean action isN∑ ∑{ ( )Ã E= λ +∣∣ˆn k2 a (k) ∣ 2 − √ N2gĥ∗ a(k) ˆn (k)}a − qNV L 0 λ . (4.22)a=1kWe now integrate out the {ˆn a (k)}, yielding an effective free energy functionF eff(λ):f(λ) ≡ F eff (λ)¯hcV= −qλ + 12L 0 V∑lnk− 14L 0 V) (λ + k22g∑ ĥ a (k) ĥa (−k)( ) . (4.23)λ +k 22gkThe order parameter m, which is the static Néel field in the case of theantiferromagnet and the static magnetization in the case of the ferromagnet,is obtained by differentiating the free energy with respect to the q = 0 Fouriercomponent of the field ĥa (k). We therefore obtainsince ĥ(0) = √ L 0 V h.m = 〈n〉 √N= − 1NL 0 V∂(NF eff/k B T )∂h= − ∂f∂h = h2λ , (4.24)To find the saddle point in λ, we set ∂f/∂λ = 0, yieldingq = m 2 +g ∑ 1L 0 V k 2 + 2gλ . (4.25)The second mean field equation,requires either (i) λ = 0 or (ii) m = 0.kh = 2λ m = 0 , (4.26)We now explore the solution to these equations as we vary dimensionalityand temperature.44

• d = 1, T = 0: In this case the integral is infrared divergent whenλ = 0. The mean field equation can always be solved with m = 0 forsome finite λ:q = 1 2∫Λd 2 k(2π) 2 1λ + k22gλ(g) == g ( )4π ln 1 + Λ22gλΛ 2 /2gexp(4πq/g) − 1(4.27)(4.28)which monotonically decreases on g ∈ [0.∞] from λ(0) = Λ 2 /8πq toλ(∞) = 0.• d > 1, T = 0: In this case there exists a quantum critical point atg = g c . The gap λ vanishes for g ≤ g c . To find g c , setq = g c∫Λd d+1 k 1(2π) d+1 k = g 2 cΩ d+1(2π) d+1Λ d−1d − 1 , (4.29)where Ω d+1 is the area of the (d + 1)-dimensional unit sphere:Thus, the critical coupling isg =√2d ad−1SΩ d+1 = 2π(d+1)/2Γ ( ) . (4.30)d+12g c Λ d−1 = 2 d π (d+1)/2 (d − 1) Γ ( d+12[ ∑j J ij (1 − η i η j )− ∑ j J ij η i η j∣∣Ri − R j∣ ∣2/a2)(4.31)] 1/2(4.32)Setting Λ = π/a, we find that the criterion for ordering depends solelyon S and the range of the coupling J ij = J(R i − R j ). For nearestneighbor coupling, one finds g = 2 √ d a d−1 /S, and, taking Λ = π/a,the system becomes ordered for S > S c , whereS c (d) = 21−d π √ (d−3)/2 d(d − 1) Γ ( ) . (4.33)d+1245

One therefore finds S c (d = 2) = √ 2≃ 0.45, which taken at face valueπsays that the square lattice antiferromagnet is ordered for all allowedvalues of S. Indeed numerical work convincingly shows that the groundstate for S = 1 is Néel-ordered, and rigorous proofs exist which show2that long-ranged Néel order exists for S ≥ 1.One might suspect, given (4.32), that by extending the range of theinteractions one can push g above g c and obtain a quantum-disordered‘spin-liquid’ ground state for the S = 1 antiferromagnet on a square2lattice. For example, if one includes next-nearest neighbor antiferromagneticcoupling J 2 as well as nearest neghibor antiferromagneticcoupling J 1 , one hasg = 2√ d a d−1S1√1 − 2J2 /J 1, (4.34)which is increased above its value when J 2 = 0. In fact, the searchfor spin liquid states has been an arduous one. On the square lattice,one generally finds that frustrating further-neighbor couplings push thesystem into another ordered state, for example one with four sublatticeantiferromagnetic order. On lattices which are highly geometricallyfrustrated, such as the Kagomé and pyrochlore lattices, the S = 1 antiferromagnetis generally believed to have a quantum-disordered spin2liquid ground state, i.e. the ground state has no long-ranged order andbreaks no lattice translation or point group symmetries.• d > 2, T > 0: In this case there is a finite temperature phase transition.Defining the Matsubara wavevectors κ n ≡ 2πn/L 0 , we use the result1 ∑H(−iκ n ) = H(0) +L 0 L 0κ n∫ ∞−∞dκπIm H(κ + i0 + )exp(κL 0 ) − 1(4.35)to obtain the finite temperature mean field equation,∫q = m 2 + gΛ{d d k 2/L 0(2π) d k 2 + 2gλ + ctnh ( 1L √ }2 0 k2 + 2gλ)√k2 + 2gλ(4.36)46

The equation for T c is obtained by setting λ = m 2 = 0:∫q = gΛ{ d d k 2(2π) d L 0 k + 1 2 k ctnh ( 1kL 2 0,c)}, (4.37)which is to be solved for T c = ¯hc/k B L 0.c , assuming g < g c , i.e. that theT = 0 (ground) state is ordered.4.2.1 Correlation FunctionsThe correlation functions are obtained via〈ˆn a (k) ˆn b (−k)〉 c = −L 0 V∂ 2 f∂ĥa (−k) ∂ĥb (k) =hence the full correlator is given by{〈n a (0) n b (x)〉 = m 2 +gL 0 V∑kgk 2 + 2gλ δab (4.38)}e ik·xδ ab . (4.39)k 2 + 2gλAt zero temperature, and in the thermodynamic limit, we haveC(x) ≡ 1 ∫N 〈ˆn(0) · ˆn(x)〉 = m2 + gΛd d+1 k(2π) d+1which at large distances takes the Ornstein-Zernike form,e ik·xk 2 + 2gλ , (4.40)C(x) ∼ e−|x|/ξ, (4.41)|x|d/2with ξ 2 = (2gλ) −1/2 . At the quantum critical point, where λ vanishes, onefinds C(x) ∼ |x| 1−d .47