Analytical and numerical studies of disordered spin-1 Heisenberg chains with aperiodic couplings
AAnalytical and numerical studies of disordered spin-1 Heisenberg chains with aperiodic couplings
H. L. Casa Grande,
1, 2
N. Laflorencie, F. Alet, and A. P. Vieira Instituto de F´ısica, Universidade de S˜ao Paulo, Caixa Postal 66318, 05314-970, S˜ao Paulo, SP, Brazil Laboratoire de Physique Th´eorique, IRSAMC, Universit´e de Toulouse, CNRS, 31062 Toulouse, France
We investigate the low-temperature properties of the one-dimensional spin-1 Heisenberg model with geomet-ric fluctuations induced by aperiodic but deterministic coupling distributions, involving two parameters. Wefocus on two aperiodic sequences, the Fibonacci sequence and the 6-3 sequence. Our goal is to understand howthese geometric fluctuations modify the physics of the (gapped) Haldane phase, which corresponds to the groundstate of the uniform spin-1 chain. We make use of different adaptations of the strong-disorder renormalization-group (SDRG) scheme of Ma, Dasgupta and Hu, widely employed in the study of random spin chains, supple-mented by quantum Monte Carlo and density-matrix renormalization-group numerical calculations, to study thenature of the ground state as the coupling modulation is increased. We find no phase transition for the Fibonaccichain, while we show that the 6-3 chain exhibits a phase transition to a gapless, aperiodicity-dominated phasesimilar to the one found for the aperiodic spin-1/2 XXZ chain. Contrary to what is verified for random spin-1chains, we show that different adaptations of the SDRG scheme may lead to different qualitative conclusionsabout the nature of the ground state in the presence of aperiodic coupling modulations.
I. INTRODUCTION
Quantum spin chains represent a suitable laboratory for thestudy of the combined effects, on many-body systems, ofquantum fluctuations and broken translation symmetry, rep-resented for instance by the presence of an inhomogeneouscoupling distribution. A paradigmatic model in this context isthe Heisenberg chain, described by the Hamiltonian H = L ∑ j = J j S j · S j + , (1)in which the constants J j > S operators located at contiguous sites.Even in the uniform limit ( J j ≡ J ), the model exhibits avariety of physical behavior, strongly dependent on the integeror half-integer character of S . According to a widely acceptedconjecture by Haldane, chains with half-integer S have agapless energy spectrum, while the ground state of chains withinteger S is separated from the first excited states by a finiteenergy gap. The most notable effects are seen in the extremequantum limit of small values of S , in which the two classesof systems are represented by S = and S =
1. In this lastcase ( S = exhibits a hidden topological order, revealed by a stringorder parameter, and the boundaries of open finite chains oflength L harbor spin- degrees of freedom.Whether the low-energy spectrum is gapless or gapped gov-erns not only the low-temperature thermodynamic behavior,but also affects the stability of the uniform ground state to-wards the breaking of translation symmetry. In the simple caseof dimerization (the introduction of alternating couplings J odd and J even along the chain), the spin- chain becomes gappedeven in the presence of an infinitesimal difference between J odd and J even , while the Haldane phase is protected by thefinite gap.Disorder effects, represented by random uncorrelated cou-plings, are even more pronounced. For the spin- chain,much information on the effects of random couplings can be obtained by using the strong-disorder renormalization-group(SDRG) scheme introduced by Ma, Dasgupta, and Hu. Thebasic idea is to eliminate high-energy degrees of freedom byidentifying strongly coupled spin pairs along the chain, whichcontribute very little to magnetic properties at low temper-atures and therefore can be decimated away, giving rise toweak effective couplings between the remaining neighboringspins. A number of studies performed during the last twodecades showed that, in the presence of any finite disor-der, the ground state turns into a random-singlet phase, whichcan be pictured as a collection of widely separated spin pairscoupled in singlet states. This is a consequence of the factthat, in the renormalization-group language, disorder inducesa flow of the probability distribution of effective couplings to-wards an infinite-randomness fixed point, in which, at a givenenergy scale, there are only a few strong effective couplings,which give rise to the singlet pairs, while the vast majority ofthe remaining couplings are much weaker. In this random-singlet phase, physical properties are quite distinct from theones in the uniform chain, being characterized by an activateddynamics, in which energy and length scales are not relatedby a power law, but by a stretched exponential form. Fur-thermore, ground-state spin-spin correlations are dominatedby the rare singlet pairs, leading to a striking distinction be-tween average and typical behaviors. The picture for random spin-1 chains looks even richer. In-vestigations based on extensions of the SDRG scheme, combined with numerical studies, point to the stabil-ity of the Haldane phase towards sufficiently weak disor-der; intermediate disorder seems to lead to a gapless Hal-dane phase, characterized by a finite string order parame-ter and exhibiting nonuniversal effects associated with Grif-fiths singularities; and sufficiently strong disorder induces arandom-singlet phase.For the spin- chain, effects partially similar to those pro-duced by randomness are induced by the presence of aperiodicbut deterministic couplings. This kind of aperiodicity is sug-gested by analogy with quasicrystals, structures exhibitingsymmetries forbidden by traditional crystallography and cor-responding to projections of higher-dimensional Bravais lat- a r X i v : . [ c ond - m a t . s t r- e l ] A p r tices onto low-dimensional subspaces. Aperiodic couplingscan be produced by letter sequences generated by substitu-tion rules such as the one associated with the Fibonacci se-quence, a → ab , b → a . The iteration of the rule leads to asequence abaababa . . . , in which there is no characteristic pe-riod. Associating different letters with different coupling val-ues J a and J b , an aperiodic chain is built. Distinct aperiodicsequences give rise to different geometric fluctuations, gaugedby a wandering exponent ω associated with the power-law de-scribing the growth of suitably defined coupling fluctuationsas the chain length increases. The case ω = emulates thefluctuations induced by random uncorrelated couplings.An adaptation of the SDRG method to the aperiodic spin- XXZ chain, a particular case of which is the Heisenbergchain, revealed that low-temperature thermodynamic proper-ties and the nature of the ground state are deeply changed byaperiodicity generated by sequences for which ω ≥
0, and be-havior reminiscent of that characterizing the random-singletphase can be observed. Notably, there is in general a stretchedexponential relation between energy and length scales, and aclear distinction between average and typical behavior of spin-spin correlation functions, in this case associated with the ex-istence of characteristic lengths emerging from the combina-tion of aperiodicity and quantum fluctuations. Furthermore,and in contrast to the random-singlet phase, correlations mayexhibit an ultrametric structure related to the inflation symme-try inherent to aperiodic sequences.In this paper, we investigate the effects of aperiodic cou-plings on the low-temperature properties of quantum spin-1Heisenberg chains. As in the case of random uncorrelatedcouplings, we expect that the Haldane phase is stable towardsthe introduction of weak aperiodic modulation (as measuredby a coupling ratio r = J b / J a (cid:39) r → r → ∞ ) may lead to an aperiodicity-dominatedgapless phase, quite similar to the one observed for the cor-responding spin- chain. We obtain analytical results in thecase of strong modulation by using different adaptations ofthe SDRG scheme. Analytical results are compared to numer-ical simulations obtained using quantum Monte Carlo (QMC)and density matrix renormalization group (DMRG) methods.The first adaptation — or approach — of SDRG is validonly in the limit of very strong modulation, and correspondsto the immediate extension to spin-1 particles of the SDRGapproach of Refs. 5 and 6. This involves identifying the moststrongly connected spin cluster in the chain, and calculatingeffective couplings between spins neighboring the cluster byassuming that the cluster is locked in its ground state. In thesimplest case in which such cluster is a spin pair, the groundstate is a singlet, and the excited states consist of a triplet anda quintuplet, all of which are only assumed to contribute tothe effective couplings via virtual excitations. Since this failsfor intermediate disorder, a number of alternative adapta-tions have been proposed. One of these — the secondapproach in the present paper — consists in ignoring only thehighest local excitations, usually introducing effective spins inthe process, and calculating effective couplings so that localgaps are preserved. In case the most strongly correlated clus-ter is a spin pair, this amounts to replacing the pair of S = S = spins, connected by a bond identi-cal to the original one. This process is known not to preserveall matrix elements in the subspace of local states kept, aproblem that can be corrected at the expense of introducingnonfrustrating ferromagnetic next-nearest neighbor couplings— the third approach.For random uncorrelated couplings, the second and thirdapproaches described above are expected to lead to the samequalitative results. However, we show here that this is not nec-essarily the case in the presence of aperiodic but deterministiccouplings. We argue that the qualitative equivalence betweenthe second and third approaches is to be expected only whengeometric fluctuations, as measured by the wandering expo-nent ω , are sufficiently strong.The remaining of this paper is as follows. For the sakeof completeness, in Sec. II we review the SDRG scheme ofMa, Dasgupta and Hu for the random-bond spin- Heisen-berg chain. The adaptation of the scheme to spin-1 chains,along the three approaches mentioned above, is described inSec. III. In Secs. IV and V we apply the three approaches tothe Heisenberg spin-1 chain with couplings modulated by theFibonacci and the 6-3 sequences, which respectively inducegeometric fluctuations characterized by ω = ω (cid:39) . chain,which the spin-1 chain may be expected to approach in thestrong-modulation limit. Results are checked against QMCand DMRG simulations. Section VI summarizes our findings,while several technical points are discussed in the appendices. II. STRONG-DISORDER RENORMALIZATION GROUPFOR THE HEISENBERG SPIN- CHAIN
The SDRG scheme of Ma, Dasgupta and Hu consists in theiterative decimation of the strongest energy parameter — usu-ally a bond connecting two spins — and its replacement by aneffective parameter calculated by perturbation theory, in orderto eliminate the highest energy degrees of freedom present inthe system. The new effective bond is always smaller thanthe decimated one. After the decimation, the process is re-peated with the next strongest bond in the chain, and so on.In the asymptotic limit of a very large number of iterations,the effective Hamiltonian generated by this method should de-scribe well the low-energy (low-temperature) thermodynamicbehavior of the system.The method was first introduced to study the random-bondspin- Heisenberg chain, and successfully reveals the ther-modynamics properties of the corresponding ground state,which has been dubbed a random-singlet phase. The first stepof the method is finding the strongest bond in the chain, say J .Assuming that the coupling distribution is sufficiently broad,at low temperatures ( T (cid:28) J in suitable units), the spin paircoupled by J can be pictured as frozen in its local groundstate (a singlet), and thus can be eliminated, its virtual ex-citations giving rise to an effective bond coupling the spinsneighboring the pair, as illustrated in Fig. 1.If we assume that the neighboring bonds J l and J r are much s l s l s r s r s s J J r J l J ′ FIG. 1. Decimation procedure for a pair of S = spins. weaker that J , we can calculate the effective bond by pertur-bation theory. Treating the interactions between the pair andthe rest of the chain, via the neighboring spins s l and s r , as aperturbation over the exact states of the pair, we can write thelocal Hamiltonian as h = h + h , with h = J s · s , h = J l s l · s + J r s · s r , where h represents the perturbation over the states of the pair s and s associated with h . (Throughout this paper, spin- operators are represented in lowercase, unless explicitly statedotherwise.) The eigenstates of h are a singlet, | Φ (cid:105) = √ ( | + −(cid:105) − |− + (cid:105) ) , (2)with energy E = − J , and a triplet, (cid:12)(cid:12) Φ + (cid:11) = | ++ (cid:105) , (cid:12)(cid:12) Φ (cid:11) = √ ( | + −(cid:105) + |− + (cid:105) ) , (cid:12)(cid:12) Φ − (cid:11) = |−−(cid:105) , (3)with energy E = J . If we assume that h sets the en-ergy scale ∆ of the system, a reasonable estimate for this is ∆ = E − E , and at lower energy scales the pair s and s iseffectively frozen in its ground state.Up to second order in perturbation theory, the effectiveHamiltonian is then written as h eff = (cid:104) Φ | h | Φ (cid:105) + ∑ i (cid:12)(cid:12)(cid:10) Φ | h | Φ i (cid:11)(cid:12)(cid:12) E − E == E (cid:48) + J (cid:48) s l · s r , (4)with the summation running over i ∈ { + , , −} . The effectiveparameters are given by E (cid:48) = − J − ( J l + J r ) J and J (cid:48) = J l J r J , (5)in which E (cid:48) represents a correction to the ground-state energyof h , and J (cid:48) is an effective coupling between the spins s l and s r . Notice that the effective bond J (cid:48) is always smaller than theoriginal bond J , so that the energy scale is consistently re-duced. The iteration of the above renormalization rule willlead to a probability distribution of effective bonds, whichgets broader and broader, suggesting that the results thus ob-tained are asymptotically exact. In fact, the fixed-point prob-ability distribution of the effective couplings has infinite vari-ance — an infinite-randomness fixed point. III. STRONG-DISORDER RENORMALIZATION GROUPFOR THE HEISENBERG SPIN- CHAIN
In this section, we review and discuss three different ap-proaches to adapting the SDRG method for spin-1 chains.
The different approaches arise from the difference betweenthe spectrum of the spin-1 and spin- pairs, and from thesearch for a decimation procedure which consistently reducesthe energy scale. Other approaches have also been consideredin the literature. A. The first approach
This approach is the direct adaptation of the calculations inthe previous section to the spin-1 case. The local Hamiltonianis defined by h = h + h , (6)with h = J S · S , h = J l S l · S + J r S · S r , (7)where h is to be treated as a perturbation over h . (Through-out this paper, spin-1 operators are represented in uppercase.)The energy levels of h are a singlet, with energy E = − J ,a triplet, with energy E = − J , and a quintuplet, with energy E = J . Discarding all excited states sets the local energyscale to ∆ = E − E = J .Applying second-order perturbation theory to the aboveHamiltonian, by summing over all excited states of h , as inEq. (4), the effective bond between S l and S r is given by therule J (cid:48) = J l J r J , (8)which is not necessarily consistent, because the conditions J l < J and J r < J are not enough to guarantee that J (cid:48) < J .However, this result should be valid if the coupling distribu-tion is sufficiently broad, i.e., if one is sure that J l , J r (cid:28) J .As discussed below, the search for a decimation rule whichis consistent when the above rule fails gives rise to two otherapproaches, in which the spin-1 pair is replaced by a spin- pair. ❙♣✐(cid:0) ✶❙♣✐(cid:0) ✶✴✷✁✂ ✁✄✁❧✁❧ ✁r✁rs✵✂ s✵✄❏☎❏☎ ❏r❏r❏❧❏❧ FIG. 2. Decimation procedure for a pair of S = B. The second approach
The second approach we discuss was used by Monthus etal. in the study of random spin-1 chains. The idea is todiscard only the quintuplet states of h , by replacing the spin-1 pair S and S by a pair of spin- effective spins s (cid:48) and s (cid:48) , also connected by a bond J , in order to reproduce thelowest energy gap of h . The effective local Hamiltonian isthen written as h eff0 = − J + J s (cid:48) · s (cid:48) . (9)It should be noted that the constant − J / h eff0 and h in Eq. (7).The local Hamiltonian is replaced by an effective localHamiltonian with the spins S l , s (cid:48) , s (cid:48) , and S r , h eff = h eff0 + h eff1 , (10)as shown in Fig. 2. Now the question is how to determinethe perturbation term h eff1 , which represents the connection be-tween the effective spin- pair and the rest of the chain. If onerequires, to first-order in perturbation theory, that both h eff1 and h in Eq. (7) yield the same matrix elements inside their re-spective singlet subspaces and inside their respective tripletsubspaces, one concludes that h eff1 = J l S l · s (cid:48) + J r s (cid:48) · S r . (11)This rule reduces the local energy scale from 3 J (the gap be-tween the singlet and the quintuplet states of h ) to J (the gapbetween the singlet and triplet states of h eff0 ).However, the effective Hamiltonian in Eq. (11) does notreproduce the matrix elements of h between states in the sin-glet and triplet subspaces. In order to achieve this, one has tointroduce next-nearest-neighbor couplings, giving rise to theexact first-order effective Hamiltonian h exact1 = J l S l · (cid:0) α + s (cid:48) + α − s (cid:48) (cid:1) + J r (cid:0) α − s (cid:48) + α + s (cid:48) (cid:1) · S r , (12)with α ± = ± α α = (cid:114) (cid:39) . . (13)As the next-nearest-neighbor bonds are ferromagnetic, α − J l , r (cid:39) − . J l , r , they do not introduce frustration in the s S s l s l S r S r s ′ J J r J l J ′ r J ′ l FIG. 3. Decimation procedure for a mixed-spin pair according to thesecond approach. s s S l S l S r S r S ′ J < J r J l J ′ r J ′ l FIG. 4. Decimation procedure for a pair of ferromagnetically cou-pled spin- objects according to the second approach. system. Note also that, although the nearest-neighbor effec-tive bonds, α + J l , r (cid:39) . J l , r , are stronger than the originalones, it can be checked that the associated gap of the four-spin cluster decreases as compared to 3 J (see Tab. II in App.B), so that the energy scale is still consistently reduced.Due to the nonfrustrating character of the ferromagneticbonds, Monthus et al. argued that it is safe to ignore them,if one is interested only in qualitative features of the physicaleffects introduced by randomness, and build a renormalizationscheme based on the effective Hamiltonian of Eq. (11). Thisforms the basis for the second approach.Since the effective perturbative term introduces spin- ob-jects, one needs to deal with renormalization steps involvingboth spin-1 and spin- operators in order to have a closedscheme for the renormalization group.There is clearly the possibility that the largest local energyscale is set by a pair composed of a spin- object s and aspin-1 object S connected by a bond J , and interacting withneighboring spins s l and S r via weaker bonds J l and J r , asshown in Fig. 3. The ground state of the pair corresponds toa doublet, giving rise to an effective spin- object s (cid:48) , and tofirst order in perturbation theory the four-spin cluster can bedescribed by an effective Hamiltonian h eff = J (cid:48) l s l · s (cid:48) + J (cid:48) r s (cid:48) · S r , (14)with J (cid:48) l = − J l and J (cid:48) r = J r . (15)Notice that this last process generates ferromagnetic bonds,but these only connect spin- objects. In case the local energyscale is set by such a bond, − | J | , connecting s and s , theunperturbed ground state is a triplet, giving rise to an effec-tive spin-1 object S (cid:48) ; see Fig. 4. A first-order perturbativecalculation leads to an effective Hamiltonian h eff = J (cid:48) l S l · S (cid:48) + J (cid:48) r S (cid:48) · S r , with J (cid:48) l , r = J l , r . (16)To summarize, in this second approach there are four kindsof bonds, and each of them requires a different decimationrule. In the same notation used in Ref. 17, these are: (i) Rule 1 : A pair of S = spins connected by an ferromag-netic bond [Fig. 4, Eq. (16)]; (ii) Rule 2 : A pair of S = spins connected by an antifer-romagnetic bond [Fig. 1, Eqs. (4) and (5)]; (iii) Rule 3 : A mixed-spin pair connected by an antiferro-magnetic bond [Fig. 3, Eqs. (14) and (15)]; (iv) Rule 4 : A pair of S = ∆ between the groundstate and the first discarded excited energy level of the spinpair, we have for the different rules ∆ = − J = | J | , ∆ = J , ∆ = J , ∆ = J . (17)The SDRG scheme for the random-bond spin-1 chain thenamounts to recursively looking for the bond associated withthe largest ∆ and applying the corresponding decimation rule.In the case of deterministic aperiodicity generated bysubstitution rules, there appear blocks composed of morethan one strong bond, as in the Fibonacci sequence( abaababaabaab . . . ) with J a > J b . In order to deal with thesecases, the set of decimation rules has to be extended, as de-scribed, in the spin- case, in Refs. 26 and 27. The start-ing point is to find the spin block yielding the largest localenergy gap, and renormalizing it to either an effective bondbetween the spins neighboring the block, or to one or twoeffective spins, according to the lowest energy levels of theoriginal block; see App. B. The effective couplings are thento be calculated by first- or second-order perturbation theory.In order to avoid such complications as much as possible, wechoose J b > J a and focus on the strong-modulation regime foranalytical calculations. Moderate modulation can be studiednumerically using SDRG, by implementing the rules for dif-ferent blocks, and results from such calculations are brieflymentioned below. C. The third approach
The third approach consists in using the exact first-orderHamiltonian h exact1 of Eq. (12) as the effective local Hamilto-nian that arises from the decimation of a spin-1 pair. Amongthe decimation rules, of concern here is a modification of rule PSfrag Spin 1Spin 1/2 S S S l S l S r S r s ′ s ′ J J J r J l α + J r α + J l α − J r α − J l FIG. 5. Decimation procedure for a pair of S = S is strongly coupled to a spin S while both are weaklycoupled to a number of other spins. Therefore, the exact first-order Hamiltonian turns into h exact1 = n l ∑ i = J ( i ) l S ( i ) l · (cid:0) α + s (cid:48) + α − s (cid:48) (cid:1) + n r ∑ i = J ( i ) r (cid:0) α − s (cid:48) + α + s (cid:48) (cid:1) · S ( i ) r , (18)where n l is the number of spins S ( i ) l to which S is weaklycoupled via J ( i ) l , and n r is the number of spins S ( i ) r to which S is weakly coupled via J ( i ) r . Thus, rule 4 now reads (iv’) Rule (cid:48) : A pair of S = n l = n r = IV. THE SPIN- FIBONACCI-HEISENBERG CHAIN
The Fibonacci sequence is produced by the iterative appli-cation of the substitution rule σ fb : (cid:26) a → abb → a , (19)starting from a single letter (either a or b ).The spin- Heisenberg chain with couplings J i ∈ { J a , J b } following the Fibonacci sequence remains critical (i.e. gap-less) for all finite values of the coupling ratio J b / J a . Sinceenforced dimerization makes the chain gapped, for generalaperiodic sequences the relevant geometric fluctuations to bemeasured are those associated with pairs of subsequent let-ters. As discussed in Ref. 27 (and references therein), thesegrow with the chain length as a power-law, with a wander-ing exponent ω related to the substitution rule for letter pairs,rather than for single letters. (For an example concerning theFibonacci sequence, see App. A )In the case of the Fibonacci sequence, this exponent is ω =
0, in contrast to the random-bond chain, for which ω = . J (0) a J (0) b J (1) b J (1) a FIG. 6. (a) Coupling distribution of the spin-1 Fibonacci-Heisenbergchain. (b) Effective chain obtained from the first SDRG approachafter a single lattice sweep.
Thus, geometric fluctuations associated with couplings cho-sen from the Fibonacci sequence are much weaker than thoseproduced by a random coupling distribution. Despite this fact,Fibonacci couplings also induce dramatic changes in the low-temperature behavior of the Heisenberg spin- chain. Aswe show below, this is not the case for the Heisenberg spin-1chain.In the following sections, we present the results of apply-ing the three different SDRG approaches defined in the previ-ous section to the problem of the Fibonacci-Heisenberg spin-1chain. We also discuss the discrepancies between the secondand the other two approaches, and present results from quan-tum Monte Carlo and DMRG calculations, which point to thefact that, in contrast to the random-bond chain, the second ap-proach does lead to qualitatively incorrect conclusions aboutthe low-temperature behavior of the system.
A. SDRG: The first approach
Figure 6(a) shows the first few bonds near the left end of theFibonacci-Heisenberg chain. As mentioned before, through-out the paper we assume J b > J a , but here, in order to applythe first approach, we assume the stronger condition J b (cid:29) J a .According to the usual recipe of the first approach, all J b bonds, which appear enclosed in Fig. 6(a), are to be deci-mated in a first SDRG lattice sweep, giving rise to effectivecouplings. Between spins 1 and 4 in Fig. 6(a) there is onespin pair connected by an isolated J b bond, and its decimationresults in an effective bond J (cid:48) b , by directly applying Eq. (8).But there is also another effective bond, J (cid:48) a , which appears forinstance between spins 4 and 9 by sequentially decimating the J b bonds connecting spins 5-6 and 7-8. Thus we have J (cid:48) a = (cid:18) (cid:19) J a J b and J (cid:48) b = J a J b . (20)Notice that if the effective bond J (cid:48) b is to be smaller than theoriginal bond J b , so that the decimations lead to a reductionof the energy scale, we must have J b > (cid:112) / J a , which con-stitutes a consistency condition for the first approach.As hinted by Fig. 6(b), decimating all original J b bondsleads to a Fibonacci modulation of the effective bonds (dis-regarding the first effective bond as a boundary effect). It is then clear that a new SDRG sweep will again generate a Fi-bonacci sequence, and so on. Therefore we can define recur-sive equations for the effective parameters, as well as for theratio between them. These are given by J ( j + ) a = (cid:18) (cid:19) (cid:104) J ( j ) a (cid:105) (cid:104) J ( j ) b (cid:105) , J ( j + ) b = (cid:104) J ( j ) a (cid:105) J ( j ) b , r ( j + ) ≡ J ( j + ) b J ( j + ) a = (cid:18) (cid:19) r ( j ) , (21)in which j labels the SDRG lattice sweep, j = chain. Thus we expect the chain to remain in the Haldane phase,but with a gap which depends on the bare coupling ratio r = J b / J a . An estimate of this gap is provided by the valueof the effective couplings at the energy scale for which the ef-fective coupling ratio becomes of order 1. This happens after j ∗ iterations of the SDRG scheme, with j ∗ = ln r ln . (22)From the above equations, we thus conclude that the gapshould behave as ∆ ( r ) ∼ r − ln r ln ( / ) J b , (23)up to a multiplicative constant. Taking logarithms on bothsides, this last result can be rewritten asln ∆ ( r ) ∼ ln J b − ln r ln ( / ) , (24)making evident that the gap vanishes asymptotically as thebare coupling ratio becomes larger and larger, with J b heldconstant.It is also possible to follow the growth of bond lengths asthe SDRG scheme proceeds. If we denote by (cid:96) ( j ) a and (cid:96) ( j ) b therespective lengths of the weak and strong bonds after j SDRGlattice sweeps, inspection of Fig. 6 leads to relations whichcan be written in matrix form as (cid:34) (cid:96) ( j + ) a (cid:96) ( j + ) b (cid:35) = (cid:20) (cid:21) · (cid:34) (cid:96) ( j ) a (cid:96) ( j ) b (cid:35) , (25)so that the asymptotic growth of the bond lengths follows (cid:96) ( j ) a ∼ (cid:96) ( j ) b ∼ τ j , (26)with τ = + √
5, the largest eigenvalue of the above matrix,corresponding to the rescaling factor of the renormalization-group transformation. Taking into account the bare lengths
Spin 1/2Spin 1(a)(b)(c) J (0) a J (0) b J (1) b J (1) a FIG. 7. Renormalization of the spin-1 Fibonacci-Heisenberg chainaccording to the second SDRG approach. (a) The original chain. (b)Spin-1 pairs connected by (strong) J b bonds are replaced by spin- pairs. (c) Spin- pairs are decimated, yielding effective couplingsbetween remaining S = (cid:96) ( ) a = (cid:96) ( ) b =
1, the asymptotic length of the strong bonds isgiven by (cid:96) ( j ) b (cid:39) + √ √ τ j ≡ c τ j . (27)An estimate for the correlation length of the spin-1Fibonacci-Heisenberg chain is provided by the length of thestrong bonds at the SDRG iteration where the effective cou-pling ratio becomes of order 1. Thus, we have ξ ∼ (cid:96) ( j ∗ ) b (cid:39) cr ν , (28)showing that the correlation length diverges at the infinite-modulation limit as a power law with a quite large exponent ν = ln τ ln (cid:39) . . (29) B. SDRG: The second approach
Now we study the conclusions we can extract from the sec-ond approach by applying it to the strong-modulation case J b (cid:29) J a .Figure 7 pictures the steps required to obtain effective cou-plings in the Fibonacci-Heisenberg chain according to the sec-ond approach. The original chain is shown in Fig. 7(a). Ap-plying rule 4 of Sec. III B to all J b bonds connecting spin-1pairs, these are replaced by spin- pairs, as shown in Fig. 7(b).As we assume J b (cid:29) J a , the next step involves decimating all J b bonds connecting spin- pairs, yielding effective couplings J (cid:48) a = (cid:18) (cid:19) J a J b and J (cid:48) b = J a J b . (30)Again, ignoring the leftmost bond in Fig. 7(c), the effectivecouplings follow a Fibonacci sequence.From the above equations, it is clear that the values ofthe effective couplings predicted by the second approach aresignificantly smaller than the ones predicted by the first ap-proach. This fact leads to errors when using effective J b J a J b ( α + ) J a ( α ) J a ( α − ) J a ( α + α − ) J a ( α − ) J a J ′ b J ′ a FIG. 8. Renormalization of the spin-1 Fibonacci-Heisenberg chainaccording to the third SDRG approach. (a) The original chain. (b)Spin-1 pairs connected by (strong) J b bonds are replaced by spin- pairs, are further-neighbor couplings are produced. (c) Spin- pairsare decimated, yielding effective couplings between remaining S = couplings to estimate the energy levels of the Fibonacci-Heisenberg chain, but also, in contrast to the first approach, itis clear that the effective coupling ratio predicted by the sec-ond approach, r (cid:48) = J (cid:48) b J (cid:48) a = J b J a = r , (31)is larger than the bare coupling ratio r . Therefore, ac-cording to the second approach, the effective coupling ratioshould become larger and larger as the SDRG scheme is it-erated, so that Fibonacci-modulated couplings should inducean aperiodicity-dominated gapless phase analogous to the oneobserved for the Fibonacci-Heisenberg spin- chain. Thisconclusion is qualitatively incorrect, since, as we will see be-low, taking into account the next-nearest-neighbor bonds ne-glected in the second approach recovers the predictions of thefirst approach for strong modulation. C. SDRG: The third approach
When applying the third approach to the Fibonacci-Heisenberg chain following the recipe of Secs. III B and III C,after replacing all spin-1 pairs connected by J b bonds by spin- pairs, there appear next-nearest- and further-neighbor bondsas illustrated in Fig. 8(b). In particular, the coupling betweenspins 5 and 8 in Fig. 8(b) appears due to the repeated applica-tion of rule 4 (cid:48) .For J b > α + J a (cid:39) . J a , the largest local gap in Fig. 8(b)is provided by the nearest-neighbor J b bonds (see App. B),which should then be decimated to yield the effective cou-plings shown in Fig. 8(c). This procedure is different for the J b bonds which are separated from other J b bonds by at leasttwo weaker J a bonds (such as the bond between spins 2 and3 in the figure) and for the J b bonds separated by a single J a bond (as in the sequence of bonds between spins 5 and 8).In the former, case we have to treat all weaker bonds (near-est and next-nearest) as perturbations over the Hamiltonian h = J b s · s , (32)following a second-order perturbative approach analogous tothe one in Eq. (4). The result is an effective bond betweenspins 1 and 4 in Fig. 8, given by J (cid:48) b = ( α + − α − ) J a J b = J a J b . (33)In the latter case, so that we avoid ambiguities arising fromthe order in which the J b bonds are decimated, we must per-form a third-order perturbative calculation in which all weakerbonds (nearest and next-nearest) are treated as perturbationsover the Hamiltonian h = J b s · s + J b s · s . (34)As detailed in App. C, this yields an effective bond connectingspins 4 and 9, given by J (cid:48) a = ( α + − α − ) J a J b = (cid:18) (cid:19) J a J b . (35)Thus, comparing the above results with Eq. (20), we seethat for J b > α + J a the third approach yields exactly the sameeffective bonds as the first approach. Therefore, properly tak-ing into account next-nearest neighbor bonds generated by theSDRG scheme fixes the qualitatively incorrect prediction ofthe second approach that strong Fibonacci modulations in-duce a gapless, aperiodicity-dominated phase in the Heisen-berg spin-1 chain.For weaker coupling ratios, 1 < J b / J a < α + , the largest lo-cal gap in Fig. 8(b) is not set by the J b bonds, and the order ofthe decimations is altered. Numerical implementations of thethird approach indicate that the distribution of effective bondsbecomes dimerized. As it is known that a dimerized spin- chain has a ground state which is adiabatically connected tothe Haldane phase , this is in qualitative agreement with theprediction of the first approach, and with the fact that the Hal-dane phase is stable towards Fibonacci modulations for anyvalue of the coupling ratio. D. Comparison with QMC simulations
According to the SDRG predictions, for strong modula-tion ( r (cid:29) J b bonds. In the spin- case, forwhich the ground state is expected to be in the aperiodic sin-glet phase, this picture should be qualitatively correct at alltemperatures, provided the modulation is strong enough. Onthe other hand, in the spin-1 case, this picture should breakdown at temperatures below the energy gap ∆ ( r ) of Eq. (23).Therefore, we can estimate the free energy of the Fibonacci-Heisenberg chain as f ( B , T ) = j ∗ ∑ j = ( n j − n j + ) F pair (cid:16) J ( j ) b ; B , T (cid:17) , (36) −4 −2 T −2 χ ( T ) r = 3 r = 10 r = 20 r = 100 FIG. 9. Magnetic susceptibility as a function of temperature for thespin-1/2 Fibonacci-Heisenberg chain. Solid lines correspond to theSDRG prediction for different coupling ratios r = J b / J a . QMC re-sults (symbols) were obtained using chains with 90 sites and openboundary conditions. Error bars are smaller than symbol size. where F pair (cid:16) J ( j ) b ; B , T (cid:17) is the free energy of a spin pair inter-acting via the Hamiltonian H pair = J ( j ) b S · S − B ( S z + S z ) (37)(with B a small magnetic field, introduced to allow the cal-culation of the magnetic susceptibility), n j is the fraction ofactive spins (those not yet decimated) at the j -th iteration ofthe SDRG scheme, and j ∗ is the iteration at which the effec-tive coupling ratio becomes of order unity. From the abovediscussion, it is clear that j ∗ = ∞ for the spin- case, while itcan be shown from Eqs. (21) that j ∗ = ln r / ln ( / ) for thespin-1 case.For the first j ∗ iterations of the SDRG scheme, as the ef-fective couplings always follow a Fibonacci sequence, thefraction of active spins satisfy the recurrence relation n j + =( − f b ) n j , where f b = ( − √ ) / (cid:39) .
382 is the fractionof letters b in the infinite Fibonacci sequence (see App. D).Thus, we obtain n j = ( − f b ) j .The susceptibility at zero field is readily obtained from thefree energy, χ ( T ) = − ∂ f ∂ B (cid:12)(cid:12)(cid:12)(cid:12) B = . (38)We first checked the SDRG predictions for the spin- chain, using the effective couplings calculated in Ref. 27,by comparing the results of Eqs. (36) and (38) with QMCsimulations, performed using the stochastic series expansionscheme with directed loop updates . As shown in Fig. 9,the SDRG prediction gets closer and closer to the QMC resultsas the modulation increases, as expected from the perturbativenature of the SDRG scheme.For the corresponding spin-1 chain, Figs. 10 and 11 showthe temperature dependence of the susceptibility according to -3 -2 -1 T -1 χ ( T ) SDRG-second approachSDRG-third approachQMC L = 35 QMC L = 45 QMC L = 55 QMC L = 65 QMC L = 85 QMC L = 145 FIG. 10. Temperature dependence of the magnetic susceptibility forthe spin-1 Fibonacci-Heisenberg chain with coupling ratio r = L . QMC error bars are smaller thansymbol size. -4 -3 -2 -1 T -1 χ ( T ) SDRG-second approachSDRG-third approachQMC L = 35 QMC L = 45 QMC L = 55 QMC L = 65 QMC L = 85 QMC L = 145 FIG. 11. Temperature dependence of the magnetic susceptibility forthe spin-1 Fibonacci-Heisenberg chain, similar as Fig. 10, but with acoupling ratio r = the second and third SDRG approaches, along with QMCdata, for coupling ratios r =
10 and r =
20, respectively.Clearly, the agreement with low-temperature numerical datais significantly better for the third SDRG approach, and im-proves as the number L of spins in the chain increases. Noticethe shoulders in the susceptibility curves (e.g., slightly to theleft of T (cid:39) and 10 − in Fig. 10) at temperatures close toenergy scales related to the effective J b bonds.As the QMC calculations involve chains with an odd num-ber of spins, the susceptibility does not vanish at low temper-atures even when the ground state is gapped. However, for thecoupling ratios used in Figs. 10 and 11, the energy scale of the gaps, according to Eq. (23), correspond to temperaturesbelow 10 − J b , much lower than the temperatures that couldbe reached in our simulations. E. Gap and string order correlations of the Fibonacci S = chain as a function of the coupling ratio In order to check whether a sufficiently large coupling ra-tio could induce a gapless aperiodic singlet phase, we nowpresent a numerical determination of the spin gap for differ-ent values of the coupling ratio and different chains lengths,using the DMRG method.
DMRG simulation details.
We simulate aperiodic S = L of spins, with open boundary condi-tions, using DMRG formulated in the matrix-product stateformalism . We use an SU ( ) -symmetric formulation , tak-ing advantage of the symmetry of the Hamiltonian (1), whichreduces considerably the number of states m to be kept in theDMRG calculation. We nevertheless find that the convergenceto the ground states in different total spin sector S T = , L and large cou-pling ratio r , which we attribute to the aperiodicity in the sys-tem. To ensure convergence, we use a specific warming proce-dure where we increase sequentially the number m of SU ( ) states kept, typically by values of 20 or 50, up to values of m where the ground-state energy no longer varies. For thelargest Fibonacci chains (here L = ( ) states was m = ( ) states. For each value of m in this warm-ing procedure, we perform a very large (sometimes more than200) number of sweeps, again checking that the energy doesnot vary. Numerical determination of gaps.
Depending on the par-ity of the chain size L , the ground-state is found to be in the S T = L ) or the S T = L ), as ex-pected. In the Haldane phase, the energy difference betweenthese two sectors is expected to decrease exponentially withincreasing L for open chains, due to the presence of spin- de-grees of freedom near the boundaries. Similar to what wasdone in the original DMRG study of uniform S = ,we compute the gap ∆ as the energy difference between thisground-state and the energy of the ground-state in the S T = ∆ = E ( S T = ) − E ( S T = / ) . We simulate chainswith sizes L = , , , , , , ,
378 correspondingto the “natural” numbers (in the Fibonacci sequence) of bonds L − = , , , , , , , L and starting with a sin-gle letter a , but we checked for small L <
70 that the samequalitative behavior is obtained when averaging results overthe L + L spins.We present in Fig. 12 the results for the gap ∆ (in units of J a ), as a function of coupling ratio r , for different system sizes L . It is clear from this figure that the gap does not vanish inthe entire range r ∈ [ , ] that we simulated, even though asexpected, it decreases quite considerably with increasing r .0Notice that we should not expect the DMRG gaps to bedirecly comparable to those predicted by Eq. (23), which isvalid in the infinite-chain, large-modulation limit, and disre-gards boundary effects. These turn out to be quite important,especially for the small chain lengths accessible via DMRG.Instead, we present in the inset of Fig. 12 a comparison be-tween the DMRG gaps for the strongest modulation for whichreliable data are available, r =
6, and the corresponding open-chain SDRG predictions (see App. E). The agreement is quitegood for small chains, but discrepancies arise for L ≥
56, dueto the fact that, as the effective coupling ratio decreases forincreasing system size [see Eq. (21)], the perturbative calcu-lations underlying the SDRG approach become less precise,leading to errors in the gap estimate. Nevertheless, for stilllarger chains ( L =
234 and L = String order.
The previous gap results indicate that the Hal-dane phase is not destroyed by imposing a Fibonnaci aperiodicsequence for the couplings. This is furthermore confirmed bythe numerical DMRG computation of the string order correla-tion function, (cid:104) O z ( i , j ) (cid:105) = (cid:104) S zi exp ( ı π j − ∏ k = i + S zk ) S zj (cid:105) , as a function of the distance x = | j − i | . The string ordercorrelation function takes non-vanishing values in the largedistance limit in the Haldane phase and is thus a good indi-cator of the continuity of the Haldane phase as the strengthof the aperiodicity is increased. We represent in Fig. 13 (cid:104) O z ( x = | j − i | ) (cid:105) with i = L / x running from 0 to L / r , for the largest L =
378 system simulated.A real-space correlation function such as (cid:104) O z ( x ) (cid:105) is inevitablynon-monotonous for such aperiodic systems, but the resultsof Fig. 13 indicate that the string order does not vanish up to r =
6, albeit it reaches smaller thermodynamic values (when x → ∞ ) as r is increased, as expected from the gap behavior.Overall, the DMRG results on the gap and string order sup-port the conclusion of SDRG (approaches 1 and 3) that thegapped Haldane phase remains robust against Fibonacci ape-riodicity. V. THE SPIN- CHAIN WITH COUPLINGS FOLLOWINGTHE 6-3 SEQUENCE
We now study the effects of geometric fluctuations inducedby couplings following the 6-3 sequence on the spin-1 Heisen-berg chain. The 6-3 sequence is defined by the substitutionrule σ : (cid:26) a → babaaab → baa , (39)starting from a single letter (either a or b ). The wandering ex-ponent characterizing pair fluctuations in the 6-3 sequence is ω = ln 2 / ln 5 (cid:39) .
43, and thus we expect for the spin- r ∆ L = 22 L = 35 L = 56 L = 90 L = 145 L = 234 L = 378
22 56 145 378 L -3 -2 -1 ∆ DMRGSDRG
FIG. 12. Spin 1 chain modulated by the Fibonacci sequence: gap ∆ = E ( S T = ) − E ( S T = / ) between the lowest-lying quintu-plet S T = E ( S T = ) and the ground-state energy E (which is either in the S T = S T = r = J b / J a , fordifferent system sizes L . x -0.4-0.3-0.2-0.10 < O z > r =1.0 r =2.0 r =3.0 r =4.0 r =5.0 r =6.0 FIG. 13. Spin 1 chain modulated by the Fibonacci sequence: stringorder correlation function (cid:104) O z ( x = | i − j | ) (cid:105) as a function of distance x taken starting from the quarter-chain point i = L / x = L /
2, for a L =
376 chain and different aperiodicitystrengths r = J b / J a . chain and for the strong-modulation spin-1 chain a dynamicalscaling characterized by the stretched exponential form ∆ ( (cid:96) ) ∼ exp ( (cid:96)/(cid:96) ) ω , (40)with r a nonuniversal constant. As described below, this isexactly what we obtain from the SDRG scheme. A. The first approach
Figure 14(a) shows the bond distribution prescribed by the6-3 sequence. Assuming again J b > J a , the first SDRG latticesweep generates two kinds of effective bonds, exactly as in1 J a J b J (0)1 J (0)2 J (0)3 (a)(b) FIG. 14. (a) Coupling distribution of the spin-1 Heisenberg chainaccording to the 6-3 sequence. (b) Effective chain obtained from thefirst SDRG approach after a single lattice sweep. J ( i )1 J ( i )2 J ( i )3 J ( i +1)1 J ( i +1)2 J ( i +1)3 (a)(b) FIG. 15. Self-similar coupling distribution obtained from the firstSDRG approach for subsequent lattice sweeps. Singlets are formedbetween spins connected by the strong effective bonds J ( i ) . the case of the Fibonacci-Heisenberg chain (see Fig. 6). Fur-thermore, the remaining J a couplings can be reinterpreted asa third kind of effective bond, so that we can write J ( ) = J a , J ( ) = (cid:18) (cid:19) J a J b , J ( ) = (cid:18) (cid:19) J a J b , (41)as long as J b > (cid:112) / J a .Subsequent SDRG lattice sweeps yield a scale-invariantcoupling distribution, as shown in Fig. 15, leading to a setof recurrence equations given by J ( j + ) = (cid:18) (cid:19) J ( j ) J ( j ) J ( j ) , J ( j + ) = (cid:18) (cid:19) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) , J ( j + ) = (cid:18) (cid:19) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) , (42)which are valid as long as J ( j ) > J ( j ) . This last condition istrue only for J b > ( / ) J a .Defining coupling ratios between the parameters J , J and J , we can write the recurrence equations ρ ( j + ) ≡ J ( j + ) J ( j + ) = (cid:20) ρ ( j ) (cid:21) , σ ( j + ) ≡ J ( j + ) J ( j + ) = ρ ( j ) , (43)making it clear that, under the condition J b > ( / ) J a , thereis a single, infinite-modulation fixed point, ρ ∞ = σ ∞ = ∞ . Spin 1/2Spin 1 J a J b J (0)1 J (0)2 J (0)3 (a)(b)(c) FIG. 16. First step of renormalization of the spin-1 Heisenberg chainwith couplings following the 6-3 sequence according to the secondSDRG approach. (a) The original chain. (b) Spin-1 pairs connectedby (strong) J b bonds are replaced by spin- pairs. (c) Spin- pairsare decimated, yielding effective couplings between remaining S = Thus, the first SDRG approach predicts that, in the strong-modulation limit, couplings following the 6-3 sequence in-duce a gapless aperiodic singlet phase, whose dynamical scal-ing form is calculated in Sec. V D. A rough estimate of thecritical point separating the Haldane phase from the gaplessphase is provided by the condition J b > ( / ) J a . B. The second approach
Figure 16 shows the results of applying the second SDRGapproach to the spin-1 Heisenberg chain with couplings mod-ulated by the 6-3 sequence. In Fig. 16(b), all spin pairs con-nected by J b bonds are replaced by spin- pairs after the firstlattice sweep, and for J b (cid:38) . J a all spin- pairs are thendecimated, leading to the configuration in Fig. 16(c), withthree effective bonds given by J ( ) = J a , J ( ) = (cid:18) (cid:19) J a J b , J ( ) = (cid:18) (cid:19) J a J b . (44)Starting from the configuration in Fig. 16(c), each subse-quent SDRG steps involve two consecutive sweeps throughthe lattice, the first one replacing all spin-1 pairs connected by J bonds by spin- pairs, which are then decimated to yieldnew effective couplings. This is illustrated in Fig. 17, andleads to the recurrence relations J ( j + ) = (cid:18) (cid:19) J ( j ) J ( j ) J ( j ) , J ( j + ) = (cid:18) (cid:19) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) , J ( j + ) = (cid:18) (cid:19) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) , (45)in which j labels the SDRG step. These equations are validas long as J ( j ) > J ( j ) , a condition that is always verified for J b (cid:38) . J a .2 Spin 1/2Spin 1 J ( i )1 J ( i )2 J ( i )3 J ( i +1)1 J ( i +1)2 J ( i +1)3 (a)(b)(c) FIG. 17. Self-similar coupling distribution obtained from the secondSDRG approach for subsequent RG steps. (a) Effective chain con-sisting only of S = pairs. (c) Spin- pairs are decimated, giving rise to a new ef-fective chain, again consisting only of spin-1 pairs, with an invariantcoupling distribution. As in the case of the first approach, we can define the cou-pling ratios ρ ≡ J / J and σ ≡ J / J , whose recurrence rela-tions read ρ ( j + ) = (cid:104) ρ ( j ) (cid:105) , σ ( j + ) = ρ ( j ) . (46)These also point to an infinite-modulation fixed point, ρ ∞ = σ ∞ = ∞ , so that predictions from the first and the second ap-proach are now in qualitative agreement, although, as for theFibonacci-Heisenberg chain, the energy levels predicted bythe two approaches (estimated from the effective J bonds)are distinct.If J b (cid:46) . J a , numerical implementations of the secondSDRG approach (not detailed here) predict the renormaliza-tion of a different set of bonds than in the first SDRG step,according to the recipe associating the energy scale with thebond clusters yielding the largest local gap. However, for1 . J a (cid:46) J b (cid:46) . J a , the distribution of effective couplingsin Fig. 17(a) is eventually reached, so that the scheme stillpredicts a gapless, aperiodic singlet phase as the ground state.For J b (cid:46) . J a , however, the distribution of effective cou-plings arrives at a dimerized spin- chain, a state equivalentto the Haldane phase. Thus, within the approximations lead-ing to the second SDRG approach, J b (cid:39) . J a correspondsto the critical point separating a gapped from an aperiodicity-dominated gapless phase. C. The third approach
Figure 18 shows the first step of the renormalization of thespin-1 Heisenberg chain with couplings following the 6-3 se-quence, according to the third SDRG approach. Forming spin- pairs from strongly connected spin-1 pairs, and assuming J b > J a , second- and third-order perturbation theory leads to PSfrag (a)(b)(c)
Spin 1Spin 1/2 J b J a ( α + ) J a ( α − ) J a ( α ) J a ( α + α − ) J a ( α − ) J a J (0)1 J (0)2 J (0)3 FIG. 18. First step of the renormalization of the spin-1 Heisenbergchain with couplings following the 6-3 sequence, according to thethird SDRG approach. (a) The original chain. (b) Spin-1 pairs con-nected by (strong) J b bonds are replaced by spin- pairs, are further-neighbor couplings are produced. (c) Spin- pairs are decimated,yielding effective couplings between remaining S = effective couplings which, along with the remaining J a bonds,define a set of effective bonds J ( ) = J a , J ( ) = ( α + − α − ) J a J b = (cid:18) (cid:19) J a J b , J ( ) = ( α + − α − ) J a J b = (cid:18) (cid:19) J a J b , (47)exactly as in the first SDRG approach. Notice that, as in theFibonacci-Heisenberg chain, further-neighbor couplings areintroduced in the middle of the RG step, but eliminated at theend for strong enough modulation. Nevertheless, they are es-sential in obtaining from the third approach the same effectivecouplings predicted by the first approach. Subsequent SDRG steps start from the coupling distribu-tion in Fig. 18(c), shown in expanded form in Fig. 19(a).After forming spin- pairs from spin-1 pairs coupled by ef-fective J bonds, these spin- pairs are decimated, taking intoaccount the presence of further neighbor bonds, to yield, fromsecond-, third- and fourth-order perturbation theory, new ef-fective couplings obeying the recurrence relations J ( j + ) = ( α + − α − ) J ( j ) J ( j ) J ( j ) = (cid:18) (cid:19) J ( j ) J ( j ) J ( j ) , J ( j + ) = ( α + − α − ) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) = (cid:18) (cid:19) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) , J ( j + ) = ( α + − α − ) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) = (cid:18) (cid:19) J ( j ) (cid:104) J ( j ) (cid:105) (cid:104) J ( j ) (cid:105) , (48)valid as long as J b > α + Ja .Thus, for sufficiently strong modulation, the first and thethird SDRG approaches yield the same quantitative predic-tions for the ground-state properties and the energy levels,3 Spin 1Spin 1/2 J ( i )1 J ( i )2 J ( i )3 α + J ( i )2 α − J ( i )2 α + J ( i )3 α − J ( i )3 α J ( i )3 α + α − J ( i )3 α − J ( i )3 J ( i +1)1 J ( i +1)2 J ( i +1)3 FIG. 19. Self-similar coupling distribution obtained from the third SDRG approach for subsequent RG steps. (a) Effective chain consistingonly of S = pairs, and further-neighbor bonds are formed. (c) Spin- pairs aredecimated, giving rise to a new effective chain, again consisting only of spin-1 pairs, with an invariant coupling distribution. while the second approach qualitatively agrees with the othertwo.In the presence of moderate or weak modulation, in whichthe perturbative calculations underlying the SDRG schemebecome increasingly inadequate, quantitative predictions areexpected to depend on finer details of the first and third ap-proaches. Indeed, for α + J a < J b < J a , numerical implemen-tations of the third approach still predict a gapless groundstate, although with a slightly different set of renormalizedbonds in the first RG step. However, for J b (cid:46) α + J a , the thirdapproach eventually leads to an effective chain composed ofspin- objects with a dimerized distribution of effective cou-plings, thus predicting a gapped phase. Of course, for such arange of coupling ratios, we do not expect any of the predic-tions for the critical coupling ratio to be precise. D. Dynamic scaling relation
In the strong-modulation gapless phase we can derive thedynamic scaling relation between energy and length scales.It is natural to assume that, as the various energy levels areestimated from the values of the strongest effective couplingsat each step of the SDRG scheme, the relevant length scalesare the corresponding effective lengths. From the recurrencerelations in Eqs. (48), and by looking at Fig. 19, it can be seenthat the lengths of the effective couplings satisfy recurrencerelations that can be written in matrix form as (cid:96) ( j + ) (cid:96) ( j + ) (cid:96) ( j + ) = · (cid:96) ( j ) (cid:96) ( j ) (cid:96) ( j ) , (49)in which again j labels the SDRG steps. The matrix appearingin the above equation has eigenvalues λ = λ = λ = (cid:96) ( j ) ∼ λ j . (50)The energy levels, being proportional to the value of thelargest bond in each iteration, scale as ∆ j ∼ J ( j ) . Thus, by solving the recurrence relations in Eqs. (43) and (48), we canwrite ∆ j ∼ (cid:96) − ( / ) ln5 j exp (cid:20) −
32 ln (cid:18) ρ ( ) (cid:19) (cid:96) ω j (cid:21) , (51)with ω = ln2ln5 . As expected, apart from unimportant constants,this is the same stretched-exponential form obeyed by thespin- Heisenberg chain with couplings following the 6-3 se-quence.
E. Comparison with QMC simulations
Using the same independent-singlet approximation de-scribed for the Fibonacci-Heisenberg chain in Sec. IV D, wecan obtain SDRG predictions for the susceptibility at zerofield when couplings follow the 6-3 sequence. Only a smalladaptation is necessary, as the self-similar coupling distribu-tion is distinct from the 6-3 sequence itself. Thus, we musttake into account that the fraction of J b bonds in the orig-inal chain is f b = , while the fraction of J bonds in theself-similar distribution is f J = . Below, the results of theindependent-singlet approximation are compared with quan-tum Monte Carlo simulations.For the spin- chain with L =
75 sites the results are shownin Fig. 20. As expected, the agreement between the SDRGprediction and QMC simulations is better for larger couplingratios r = J b / J a .In Figs. 21 and 22 we show the results for the spin-1 chainwith coupling ratios r = r =
10 respectively. As in thecase of the Fibonacci-Heisenberg chain, it is clear that theQMC results are in better agreement with the predictions ofthe third SDRG approach. Again notice the shoulders in thesusceptibility curves close to temperatures corresponding toenergy scales related to the effective J b bonds.4 −4 −2 T −2 χ ( T ) r = 3 r = 10 r = 20 r = 100 FIG. 20. Temperature dependence of the magnetic susceptibilityfor the spin- Heisenberg chain with aperiodic couplings followingthe 6-3 sequence. Solid lines are the SDRG predictions for variouscoupling ratios r = J b / J a , while symbols indicate the correspondingQMC results obtained for L =
75 sites. -3 -2 -1 T -1 χ ( T ) SDRG-second approachSDRG-third approachQMC L = 25 QMC L = 45 QMC L = 65 QMC L = 85 QMC L = 105 FIG. 21. Temperature dependence of the magnetic susceptibilityfor the spin-1 Heisenberg chain with aperiodic couplings followingthe 6-3 sequence with coupling ratio r =
5. Solid (dashed) line cor-responds to the SDRG prediction according to the second (third) ap-proach, while symbols correspond to QMC results for different chainsizes L . QMC error bars are smaller than symbol size. F. Gap and string order of the 6-3 S = chain as a function ofthe coupling ratio We use the same DMRG procedure described in Sec. IV Eto compute the spin gap ∆ (in units of J a ) for open S = L = , , , , ,
376 and display the results inFig. 23. Our calculations reveal that the spin gap ∆ clearlyvanishes for sufficiently large systems, when the couplingratio r is large enough. For the largest systems considered -4 -3 -2 -1 T -1 χ ( T ) SDRG-second approachSDRG-third approachQMC L = 25 QMC L = 45 QMC L = 65 QMC L = 85 QMC L = 105 FIG. 22. Temperature dependence of the magnetic susceptibility forthe spin-1 Heisenberg chain with aperiodic couplings following the6-3 sequence, similar as Fig. 21, but with a coupling ratio r = ( L =
325 and L = r ≥
4, but the behavior at smaller r and forsmaller L clearly indicates that the gap must vanish for thesecases.These results are again in agreement with the SDRG cal-culations, which indicate that the Haldane gap must vanishabove a critical modulation r c , to give rise to the gapless ape-riodic singlet phase. The critical value r c at which this quan-tum phase transition takes place is difficult to estimate pre-cisely due to strong finite-size effects arising from a small gap.Considering the largest system available, we can neverthelessascertain that the system is gapless at r = .
4. The inset ofFig. 23 displays the spin gap ∆ as a function of inverse systemsize 1 / L , for values of the coupling ratio close to the tran-sition. We can tentatively deduce a value r c ≈ . ( ) , eventhough this phenomenological determination has to be takenwith care. Even though the SDRG prediction r c (cid:39) .
73 (fromapproach 3) is different, it is subject to a large uncertainty,since for such small values of the bond ratio the perturbativecalculations become much imprecise, and we can neverthelessconclude that the numerical calculations of the spin gap sup-port the SDRG prediction of a gapless phase at large enough(but finite) value of the coupling ratio r .We finally confirm this finding by computing the string or-der correlation function (cid:104) O z ( x = | j − i | ) (cid:105) using the same setupas presented in Sec. IV E. We present in Fig. 24 the resultsof our simulations for the maximum chain size L =
376 wherewe could reach convergence for r = , , L =
105 where convergence was ensured upto r = r . For L = (cid:104) O z ( x ) (cid:105) indicates that the Hal-dane phase is still present in this finite-size sample up to r = r , in agreement with the small gap value found for this sys-tem. On the other hand, for r = , ,
6, results for the smallersample L =
105 already clearly indicate that the string order5 r ∆ L = 45 L = 75 L = 85 L = 105 L = 325 L = 376 L ∆ r = 2.6 r = 2.7 r = 2.8 r = 2.9 r = 3.0 r = 3.2 FIG. 23. Spin-1 chain modulated by the 6-3 sequence: gap ∆ = E ( S T = ) − E ( S T = ) between the lowest-lying quintuplet S T = E ( S T = ) and the ground-state energy E (which iseither in the S T = S T = r = J b / J a , for differentsystem sizes L . x -1-0.50 < O z > r =1.0 r =2.0 r =3.0 r =4.0 r =5.0 r =6.0 x -0.5-0.4-0.3-0.2-0.10 < O z > L=376L=105
FIG. 24. Spin-1 chain modulated by the 6-3 sequence: string ordercorrelation function (cid:104) O z ( x = | i − j | ) (cid:105) as a function of distance x takenstarting from the quarter-chain point i = L / x = L /
2, for a L =
376 chain (top) and a L =
105 chain (bottom)and different aperiodicity strengths r = J b / J a . vanishes in the long-distance limit, nicely confirming that theHaldane phase has disappeared. Due to the irregular behav-ior in x , we did not attempt to perform finite-size scaling onthe string-order correlation function for different system sizesto estimate the critical coupling value r c , but our results forthe largest sample L =
376 are consistent with the estimate r c = . ( ) obtained from the gap estimate. VI. DISCUSSION AND CONCLUSIONS
In this paper, we investigated the effects of aperiodic but de-terministic bond modulations on the zero and low-temperatureproperties of the spin-1 Heisenberg chain. We presented ex- plicit results for aperiodic bonds generated by two differentbinary substitution rules, associated with the Fibonacci andthe 6-3 sequences.For the Fibonacci-Heisenberg chain, whose geometric fluc-tuations are gauged by a wandering exponent ω =
0, calcu-lations based on different adaptations of the SDRG schemeyielded conflicting results. While the SDRG approach ofMonthus et al. , which allows for the appearance of effec-tive S = spins as the transformation proceeds, predicts thatfor strong bond modulation the ground state corresponds toa gapless, aperiodicity-dominated phase, the inclusion of non-frustrating next-nearest-neighbor effective bonds in the SDRGscheme points to a gapped ground state, and to the stabilityof the Haldane phase towards any finite Fibonacci modula-tion. This is the same prediction as obtained from the sim-plest SDRG scheme which only involves S = ω (cid:39) . r between the strongand weak bonds J b and J a ), while strong bond modulation( r (cid:29)
1) drives the ground state towards a gapless aperiodicity-dominated phase, similar to the one obtained for the analogous S = Heisenberg chain. Again, this prediction is nicely sup-ported by quantum Monte Carlo and DMRG calculations.Although we only presented explicit calculations for twoaperiodic sequences, we can draw more general conclusionsfor the strong-modulation regime based on known results forthe S = Heisenberg chain. For virtually all binary substitu-tion sequences characterized by a wandering exponent ω ≥ r ( j + ) = γ (cid:104) r ( j ) (cid:105) k , (52)where γ is a constant, r ( j ) is the bond ratio calculated at the j th iteration of the SDRG transformation, and k is an integerrelated to the wandering exponent ω and to the rescaling factor τ of the transformation by ω = ln k ln τ . (53)While for the SDRG approach of Monthus et al . the con-stant γ is greater than 1, the other approaches predict 0 < γ < k ≥ ω >
0) the effective bond ratio diverges along theiterations, as long as the bare bond ratio r ( ) is large enough,irrespective of the value of γ , thus always driving the sys-tem towards a gapless phase in the strong-modulation limit.On the other hand, for k = ω = γ defineswhether the flow of the effective bond ratio is directed towardsthe Haldane phase of unit effective bond ratio or to the oppos-ing aperiodicity-dominated phase. Therefore, we predict thatonly for sequences for which the wandering exponent is zero6the different SDRG approaches will offer conflicting qualita-tive results.In general, the presence of aperiodic bonds characterized bya positive wandering exponent will lead to a phase transitionbetween the Haldane phase and a gapless phase as the bondmodulation increases. In contrast to the random-bond spin-1chain, however, we do not expect an intermediate phase asso-ciated with Griffiths singularities. This is due to the fact thatthe inflation symmetry of substitution sequences precludes theappearance of arbitrarily large regions in which the systemis locally in the opposite phase as compared to the infinitechain. This is in agreement with the behavior of the aperiodicquantum Ising chain and also, in the context of nonequilib-rium transitions to an absorbing state, of the aperiodic contactprocess. Furthermore, due to the fact that the critical pointcorresponds to a bare bond ratio of order unity, estimates ofthe critical exponents of the transition from the (perturbative)SDRG scheme are both technically quite difficult and unreli-able. Any calculations of such quantities by numerical meth-ods are left for future work.
ACKNOWLEDGMENTS
We thank J. A. Hoyos for insightful discussions. Wewarmly thank I. McCulloch for providing access to his code used to perform the DMRG calculations. QMC calculationswere partly performed using the SSE code from the ALPSproject . This work was performed using HPC resourcesfrom GENCI (grants x2013050225 and x2014050225) andCALMIP (grants 2013–P0677 and 2014–P0677) and is sup-ported by the French ANR program ANR-11-IS04-005-01, by the Brazilian agencies FAPESP(2009/08171-3 and2012/02287-2), CNPq (530093/2011-8 and 304736/2012-0),and FAPESB/PRONEX, and by Universidade de S˜ao Paulo(NAP/FCx). Appendix A: The wandering exponent for letter pairs
For the Fibonacci sequence, the substitution rule for letterpairs is built by applying three times the substitution rule σ fb of Eq. 19, yielding σ : (cid:26) a → abaabb → aba . Noting that the pair bb does not occur in the sequence, it fol-lows that σ ( ) fb : aa → ( ab )( aa )( ba )( ba )( ab ) ab → ( ab )( aa )( ba )( ba ) ba → ( ab )( aa )( ba )( ab ) . For a general pair inflation rule σ ( ) , an associated substitu-tion matrix can be defined as M ( ) = m aa ( w aa ) m aa ( w ab ) m aa ( w ba ) m aa ( w bb ) m ab ( w aa ) m ab ( w ab ) m ab ( w ba ) m ab ( w bb ) m ba ( w aa ) m ba ( w ab ) m ba ( w ba ) m ba ( w bb ) m bb ( w aa ) m bb ( w ab ) m bb ( w ba ) m bb ( w bb ) , where m αβ ( w γδ ) denotes the number of pairs αβ in the wordassociated with the pair γδ . The leading eigenvalues λ and λ of M ( ) define a wandering exponent ω = ln | λ | ln λ , which governs the fluctuations of the letter pairs (see Ref. 27and references therein). Appendix B: Local gaps
At moderate modulation, it is important to identify whichspin blocks lead to the largest local energy gap, since theseare the blocks to be renormalized according to the SDRG pre-scription. In the following tables, we list the local gaps cor-responding to the various blocks produced by the aperiodicsequences used in this paper. Table I lists the blocks relevantfor the first SDRG approach, while Tab. II is relevant for thesecond and third approaches. The last column in each tableshows the renormalized blocks, a single straight line corre-sponding to an effective coupling between the spins neigh-boring the original block. Additional effective couplings mayappear; see Figs. 1 to 5.
TABLE I. Local gaps ∆ , in units of the bond J connecting spins ineach of the various blocks relevant for the first approach. The lastcolumn shows the corresponding renormalized block.n (block size) configuration ∆ / J (gap) renorm. block2 ◦ — ◦ ◦ — ◦ — ◦ ◦ ◦ — ◦ — ◦ — ◦ ◦ = spin 1TABLE II. Local gaps ∆ , in units of the bond J connecting spinsin each of the various blocks relevant for the second and third ap-proaches. The last column shows the corresponding renormalizedblock, with spins connected by a bond J (cid:48) .n (block size) configuration ∆ / | J | (gap) renorm. block2 ◦ — ◦ • — • ( J (cid:48) > • — ◦ • • — • ( J > ) J (cid:48) > • — • ( J < ) ◦ ◦ — ◦ — ◦ • — • ( J (cid:48) < • — ◦ — • J (cid:48) > ◦ — ◦ — • • • — • — • • ◦ — ◦ — ◦ — ◦ • — • ( J (cid:48) > • — ◦ — ◦ — • • — • ( J (cid:48) > ◦ — ◦ — ◦ — • • notation: ◦ = spin 1; • = spin 1/2 Appendix C: Third-order perturbative calculations of effectivecouplings in the third SDRG approach
We consider, as a perturbation over the local Hamiltonian h in Eq. (34), the Hamiltonian h exact1 = J a S · ( α + s + α − s ) ++ J a s · (cid:0) α + α − s + α − s (cid:1) ++ J a s · (cid:0) α + s + α + α − s (cid:1) ++ J a ( α − s + α + s ) · S , (C1)which includes both nearest- and next-nearest bonds to thespins in h .Following degenerate perturbation theory, we find that first-and second-order corrections to h are identically zero, whilethe third-order corrections arise from the eigenvalues of thematrix h eff = ∑ i (cid:54) = , j (cid:54) = (cid:104) Ψ | h exact1 | Ψ i (cid:105) (cid:10) Ψ i | h exact1 | Ψ j (cid:11) (cid:10) Ψ j | h exact1 | Ψ (cid:11) ( E i − E )( E j − E ) + − (cid:10) Ψ (cid:12)(cid:12) h exact1 (cid:12)(cid:12) Ψ (cid:11) ∑ i (cid:54) = |(cid:104) Ψ | h exact1 | Ψ i (cid:105)| ( E i − E ) , (C2)in which the states are obtained from direct products of theeigenstates of the spin pairs 5-6 and 7-8. Those states are: theground state | Ψ (cid:105) = | Φ (cid:105) ⊗ | Φ (cid:105) , (C3)formed by combining both pairs in the singlet states defined inEq. (2), and excited states | Ψ i (cid:105) which are formed by singlet-triplet or triplet-triplet combinations of the states | Φ (cid:105) , | Φ + (cid:105) , | Φ (cid:105) , and | Φ − (cid:105) ; see again Eq. (3).Expanding the summations we arrive at an effective bondbetween spins 4 and 9 given by Eq. (35). Appendix D: Fractions of letters in an infinite aperiodicsequence
Let us consider a general two-letter substitution rule σ : (cid:40) a → w a b → w b , (D1)in which w a and w b are words formed by arbitrary combi-nations of letters a and b . If the numbers of letters a and b are respectively n a and n b , after applying the substitution rulethese numbers change to n (cid:48) a and n (cid:48) b , such that (cid:34) n (cid:48) a n (cid:48) b (cid:35) = (cid:34) m aa m ab m ba m bb (cid:35) (cid:34) n a n b (cid:35) , (D2) m αβ being the number of letters α in the word w β .After many iterations of the substitution rule, assuming theconvergence of the fractions of letters a and b , f a = n a / ( n a + J (0) a J (0) b J (1) a J (1) b (a)(b) FIG. 25. SDRG approach as applied to the spin-1 Fibonacci-Heisenberg chain with L =
14 sites. After sweeping over the effectivechain in (b), all spins are eliminated. J (0) a J (0) b J (1) a J (1) b (a)(b)(c) FIG. 26. SDRG approach as applied to the spin-1 Fibonacci-Heisenberg chain with L =
22 sites. Sweeping over the effectivechain in (b) removes all effective J b bonds, leaving a single effective J a bond, as shown in (c). n b ) and f b = n b / ( n a + n b ) , it follows from the above matrixequation that we can write f b = m ba + f b ( m bb − m ba ) m aa + m ba + f b ( m ab + m bb − m aa − m ba ) , (D3)with f a = − f b .For the Fibonacci sequence, whose substitution rule isgiven by Eq. (19), we have m aa = m ab = m ba = m bb = f b = ( − √ ) /
2. For the 6-3 sequence,with the substitution rule in Eq. (39), we have m aa = m ab = m ba = m bb =
1, so that f b = . Appendix E: Finite-chain SDRG gaps for theFibonacci-Heisenberg chain
An SDRG estimate of the gap for finite open chains canbe obtained by stopping the RG scheme at the lowest energyscale for which at least two spins are still active. Figures 25through 27 illustrate this for chains with L = L =
22, and L =
35 spins. Since we want to obtain estimates to comparewith the DMRG results of Sec. IV E, we need to considerthe gap between the ground state, with total spin S T = S T =
1, and the lowest energy level with S T =
2. For L = L =
22, the ground state is a singlet ( S T = L =
35, with an odd number of spins, the ground state has totalspin S T = L =
14 spins, the lowest S T = S = PSfrag J (0) a J (0) b J (1) a J (1) b (a)(b) FIG. 27. SDRG approach as applied to the spin-1 Fibonacci-Heisenberg chain with L =
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22, as depicted in Fig. 26, the lowest S T = S = J ( ) a bond, yielding a gap of 3 J ( ) a . Finally, since theground state for L =
35 has total spin S T =
1, the lowest S T = J ( ) b , as shown in Fig. 27. Noticethat this explains the nonmonotonic behavior of the gaps withincreasing system size, for r (cid:38)
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