Spin Coherent States And Statistical Physics
Spin Coherent States and Statistical PhysicsRobert Jones MIT Department of Physics(Dated: May 20, 2019)I review the basic properties of spin coherent states, and explain how they may be used to relate1 dimensional quantum chains to 2 dimensional classical statistical physical models. In particular,I review the map from quantum antiferromagnets to the O(3) NLSM with a topological term, anddiscuss the Haldane conjecture.I.INTRODUCTIONBecause of the enhanced role of fluctuations, lowdimensional quantum and thermal systems often behavevery differently than their high dimensional counterparts.An important example of this is the Mermin-Wagner theorem, which states that continuous symmetry-breakingorder is not possible at finite temperature in two dimensions. While this rules out a symmetry-breaking phasetransition in 2d, a more exotic phase transition driven bythe proliferation of vortices, the BKT transition, is possible at finite temperature. Since the transfer matrix formalism may be used to relate classical statistical physicsand quantum mechanics in one lower space dimension, itis interesting to ask what these results can tell us aboutone dimensional quantum systems. Studying this is thepurpose of this note.This connection appears in many ways. For instance,the one dimensional spin-1/2 XXZ model, with Hamiltonian:XyxzHXXZ JSnx Sn 1 Sny Sn 1 Snz Sn 1nmay be mapped to the classical six-vertex model [1], andthe isotropic antiferromagnetic point 1, J 0 on thequantum side corresponds to the BKT critical point onthe classical side [2]. But the result this note aims for isAffleck’s argument [3] for the Haldane conjecture [4]. TheHaldane conjecture [5] is that one dimensional Heisenberg antiferromagnets display drastically different behavior depending on whether the spins are integral or halfintegral, with the former leading to a gapped groundstate and the latter leading to a gapless ground state.A crucial step in the argument is mapping the quantumsystem onto a 2d O(3) nonlinear sigma model.This mapping may be accomplished in a variety ofways, but following [6], we will use the formalism ofspin coherent states. To that end, after reviewing thequantum-classical correspondence provided by the transfer matrix method, we will introduce spin coherent statesand their basic properties. Then, we will use spin coherent states to construct classical systems correspondingto a single spin in a magnetic field, and a pair of spins jonesr@mit.eduinteracting via an isotropic exchange interaction. Next,we will show how the Berry phase term for a single spinbecomes a topological term for the action of a spin chain,and we will obtain the NLSM for a 1d antiferromagneticchain. Finally, we will discuss how to use this NLSMframework to establish the Haldane conjecture.II. TRANSFER MATRICES RELATEQUANTUM AND CLASSICAL PHYSICSIn this section, we review how the transfer matrix method provides a connection between d-spacedimensional quantum mechanics and d 1-spacedimensional classical statistical physics. This has beentaken up by many authors (see, for instance, [1]), including past 8.334 projects [7]. We wish to emphasize the roleof anisotropy on the classical side and the specific relationship between quantum and thermal fluctuations (andwhat that means for, e.g., the Mermin Wagner theorem).Suppose we have a classical Hamiltonian which depends on some parameters g and the configuration σ as:H(g, σ) NXH[g; σi , σi 1 ],i 1where σi is the configuration of the i’th “row” of thesystem. Suppose also that H is symmetric in the σ’s.Then if we introduce T (g, β) via its matrix elements as:0hσ 0 T (g, β) σi e βH[g;σ ,σ]then we can write:X Z e βH tr T (g, β)N tr e βq Hq .{σ}Thus, the equation:0e βH[g;σ ,σ] hσ 0 e βq Hq /N σi(1)provides a mapping from classical Hamiltonians H andquantum hamiltonians Hq which relates their partitionfunctions. Computing expectation values for classical observables corresponds to inserting diagonal operators intothe trace on the quantum side.Let us first consider 0 1 dimensional quantum sys-
2tems. When the quantum mechanical system is zerodimensional and the classical mechanical system is onedimensional, (1) is often satisfactory as is. For example, it may be used to provide a correspondence betweena quantum-mechanical spin-1/2 particle in an externalfield and the classical Ising chain. For such a particle, wemay compute: cosh γ sinh γexp (γσx ) ,sinh γ cosh γwhich tells us:1βH[σ, σ 0 ] log2 11sinh(2γ) log(tanh(γ))σσ 0 ,22which is a classical Ising chain. Notice that the bigger γis, the smaller log(tanh(γ)) is, and thus the stronger thethermal fluctuations on the classical side become. Thismakes sense since on the quantum side i has maximum uncertainty for Sz , so the larger γ is, the strongerthe quantum fluctuations of Sz .For generic quantum-mechanical systems with a spatial extent, though, equation (1) is unsatisfactory because the exponentials involved do not preserve the shortrangedness of the interaction. In other words, a classical Hamiltonian with only nearest-neighbor interactions would nevertheless produce a (very complicated)quantum-mechanical Hq with nonzero interactions at arbitrarily long range, and vice-versa. This problem maybe averted by working in a highly anisotropic region ofthe classical parameter space: if there is a very high energy cost for σ to differ from σ 0 , then the transfer matrixwill be close to the identity, and can thus be safely Taylorexpanded. This is the procedure used to accomplish thepreviously mentioned map between the six-vertex modeland the spin-1/2 XXZ model.Let us examine what happens to the Mermin-Wagnertheorem under this correspondence. In classical statistical physics, in the form presented in 8.334, the MerminWagner theorem rules out the possibility of symmetrybreaking order for a continuous symmetry in 2 spatial dimensions. The order is destroyed by thermal excitationof the Goldstone modes. One might thus suspect that,as a result of the previously discussed correspondence,quantum fluctuations would rule out the possibility ofcontinuous symmetry breaking order in quantum chains.However, it is easy to devise a counterexample, like theHeisenberg ferromagnet:X i · S i 1 .H JSiAny product state where all the spins have maximal projection along the same axis is a ground state of this system, which therefore spontaneously breaks rotation symmetry.What has gone wrong? I wish to argue, elaborating ona brief comment to [8], that when this occurs, the corre-sponding classical system is at zero temperature (meaning all configurations with a nonzero weight have thesame weight). However, unlike [8], I do not believe thatthe unentangled nature of the Heisenberg ground stateis relevant, and in fact something similar can occur in0 1-dimensional quantum systems where entanglementisn’t even possible.Making a quantum-classical correspondence using atransfer matrix requires a choice of basis on the quantumside, and thermal fluctuations on the classical side resultfrom the fact that the quantum-mechanical ground stateis a superposition of various vectors in this basis. But ifthe ground state belongs to the basis, this superpositionis trivial, and the classical system is at zero temperature.For instance, if we had used σz instead of σx in our earlierexample of a single quantum spin, we would have foundthat the corresponding classical system assigns a finiteweight only to the state where all spins are up.There are more interesting gapless states than theaforementioned Heisenberg ferromagnet ground state,like the Heisenberg antiferromagnet ground state alludedto in the introduction. But before studying that, we mustintroduce the relevant piece of mathematical machinery,namely spin coherent states.III.SPIN COHERENT STATESIn the previous section, we explained how to rewrite aclassical partition function as a quantum-mechanical oneby introducing a quantum Hamiltonian equal to the logarithm of the transfer matrix. How do we go in the otherdirection?P If we insert many resolutions of the identity1 n nihn into the quantum-mechanical partitionfunction, the sums in the resolution of the identity become a sum over classical configurations, and products ofthe matrix elements of e βq Hq /N become classical Boltzmann weights. Thus, any way of resolving the identityprovides a way to go from a quantum Hamiltonian to aclassical one.A particularly convenient resolution of the identity forstudying spin systems is provided by the spin coherentstates. In this section, we establish the properties of thespin coherent states (loosely following [6]) and use themto study the classical analogue of a quantum particle ina magnetic field. We notice a complication: there is noguarantee that the Boltzmann weights obtained by thisprocedure are real. As also observed in [7], these Berryphases are physically very important.Following [6], we define the spin coherent state associated with a particular direction n as the state obtainedby rotating the highest-weight Sz eigenstate to point inthe n direction. We accomplish this by rotating by anamount θ along the axis parallel to n ẑ[9]: ẑi. ni exp iθ( n[ ẑ) · SIn the same way that coherent states for the harmonic
3oscillator may be pictured as a sharply peaked (area )Gaussian in the (x, p) plane, the spin coherent statespoint in a particular direction to the greatest extent allowed by the angular momentum commutation relations.In particular, as the spin of the particle gets larger, theangular uncertainty decreases.where we have introduced:µ eiφ tan θ.2To compute the overlap:h n1 n2 i 1 λ 2 s1 µ 2 s hẑ eλS µS e ẑi,we may perform a similar maneuver to obtain:h n1 n2 i 1 λ 2Let us evaluate the overlap of two spin coherent states.This will help clarify the preceding remark, and it will beuseful in later calculations. The result is: s1 n1 · n2. h n1 n2 i 2As s gets big, the overlap falls off more rapidly as n1 and n2 get father apart. We sketch a proof, following [10, 11].First, we re-express the coherent state in terms of theangular momentum raising and lowering operators S Sx iSy : θ ni exp(S eiφ S e iφ ) ẑi.2Since S annihilates ẑi, we want to use a BakerCampbell-Hausdorff style argument to get it out of theexponential. This is done in [11] by proving the followingfact in the spin-1/2 representation, which thus holds inall representations since the spin-1/2 rep is faithful: 2 θiφ iφexp(S e S e ) eµS e log(1 µ )Sz e µ S .2h n1 e γSz seµS ẑi1 µ 2 s(1 λ µ)2s .The modulus of this gives the result quoted above, whilethe phase is eisΦ( n1 , n2 ,ẑ) , where Φ gives the area of thespherical triangle defined by the 3 points.The last property of the spin coherent states we need isthat they resolve the identity. This appears to be done byusing (2) to obtain the components of ni in the Sz eigenbasis, and then explicitly evaluating the integral over S 2 .The result is:Z2s 11 dΩ nih n .4πHaving reviewed some important properties of thespin coherent states, let us now apply them to some0-dimensional quantum systems of interest. We beginwith a particle in an external magnetic field. In [6], theauthor considers a continuous classical analogue, but Iwould like to consider a discrete one (i.e. a chain). Thiswill allow us to avoid making potentially hard-to-justifyapproximations. We are thus interested in evaluatingh n1 e βq Hq n2 i, where βq Hq γSz .The trick here is to Wick rotate: if we let γ it, theexponential becomes a rotation matrix, which acts on thecoherent state to produce another coherent state (and aphase). The overlap can then be computed using thepreviously established formula. When this is done, wereplace it by γ and we are done. In particular, we have:Using that ẑi is an eigenstate of Sz gives: ni 1 µ 2 se itSz n2 i e its R n2 i,(2)which allows us to compute: 2sθ1θ2θ1θ2 γ/2i(φ2 φ1 ) γ/2 n2 i ecoscos esinsin2222Taking the log of this gives our classical H analogue, which is complex. Nevertheless, it is amusing to take this modelseriously and to evaluate e.g. correlation functions in it. We have:Z1 X βH1hS0,i Sn,j i eS0,i Sn,j S0,i hS0 e γSz S1 ihS1 e γSz · · · Sn iSn,j hSn · · · SN ihSN e γSz S0 i,ZZ {Si }cfgsZ1 S0,i Sn,j hS0 e nγSz Sn ihSn e (N n)γSz S0 i.Z S0 ,SnIn the limit N , we have: Ae nγ iAe nγ 0hS0,i Sn,j i iAe nγ Ae nγ 0 ,00Bwhere:2s,(2s 2)2(2s)2B .(2s 2)2A
4The B term is due to the disconnected expectationhS0,i ihSn,j i which comes from the external potential,while the upper left block comes from correlations in thequantum fluctuations. Note that in the large s limit werecover classical behavior, as we would expect.We may also consider two spins interacting by a (1) · S (2) . I could not find a simpleHeisenberg term Sclosed form for the classical Hamiltonian H. For correlation functions, let us consider the case s 1/2 andcalculate:DE(N n)γ enγ (1) S (1) δij 2 e.S0,i n,j27 9eN γThe qualitative behavior as we send N to depends onwhether γ 0 (i.e. antiferromagnetic) or γ 0 (ferromagnetic). In the antiferromagnetic case, the termsmultiplied by eN γ dominate, and we have:DEe nγ (1) S (1)S.0,i n,j δij9In the ferromagnetic case, the terms multiplied by eN γare suppressed, and we have:DE2 e n γ (1) S (1)S.0,i n,j δij27These are qualitatively very different behaviors, withthe classical analogue of the ferromagnet exhibiting longrange order and the classical analogue of the antiferromagnet being disordered. The reason for this is that aproduct spin coherent state is a ground state for the ferromagnet, but not the antiferromagnet—the only groundstate of the antiferromagnet is in fact maximally entangled. Nevertheless, the classical ensemble correspondingto the ferromagnetic pair of spins is still at a finite temperature. This appears to contradict the argument I gavein the introduction that if the ground state of the system is one of the states in the basis used to constructthe classical analogue, the classical analogue must be atzero temperature. But that argument assumed that thestates used to resolve the identity were all orthogonal toeach other, which the spin coherent states are not—thefinite temperature comes from the nonzero overlap of thespin coherent states.IV.SPIN CHAINSgapped (the Haldane conjecture).Instead of considering discrete imaginary time, as before, imagine inserting enough resolutions of the identitythat the classical configuration may be thought of as acontinuously evolving spin. Then, following Fradkin, itis a reasonable approximation that the classical Hamiltonian as a function of n(t) is just the expectation of thequantum Hamiltonian in the ni state, plus the log ofh n(t) n(t δt)i. From the previously evaluated overlap,this log will have a real part equal to: 1 n(t) · n(t δt)re logh n(t) n(t δt)i s log,2which in the continuum limit will produce a kinetic termwhich penalizes dramatic changes in n. The imaginarypart, the so-called Wess-Zumino term, gives the areaswept out by the line connecting the spin to the northpole:im logh n(t) n(t δt)i Φ( n(t), n(t δt), ẑ).To summarize, the spin coherent state formalism mapsany quantum spin system onto a classical system in onehigher dimension, with a term coming from the quantum Hamiltonian, a kinetic term for the imaginary timedirection, and the imaginary Wess-Zumino term:H HQ Hkin HWZ .Let us consider what happens to the Wess-Zuminoterm on a 1d antiferromagnetic chain. This argumentclosely follows [6], but I have attempted to be more wordysince I found that treatment quite telegraphic. For a single spin, the Wess-Zumino term gave us the area sweptout by the trajectory of the spin in imaginary time [12].For the chain, since the spins are staggered (i.e. it is favorable for the spins to be antialigned), the area sweptout by one spin will nearly cancel the area swept outby its neighbor. To make this precise, let us characterize the state of our chain using unit vectors ni which arestaggered relative to the physical orientation of the spins.This means, for example, that a constant ni configuration corresponds to the Néel state. Then the topologicalportion of the action may be written:Stop SNX( 1)j SWZ [ n(j)],j 1Now that we have some experience with spin coherentstates and their application to the quantum-classical correspondence, we consider the application of these techniques to the problem of 1d antiferromagnetic Heisenbergspin chains. We will show that these map onto an O(3)nonlinear sigma model with a topological term. Then,we review an argument, first due to Affleck [3], that usesthis language to explain why half-integer antiferromagnetic spin chains are gapless while integer-spin ones arewhere SWZ is the area swept out by n as it evolves andthe ( 1)j comes from the fact that the n’s are staggered.We now group this sum into pairs of terms to obtain:XStop SSWZ [ n(j 1)] SWZ [ n(j)],j oddIf we let δ n(j) be the (assumed small) difference between n(j) and n(j 1), we may evaluate this area difference by
5accumulating small quadrilaterals with sides δ n(j) and 0 n(j), where x0 is time. The area of each small quadrilateral may be obtained by dotting the sides after rotating one by 90 by crossing with n(j):modes:ZD eiZ1 [m] iStop SX Zj odddx0 δ n(j) · ( n(j) 0 n(j)).(3)0 n(j) m(j) ( 1)j a0 (j),δ n(j) n(j 1) n(j) (m(j 1) a0 (j 1)) (m(j) a0 (j)) a0 ( 1 m)(j) 2a0 (j)PPlugging this into (3) and replacingj oddR(2a0 ) 1 dx1 gives: Z12 Stop S d x( 1 m) · (m 0 m), 2SLtop ( 1 m) · (m 0 m) S · (m 0 m). 27 Having expressed the Wess-Zumino contribution to theLagrangian in terms of the m and fields, we would nowlike to add the term coming from the antiferromagneticexchange interaction, which on the lattice is:Lmag JS 2 n(j) · n(j 1) ei12 4a0 Jd2 x 21 4a0 JS 2 2 S ·(m 0 m)S 2 (m m 0 m)·(0 m),,2( 0 m) ,where we have used that m ·m 1, which also impliesm · 0 m 0. This gives:1 1a0 JS 222( 0 m) ( 1 m) 2 4a0 J2S m · ( 0 m 1 m) ,2 1122 ( 0 m) vs ( 1 m) 2g vsθ m · ( 0 m 1 m) .4πL(m) where m is normalized and is small. The reason forthis is that the staggering transformation we have performed, n(j) 7 ( 1)j n(j), does not preserve the quantum mechanical angular momentum commutation relations: [Si , Sj ] i εijk Sk is even under inversion on theL.H.S. but odd on the R.H.S. Thus, the “unstaggered”field must be used to capture quantum mechanical fluctuations in the angular momentum. Using this, we have:1 22JS (δ n(j)) .2Rewriting in terms of m and gives: 1 2 2JS a0 ( 1 m) 2 4a20 2 .2Putting these together (and using the cyclic property ofthe scalar triple product) gives:L(m, ) 2a0 JS 2 2 S · (m 0 m) 1 e 2 4a0 JS2TAt this point, following Affleck [13], Fradkin rewrites nas a sum of two terms:Lmag Ra0 JS 2S2( 1 m) m · ( 0 m 1 m) ,22compare (7.67) in Fradkin. Now, we integrate out the where we have introduced g, vs , and the topological angleθ 2πS. Now, when we Wick rotate back to imaginarytime, each time derivative 0 picks up a factor of i, sothat the kinetic term becomes negative and the topological term becomes imaginary. The topological term isa pullback of the area form on the target S 2 , so if wecompactify the domain spacetime of the nonlinear sigmamodel into a sphere, it counts the degree of the configuration seen as a map from S 2 to S 2 . Since this is alwaysan integer, the partition function seen as a function of θis periodic with period 2π, which is why the symbol θ isconventionally used for this parameter. Since θ 2πS,this means we only care about whether S is an integer orhalf-integer. The behavior in each of these two cases iswhat concerns us next.With our mapping between 1d quantum antiferromagnets and classical O(3) nonlinear sigma models with a θterm in hand, we are in a position to discuss the Haldaneconjecture, which says that half integer spin Heisenbergantiferromagnets are gapless and integer spin ones aregapped. Let us consider integer spin first. The topological term is zero in this case, and we just have a classical NLSM. While the 2d NLSM is scale invariant atthe dimension counting level, its conformal symmetry isanomalous and the model acquires a correlation length(which on the quantum side translates to a gap).To see what happens to half-integer spins, Fradkinnotes that the topological term does not change underRG flow, while the coupling strength g increases (corresponding to decreasing S). Thus, we must only understand what happens to the spin-1/2 chain to understand what happens to all half-integer antiferromagneticHeisenberg chains. The spin-1/2 chain is integrable bythe Bethe ansatz, which reveals gapless spinon excitations.Affleck [3] presents a less rigorous but perhaps moredirect argument [14]. Imagine adding an anisotropy m21to the O(3) sigma model energy. For very large , spins
6would be heavily penalized for pointing out of the plane,and the model would be an XY model. We know the XYmodel has two phases: a quasi-long-range ordered phaseconsisting of bound vortices and a disordered phase whichis a plasma of unbound vortices. If we examine the phasediagram in the (g, ) plane, the entire 0 line belongsto the same phase as the vortex-plasma phase of the XYmodel. The gist of the argument is that for half integerspin, the topological term suppresses vortex proliferation,and thus the mechanism which gives the O(3) model afinite correlation length.To show this, note that at any non-infinite value of , there are two kinds of vortices: vortices where thespin moves up and vortices where the spin moves down.Imagine two configurations that are identical except forwhich direction the spin at the vortex core points. Because each vortex configuration covers half of the targetS 2 , and θ π, these two configurations will have Boltzmann weights which differ by a factor of i/( i) 1,and will thus cancel.CONCLUSIONIn this note, we have developed the spin coherent stateformalism, and applied it to obtain discrete-imaginarytime classical analogues for a single spin in a magneticfield and two spins with an isotropic exchange interaction. We computed some correlation functions of interestin those models and commented on their physical interpretation. Then, we noted that for continuous imaginarytime, the spin coherent state formalism produces a classical analogue Hamiltonian which consists of a term com-[1] Paul Fendley. Modern Statistical Mechanics.[2] Hans-Jrgen Mikeska and Alexei K. Kolezhuk. Onedimensional magnetism. In Ulrich Schollwck, JohannesRichter, Damian J. J. Farnell, and Raymod F. Bishop,editors, Quantum Magnetism, Lecture Notes in Physics,pages 1–83. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004.[3] Ian Affleck. Mass generation by merons in quantum spinchains and the O(3) \ensuremath{\sigma} model. Physical Review Letters, 56(5):408–411, February 1986.[4] I believe the result I refer to in this note as the “Haldaneconjecture” is well-established, but it appears to be thecommonly used name for this result.[5] F.D.M. Haldane. Continuum dynamics of the 1-d heisenberg antiferromagnet: Identification with the o(3) nonlinear sigma model. Physics Letters A, 93(9):464 – 468,1983.[6] Eduardo Fradkin. Field Theories of Condensed MatterPhysics. Cambridge University Press, February 2013.Google-Books-ID: x7 6MX4ye wC.ing from the quantum Hamiltonian, a kinetic term, anda WZ term. We explained how the WZ term gives riseto the topological term in the O(3) description of antiferromagnetic spin chains, and argued how this topologicalterm supresses the generation of a mass at θ π (i.e.half integer spin).There are some things which still puzzle me. One issue which was raised at several points was the origin ofclassical temperature in the quantum-mechanical model.The fact that the ferromagnetic spin pair exhibited longrange order while still being at a finite temperature issurprising, and suggests that I haven’t yet completelyunderstood what is going on (or made an error in thecalculation). Another thing I wish I understood betterwas where Affleck’s parameterization of n m a0 comes from in the spin coherent state language. If n, m, and are operators, as they are in Affleck’s derivation(he doesn’t use spin coherent states), then this makessense, because as mentioned in the main text one cannot negate all the components of n without violating thesu(2) algebra. But if n is just a label on spin coherentstates, it seems like there should be nothing wrong withstaggering them.ACKNOWLEDGMENTSI would like to thank Professor Max Metlitski for introducing me to this topic, and Professor Mehran Kardarfor giving interesting problems relevant to this topic. Inthe course of being a writing TA for 8.06 I also benefitted from conversations with Hikari Iwasaki, who wrotea term paper on the quantum-classical correspondencewith emphasis on the Ising model.[7] Timothy H Hsieh. From d-dimensional Quantum to d 1-dimensional Classical Systems.[8] octonion om/q/404586(version: 2018-05-07).[9] This differs from the equation given in Fradkin slightly,but I think what I have written here is right.[10] J M Radcliffe. Some properties of coherent spin states.Journal of Physics A: General Physics, 4(3):313–323,May 1971.[11] Anna Vershynina. NOTES ON COHERENT STATES.[12] Actually, for this portion of the argument, I work in realtime to follow Fradkin, but this is not a major issue.[13] I. Affleck. Quantum spin chains and the Haldane gap.Journal of Physics: Condensed Matter, 1(19):3047–3072,May 1989.[14] It is very related to Problem 9 on Exam Review 3.
May 20, 2019 · Let us examine what happens to the Mermin-Wagner theorem under this correspondence. In classical statisti-cal physics, in the form presented in 8.334, the Mermin-Wagner theorem rules out the possibility of symmetry-breaking order for a continuous symmetry in 2 spatial di-mensions. The order i
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Spin current operator is normally defined as: Issues with this definition: 1. Spin is not conserved so the spin current does not satisfy the continuity equation 2. Spin current can occur in equilibrium 3. Spin currents also can exist in insulators 4. Spin current cannot
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Chapter 5 Many-Electron Atoms 5-1 Total Angular Momentum Spin magnetic quantum number: ms 2 1 because electrons have two spin types. Quantum number of spin angular momentum: s 2 1 Spin angular momentum: S s s ( 1) h h 2 3 Spin magnetic moment: μs -eS/m The z-component of spin angular momentum: Sz ms h 2 1 h
Of the many non-linear optical techniques that exist, we are interested in the coherent Raman rl{ effect known as Coherent Anti-Stokes Raman Scattering (CNRS). The acronym CARS is also used to refer to Coherent Anti-Stokes Raman Spectroscopy. CA RS is a four-wave mixing process where three waves are coupled to produce coherent
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in the relevant elements of the SPIN Suite: that is to say, the sales team in SPIN Selling; the marketing team in SPIN Marketing; and the managers in SPIN Selling and SPIN Coaching - to make sure that everyone had the skills they'd need, and many of the reinforcement tools too. To these skills we added
