Physica B 406 (2011) 1395–1402

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

Lebanes

E-m

richert@

journal homepage: www.elsevier.com/locate/physb

Study of the spectral properties of spin ladders in different representationsvia a renormalization procedure

Tarek Khalil a,b,�, Jean Richert c

a Department of Physics, School of Arts and Sciences, Lebanese International University, Beirut, Lebanonb Department of Physics, Faculty of Sciences(V), Lebanese University, Nabatieh, Lebanonc Institut de Physique, Universite de Strasbourg, 3, rue de l’Universite, 67084 Strasbourg Cedex, France

a r t i c l e i n f o

Article history:

Received 24 November 2008

Received in revised form

4 January 2011

Accepted 13 January 2011Available online 21 January 2011

Keywords:

Effective theories

Renormalization

Strongly interacting systems

Quantum spin systems

26/$ - see front matter & 2011 Elsevier B.V. A

016/j.physb.2011.01.036

esponding author at: Department of Physics,

e International University, P.O. Box: 146404,

ail addresses: tarek.khalil@liu.edu.lb (T. Khali

fresnel.u-strasbg.fr (J. Richert).

a b s t r a c t

We implement an algorithm which is aimed to reduce the dimensions of the Hilbert space of quantum

many-body systems by means of a renormalization procedure. We test the role and importance of

different representations on the reduction process by working out and analyzing the spectral properties

of strongly interacting frustrated quantum spin systems.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Most microscopic many-body quantum systems are subject tostrong interactions which act between their constituents. Ingeneral, there exist no analytical methods to treat exactlystrongly interacting systems apart from the assumption of trialwave functions able to diagonalize the Hamiltonian of integrablemodels, like the Bethe Ansatz (BA) for specific one dimensional(1D) systems [1], the BCS hypothesis which explains the supra-conductivity and others like the Neel state, the Resonant ValenceBond (RVB) spin liquid states proposed by Anderson [2], theValence Bond Crystal (VBC) states, etc. They are aimed to describe2D systems but there remains the problem of their degree ofrealism, i.e. their ability to include the essentials of the interactionin strongly interacting systems [3]. The complexity of the struc-ture of such systems leads to the diagonalization of the Hamilto-nian numerically which must in general be performed in verylarge Hilbert space although the information of interest isrestricted to the knowledge of a few low-energy states generallycharacterized by collective properties. Consequently it is neces-sary to manipulate very large matrices in order to extract areduced quantity of informations.

ll rights reserved.

School of Arts and Sciences,

Mazraa, Beirut, Lebanon.

l),

Non-perturbative techniques are needed. During the lastdecades a considerable amount of procedures relying on therenormalization group concept introduced by Wilson [4] havebeen proposed and tested. Some of them are specifically devisedfor quantum spin systems, like the Real Space RenormalizationGroup (RSRG) [5,6] and the Density Matrix RenormalizationGroup (DMRG) [7,8].

We propose here a non-perturbative approach which tacklesthis question [9]. The procedure consists of an algorithm whichimplements a step by step reduction of the size of Hilbert spaceby means of a projection technique. It relies on the renormaliza-tion concept following in spirit former work based on thisconcept [10–12]. Since the reduction procedure does not act inordinary or momentum space but in Hilbert space, it is universalin the sense that it works for any kind of many-body quantumsystem.

The properties of physical systems can be investigated indifferent representations. In the present work which deals withfrustrated spin ladders the common SU(2)-representation is con-fronted with the SO(4)-representation [13–15] in order to workout the energies of the low-lying levels of the spectrum of thesesystems. The efficiency of one or the other representation interms of the number of relevant basis states characterizing theground state wave function is tested in connection with thereduction process.

The outline of the paper is the following. In Section 2 wepresent the formal developments leading to the secular equationin the reduced Hilbert space. Section 3 is devoted to the

T. Khalil, J. Richert / Physica B 406 (2011) 1395–14021396

application of the algorithm to frustrated quantum spin ladderswith two legs and one spin per site. We analyze the outcome ofthe applied algorithm on systems characterized by differentcoupling strengths by means of numerical examples, in bases ofstates developed in the SU(2) and SO(4)-representations andcompare the results obtained in both cases. Conclusions andfurther possible investigations and developments are presentedin Section 4.

2. The reduction algorithm

2.1. Reduction procedure and renormalization of the coupling

strengths

We consider a system described by a Hamiltonian dependingon a unique coupling strength g which can be written as a sum oftwo terms

H¼H0þgH1 ð1Þ

The Hilbert space HðNÞ of dimension N is spanned by anorthonormalized arbitrary set of basis states fjFiS,i¼ 1, . . . ,Ng.In this basis an eigenvector jCðNÞl S takes the form

jCðNÞl S¼XN

i ¼ 1

aðNÞli ðgðNÞÞjFiS ð2Þ

where the amplitudes faðNÞli ðgðNÞÞg depend on the value g(N) of g in

HðNÞ.Using the Feshbach formalism [16] the Hilbert space may be

decomposed into subspaces by means of the projection operatorsP and Q,

HðNÞ ¼ PHðNÞ þQHðNÞ ð3Þ

In practice the subspace PHðNÞ is chosen to be of dimensiondim PHðNÞ ¼N�1 by elimination of one basis state. The projectedeigenvector PjCðNÞl S obeys the Schrodinger equation

Heff ðlðNÞl ÞPjC

ðNÞl S¼ lðNÞl PjCðNÞl S ð4Þ

where Heff ðlðNÞl Þ is the effective Hamiltonian which operates in the

subspace PHðNÞ. It depends on the eigenvalue lðNÞl which is theeigenenergy corresponding to jCðNÞl S in the initial space HðNÞ. Thecoupling strengths g(N) which characterize the Hamiltonian H(N) inHðNÞ is now aimed to be changed into gðN�1Þ in such a way that theeigenvalue in the new space HðN�1Þ is the same as the one in thecomplete space

lðN�1Þl ¼ lðNÞl ð5Þ

The determination of gðN�1Þ by means of the constraint expressedby Eq. (5) is the central point of the procedure. It is the result of arenormalization procedure induced by the reduction of the vectorspace of dimension N to N�1 which preserves the physicaleigenvalue lðNÞl .

In the sequel PjCðNÞ1 S is chosen to be projection of the groundstate eigenvector jCðNÞ1 S (l¼1) and lðNÞ1 ¼ lðN�1Þ

1 ¼ l1 the corre-sponding eigenenergy. In Ref. [9] it is shown how gðN�1Þ can beobtained as a solution of an algebraic equation of the seconddegree. One gets explicitly a discrete quadratic equation

aðN�1ÞgðN�1Þ2þbðN�1ÞgðN�1Þ þcðN�1Þ ¼ 0 ð6Þ

where

aðN�1Þ ¼ G1N�HNNF1N ð7Þ

bðN�1Þ ¼ aðNÞ11 HNNðlðNÞ1 �a1ÞþF1Nðl

ðNÞ1 �aNÞ ð8Þ

cðN�1Þ ¼ �aðNÞ11 ðlðNÞ1 �a1Þðl

ðNÞ1 �aNÞ ð9Þ

with

F1N ¼XN�1

i ¼ 1

aðNÞ1i /F1jH1jFiS

G1N ¼H1N

XN�1

i ¼ 1

aðNÞ1i /FNjH1jFiS

Hij ¼/FijH1jFjS

and

ai ¼/FijH0jFiS, i¼ 1, . . . ,N

The reduction procedure is then iterated in a step by stepdecrease of the dimensions of the vector space,N/N�1/N�2/ � � � leading at each step k to a couplingstrength gðN�kÞ which can be given as the solution of a flowequation in a continuum limit description of the Hilbert space.The procedure can be generalized to Hamiltonians depending onseveral coupling constants which experience a renormalizationduring the reduction procedure under further constraints [18].

2.2. Outline of the reduction algorithm

We sketch here the different steps of the procedure.

1.

Consider a quantum system described by a Hamiltonian H(N)

which acts in an N-dimensional Hilbert space.

2. Compute the matrix elements of the Hamiltonian matrix H(N)

in a definite basis of states fjFiS,i¼ 1, . . . ,Ng. The diagonalmatrix elements fei ¼/FijH

ðNÞjFiSg are arranged either inincreasing order with respect to the feig or in decreasing orderof the absolute values of the ground state wave functionamplitudes jaðNÞ1i ðg

ðNÞÞj [17].

3. Use the Lanczos technique to determine lðNÞ1 and jCðNÞ1 ðg

ðNÞÞS[19,20].

4.

Fix gðN�1Þ as described in Section 2.1. Take the solution of thealgebraic second order equation closest to g(N) (see Eq. (6)).

5.

Construct HðN�1Þ ¼H0þgðN�1ÞH1 by elimination of the matrixelements of H(N) involving the state jFNS.

6.

Repeat procedures 3–5 by fixing at each step k fk¼ 1, . . . ,N�Nming, l

ðN�kÞ1 ¼ lðNÞ1 ¼ l1. The iterations are stopped at Nmin

corresponding to the limit of space dimensions for which thespectrum gets unstable.

2.3. Some remarks

�

The procedure is aimed to generate the energies of the low-energy excited states of strongly interacting systems andpossibly the calculation of further physical quantities. � The implementation of the reduction procedure asks for the

knowledge of l1 and the corresponding eigenvector jCðN�kÞ1 S

at each step k of the reduction process. The eigenvalue l1 ischosen as the physical ground state energy of the system.Eigenvalue and eigenvector can be obtained by means of theLanczos algorithm [8,19,20] which is particularly well adaptedto very large vector space dimensions. The algorithm fixeslðN�kÞ

1 ¼ lðNÞ1 and determines jCðN�kÞ1 S at each step.

�

The process does not guarantee a rigorous stability of theeigenvalue l1. jCðN�k�1Þ

1 S which is the eigenvector in the spaceHðN�k�1Þ and the projected state PjCðN�kÞ

1 S of jCðN�kÞ1 S into

HðN�k�1Þ may differ from each other. As a consequence it maynot be possible to keep lðN�k�1Þ

1 rigorously equal to lðN�kÞ1 ¼ l1.

In practice the degree of accuracy depends on the relative sizeof the eliminated amplitudes faðN�kÞ

1ðN�kÞðgðN�kÞÞg. This point will be

T. Khalil, J. Richert / Physica B 406 (2011) 1395–1402 1397

tested by means of numerical estimations and furtherdiscussed below.

� The Hamiltonians of the considered ladder systems are char-

acterized by a fixed total magnetic magnetization Mtot. Wework in subspaces which correspond to fixed Mtot. The totalspin Stot is also a good quantum number which defines smallersubspaces for fixed Mtot [21]. We do not introduce them herebecause projection procedures on Stot are time consuming.Furthermore we want to test the algorithm in large enoughspaces although not necessarily the largest possible ones inthis preliminary tests considered here.

3. Application to frustrated two-leg quantum spin ladders

3.1. The model

3.1.1. SU(2)-representation

Consider spin� 12 ladders [22,23] described by Hamiltonians of

the following type and shown in Fig. 1

Hðs,sÞ ¼ Jt

XL

i ¼ 1

si1 si2þ Jl

X/ijS

si1 sj1þ Jl

X/ijS

si2 sj2þ J1c

XðijÞ

si1 sj2þ J2c

XðijÞ

si2 sj1

ð10Þ

The index 1 or 2 labels the spin 12 vector operators sik acting on

the site i on both ends of a rung, in the second and third terms i

and j label nearest neighbours, here j¼ i+1 along the legs of theladder. The fourth and fifth terms correspond to diagonal inter-actions between sites located on different legs, j¼ i+1. L is thenumber of sites on a leg (Fig. 1) where J1c ¼ J2c ¼ Jc . The couplingstrengths Jt, Jl, Jc are positive.

As stated above the renormalization is restricted to a uniquecoupling strength, see Eq. (1). It is implemented here by puttingH0¼0 and HðNÞ ¼ gðNÞH1 where g(N)

¼ Jt and

H1 ¼XL

i ¼ 1

si1 si2þgtl

X/ijS

ðsi1 sj1þsi2 sj2

Þþgc

X/ijS

ðsi1 sj2þsi2 sj1

Þ ð11Þ

where gtl ¼ Jl=Jt , gc ¼ Jc=Jt . These quantities are kept constant andg(N)¼ Jt will be subject to renormalization during the reduction

process.One should point out that the renormalization does not change

if one chooses another coupling parameter as a renormalizableparameter, here Jl or Jc, because they are related to each other atthe beginning of the reduction procedure by the ratios gtl ¼ Jl=Jt

and gc ¼ Jc=Jt .The basis of states fjFpS,p¼ 1, . . . ,Ng is chosen as

jFpS¼ 1=2 m1, . . . ,1=2 mi, . . . ,1=2 m2L,X2L

i ¼ 1

mi ¼Mtot ¼ 0

�����+

with fmi ¼ þ1=2,�1=2g.

Fig. 1. Top: the original spin ladder. The coupling strengths are indicated as give

3.1.2. SO(4)-representation

Different choices of bases may induce a more or less efficientreduction procedure depending on the strength of the couplingconstants Jt, Jl, Jc. This point is investigated here by choosing also abasis of states which is written in an SO(4)-representation.

We replace ðsi1,si2Þ corresponding to dimers by ðSi,RiÞ. Bymeans of a spin rotation [14,15]

si1 ¼12ðSiþRiÞ ð12Þ

si2 ¼12ðSi�RiÞ ð13Þ

The Hamiltonian, Eq. (10), can be expressed in the form

HðS,RÞ ¼Jt

4

XL

i ¼ 1

ðS2i �R2

i Þþ J1

X/ijS

SiSjþ J2

X/ijS

RiRj ð14Þ

The structure of the corresponding system is shown in the lowerpart of Fig. 1. Here J1 ¼ ðJlþ JcÞ=2, J2 ¼ ðJl�JcÞ=2 and as beforeJ1c ¼ J2c ¼ Jc. The components Sðþ Þi ,Sð�Þi ,SðzÞi and Rðþ Þi ,Rð�Þi ,RðzÞi of thevector operators Si and Ri are the SO(4) group generators and /ijSdenotes nearest neighbour indices.

In this representation the states fjSiMiSg are defined as

jSiMiS¼X

m1 ,m2

/1=2 m1 1=2 m2jSiMiSj1=2 m1Sij1=2 m2Si

along a rung are coupled to Si¼0 or 1. Spectra are constructed inthis representation as well as in the SU(2)-representation and thestates fjFpSg take the form

jFpS¼ S1M1, . . . ,SiMi, . . . ,SLML,XL

i ¼ 1

Mi ¼Mtot ¼ 0

�����+

3.2. Test observables

In order to quantify the accuracy of the procedure we intro-duce different test quantities in order to estimate deviationsbetween ground state and low excited state energies in Hilbertspaces of different dimensions. The stability of low-lying statescan be estimated by means of

pðiÞ ¼ðeðNÞi �eðnÞi Þ

eðNÞi

����������� 100 with i¼ 1, . . . ,4 ð15Þ

where eðnÞi ¼ lðnÞi =2L with n¼(N�k) corresponds to the energy persite at the ith physical state starting from the ground state at thekth iteration in Hilbert space. This quantity provides a percentageof loss of accuracy of the eigenenergies in the different reducedspaces.

A global characterization of the ground state wavefunction indifferent representations can also be given by the entropy per site

n in the text. Bottom: The ladder in the SO(4)-representation. See the text.

T. Khalil, J. Richert / Physica B 406 (2011) 1395–14021398

in a space of dimension n

s¼�1

2L

Xn

i ¼ 1

PilnPi with Pi ¼ j/FðnÞi jC

ðnÞ1 Sj2 ¼ jaðnÞ1i j

2 ð16Þ

which works as a global measure of the distribution of theamplitudes faðnÞ1i g in the physical ground state.

In the remaining part we work out the spectra of differentsystems and compare results obtained in the two representationsintroduced above.

3.3. Spectra in the SU(2)-representation

Results obtained with an SU(2)-representation basis of statesare shown in Figs. 2 and 3.

3.3.1. First case: L¼6, Jt¼15, Jl¼5, Jc¼3

We choose the basis states in the framework of the M-schemecorresponding to subspaces with fixed values of the total projec-tion of the spin of the fjFiSg, Mtot¼0.

In the present case Jt 4 Jl,Jc . The dimension of the subspace isreduced step by step as explained above starting from N¼924. Asstated in Section 2.2 the basis states fjFiSg are ordered withincreasing energy of their diagonal matrix elements feig andeliminated starting from the state with largest energy eN .

Deviations of the energies of the ground and first excited statesfrom their initial values at N¼924 can be seen in Figs. 2a and bwhere the pðiÞ’s defined above represent these deviations in termsof percentages. As seen in Fig. 2a the ground state of the systemstays stable down to n� 50 where n is the dimension of thereduced space. The coupling constant Jt also does not move downto n� 300, see Fig. 2c. Figs. 2a and b show the evolution of thefirst excited states which follows the same trend as theground state.

For nr50 the spectrum gets unstable, the renormalization ofthe coupling constant can no longer correct for the energy of the

Fig. 2. SU(2)-representation. n is Hilbert space dime

lowest state. Indeed the coupling constant Jt increases drasticallyas seen in Fig. 2c. The reason for this behaviour can be foundin the fact that at this stage the algorithm eliminates stateswhich have an essential component in the state of lowestenergy. The same message can be read in Fig. 2d, the drop inthe entropy per site s is due to the elimination of sizableamplitudes {a1i}.

3.3.2. Second case: L¼6, Jt¼5.5, Jl¼5, Jc¼3

Contrary to the former case the coupling constant Jt alongrungs is now of the order of strength as Jl, Jc. Results are shownin Fig. 3. The lowest energy state is now stable down to n� 100.This is also reflected in the behaviour of the excited states whichmove appreciably for nr200. Fig. 3c shows that the couplingconstant Jt starts to increase sharply between n¼300 and 200. It isable to stabilize the excited states down to about n¼200 and theground state down to n¼70. The instability for nr70 reflects inthe evolution of the pðiÞ’s, Figs. 3a and b which get of the order of afew percent. The entropy Fig. 3d follows the same trend.

Comparing the two cases above and particularly theentropy Figs. 3d and 2d one sees that the stronger the Jt, themore the amplitude strength of the ground state wavefunction isconcentrated in a smaller number of basis state components. Theelimination of sizable components of the wavefunction leads todeviations which can be controlled down to a certain limit bymeans of the renormalization of Jt. One sees that large values of Jt

favour a low number of significative components in the lowenergy part of the spectrum in an SU(2)-representation.

3.3.3. Remark

In Figs. 2a and b it is seen that the ground and first excitedstates show ’’bunches’’ of energy fluctuations. In the ground statethe peaks are intermittent, they appear and disappear during thespace dimension reduction process. They are small in the casewhere Jt¼15 but can grow with decreasing Jt as it can be observed

nsion. L¼6 sites along a leg. Jt¼15, Jl¼5, Jc¼3.

Fig. 3. SU(2)-representation. n is Hilbert space dimension. L¼6 sites along a leg. Jt¼5.5, Jl¼5, Jc¼3.

T. Khalil, J. Richert / Physica B 406 (2011) 1395–1402 1399

in Figs. 3a and b for Jt¼5.5. The subsequent stabilization of theground state energy following such a bunch shows the effective-ness of the coupling constant renormalization which acts in aprogressively reduced and hence incomplete basis of states.

These bunches of fluctuations are correlated with the changeof the number of relevant amplitudes (i.e. amplitudes largerthan some value e as explained in Fig. 6) during the reductionprocess.

Consider first the case where Jt 4 Jl,Jc . One notices in Fig. 6athat down to n� 300 the number of relevant amplitudes definedin Fig. 6 stays stable like the ratios {p(i)} in Figs. 2a and b. For158ono300 these ratios change quickly. A bunch of fluctuationsappears in this domain of values of n as seen in Figs. 2a and b andcorrespondingly the number of relevant amplitudes decreasessteeply. For 60onr158 the ratios {p(i)} stay again stable as wellas the number of relevant amplitudes. The {p(i)} in Figs. 2a and balmost decrease back to their initial values. The analysis showsthat these bunches of fluctuations signal the local elimination ofrelevant contributions of basis states to the physical states in thespectrum. The stabilization of the spectra which follows duringthe elimination process shows that renormalization is able to curethese effects.

In the case where Jt � Jl,Jc shown in Fig. 6b the relevant andirrelevant amplitudes move continuously during the reductionprocess and the corresponding {p(i)} do no longer decrease to thevalues they showed before the appearance of a bunch of energyfluctuations as seen in Figs. 3a and b. It signals the fact that thecoupling renormalization is no longer able to compensate for thereduction of the Hilbert space dimensions.

One should mention that the evolution of the spectrumdepends on the initial size of Hilbert space. The larger the initialspace the larger the ratio between the initial number of states andthe number of states corresponding to the limit of stability of thespectrum, see Ref. [18] for explicit numerical examples.

3.4. Spectra in the SO(4)-representation

The reduction algorithm is now applied to the systemdescribed by the Hamiltonian HðS,RÞ given by Eq. (14) with a basisof states written in the SO(4)-representation. Like above weconsider two cases corresponding to large and close values of Jt

relative to the strengths of the other coupling parameters.

3.4.1. Reduction test for L¼6, Jt¼15, 5.5 and Jl¼5, Jc¼3

Figs. 4 and 5 show the behaviour of the spectrum for a systemof size L¼6. A large value of Jt, (Jt¼15), favours the dimerstructure along rungs in the lowest energy state and stabilizesthe spectrum down to small Hilbert space dimensions. This effectis clearly seen in Fig. 4a, the ground state is very stable. Theexcited states are more affected, see Fig. 4b, although they do notmove significantly. The renormalization of the coupling strength Jt

starts to work for nC50, Fig. 4c.The situation changes progressively with decreasing values of

Jt. Fig. 5 shows the case where Jt¼5.5. The ground state energyexperiences sizable bunches of fluctuations like in the SU(2)-representation, but much stronger than in this last case. The sameis true for the excited states which are reflected through all thequantities shown in Figs. 5a and b, in particular Jt, Fig. 5c. Theentropy Figs. 4d and 5d follow the same trend like in the SU(2)-representation by changing the values of Jt from 15 to 5.5. Theanalysis of the bunches of energy fluctuations through thenumber of relevant–irrelevant amplitudes is shown for Jt¼15,Jl¼5, Jc¼3 in Fig. 6c.1 and c.2 and for Jt¼5.5, Jl¼5, Jc¼3 in Fig. 6d.

The results show that the renormalization procedure is quitesensitive to the representation chosen in Hilbert space. It isexpected that essential components of the ground state wave-function get eliminated early during the process when the rungcoupling gets of the order of magnitude or smaller than the othercoupling strengths.

Fig. 4. SO(4)-representation. n is Hilbert space dimension. L¼6 sites along the chain. Jt¼15, Jl¼5, Jc¼3.

Fig. 5. SO(4)-representation. n is Hilbert space dimension. L¼6 sites along the chain. Jt¼5.5, Jl¼5, Jc¼3.

T. Khalil, J. Richert / Physica B 406 (2011) 1395–14021400

By comparing the two representations in Fig. 6, one noticesthat the stability of the low-energy properties of the system in thereduced Hilbert space is well characterized by the number ofrelevant–irrelevant amplitudes as defined in Fig. 6 and thedistribution of the amplitudes in Hilbert space, see Ref. [17].

3.5. Summary

The present results lead to two correlated remarks. The efficiencyof the algorithm is different in different sectors of the couplingparameter space. In the case of the frustrated ladders considered

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700 800 900n n

n

n

n

relevant amplitirrelevant amplit

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700 800 900

relevant amplitirrelevan amplit

0 100 200 300 400 500 600 700 800 900

0 100 200 300 400 500 600 700 800 900

relevant amplitirrelevant amplit

0

10

20

30

40

50

0 10 20 30 40 50

relevant amplitirrelevant amplit

0

100

200

300

400

500

600

700

800

0 100 200 300 400 500 600 700 800 900

ampl

itude

sam

plitu

des

ampl

itude

s

ampl

itude

sam

plitu

des

relevant amplitirrelevant amplit

Fig. 6. n is the dimension of the Hilbert space. Amplitudes show the number of relevant–irrelevant amplitudes in the ground state eigenfunction (relevant–irrelevant

amplitude in the figure). Relevant amplitudes are those for which fja1ij4e, ðhere e¼ 10�2Þ, i¼ 1, . . . ,ng. The number of sites along a leg is L¼6. SU(2)-representation:

(a) corresponds to Jt¼15, Jl¼5, Jc¼3. (b) Corresponds to Jt¼5.5, Jl¼5, Jc¼3. SO(4)-representation: (c.1) and (c.2) correspond to Jt¼15, Jl¼5, Jc¼3. (d) Corresponds to Jt¼5.5,

Jl¼5, Jc¼3.

T. Khalil, J. Richert / Physica B 406 (2011) 1395–1402 1401

here the algorithm is more efficient the stronger the couplingbetween rung sites Jt. Second, this behaviour is strongly related tothe representation in which the basis of states is defined. The SU(2)-representation leads to a structure of the wave functions (i.e. thesize of the amplitudes of the basis states) which is very differentfrom the one obtained in the SO(4)-representation. For large valuesof Jt the spectrum is more stable in the SO(4)-representation. Forsmall values of Jt the stability is better realized in the SU(2)-representation. Finally, in the regime where Jt 4 Jl,Jc , one observesthat the reduction procedure is more efficient the closer Jl is to Jc.This effect can be understood and related to previous analyticalwork in the SO(4)-representation [25].

4. Conclusions and outlook

In the present work we tested and analysed the outcome of analgorithm which aims to reduce the dimensions of the Hilbertspace of states describing strongly interacting systems. The

reduction is compensated by the renormalization of the couplingstrengths which enter the Hamiltonians of the systems. Byconstruction the algorithm works in any space dimension andmay be applied to the study of any microscopic N-body quantumsystem. The robustness of the algorithm has been tested onfrustrated quantum spin ladders.

The analysis of the numerical results obtained in applicationsto quantum spin ladders leads to the following conclusions.

�

The present numerical applications are essentially realistictests of the renormalization algorithm restricted to rathersmall Hilbert spaces which do not necessarily necessitate theuse of a Lanczos procedure. We introduced this procedure inorder to be able to extend our present work to much largersystems for which ordinary full diagonalization cannot beperformed. We tested it successfully by comparing the out-come with that of complete diagonalization. Preliminaryextended calculations have also been performed on higherdimensional ladders. We expect to present and discuss the

T. Khalil, J. Richert / Physica B 406 (2011) 1395–14021402

physics of their application to frustrated systems infurther work.

� The stability of the low-lying states of the spectrum in the

course of the reduction procedure depends on the relativevalues of the coupling strengths. The ladder favours a dimerstructure along the rungs, i.e. stability is the better the largerthe transverse coupling strength Jt. This is the reason why theSO(4)-representation is favoured when compared to the SU(2)-representation in this case. It leads to a more efficient basis ofstates as commented below (see next paragraph).

� The efficiency of the reduction procedure depends on the

representation frame in which the basis of states is defined.It appears clearly that the evolution of the spectrum describedin an SU(2)-representation is significantly different from theevolution in an SO(4)-representation. This is understandablesince different representations partition Hilbert space in dif-ferent ways and favour one or the other representationdepending on the relative strengths of the coupling constants.It is always appropriate to work in a basis of states whosesymmetry properties are closest to the symmetry properties ofthe physical system so that the contributions of the non-diagonal couplings are the smallest. The choice of an optimalbasis may not always be evident. In the case of the ladder weconsider here the SO(4)-representation.

� Local spectral instabilities appearing in the course of the

reduction procedure are correlated with the elimination ofbasis states with sizable amplitudes in the ground statewavefunction. One or another representation can be moreefficient on the reduction process for a given set of couplingparameters because it leads to physical states in which theweight on the basis states is concentrated in a differentnumber of components. This point is strongly related to thecorrelation between quantum entanglement and symmetryproperties which have been under intensive scrutiny, see f.i.Ref. [26] and references therein.

Further points are worthwhile to be investigated:

�

Extension to larger ladders and systems of higher spacedimensions with the help of more sophisticated numericalalgorithms [27,28].

�

Extension of the renormalization procedure to systems atfinite temperature [24] and more than one coupling constantrenormalization [18].

References

[1] I. Bose, in: A. Kundu (Ed.), Classical and Quantum Nonlinear IntegrableSystems Theory and Application, Taylor and Francis, 2003 (Chapter 7), PrintISBN: 978-0-7503-0959-2 /arxiv:cond-mat/0208039v1S.

[2] P.W. Anderson, Science 235 (1987) 1196.[3] C. Lhuillier, G. Misguish, Frustrated quantum magnets, Lecture Notes in

Physics, vol. 595/2002, 2002, pp. 161–190.[4] K.G. Wilson, Phys. Rev. Lett. 28 (1972) 548;

K.G. Wilson, Rev. Mod. Phys. 47 (1975) 773.[5] J.-P. Malrieu, N. Guithery, Phys. Rev. B 63 (1998) 085110.[6] S.R. White, R.M. Noack, Phys. Rev. Lett. 68 (1992) 3487.[7] S.R. White, Phys. Rev. Lett. 69 (1992) 2863.[8] M. Henkel, Conformal Invariance and Critical Phenomena, Springer Verlag,

1999.[9] T. Khalil, J. Richert, J. Phys. A: Math. Gen. 37 (2004) 4851.

[10] S.D. Glazek, K.G. Wilson, Phys. Rev. D 57 (1998) 3558.[11] H. Mueller, J. Piekarewicz, J.R. Shepard, Phys. Rev. C 66 (2002) 024324.[12] K.W. Becker, A. Huebsch, T. Sommer, Phys. Rev. B 66 (2002) 235115.[13] A. Bohm, Quantum Mechanics Foundations and Applications, Springer edi-

tion, 1993, pp. 205–222.[14] K. Kikoin, Y. Avishai, M.N. Kiselev, in: A.S. Alexandrov, R.S. Williams (Eds.),

Molecular Nanowires and Other Quantum Objects, NATO Science Series II,vol. 148, 2004, pp. 177–189 arXiv:cond-mat/0309606.

[15] K.A. Kikoin, M.N. Kiselev, Y. Avishai, Dynamical symmetries in nanophysics,in: K.S. Gehar (Ed.), Nanophysics, Nanoclusters and Nanodevices, NOVAScience Publisher, New York, USA, 2006, pp. 39–86 ISBN: 1-59454-852-8,arXiv:cond-mat/0407063.

[16] H. Feshbach, Nuclear Spectroscopy, Part B, Academic Press, 1960.[17] T. Khalil, J. Richert, Phys. Lett. A 372 (2008) 2217.[18] T. Khalil, Ph.D. Thesis, ULP/Strasbourg, 2007 /http://eprints-scd-ulp.u-strasbg.

fr:8080/761/01/khalil2007.pdfS.[19] N. Laflorencie, D. Poilblanc, in: Lecture Notes in Physics, vol. 645, 2004,

pp. 227–252.[20] J.K. Cullum, R.A. Willoughby, Lanczos Algorithms for Large Symmetric

Eigenvalue Computations, Vol. I: Theory, Classics in Applied Mathematics,SIAM, 1985.

[21] P. Lowdin, Phys. Rev. 97 (1955) 1509.[22] Z. Weihong, V. Kotov, J. Oitmaa, Phys. Rev. B 57 (1998) 11439.[23] H.Q. Lin, J.L. Shen, H.Y. Schick, Phys. Rev. B 66 (2002) 184402.[24] J. Richert, arxiv:quant-ph/0209119.[25] J. Richert, O. Guhne, Phys. Status Solidi (b) 245 (8) (2008) 1552.[26] J.K. Korbicz, M. Lewenstein, Phys. Rev. A 74 (2006) 022318.[27] R.R. Whitehead, A. Watt, B.J. Cole, I. Morrison, in: M. Baranger, E. Vogt (Eds.),

Advances in Nuclear Physics, vol. 9, Plenum, New York, 1977.[28] E. Caurier, F. Nowacki, Acta Phys. Pol. B 30 (1999) 705.