The Heisenberg Antiferromagnet and the Lanczos Algorithm Abstract

**The** **Heisenberg** **Antiferromagnet** **and** **the** **Lanczos** **Algorithm**B.Sc. Final Year ReportMat**the**w HolmesDepartment of PhysicsUniversity of SurreyGuildford GU2 7XH**Abstract****The** Hamiltonian for **the** antiferromagnetic, S=1/2 spin chain is constructed using **the** **Heisenberg**Hamiltonian. A largely quantitative analysis of **the** Hamiltonian is carried out using **the** **Lanczos**method. **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 **the**comparison to an 8 spin chain on **the** author's computer.**The** loss or orthogonality is reduced by using **the** modified **Lanczos** method, which maintains **the**overlap 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 **the**computer used, demonstrating **the** efficacy of **the** **Lanczos** method.

BSc Final Year Report - Mat**the**w HolmesContents**Abstract** 1Introduction 3**The**ory of Magnetic OrderingMagnetic Domains 3**The** Exchange Interaction 3**The** **Heisenberg** Hamiltonian 4Quasi-1D Spin ChainsModelling a 1D Spin Chain**The** Spin Chain HamiltonianLadder OperatorsComputing **The** HamiltonianGround EigenstatesBlock Diagonalisation**The** **Lanczos** **Algorithm**SummaryAdvantages of **the** **Lanczos** MatrixInitiating **the** ProcedureApplication & AnalysisOrthogonalityGround State Divergence**The** Variational Approach: Modified **Lanczos**Eigenvalue Spacing & DegeneracySpin Chains in External FieldsGround State ConvergenceAccessing Larger SystemsLimitations of **the** Modified **Lanczos** MethodConclusionAcknowledgmentsReferences2

BSc Final Year Report - Mat**the**w 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 o**the**r.**The** origin of ferromagnetism is reviewed before considering **the** **Heisenberg** Hamiltonian as **the** basisof quantum magnetic ordering, with reference to real systems. From **the**se discussions a model of **the**one 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 **and**particular attention paid to **the** use of magnetic fields for improving **the** variational eigenvalues. **The**report concludes by highlighting…cheesecake.**The**ory of Magnetic OrderingMagnetic DomainsBelow certain critical temperatures, ferromagnetic materials exhibit spontaneous magnetic ordering[17]. **The**re 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 **the**exchange interaction produces (a)ferromagnetism with non-zeromagnetisation M, (b) antiferromagnetismwith zero M **and** (c) ferrimagnetism inwhich sub-lattices of opposing M **and**unequal magnitude give a net M. **The**arrows 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 interaction**The** 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 withst**and** **the** **the**rmaldisordering of kT room ~25meV. We account for **the** ordering by a combination of Coulombic interaction**and** 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. **The**n **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** is**the** 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 - Mat**the**w 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'. Throughout**the** 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 **the**n, 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 wi**the**ach o**the**r.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 o**the**rs. A measure of **the** isolation is given by **the**ratio 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 - Mat**the**w HolmesModelling a 1D Spin Chain**The** 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). **The**reare 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 **and**must be diagonalised to find **the** eigenvalues.Ladder Operators**The** 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 **the**y can annihilate states [2], whereas x or y always flip **the**state. Polarised pairs are always destroyed by **the** ladder terms in Eq. (13), as shown in Eq. (14).5

BSc Final Year Report - Mat**the**w 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** Hamiltonian**The** 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) **and**IBSET(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 **the**Hamiltonian. **The**procedure followsnaturally from Eq. (8).Where states areflipped, IBCLR sets to 0**and** 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 - Mat**the**w 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 Eigenstates**The** 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 - Mat**the**w 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** Matrix**The** 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 - Mat**the**w HolmesIf only **the** ground or lower levels are required **the** **Lanczos** method enables much larger systems to betreated than would o**the**rwise 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** Procedure**The** initial vector v 1 , must have some projection along **the** desired eigenstate v. This is usually satisfiedby r**and**omising **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 **the**n,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 **the**mselves 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 - Mat**the**w 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 **the****Lanczos** 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 **the**10

BSc Final Year Report - Mat**the**w 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 fur**the**r analysis. **The**influence of finite precision is also seen to be negligible.Ground State Energy, J0-10-20-30Original Hamiltonian-40 **Lanczos** Hamiltonian: 18 Place Precision**Lanczos** Hamiltonian: 7 Place Precision-502 3 4 5 6 7System Size, NFig. 6 Divergence of**the** ground stateenergies. If overlapis ignored, **the** valuescomputed from **the****Lanczos** matrixbegin, very quickly,to diverge from **the**true 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. Ano**the**r option is to suspend **the** procedure when **the** overlap becomes too large. We **the**nuse **the** current eigenvalue of interest k to construct a new vector, with greater projection along **the**desired 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 **and**eigenvectors of current matrix4. Calculate overlap5. If overlap > **the**n build new v 1 from k **and** cycle to 1.Fig. 7 Flow scheme forevolving **the** **Lanczos**basis whilst maintainingorthogonality.We terminate before **the**overlap reaches , **the**value at whichunacceptable errorsappear. Using **the** currenteigenvalue of interest k anew vector v 1 isconstructed for restarting**the** 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 **the**n used to initiate a separate run, building a new starting vector v 1 for step 1. of Fig. 7.**The** Variational Approach: Modified **Lanczos****The** 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 **the**memory. 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 - Mat**the**w 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** Modified**Lanczos** MethodOriginal HamiltonianMod. **Lanczos** Hamiltonian-4.52 3 4 5 6 7System Size, NFig. 8 Ground stateenergies using **the**modified **Lanczos**method. Eigenvaluescomputed using **the**modified are muchimproved on **the**runaway 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 **the**oscillations slowly decrease but are apparently constant for N=7. Omitted for clarity, **the** oscillationsfor N=7 continue even after thous**and**s 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 **the**system 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 one**the** 2x2 matrices would start **the** oscillatory behaviour. **The** eigenvector constructed from one of **the**pair 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** o**the**r eigenvalue is12

BSc Final Year Report - Mat**the**w Holmesalways positive **and** converges to **the** ferromagnetic ground state. **The** pair are always well separated**the**n.**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, **the**construction 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. Ano**the**r 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 **Lanczos**scheme, 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 placing**the** 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 **the**computation.zH = −JSˆ ⋅ Sˆ− gµB Sˆ(33)i< i,j>jBiiHere g is **the** L**and**é splitting factor, ~2 for pure spin, **and** µ B is **the** Bohr magneton, ~58µeVT -1 . **The**field 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 **the**chosen 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. **The**variational estimatesapproach **the** truevalues as **the** fieldstrength is increasedtoward ~10T, **the**lowest 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 - Mat**the**w 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** **the**exact 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 toge**the**r. By pulling **the**m 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 - Mat**the**w HolmesGround State Energy, J-2-4-6-8-10-12Ground State Convergence for h=1.5Fig. 12 Ground stateconvergence after~35 iterations **and**sooner 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 Iterations**The** previous oscillations have now vanished, permitting a fast **and** accurate convergence. Even up toN=12, convergence is now possible. We cannot say yet whe**the**r this level is accurate however, as **the**N=4,5 systems in Fig. 8 show. Despite **the**ir fast convergence, Fig. 9, **the**y 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. **The**apparent 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 that**the** modified **Lanczos**method allows largersystems to be treated.Memoryrequirements preventcompilation for N>12but **the** estimates forN>8 show excellentagreement with **the**preceding linearity.-12 3 4 5 6 7 8 9 10 11 12System Size, N15

BSc Final Year Report - Mat**the**w 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 **the**variational 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** Method**The** Modified **Lanczos** method is confined to making estimation of **the** ground state energies, for both**the** 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 **the**oretical consideration of**the** 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 **Lanczos**algorithm, **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** **The**ory of Ferromagnetism, Oxford (2000).[2] Brussard P. J. & Glaudemans P. W. M., Shell Model Applications in Nuclear Spectroscopy,North-Holl**and** 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 **The**oretical Magnetism, Iliffe (1968).[12] Lake B. et al, ISIS Science, www.isis.rl.ac.uk/science/index.htm, Ru**the**rford AppletonLaboratory, (2001).[13] Lieb E. H. et al., Ma**the**matical Physics in One Dimension, Academic Press (1996).16

BSc Final Year Report - Mat**the**w 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 **The**ory of Solids, Taylor & Francis (2002).[18] S**and**vik A. W., 'Computational Studies of Quantum Spin Systems' & 'Numerical Solutions of**the** 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 **The**ory of Magnetisation, Plenum Press (1967).17