We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians
with dramatically reduced memory requirements, {\em without restricting to
variational ansatzes}. The lattice of size $N$ is partitioned into two
subclusters. At each iteration the Lanczos vector is projected into two sets of
$n_{{\rm svd}}$ smaller subcluster vectors using singular value decomposition.
For low entanglement entropy $S_{ee}$, (satisfied by short range Hamiltonians),
the truncation error is expected to vanish as $\exp(-n_{{\rm
svd}}^{1/S_{ee}})$. Convergence is tested for the Heisenberg model on Kagom\'e
clusters of 24, 30 and 36 sites, with no lattice symmetries exploited, using
less than 15GB of dynamical memory. Generalization of the Lanczos-SVD algorithm
to multiple partitioning is discussed, and comparisons to other techniques are
given.Comment: 7 pages, 8 figure

/SLAC

/Stanford U., Phys. Dept. /Technion

/Technion

Auerbach, Assa

Chandra, V.Ravi

Weinstein, Marvin

English

arXiv

Marvin Weinstein

Assa Auerbach

V. Ravi Chandra

Crossref

Physical Review E

Reducing memory cost of exact diagonalization using singular value decomposition

We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramatically reduced memory requirements. The lattice of size N is partitioned into two subclusters. At each iteration the Lanczos vector is projected into a set of n{sub svd} smaller subcluster vectors using singular value decomposition. For low entanglement entropy S{sub ee}, (satisfied by short range Hamiltonians), we expect the truncation error to vanish as exp(-n{sup 1/S{sub ee}}{sub svd}). Convergence is tested for the Heisenberg model on Kagome clusters of up to 36 sites, with no symmetries exploited, using less than 15GB of memory. Generalization to multiple partitioning is discussed

Chandra, V. Ravi

UNT (University of North Texas) Digital Library

SLAC-PUB-14373Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition∗Marvin Weinstein1, Assa Auerbach2,3, and V. Ravi Chandra31SLAC National Accelerator Laboratory, Stanford, 2575 Sand Hill Road, CA 94025, USA.2Department of Physics, Stanford University, Stanford CA 94306, USA. and3Physics Department, Technion, Haifa, 32000, Israel(Dated: April 29, 2011)We present a modified Lanczos algorithm to diagonalize lattice Hamiltonians with dramaticallyreduced memory requirements. The lattice of size N is partitioned into two subclusters. At eachiteration the Lanczos vector is projected into a set of nsvd smaller subcluster vectors using singularvalue decomposition. For low entanglement entropy See, (satisfied by short range Hamiltonians),we expect the truncation error to vanish as exp(−n1/Seesvd ). Convergence is tested for the Heisenbergmodel on Kagomé clusters of up to 36 sites, with no symmetries exploited, using less than 15GB ofmemory. Generalization to multiple partitioning is discussed.PACS numbers: 05.30.-d, 02.70.-c, 03.67.Mn, 05.50.+qNumerical (”Exact”) diagonalization (ED) of quan-tum many-body Hamiltonians on finite clusters is of-ten used to advance our understanding of larger lattices.For example, Contractor Renormalization [1–3] uses EDto compute the short range interactions of the effectivehamiltonian. Approximation schemes, mean field theo-ries and variational wavefunctions, are routinely testedon small clusters by comparisons to ED. ED also yieldsshort wavelength dynamical response functions[4] andChern numbers for computing Hall conductivity [5].Lanczos algorithms [6, 7] are particularly efficient forsparse Hamiltonians, such as the Hubbard and Heisen-berg models. Much effort has been devoted to speedingup Lanczos convergence, and avoiding error accumula-tion. However, for a lattice of size N , with m statesper site, the dimension of the Lanczos vectors (which arestored in the dynamical memory) goes as mN . There-fore, increasing ED to larger lattice sizes is preventedprimarily by memory limitations, rather than processorspeed.The central idea of this paper is to significantly re-duce the dynamical memory cost of ED. The lattice isdivided into two subclusters. We use singular value de-composition (SVD) to reduce ”large” vectors to a setof nsvd ”small” subcluster vectors. High accuracy canbe guaranteed if the truncation error is a rapidly de-creasing function of nsvd. Significant memory compres-sion is achieved when nsvd << mN/2, which in turndepends on having low entanglement entropy See [8] ofthe target eigenstates. Indeed, many of the interestingshort range models in one and two dimensions [9–12] areknown to have low entanglement entropy. This explainsthe remarkable success of density matrix renormalization∗This work was supported by the U. S. DOE, Contract No. DE-AC02-76SF00515.36	  sites	  24	  sites	  FIG. 1. Partitioning Kagomé clusters for application of theLanczos-SVD algorithmgroup [13], matrix product states [9, 14] , and multiscaleentanglement renormalization approaches [15] for thesemodels. In higher dimensions d, the area law scalingSee ∼ cN (d−1)/d is expected [8]. This allows us to chooseSee << log(nsvd) << N/2 log(m) and still obtain a smalltruncation error, as we shall demonstrate below.The paper is organized as follows. The SVD projectionof a large vector into small vectors is defined. The con-vergence of the truncation error as a function of nsvd andSee is obtained for a generic model of entanglement spec-trum. We describe a Lanczos-SVD iteration step with itsorthonormalizations and diagonalizations. We test theLanczos-SVD algorithm for the spin half Heisenberg an-tiferromaget on the Kagomé clusters of 24 and 36 sites,which are depicted in Fig. 1. The results agree with our2estimate of the truncation error. The ground state energyof 36 sites converges to a relative error of ≈ 7× 10−5 ascompared to the exact result [16], where we use a desktopcomputer with less than 15GB of memory and no latticesymmetries. Finally, we discuss a possible extension ofthis approach to multi-partitioning, and estimate the op-timal reduction in memory cost that could be achieved.The SVD projection. Any state |ψ〉 of the full clustercan be represented in a unique SVD form as|ψ〉 =∑αλα|α〉1|α〉2,∑αλ2α = 1, 〈α|α′〉i = δαα′ , (1)where the λα are positive, and |α〉i are ”small” basisvectors of subclusters i = 1, 2. Truncating the sum intothe largest nsvd terms defines the SVD projector,Psvd|ψ〉 =nsvd∑α=1λα|α〉1|α〉2, (2)which introduces a wavefunction error ε = ||(Psvd −1)|ψ〉||2.To get an idea of how ε(nsvd) converges, we must knowthe ”entanglement spectrum” {sα} defined by λ2α ≡ e−sα .A generic density of states can be modelled by a powerlaw formρp(s) =∑αδ(s− sα) =spΓ(p+ 1), p > −1, (3)which describes the many-body density of states of a clas-sical gas with constant (Dulong-Petit) specific heat [20].p counts with the number of entangled degrees of free-dom. The corresponding entanglement entropy is easy toevaluate,See = −∑αλ2 log(λ2) =∫ ∞0dssρp(s)e−s = p+ 1. (4)Choosing a cut-off exponent sc such that nsvd =∫ sc0dsρp(s), we arrive at the error estimate at large nsvd,ε ∼ e−n1/Seesvd . (5)Hence a choice of nsvd > eSee will ensure an exponentiallysmall truncation error.Lanczos-SVD step. Lanczos-SVD economizes on thestorage space by applying an SVD projection after eachapplication of the Hamiltonian on the Lanczos vector,|ψ〉′ = PsvdH|ψ〉. (6)The projection entails the following computational steps.H can be written as a sum of products of the two sub-cluster operators,H = H01 ⊗ I2 + I1 ⊗H02 +M∑µ=3Hµ1 ⊗Hµ2 . (7)For example, a nearest neighbor Heisenberg model(∑ij Si·Sj) with K bonds connecting the two subclustershas M = 2 + 3K terms.Acting with H on |ψ〉 produces a new state,H|ψ〉 =nsvdM∑ν=1|ν)1|ν)2, (8)where the new (non orthonormal) small vectors are la-belled by |ν)i = Hµi |α〉i. To project the new state byPsvd, we must first orthonormalize |ν)i. We diagonalizethe two Hermitian overlap matrices (of row dimensionsnsvdM)〈ν′|ν〉i =(V †i DiVi)νν′i = 1, 2|β〉i =∑ν(D− 12i Vi)βν|ν)i, (9)where Di are diagonal and positive semidefinite, and Viare unitary. |β〉i, i = 1, 2 are orthonormal sets in theirrespective spaces. Thus, the new vector is given byH|ψ〉 =nβnβ′∑ββ′Cββ′ |β〉1|β′〉2C =√D1V∗1 V†2√D2. (10)Now we perform an SVD on the matrix C,C = ν2cUt1Λ′U2, (11)where νc is the normalization. Ut1, U†2 are unitary matri-ces which diagonalize the Hermitian products CC† andC†C respectively. After computing Λ, U1, U2 we obtainthe SVD form of the new state. Λ is diagonal and nor-malized to TrΛ2 = 1 with positive eigenvalues λ′α. Wekeep only the nsvd largest λ′α and obtainPsvdH|ψ〉 =nsvd∑α=1λ′α|α〉′1|α〉′2, (12)where the new small vectors of i = 1, 2 are,|α〉′i =nsvdM∑ν=1(UiD− 12Vi)αν|ν)i. (13)Convergence to eigenstates The Lanczos algorithm ro-tates a set of basis states |ψn〉 into the lowest energyeigenstates with which the basis has a finite overlap. Ifwe choose ε to be much smaller than the lowest relativeenergy gap, the Lanczos-SVD vectors converge to a stateswhich are within ε distance from the SVD projection ofthe corresponding exact eigenstate.ED of Kagomé clusters. We test the convergence ofthe Lanczos-SVD algorithm for the spin-half HeisenbergantiferromagnetH =∑〈ij〉Si · Sj (14)310	   50	   100	  log(￿)log(∆E0 /E0 )-­‐10	  -­‐15	  -­‐15	  -­‐10	  energy	  error	  wave	  func2on	  	  error	  nsvdFIG. 2. Lanczos-SVD truncation errors, for the 24 sitesKagomé cluster. nsvd is the number of retained products inthe SVD representation of the ground state (1). ε is the wave-function error Eq. (2) and following text). ∆E0/E0 is therelative error in the the Lanczos-SVD ground state energy ascompared to the exact result. The low entanglement entropyof the subclusters (See = 1.510) is the reason for the rapiddecay of the errors in the regime nsvd << 212.on Kagomé clusters of N=24,36 sites, as depicted inFigs.1 . The Lanczos-SVD routine proceeds as follows:We initialize |ψ〉(0) as a direct product of the two sub-cluster states. We compute (PsvdH)n|ψ(0)〉 = |ψ(n)) asdescribed above. Since our method is economical in mem-ory, we can afford to retain L sequential Lanczos vectors|ψ(n)), |ψ(n+1)), . . . |ψ(n+L)), which speeds up the conver-gence with iteration number considerably. (If memory isscarce, one could use the slower method of keeping onlytwo Lanczos vectors).Now, we compute the overlap matrix and orthonormal-ize this set of Lanczos vectors. This produces a ”rotatingbasis” of dimension L|ϕ(i)〉 =n+L∑n′=nAin′ |ψ(n′)〉, 〈ϕ(i)|ϕ(j)〉 = δij , (15)where A are the coefficients determined by diagonalizingthe overlap matrix (see e.g. Eq. (9)).Subsequently, we compute the matrix elements of thereduced Hamiltonian,Hij = 〈ϕ(i)|H|ϕ(j)〉, i, j = 1, . . . L. (16)The reduced Hamiltonian matrix is diagonalized. Itslowest eigenvalue and eigenvector yield the best varia-tional approximations to the ground state energy andwavefunction at this level of iteration [17].Finally, we bring the resulting wavefunction to theSVD form truncated into nsvd terms. This provides thenew initial state |ψ(n+L+1)〉 for further Lanczos steps.Results. For 24 sites, we compare standard Lanczos toLanczos-SVD in order to investigate the dependence of10	   20	   30	   40	  -­‐4	  -­‐5	  -­‐6	  -­‐7	  -­‐8	  itera1on	  log(∆E0(i)/E0)iFIG. 3. Lanczos-SVD convergence of the ground state energyof the 36 site Kagomé cluster. ∆E0(i) is difference betweenthe energy at iteration i and the ED result of Ref. [16]. Weused nsvd = 100, and L = 4 Lanczos vectors.truncation error on nsvd. The ground state entanglemententropy is See = 1.51. In Fig.2 we plot the wave functionand ground state energy errors versus nsvd, and find thatthey decrease exponentially as expected by Eq. (5). Wealso verified that the SVD projection Psvd does not slowdown the energy convergence with iteration number.The 36 site Kagomé cluster was split into two 18 siteclusters as shown in Fig. 1. The Hamiltonian has 7 in-terconnecting bonds, which implies M = 23. The num-ber of retained small vectors before orthonormalization istherefore 2Mnsvd. In Fig 3 we plot ground state energyof Lanczos-SVD versus iteration, using nsvd = 100, andL = 4, and compare it to the exact ground state energy asdetermined by standard Lanczos E0(36) = −14.859397[16]. The entanglement entropy was measured to beSee ≈ 2.5. The calculation converges to relative energyaccuracy of 10−4, and an SVD truncation error of similarmagnitude.Comparing ε and See of 36 sites and 24 sites for thesame nsvd = 100 is consistent with the errors being 3.8×10−8 and 6.6×10−5 respectively, as expected by Eq. (5).Extension to larger lattices. The Lanczos-SVD com-presses the memory requirement by a single division ofthe cluster into two subclusters i = 1, 2. This idea couldbe extended to recursive partitioning. Consider that eachsmall vector (e.g. |α〉i in Eq. (1) gets further decomposedinto nsvd products of even smaller subcluster vectors. Ifthe SVD is thus iterated p times, one obtains a represen-tation in terms of small vectors of P = 2p subclusters.|ψ〉 is thus stored in terms of a set of the smallest vec-tors. Let us estimate the memory cost of storing such adecomposition and applying to it the Hamiltonian.For concreteness, consider a two dimensional disk ofradius R >> 1, containing N ' πR2 sites of spin half,divided into P equal sections as shown in Fig.4. The sec-tions are labelled by a binary number i = (i1, i2, . . . ip),ik = 0, 1. The recursive SVD decomposition yields the4000	  001	  010	  011	   100	  101	  110	  111	  λα1λ01α1α2α3λ11α 1α 2α 3λ 00α1α2α3λ 10α1α2α3λ1α1α2λ0α1α2i =P = 8, p = 3FIG. 4. Multiple subclustersexpression|ψ〉 =∑α1,α2...αpλα1λi1α1,α2 · ·λi1,...ip−1α1,...αp∏i|α1, . . . αp〉i.(17)The SVD weights λi are labelled according to the bound-aries they describe, as shown in Fig. (4). Each αi runsover nsvd numbers, which means that each section i isrepresented by npsvd vectors of dimension 2N/P . By the”area law” See ∝ R on each boundary. As shown before,we must retain nsvd ∼ ecR terms in each SVD, wherec(ε) > 1 in order to achieve a desired truncation error ε.After applying H to |ψ〉, we generate a factor of M ≈6R more small vectors. Thus we should store 6PRnp)svdsmall vectors in the memory. Thus the memory cost isMc ≈ 6PR exp(cR log(P ) +πR2 log(2)P). (18)Minimizing Mc(P ) one finds the optimal partitioningP opt, and the optimal memory cost Mopt at large N toscale asP opt ≈ π log(2)Rc, Moptc ∼ Nec2(√N/π log(N/π)−2).(19)This would amount to a significant compression of mem-ory as compared to standard Lanczos M∼ 2N . The re-maining challenge is to speed up the significantly largercomputational time needed to orthonormalize and SVDlarge sets of small vectors.Acknowledgements We thank Dan Arovas, Yosi Avron,Sylvain Capponi, Steve Kivelson, Israel Klich and Ne-tanel Lindner for conversations. AA acknowledges sup-port from israel Science Foundation, and US Israel Bina-tional Science Foundation, and the hospitailty of AspenCenter For Physics.[1] C. J. Morningstar, M. Weinstein Phys.Rev. D 54, 4131(1996); Marvin Weinstein Phys.Rev. B63, 174421 (2001);M.Stewart Siu, Marvin Weinstein Phys. Rev. B 77,155116 (2008).[2] E. Altman and A. Auerbach, Phys. Rev. B 65, 104508(2002); E. Berg, E. Altman, and A. Auerbach, Phys. Rev.Lett. 90, 147204 (2003); R. Budnik and A. AuerbachPhys. Rev. Lett. 93, 187205 (2004).[3] S. Capponi, A. Läuchli, and M. Mambrini Phys. Rev. B70, 104424 (2004).[4] N. H. Lindner and A. Auerbach, Phys. Rev. B 81, 054512(2010).[5] J. E. Avron and R. Seiler, Phys. Rev. Lett. 54, 259(1985); N. Lindner, A. Auerbach, D. P. Arovas, Phys.Rev. B 82 134510 (2010).[6] C. Lanczos. J. Res. Nat. Bur. Stand 45, 255 (1950)[7] J. K. Cullum and R. A. Willoughby, Lanczos Algo-rithms for Large Symmetric Eigenvalue Computations,Birkhäuser (1985)[8] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev.Mod. Phys. 80, 517 (2008) (and references therein).[9] F. Verstraete, M. A. Martin-Delgado, and J. Cirac, Phys.Rev. Lett. 92, 087201 (2004).[10] M. B. Hastings,Phys. Rev. B 69, 104431 (2004);[11] E. Fradkin, and J. E. Moore, Phys. Rev. Lett. 97, 050404(2006).[12] I. Klich, G. Refael, and A. Silva, Phys. Rev. A 74, 032306(2006).[13] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys.Rev. B 48, 10345 (1993).[14] S. Ostlund and S. Rommer, Phys. Rev. Lett. 75, 3537(1995).[15] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007); G. Evenblyand G. Vidal, Phys. Rev. Lett. 104, 187203 (2010).[16] S. Capponi, Private communication[17] M. Weinstein, Phys. Rev. D 47, 5499 (1993).[18] M. Weinstein, A. Auerbach, V. Ravi Chandra (unpub-lished)[19] A.M. Läuchli, J. Sudan, E.S. Sorensen, arXiv:1103.1159.[20] A power law can fit the numerical entanglement spectraasymptotics of two dimensional systems computed by e.g.M-C. Chung and I. Peschel , Phys. Rev. B 64, 064412(2001). H. Li and F. D. M. Haldane, Phys. Rev. Lett.101, 010504 (2008).

Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition

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Reducing Memory Cost of Exact Diagonalization using Singular Value
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Reducing Memory Cost of Exact Diagonalization using Singular Value Decomposition

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