The Heisenberg Antiferromagnet and the Lanczos Algorithm Abstract

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The Heisenberg Antiferromagnet and the Lanczos Algorithm Abstract

The Heisenberg Antiferromagnet and the Lanczos AlgorithmB.Sc. Final Year ReportMatthew HolmesDepartment of PhysicsUniversity of SurreyGuildford GU2 7XHAbstractThe Hamiltonian for the antiferromagnetic, S=1/2 spin chain is constructed using the HeisenbergHamiltonian. A largely quantitative analysis of the Hamiltonian is carried out using the Lanczosmethod. The method is used to calculate the ground state eigenvalues of the Hamiltonian. The valuesfound with the Lanczos method strongly diverge from the exact values, due to the loss of orthogonalityin the Lanczos basis. The exact answers are obtained using a Jacobi method, which limited thecomparison to an 8 spin chain on the author's computer.The loss or orthogonality is reduced by using the modified Lanczos method, which maintains theoverlap in successive Lanczos basis states to 10 -20 . This improves accuracy but accurate eigenvaluesare still elusive beyond a 3 spin chain. Complete accuracy is achieved be placing the chain in amagnetic field, pulling apart the eigenvalues and allowing an accurate treated of up to 12 spins on thecomputer used, demonstrating the efficacy of the Lanczos method.


BSc Final Year Report - Matthew HolmesContentsAbstract 1Introduction 3Theory of Magnetic OrderingMagnetic Domains 3The Exchange Interaction 3The Heisenberg Hamiltonian 4Quasi-1D Spin ChainsModelling a 1D Spin ChainThe Spin Chain HamiltonianLadder OperatorsComputing The HamiltonianGround EigenstatesBlock DiagonalisationThe Lanczos AlgorithmSummaryAdvantages of the Lanczos MatrixInitiating the ProcedureApplication & AnalysisOrthogonalityGround State DivergenceThe Variational Approach: Modified LanczosEigenvalue Spacing & DegeneracySpin Chains in External FieldsGround State ConvergenceAccessing Larger SystemsLimitations of the Modified Lanczos MethodConclusionAcknowledgmentsReferences2


BSc Final Year Report - Matthew HolmesIntroductionMagnetic ordering in ferromagnetic solids is due to the exchange interaction between spins. Orderingin one dimension, along a ‘spin-chain,’ takes place if the exchange interaction energy between nearestneighbours along that dimension is greater than along any other.The origin of ferromagnetism is reviewed before considering the Heisenberg Hamiltonian as the basisof quantum magnetic ordering, with reference to real systems. From these discussions a model of theone dimensional spin chain is developed. Computation of the closed chain Hamiltonian and itssubsequent evaluation is outlined.The later half of the report focuses on the Lanczos algorithm as a means of evaluating the lowereigenstates of the Hamiltonian. The emphasis is on the modified method; a variational approximationto the full algorithm. The importance of orthogonality in the Lanczos basis is investigated andparticular attention paid to the use of magnetic fields for improving the variational eigenvalues. Thereport concludes by highlighting…cheesecake.Theory of Magnetic OrderingMagnetic DomainsBelow certain critical temperatures, ferromagnetic materials exhibit spontaneous magnetic ordering[17]. There are three types of order, according to the relative orientation and magnitude of interactingmagnetic moments µ, as shown in Fig. 1.(a) M 0(b) M = 0(c) M 0Fig. 1 [17] Spin coupling via theexchange interaction produces (a)ferromagnetism with non-zeromagnetisation M, (b) antiferromagnetismwith zero M and (c) ferrimagnetism inwhich sub-lattices of opposing M andunequal magnitude give a net M. Thearrows show µ in each type of order.Here we are mostly concerned with antiferromagnetism, although the exchange interaction isresponsible for all spontaneous ordering.The exchange interactionThe exchange interaction is a quantum mechanical effect with no classical analogue [1]. It does notarise from the dipolar interaction, ε=µ B B, as a simple calculation can show [17]. For a typical latticespacing r~3Å for two atoms each with µ~µ B ,µ µ4πr20 Bε ~ , (1)3where µ B is the Bohr magneton, we have ε ~10 -6 eV. This is insufficient to withstand the thermaldisordering of kT room ~25meV. We account for the ordering by a combination of Coulombic interactionand Pauli's exclusion principle. The wavefunction describing two electrons is antisymmetric whenall coordinates are exchanged [17],( r , s ; r , s ) = −ψ( r , s ; r s ) , (2)1 1 2 22 2 1,1where r and s are the position and spin vectors respectively. Then the probability density || 2 is zero atr 1 = r 2 for like spins and non-zero for unlike spins. The mean separation is greater for like spins, and isthe origin of the exchange energy. The difference in Coulombic repulsion energies for like and unlikespins provides a potential in which magnetic ordering can occur.3


BSc Final Year Report - Matthew HolmesWe may now introduce the exchange energy term, Eq. (3) for spin interactions [1],E = −2Js ⋅s, (3)12where s z =±/2. Conventionally we work in the z-basis, and +1/2 is described as 'spin up'. Throughoutthe report /2 is absorbed into J. The exchange parameter J is an integral describing the type ofinteraction. The negative sign is included by convention so that in ferromagnets J>0, favouringpolarised pairs, whilst Jˆi⋅Sˆˆ x y z= Sˆi + Sˆj + SˆkS , where [*San*], (4)jS (5)Where i, j, k are orthogonal unit vectors and indicates summation over nearest neighbour pairs.Interactions do occur over a longer range but diminish exponentially with distance and can bedisregarded [9].Anisotropy in the exchange parameter J ij describes interpenetrating sub-lattices of characteristicexchange type. Thus J is different for inter- and intra- lattice interactions [10]. Within a particularlattice then, J ij is isotropic and can be taken outside the summation. Here we consider only isotropicantiferromagnetic interactions along one dimension.The spin operators in Eq. (5) are the Pauli spin matrices [15],01S ˆ x = , ˆ 0− iS y = , ˆ 10 z = 10 i 0 0−1S . (6)We work in the z-basis where the spin vectors (spinors), Eq. (7), are eigenvectors of z [15].+ 1− 0s = , = 01s . (7)Once a basis is chosen we must stick to it due to the non-commutation of the components of witheach other.Real Materials: Quasi-1D Spin ChainsIf magnetic order is strongly confined to particular directions it can be considered to exist on a distinctsub-lattice, and can be modelled in isolation from others. A measure of the isolation is given by theratio J/J´ of the exchange constants within and between sub-lattices [10,12].Purely 1D (or 2D) long range order cannot occur via Eq. (4) [1,7] but some materials with localisedspin-1/2 and appropriate exchange ratios, permit accurate 1D approximations. The quasi-1Dantiferromagnet model can be applied when J/J´~0.05 [12,21]. The antiferromagnetic couplingstrength J has a wide range, from ~1meV in copper benzoate [8], through ~30meV in KCuF 3 [5,3], to~100meV in La 2 CuO 4 [19].4


BSc Final Year Report - Matthew HolmesModelling a 1D Spin ChainThe Spin Chain HamiltonianFor a chain of N spins, the Hamiltonian is an x square matrix, given by Eq. (8) [1], where henceforth=2 N , the number of basis vectors (spin configurations), .H ψ H ˆ ψ ; (8)k ' k=zk 'zkTo obtain the Hamiltonian we must choose a model on which to base the summation of Eq. (4). Thereare two general chain topologies to follow, open or closed [20,5]. We follow the closed topology, Eq.(9) [4], modelling the chain as a ring.N 1= − ˆ + Hˆ>HH (9)< i,ji,jN ,1Thus, for an N=5 chain say, with an initial state 1 =| 1 2 3 4 5 >, we cycle through the basis, actingwith in each case. At the chain's end, when i=N, we set j=1, and consider the interaction betweenend states. The spinors are not eigenvectors of x or y , so that H contains off-diagonal elements andmust be diagonalised to find the eigenvalues.Ladder OperatorsThe computation is simplified by substituting for x and y with terms containing the ladder operators + and - [5,2]. The elements of + and - may be guessed but are derived [20] for clarity. Consider +acting on s - (flipping the state) and on s + (destroying the state),ˆs 11 12 ( ↓) = = +S ; s 12 = 1, s 22 = 0 (10i)ˆ ss21ss122011 11 ( ↑) = = 0 s210010+S ; s 11 = s 21 = 0 (10ii)And, similarly for - , we obtain the ladder operators,+ 01− 00Ŝ = , = 0010From Eqs. (11) the spin operators are,( ˆ ˆ −ˆ S )+ + S( SˆSˆ−)S x =, Sˆ+ y −=2Ŝ (11)Substituting Eqs. (12) into Eq. (4), the Heisenberg Hamiltonian becomes,H = −J2iz z + − − +( Sˆ⋅ Sˆ+ Si⋅ Sj+ Si⋅ Sj)i j1 ˆ ˆ 1 ˆ ˆ22< i,j>. (12)ˆ . (13)The ladder operators are useful because they can annihilate states [2], whereas x or y always flip thestate. Polarised pairs are always destroyed by the ladder terms in Eq. (13), as shown in Eq. (14).5


BSc Final Year Report - Matthew Holmesˆ⋅ Sˆ0111 ( ↑ ↑ ) = 01 + −2 1 2 1 2=2 000 1 0 011220S (14)And likewise for the opposite polarisation. Similarly, for those staggered pairs in which just one spin isdestroyed, we destroy the pair because of the resultant multiplication by zero.Computing The HamiltonianThe basis { } and ladder operations are conveniently provided for [20] in Fortran with the intrinsic bitprocedures. The present state , is represented using IBITS(I,POS,LEN), with IBCLR(I,POS) andIBSET(I,POS) emulating the ladder operators. In Fig. 2, an example state from an N=3 chaindemonstrates how IBITS is used.01ψ4= ↑2↓1↓0= 100 =IBITS(4,POS,N=3)4Fig. 2 Binary representation in Fortran of a 1D spin chain configuration 4 . The bit positionsPOS are numbered in subscript from the right (from 031 in a 32-bit number). The requiredsegment (chain length, N) of the full number is selected from the right with LEN.The Hamiltonian is computed according to Eq. (8), which can be implemented by following the flowscheme in Fig. 3.DO k1. IBITS=IBITS(i,POS,N)2. Move along IBITS with POS=0N, applying Eq. (13) foreach pair3. If states are flipped, store copy of altered state for use in4.Fig. 3 Flow scheme forconstructing theHamiltonian. Theprocedure followsnaturally from Eq. (8).Where states areflipped, IBCLR sets to 0and IBSET sets to 1.DO k'DO until k'=DO until: k=4. Apply orthonormality relation5. Calculate matrix elementIn this way matrix operations are unnecessary and H can be generated by logic statements. It isinstructive to consider a two spin chain, which is, essentially, the 'unit' on which acts.H ˆ 00 =1 00 + 0 , no ladder term contribution.4ˆ1H 01 = −1 01 + 10 , contribution from 1st ladder term.42ˆ1H 10 = −1 10 + 01 , contribution from 2nd ladder term.42(15)H ˆ 11=1 11 + 04Applying the orthonormality relations to Eqs. (15) we obtain the Hamiltonian for the N=2 chain whichis real symmetric, as expected.6


BSc Final Year Report - Matthew Holmes1 40= −J⋅000−14120012−14000014 H (16)We do not use a closed chain for two spins as this would involve two interactions between a pair,which is unphysical.Ground EigenstatesThe S=1/2 antiferromagnet is one of the few systems for which a non-trivial ground state is exactlyknown [4]. The ferromagnetic, J>0, ground states, obvious from Eq. (16), are always the fullypolarised states. With J. v 1 must not be orthogonal to the desired state, nor can it have equal coefficients. A neworthogonal vector v 2 is produced by subtracting from H|v 1 > the projection along v 1 .ν2ν1−α1ν1= H (18)Thus v 2 is orthogonal to v 1 , < v 1 |v 2 > = 0, whereby,0 = ν H ν −αν ν , and, (19)α11ν Hν1111 11= (20)ν1ν1For the next state, v 3 , we have,ν23ν2−α2ν2− β1ν1= H (21)7


BSc Final Year Report - Matthew HolmesWhere v 3 is orthogonal to the previous two states, < v 1 |v 3 > = < v 2 |v 3 > = 0, requiring that,αν Hν2 22= , (22)ν2ν2νν2 2 2β1= . (23)ν1ν1In this way we express the Hamiltonian in the Lanczos basis. Generally,H ν, (24)n2= νn+ 1−αnνn− βn−1νn−1with coefficients,αν Hνn nn= , (25)νnνnννβ . (26)2= n nn− 1νn−1νn−1H is now expressed in a new orthogonal basis. After normalising the basis, the diagonals n , areunchanged, whilst the sub-diagonals equal n . The generalised Lanczos matrix is given by Eq. (27),Lα1 β1 0 = 0βαβ⋅1220βα⋅23β⋅β⋅n−1⋅3⋅α⋅n⋅β⋅0nβ⋅⋅η−10 0 βη−1 α η H (27)Advantages of the Lanczos MatrixThe speed at which lower lying eigenstates are found is the key feature of the Lanczos method [20].This follows from the repeated operations with H, such that eigenvalues of greatest magnitude areprominent in the expansion of Eq. (28) [5],N2nnν1= aiHi=1i2NH ψ = λ aψ(28)i=1niiiIn Eq. (28) n is the current number of iterations and a i the coefficients of the initial vector v 1 . For eachiteration, the current nxn segment of the full matrix is diagonalised. The procedure terminates withconvergence to the eigenvalue of interest i which, from Eq. (28), occurs long before the number ofiterations exhausts the configuration space [5].Also, we need not store the Lanczos basis. We can construct the eigenstate v i corresponding to i byrunning the procedure again. To initiate the second run the ith vector (corresponding to the currentapproximation to i ) of the transformation matrix of the nxn segment is used [19]. This recoverymechanism is outlined in Fig. 7.8


BSc Final Year Report - Matthew HolmesIf only the ground or lower levels are required the Lanczos method enables much larger systems to betreated than would otherwise be the case. That said, the treatment of 1D chains is still limited to N~20spins [19] even on powerful systems [20]. At first glance this may seem small but it presents a~10 6 x10 6 matrix, which is large!Initiating the ProcedureThe initial vector v 1 , must have some projection along the desired eigenstate v. This is usually satisfiedby randomising the coefficients [20,19] of v 1 ,CALL RANDOM_NUMBER(v1)v1= v1/SQRT(DOT_PRODUCT(v1,v1))Remember, v 1 is a vector in the original basis { i } and has coefficients. Generally then,v 1 =c 1 1 +c 2 2 +…c ,. It is tempting to choose equal coefficients (as the author did) in the hope someprojection along the desired state. The Lanczos method will not work if this is done [20] as v 2 will bezero and the iteration fails. A 'proof' is proposed below.Consider the 'symmetric vector' (of equal projection, c, in all basis states) v, = c{ ψ i}; i = 1. (29)Normalising, we obtain the initial vector v 1 ,c= { ψ } = { ψ } { ψ }1 i2 i=cηc1 , (30)ηias the i are themselves normalised such that |v|=[c 2 |{ i }| 2 ] 1/2 =c 1/2 . Substituting Eq. (30) into Eq. (20),we have for 0 ,α1η{ } H { }ψ iψ i0 = (31i)εεα0=ii=ηη2{ ψ } { ψ } = { ψ } εi(31ii)Where ε is the sum along any row or column of H. Substituting (Eq. 31ii) into Eq. (18), we obtain,ε1{ ψ } −ε{ ψ } 02=ii= . (32)ηηFrom Eq. (32) a new vector v 2 cannot be produced and we conclude that an initial symmetric vector isunsuitable when representing a symmetric matrix. Such a vector is orthogonal to all eigenstates. Thisresult is physically intuitive, as a symmetric vector represents a state of zero magnetisation.Application & AnalysisOrthogonalityWhen using the Lanczos method it is vital that the basis is orthogonal. Fig. 4 shows the increasingoverlap between successive states in the absence of any remedial action. Without such action, rogueeigenvalues are introduced to the spectra [14]. This makes it difficult to check for accuracy with anexact routine when setting up the Lanczos method.9


BSc Final Year Report - Matthew Holmes1.E-071.E-171.E-27Overlap1.E-371.E-471.E-571.E-671.E-772345671.E-871 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18No. of IterationsFig. 4 Orthogonality between successive Lanczos basis states. Above N=4, orthogonality islost, whereas below N=4 it improves. The data were obtained using 18 place precision.Finite precision arithmetic [14] was felt to be of no importance for the smaller systems under study. Toestablish its influence, the overlap in the N=7 system, which from Fig. 4 increases throughout, wasplotted for different precisions as shown in Fig. 5.Overlap1.E-031.E-051.E-071.E-091.E-11Loss of Orthoganality in The Lanczos BasisFig. 5 The effect offinite precision onoverlap in theLanczos basis.Using 18 places ofprecision reducesoverlap but fails tohalt its growth.1.E-131.E-157 Place Precision1.E-1718 Place Precision1.E-191 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18No. of IterationsFrom Fig. 5, the overlap grows throughout and the effect of the different precision is slight. We willsee shortly that purely numerical errors have little impact when using the Lanczos method in smallersystems.Ground State DivergenceWhilst the overlap may seem small, it has a dramatic effect on eigenvalue calculations. Fig. 6 showshow the ground state eigenvalue, computed in the Lanczos basis, differs from that obtained directlyfrom the original Hamiltonian. To obtain the eigenvalues directly, a complete diagonalisation wasperformed using the Jacobi method. Obtaining eigenvalues is one the fundamental problems inquantum mechanics, due to the complexity of evaluating the whole matrix. Comparison with the10


BSc Final Year Report - Matthew HolmesJacobi results are for illustrative purposes and, in general, methods such as the Lanczos schemeconsidered here are preferred, and often essential.The ground state energy should be roughly linear with N [7]. Our direct evaluation is reassuringly ontrack. The observed divergence of the Lanczos values however, requires further analysis. Theinfluence of finite precision is also seen to be negligible.Ground State Energy, J0-10-20-30Original Hamiltonian-40 Lanczos Hamiltonian: 18 Place PrecisionLanczos Hamiltonian: 7 Place Precision-502 3 4 5 6 7System Size, NFig. 6 Divergence ofthe ground stateenergies. If overlapis ignored, the valuescomputed from theLanczos matrixbegin, very quickly,to diverge from thetrue ground states.The N=4 estimate is~0.1J in error for 18place precision,where J are units ofexchange energy.The negligibleinfluence of finiteprecision arithmeticis clear.If accurate eigenstates are to be obtained, the overlap must improve. Simply increasing the precision isinsufficient. Another option is to suspend the procedure when the overlap becomes too large. We thenuse the current eigenvalue of interest k to construct a new vector, with greater projection along thedesired state, with which to restart the procedure. The flow scheme below shows how this may beimplemented.DO1. Start with normalised vector, v 12. Find next state v 23. Calculate eigenvalues andeigenvectors of current matrix4. Calculate overlap5. If overlap > then build new v 1 from k and cycle to 1.Fig. 7 Flow scheme forevolving the Lanczosbasis whilst maintainingorthogonality.We terminate before theoverlap reaches , thevalue at whichunacceptable errorsappear. Using the currenteigenvalue of interest k anew vector v 1 isconstructed for restartingthe procedure.DO until: new k ~ old kWe need not store the Lanczos basis [19,13], when running the procedure. If step 5 of Fig. 7, isrequired, we select f k (the kth column of the transformation matrix found in step 3) corresponding to k .f k is then used to initiate a separate run, building a new starting vector v 1 for step 1. of Fig. 7.The Variational Approach: Modified LanczosThe idea just introduced may be taken to its extreme by continually restarting after just one iteration[13,12]. In this way we only ever evaluate 2x2 matrices and the current ground state is always in thememory. This approach, known as the modified Lanczos method, is a variational technique [8].Although more pedestrian than the regular routine [13], for smaller systems it is convenient foreliminating overlap, confining it to ~10 -20 for all system sizes. Fig. 8 shows how this effects the groundstate calculations.11


BSc Final Year Report - Matthew HolmesGround State Energy, J0.0-0.5-1.0-1.5-2.0-2.5-3.0-3.5-4.0Ground State Discrepancy for N>3 With The ModifiedLanczos MethodOriginal HamiltonianMod. Lanczos Hamiltonian-4.52 3 4 5 6 7System Size, NFig. 8 Ground stateenergies using themodified Lanczosmethod. Eigenvaluescomputed using themodified are muchimproved on therunaway divergenceseen when overlap isignored. Agreementbeyond N=3however, is still aproblem.Using the modified method, overlap is maintained at ~10 -20 throughout. The approximated eigenvaluesin Fig. 8 are much improved and approach the true values from above, as expected of a variationalmethod.It was hoped that the disparities beyond N=3 might be better understood with a plot of the convergence.This is shown in Fig 9. It is apparent that oscillatory behaviour begins at N=6, along with a significantincrease the number of iterations required for convergence. For an N=6 chain, the magnitude of theoscillations slowly decrease but are apparently constant for N=7. Omitted for clarity, the oscillationsfor N=7 continue even after thousands of iterations.-1Convergence With The Modified Lanczos MethodJ, Exchange Energy-1.5-2-2.5-3-3.50 10 20 30 40 50 60No. of IterationsFig. 9 Ground state eigenvalue convergence using the modified Lanczos method. As thesystem size increases beyond N=5, oscillatory convergence sets in, increasing the timerequired. Indeed, for N=7 (and N>7 - omitted for clarity) the system is still oscillating beyond10,000 iterations.The reason for the oscillations is unclear. It was assumed that a poorly spaced eigenvalue pair in onethe 2x2 matrices would start the oscillatory behaviour. The eigenvector constructed from one of thepair would not necessarily be a better approximation as it could correspond to a different eigenstate.However, the eigenvalue pairs are always well spaced. We will see later that the other eigenvalue is12


BSc Final Year Report - Matthew Holmesalways positive and converges to the ferromagnetic ground state. The pair are always well separatedthen.The modified method works by iteratively improving the initial trial wavefunction v 1 , such that it hasincreasing projection along the ground state. Although the eigenvalue pairs are not to blame, theconstruction of new trial wavefunctions with less ground state overlap could still be the cause.Oscillatory convergence is not believed to be solely, or even at all, responsible for any inaccuracyhowever. The eigenvalues of the N=4,5 systems readily converge, and are more accurate than largersystems (Fig. 8) yet fail to show the same accord as the N3 systems. Another factor was sought.Eigenvalue Spacing & DegeneracyHaving maintained an overlap of approximately zero, ~10 -20 , it was hoped that the variational estimateswould improve on the apparent N=3 limit of the regular method (Fig. 6). Whilst accuracy is clearlyincreased in Fig. 8, ground state estimates of a different physical model, the Mathieu equation [8], arevirtually concurrent. This suggested that the overlap is a feature of the 1D spin chain system.Although no finely spaced eigenvalue pairs ever appear in the 2x2 matrices of the modified Lanczosscheme, it is suggested [20,18,9] that eigenvalue spacing, or degeneracy, in the full spectra may inhibitaccuracy for N>3.As eigenvalue separations generally decrease with size in quantum systems [20], the loss of accuracywith N suggests that poor spacing may be a factor.Spin Chains in Magnetic FieldsIf poorly spaced eigenvalues are responsible for the discrepancies, this should be resolved by placingthe chain in an external magnetic field B, pulling apart the eigenvalues in a similar fashion to Zeemansplitting. By aligning the field along the z-axis, as described by Eq. (33) [8], we simplify thecomputation.zH = −JSˆ ⋅ Sˆ− gµB Sˆ(33)i< i,j>jBiiHere g is the Landé splitting factor, ~2 for pure spin, and µ B is the Bohr magneton, ~58µeVT -1 . Thefield term is summed over i, it being independent of exchange interactions. From Eq. (33), magneticordering via the exchange interaction now competes with the tendency toward dipole-field alignmentvia µ B B. For positive B, the arbitrary negative sign of the field term favours spin-up states in thechosen coordinate system. For strong enough fields, the exchange coupling should be sufficientlyfrustrated that we observe an antiferromagnetic, fully polarised spin-up ground state, of oppositepolarity to the ferromagnetic ground state in the field.Ground State Energy, J0-1-2-3-4Variational Estimates at B~10THamiltonian Ground StateFig. 10 Concurrentground stateenergies. Thevariational estimatesapproach the truevalues as the fieldstrength is increasedtoward ~10T, thelowest field at whicha complete accordexists.-5Modified Lanczos GroundState-62 3 4 5 6 7 8System Size, NAs the field strength is increased, the variational eigenvalues approach the true values. Completeagreement up to N=8, first occurs at a field strength of ~10T as shown in Fig. 10. This comparative13


BSc Final Year Report - Matthew Holmesanalysis is limited to 8 spins on the author's computer, due to the cost of evaluating the fullHamiltonian. The field is calculated assuming a value of order ~meV for an antiferromagnetic J, asdiscussed earlier.We are not concerned with the quantitative dependence of eigenvalue splitting, except to say that it isreasonable to study the chain in such strong fields [5]. Of interest here is the success of the field term,in combination with the modified approach, of obtaining accurate estimates of the eigenvalues. Aqualitative analysis is sufficient to demonstrate how a field term improves accuracy. Fig. 11 offers aneat summation of the available accuracy in varying field strengths as the chain size increases.Fig. 11 Improvingvariational accuracy.The difference Jbetween variationalground stateeigenvalues and theexact energies, forvarious fieldstrengths B and chainlengths N. The fieldis oriented along the+z axis, favouringspin-up as the energyminimising state.In Fig. 11, the clear trend for J to increase with N and for J0 with increasing field strength,supports the assertion that eigenvalue spacing affects the attainable accuracy for N>3. As the chainlength increases, we should expect the energy levels to move closer together. By pulling them apart tocompensate, accuracy is maintained.Ground State ConvergenceIt remains to show that the new estimates are efficiently produced and that oscillating convergence hasbeen eradicated. Previously, Fig. 9, convergence was seemingly impossible for N7, and slowed to acrawl for N=6. The accuracy for all N is evident from Fig. 10 and, convergence not having been aproblem for small chains (N


BSc Final Year Report - Matthew HolmesGround State Energy, J-2-4-6-8-10-12Ground State Convergence for h=1.5Fig. 12 Ground stateconvergence after~35 iterations andsooner for N=6.Here h=g BB/J. Byplacing the chain in amagnetic field, a fastconvergence isachieved, even atN=12.-14-160 10 20 30 40 50 60 70No. of IterationsThe previous oscillations have now vanished, permitting a fast and accurate convergence. Even up toN=12, convergence is now possible. We cannot say yet whether this level is accurate however, as theN=4,5 systems in Fig. 8 show. Despite their fast convergence, Fig. 9, they were still inaccurate.The actual time for convergence still increases with N due to the number of operations that must beperformed with a large H. In Fig. 12, convergence for N=12 at h=1.5 takes ~5mins on the author'ssystem, the time seemingly dropping exponentially as N is reduced, being virtually instantaneous belowN=9. This is, of course, an inescapable correlation but it is encouraging that the number of iterationsrequired is in keeping with smaller systems. It is also probable that more efficient coding techniquesthan those employed, might reduce the convergence time.Accessing Larger SystemsDue to its rapid convergence toward the lower levels, interest in the Lanczos method resides in itsability to treat larger systems, with correspondingly better approximations to physical systems [*Cloi*].The convergence for N=12 gives some indication that the method is valid in this regime but does notguarantee accuracy. For this a comparison is needed. At this size a complete diagonalisation usingJacobi is too costly. An application to a physical system with subsequent comparison to experimentaldata is the preferred test, but lies outside the scope of this report.We can instead, and more conveniently, rely on the apparent tendency of the ground state levels toincrease linearly with N in high fields. Fig. 13 indicates that linearity continues beyond N=8. Theapparent discontinuity at N=3 is where the ring model begins. Applying the model for N=2 would beunphysical, as mentioned previously, as it doubles up the exchange interaction.Ground State Energy, J-15-13-11-9-7-5-3Ground State Energies for Various hFig. 13 Ground stateenergies for h=1.5,demonstrating thatthe modified Lanczosmethod allows largersystems to be treated.Memoryrequirements preventcompilation for N>12but the estimates forN>8 show excellentagreement with thepreceding linearity.-12 3 4 5 6 7 8 9 10 11 12System Size, N15


BSc Final Year Report - Matthew HolmesPrevious comparisons with exact values, in Figs. 8 & 10, hinted at the trend toward linearity. With nofield, Fig. 8, the ground state increases haphazardly with N. In a field of 10T (Fig. 10), when thevariational estimates first agree for all N, the rise in energies shows more linearity. The field strengthin Fig. 10 is similar to the h=0.9 line in Fig. 13, above. For confidence in the estimations beyond 8spins, the field parameter h is increased for safety. In this way we can be confident that the degree ofaccuracy of the modified Lanczos estimates for small chains, is retained beyond N=8.Limitations of The Modified Lanczos MethodThe Modified Lanczos method is confined to making estimation of the ground state energies, for boththe ferro- and antiferromagnetic cases. It is thus useful only in the zero temperature limit and cannotevaluate excited states of the chain, for which the full algorithm is required.ConclusionBy applying the Heisenberg Hamiltonian to antiferromagnetically coupled spins along a lineardimension, a model of a closed 1D spin chain has been developed. The justification for such a model ismade with reference to experimental observations of quasi-1D chains and a theoretical consideration ofthe nature of antiferromagnetic exchange interactions.The ground state eigenvalues of the Hamiltonian, calculated with the Lanczos method were found to bestrongly dependent on the eigenvalue spacing. Accuracy was found to be significantly increased byusing the modified Lanczos method, a variational technique. To achieve complete accuracy, however,required that the chain be placed in a magnetic field. With a field of ~10T a complete agreementbetween the Lanczos estimates and a direct evaluation with the Jacobi method, was found up to chainlength of 8 spins.Direct evaluation with Jacobi, for a chain of 8 spins was the limit on the author's computer. The fieldhad the effect of pulling apart the eigenvalues and allowing complete accuracy up to a chain of 8 spins.It also permitted larger systems to be treated than the 8 spin limit using the Jacobi method.The ability for accurate eigenvalue estimation with the Lanczos method, and its ability to treat largersystems, has thus been demonstrated.It is suggested that the excited states of the Hamiltonian be investigated using the full Lanczosalgorithm, the modified method being confined to ground state estimation. The ground stateeigenvector might also be used to construct the operators with which to investigate a wider range ofphysical properties than have been considered herein.AcknowledgmentsI would like to thank my project supervisor, Dr Paul Stevenson, for his excellent guidance (and infinitepatience!) throughout. Now is also the time to acknowledge my debt to the sun; for shining throughmy window one day.References[1] Aharoni A., Introduction to the Theory of Ferromagnetism, Oxford (2000).[2] Brussard P. J. & Glaudemans P. W. M., Shell Model Applications in Nuclear Spectroscopy,North-Holland Publishing Company, Amsterdam, (1977).[3] Chatelin F., Eigenvalues of Matrices, Wiley (1993).[4] Cloizeaux J. & Pearson J. J., Phys. Rev. 128, 2131 (1962).[5] Coldea R. et al., Phys. Rev. Lett. 86, 5377 (2001).[6] Dagotto E. & Moreo A., Phys. Rev. D 31, 865 (1985).[7] Dender D. C. et al., Phys. Rev. B 53, 2583 (1996).[8] Ellis T. M. R. et al., Fortan 90 Programming, Addison-Wesley (1994).[9] Hihilashvili R., Derivation of the Heisenberg Hamiltonian, http://phjoan5.technion.ac.il/~riki/,Israel Institute of Technology Dept. of Physics, (2003).[10] Hutchings M. T., Ikeda H., Milne J. M., J. Phys. C 12, L739 (1979)[11] Krupi ka S. et al, Elements of Theoretical Magnetism, Iliffe (1968).[12] Lake B. et al, ISIS Science, www.isis.rl.ac.uk/science/index.htm, Rutherford AppletonLaboratory, (2001).[13] Lieb E. H. et al., Mathematical Physics in One Dimension, Academic Press (1996).16


BSc Final Year Report - Matthew Holmes[14] Malvezzi A. L., Brazilian J. Phys. 33, 55 (2003).[15] Merzbacher E., Quantum Mechanics, Wiley (1998).[16] Nepomechie R. I., A Spin Chain Primer, www.physics.miami.edu/nepomechie/primer.pdf,University of Miami Phys. Dept. (1998).[17] O'Reilly E., Quantum Theory of Solids, Taylor & Francis (2002).[18] Sandvik A. W., 'Computational Studies of Quantum Spin Systems' & 'Numerical Solutions ofthe Schrödinger Equation', PY502 Computational Physics, Dept. of Physics, BostonUniversity (2004).[19] Scaife B. K. P. et al, Studies in Numerical Analysis, Academic Press (1974)[20] Stevenson P. D., University of Surrey Phys. Dept., Private communication (2005).[21] Tyablikov S. V., Methods in The Quantum Theory of Magnetisation, Plenum Press (1967).17

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