Boundary-induced spin density waves in linear Heisenberg antiferromagnetic spin chains with \mathbf{S\geq 1}

Boundary-induced spin density waves in linear Heisenberg antiferromagnetic spin chains with

Dayasindhu Dey S. N. Bose National Centre for Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700098, India    Manoranjan Kumar S. N. Bose National Centre for Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700098, India    Zoltán G. Soos Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA
August 1, 2019

Linear Heisenberg antiferromagnets (HAFs) are chains of spin- sites with isotropic exchange between neighbors. Open and periodic boundary conditions return the same ground state energy in the thermodynamic limit, but not the same spin when . The ground state of open chains of spins has or , respectively, for even or odd . Density matrix renormalization group (DMRG) calculations with different algorithms for even and odd are presented up to for the energy and spin densities of edge states in HAFs with , 3/2 and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with for . The SDWs are in phase when is odd, out of phase when is even, and have finite excitation energy that decreases exponentially with for integer and faster than for half integer . The spin densities and excitation energy are quantitatively modeled for integer chains longer than spins by two parameters, the correlation length and the SDW amplitude, with for and 49.0 for . The BI-SDWs of chains are not localized and are qualitatively different for even and odd . Exchange between the ends for odd is mediated by a delocalized effective spin in the middle that increases and weakens the size dependence. The nonlinear sigma model (NLM) has been applied the HAFs, primarily to with even , to discuss spin densities and exchange between localized states at the ends as . chains with odd are fully consistent with the NLM; chains have two gaps with the same as predicted whose ratio is 3.45 rather than 3; the NLM is more approximate for chains with even and is modified for exchange between ends for odd .


I Introduction

The Hilbert space of a system of spins has dimension . The total spin and its components are conserved for isotropic (Heisenberg) exchange interactions between spins. The simplest case is a chain with equal exchange between nearest neighbors. A great many theoretical and experimental studies have been performed on the linear Heisenberg antiferromagnet (HAF), Eq. 1 below, with and . There are multiple reasons why. First, there are good physical realizations of spin-1/2 chains in inorganic crystals with localized spins on metal ions and in organic crystals based on one-dimensional (1D) stacks of radical ions. Second, the Hilbert space is smallest for for any choice of exchange interactions, small enough to access the full spectrum and thermal physics for comparison with experiment. Third, long ago Bethe and Hulthen obtained the exact ground state Bethe (1931); *hulthen38 of the infinite chain with antiferromagnetic exchange between nearest neighbors, a prototypical gapless many-body system with quasi-long-range order.

HAFs and related chains with came to the fore with Haldane’s conjecture based on field theory that integer chains are gapped. Haldane (1982) Shortly thereafter, White introduced the density matrix renormalization group (DMRG) method that made possible accurate numerical calculation of the ground state properties of chains. White (1992); *white-prb93 The thermodynamic limit of spin chains with exchange interactions leads to quantum phase diagrams with many interesting correlated phases. According to the valence bond solid (VBS) analysis, Affleck et al. (1988) integer chains have localized edge states with spin . DMRG studies of finite chains have confirmed edge states in both integer White and Huse (1993); Schollwöck et al. (1996) and half integer Qin et al. (1995); Fáth et al. (2006) chains. Machens et al., Machens et al. (2013) have recently discussed short HAFs with comparable energies for bulk excitations and edge states. They summarize previous studies such as the relation of HAFs to the nonlinear model (NLM), its application to edge states, the VBS model and its valence bond diagrams. Qin et al., Qin et al. (1995) applied DMRG to HAFs up to 100 spins to discuss the energies of edge states and to distinguish between chains of integer and half integer . DMRG is quantitative for HAFs of spins with correlation length and large Haldane gap. Longer chains are necessary for the HAF with or for the gapless HAF.

In this paper we consider edge states of HAFs with , 3/2 and in systems of up to 500 spins. We use conventional DMRG for chains with an even number of spins and another algorithm for chains with an odd number of spins. We compute and model the spin densities of edge states as well as their excitation energies. The Hamiltonian of the spin- HAF chain with open boundary conditions (OBC) is


The spin at site is , the total spin and its component are conserved, and is a convenient unit of energy.

The terminal spins and are coupled to only one spin in Eq. 1. Periodic boundary conditions (PBC) also has between sites 1 and . Every spin is then coupled to two neighbors, the system has translational symmetry, and the smallest is expected in the ground state (GS) for AF exchange. Indeed, the GS of PBC chains is a singlet, spin , except for odd and half integer , when . The sectors of integer and half integer are disjoint, and even is conventionally taken for the thermodynamic limit. As noted by Faddeev and Takhtajan, the thermodynamic limit of the HAF with odd is not well understood. Faddeev and Takhtajan (1981)

HAFs with OBC are fundamentally different because there is no energy penalty for parallel spins at sites 1 and . The GS of Eq. 1 remains a singlet for even , but it becomes a multiplet with and Zeeman degeneracy for odd . The lowest-energy triplet is necessarily an excited state when is even. For integer , the singlet is an excited state when is odd, while for half integer , the doublet is an excited state when is odd. Except in the case, depends on the boundary conditions for arbitrarily large systems. It follows that HAFs with OBC support edge states with whose energies become degenerate in the thermodynamic limit with those of PBC systems with or 1/2.

We define the energy gaps of edge states in chains of spins as


where is the lowest energy in the sector with total spin . Even leads to . Odd leads to for integer and to relative to for half integer . Since DMRG algorithms conserve rather than , the most accurate results are the GS in sectors with increasing and . Otherwise, the singlet or doublet is an excited state in the sector for integer or in the sector for half integer . The size dependence of is faster than , which distinguishes gap states from bulk excitations that may also have zero gap in the thermodynamic limit.

We shall characterize edge states using spin densities and call them boundary-induced spin density waves (BI-SDWs). BI-SDW is more descriptive than edge state and is more accurate than localized state, since BI-SDWs are not localized in half integer chains. By convention, we choose the Zeeman level when and define the spin density at site as


The expectation value is with respect to the state of interest. Singlet states have at all sites. SDWs with have equal spin density at and by symmetry in chains, by construction and . It is advantageous to focus on spin densities rather than energy gaps. Spin densities are exclusively associated with states while the in Eq. 2 are small differences between extensive energies.

The NLM is a good approximation for HAFs, and theoretical discussions have focused as much on field theory as on spin chains. Affleck (1986a); *affleck86b; Schulz (1986); Affleck et al. (1989) The model for integer chains relates edge states to an effective Hamiltonian between spins at the ends, Machens et al. (2013)


The correlation length and exchange are fit to DMRG results for . An interesting point is that refers to the bulk, the singlet GS in the thermodynamic limit, as has been confirmed within numerical accuracy in chains. White and Huse (1993) The chain has two gap states that afford more stringent tests of Eq. 4. For example, the ratio of the two gaps is necessarily 3:1 for . Edge states in HAFs with half integer have been discussed Ng (1994); Qin et al. (1995); Machens et al. (2013) using with effective spins and effective exchange that decreases faster than but not exponentially.

Our principal goal is the quantitative description of edge states in HAFs that are sufficiently long to neglect bulk excitations in or 2 chains. The paper is organized as follows. Section II summarizes conventional DMRG algorithm for even and a different algorithm for odd that is related to junctions. Section III presents BI-SDWs spin densities and gaps for and chains with finite Haldane gaps and finite correlation lengths . DMRG returns for chains, in agreement with 6.03(1) reported previously, White and Huse (1993) and for chains. DMRG spin densities are fit quantitatively by BI-SDWs that are in phase for odd , out of phase for even . The coupling between ends is quantitative for chains and is semi quantitative for chains, in qualitative agreement with the VBS picture of localized spins. Section IV presents the BI-SDWs and gaps of the chain. The BI-SDWs are not localized in this case. The singlet-triplet gap for even decreases faster than , as anticipated by Ng. Ng (1994) The gap for odd requires a modified with a delocalized spin in central part in addition to spins at the ends. The delocalized spin rationalizes and a weaker size dependence. The Discussion summarizes the limited nature of connections to the NLM or to VBS.

Ii DMRG algorithms

By now, DMRG is a mature numerical method for 1D systems. Schollwöck (2005); Hallberg (2006) It gives excellent GS properties and has been widely applied to spin chains. Conventional DMRG starts with a superblock that consists of four sites: one site in the left block, one in the right block and two new sites, the central sites. The left and right blocks increase by one site as two new sites are added at every step. The procedure generates a chain with OBC and an even number of sites . The vast majority of DMRG calculations been performed on chains with even . White has discussed White (2005) an algorithm with one rather than two central sites that speeds up the computational time by a factor of two to four. The method was tested on an HAF of 100 spins.

We use conventional DMRG for spin chains with even and adapt an algorithm for odd that was developed for junctions. Kumar et al. (2016) junctions of spins have three arms of spins plus a central site for which we recently presented an efficient DMRG algorithm. Fig. 2 of Ref. Kumar et al., 2016 shows the growth of the infinite DMRG algorithm. A chain of spins can be viewed as two arms of spins plus a central site. The algorithm takes the system as an arm plus the central site and the environment as the other arm. Since the system of spins at step becomes the environment at the next step, the chain grows by two spins at each step. The procedure described for junctions Kumar et al. (2016) holds with one fewer arm for chains of spins.

The accuracy of the algorithm for odd is comparable to conventional DMRG, as has already been shown for functions. Kumar et al. (2016) In either algorithm, new sites are coupled to the most recently added sites and the superblock Hamiltonian contains only new and once renormalized operators. Table 1 has representative DMRG results for the ground states of and 3/2 chains with spins. The index is the number of states kept per block. The truncation error is where the sum is over the eigenvalues of the density matrix. Several sweeps of finite DMRG calculations are required for or 2, with calculations per sweep, and finite DMRG is necessary for accurate spin densities. Increasing rapidly increases the required computer resources for long chains and involves trade offs. We have checked our results against previous studies in Table 1 as well as against chains and find comparably small or smaller that amount to evolutionary improvements for even . The algorithm for odd returns equally small .

N, [Ref.]
270 [Schollwöck et al., 1996] 2 210 1E-7
150 [Qin et al., 1997] 2 250 1E-6
100 [Qin et al., 1995] 3/2 120 3E-6
192 [Fáth et al., 2006] 3/2 500–800 1E-7 – 1E-8
400 2 600 1.1E-7
399 2 500 8.5E-9
400 3/2 460 8.4E-8
399 3/2 460 1.4E-8
Table 1: Representative previous and present DMRG calculations for HAF chains with spin or 3/2 and sites. The truncation error is when states are kept per block.

In the following we have set according to Table 1 and performed 5-10 sweeps of finite DMRG for and 3/2 chains. We estimate that GS energies per site are accurate to for chains, to for and to for . The energy gaps between the GS in sectors with different total spin are accurate to for and to for or 2. The spin densities are estimated to be accurate to better than based, for example, on DMRG calculations with different algorithms for and . Accurate are readily obtained in large systems whose are not accessible.

Iii Integer spin, and 2

We start with the extensively studied HAF with OBC and even . The large Haldane gap White and Huse (1993); Nakano and Terai (2009) reduces the computational effort. The singlet-triplet gap in Eq. 2 decreases rapidly with system size. The GS alternates between and 1 for even and odd , respectively. We evaluate for even as the difference of the total energy in the sectors and 1. In addition, we also obtain for odd using the first excited state in the sector. The excited state is accurate to for . As shown in Fig. 1, upper panel, with different symbols for even and odd , decreases as with . The effective exchange between the ends is in Eq. 4 with spins . The effective Hamiltonian is quantitative for chains. The gap at is , which still exceeds the estimated numerical accuracy. The inset shows the relevant VBS valence bond diagram. Affleck et al. (1988) Each line is a singlet pair, , between spins, two per site, and the circles are unpaired spins at the ends.

Comparable DMRG accuracy for chains with even has been discussed previously. Srensen and Affleck found for and 6.028(3) for spin densities. Sørensen and Affleck (1994) White and Huse obtained White and Huse (1993) the GS energy per site very accurately and reported for the spin densities of a 60-site chain with an auxiliary spin-1/2 at one end (site in Eq. 1). Schollwöck et al., Schollwöck et al. (1996) discussed the same procedure for chains with even and an auxiliary spin-1 at one end. Auxiliary spins at both ends with adjustable exchange to sites 1 and can be used to study bulk excitations. White and Huse (1993) In this paper, we shall not resort to auxiliary spins. We always consider BI-SDWs at both ends of chains.

Figure 1: Upper panel: Singlet-triplet gap of chains with spins, even or odd, in Eq. 1; inset: VBS valence bond diagram for . Lower panel: Spin densities of the chain with 65 spins.

The spin densities in Fig. 1, lower panel, are for the GS of the 65-spin chain. We take and obtain positive at odd numbered sites and negative at even numbered sites, respectively. All chains with have , which is why call them as BI-SDWs. Table 2 lists the spin densities of the first 10 sites in chains of 66/65 spins and 48/47 spins. The 66/65 spin densities clearly refer to the same triplet and speak to the numerical accuracy since different algorithms are used. The spin density at site 1 is slightly greater than 1/2, and so is the total spin density to odd-numbered sites. The total spin density to an even-numbered site approaches 1/2 from below and exceeds 0.45 at . The apparent exponential decrease of does not hold for the first few sites since, for example, . The triplets are identical near the ends but of course differ at the middle of the chain, where becomes . The 48/47 data illustrate the weak size dependence of spin densities at the ends. Well-defined edge states must become size independent. The first 10 sites of or 66 chains are close to the thermodynamic limit of BI-SDWs.

Site N = 66 N = 65 N = 48 N = 47
1 0.53204 0.53204 0.53198 0.53211
2 -0.32090 -0.32091 -0.32081 -0.32102
3 0.37324 0.37326 0.37311 0.37342
4 -0.26515 -0.26517 -0.26495 -0.26541
5 0.26242 0.26245 0.26216 0.26275
6 -0.19992 -0.19996 -0.19958 -0.20037
7 0.18544 0.18549 0.18502 0.18599
8 -0.14688 -0.14694 -0.14635Â -0.14757
9 0.13166 0.13173 0.13101  0.13249
10 -0.10675 -0.10685 -0.10597Â -0.10777
Table 2: DMRG results for spin densities at the first ten sites of chains of spins.

To minimize the even-odd variations of spin densities and to divide out an overall scale factor, we consider the function


is odd with respect to the chain’s midpoint while is even. Figure 2 shows for chains up to the middle, . The DMRG points near the edge become size independent. Except for the first few () sites, is constant up to about . The difference between even and odd is clearly seen in the middle region, and for even, odd pairs are a convenient way to present spin densities directly without making any assumptions about the appropriate model or interpretation. It follows that the thermodynamic limit is . The lines are fits as discussed below using the correlation length from the gap , in accord with the NLM’s expectation of equal for gaps and spin densities.

Figure 2: Symbols are DMRG spin densities in , Eq. 5, to the middle of chains of spins. The lines are Eq. 8 with correlation length . The horizontal dashed line is the thermodynamic limit.
Figure 3: Symbols are DMRG results for to the middle of chains. Lines are Eq. 7 with and . Even and odd deviate from near the middle of chains.

The magnitudes of the spin densities are shown in Fig. 3 as a function of up to the middle of the chains. They decrease as and deviate upward in the middle for odd , downwards for even . The amplitude is independent of system size when .

To model the spin densities of integer chains, we introduce SDWs at the left and right ends,


The SDWs are in phase for odd when all odd-numbered sites have ; they are out of phase for even with equal at sites and . Except for Ref. Sørensen and Affleck, 1994, the spin densities have been assumed to decrease exponentially, thereby ignoring contributions from the other end. While that is the case in the thermodynamic limit, is minimally required to neglect contributions from the other BI-SDW in the middle. Since the system size in DMRG calculations rarely exceeds , it is advantageous to consider both ends. We have


The postulated BI-SDWs lead to


The relative phase of the SDWs matters within of the middle. The range of is the same for and when is even.

The lines in Fig. 2 are for and continuous in Eq. 8. The thermodynamic limit is , the dashed line in Fig. 2, and it reduces to for a continuous chain. The correlation length is accurately obtained using both even and odd chains. The spin densities indicate a gap of that is far below the accuracy of the energy difference. DMRG spin densities are also limited, however, to less than 149/150; there the show considerable scatter where has even-odd variations. The SDW amplitude in Fig. 3 accounts quantitatively for spin densities aside from the first few. The parameters and suffice for all fits in Figs. 2 and 3.

The chain has a smaller Haldane gap Nakano and Terai (2009) of . Numerical analysis is more difficult since (i) there are more degrees of freedom per site; (ii) requires longer chains; and (iii) gaps also require longer chains to distinguish between edge and bulk excitations. Results are fewer and less accurate. The nature of BI-SDWs in or 3/2 chains was the motivation for DMRG calculations on even and odd chains of hundreds of spins. According to the NLM, edge states for are associated with spin in , Eq. 4. Even chains have a singlet GS and gaps to two edge states, to the triplet () and to the quintet (). The correlation length is the same for both and . The VBS valence bond diagram corresponds to two diagrams in Fig. 1(a): There are four lines per interior site and two lines, two unpaired spin at the ends. The BI-SDW analysis of chains is equally applicable to integer chains. Increasing leads to longer and to gaps whose relative magnitudes are fixed in advance by Eq. 4.

A chain with and 200 spins or 400 spins has a GS energy of roughly . The corresponding in Table 3 are less than and their estimated accuracy is . Our and 3/2 gaps are consequently limited to and 450, respectively. They are differences between total energies. Spin densities, by contrast, are exclusively related to the GS in a sector with . The representative gaps in Table 3 cover more than a decade. We studied the dependence of gaps in and 3/2 chains, summarized in Table 1, in order to find the largest accessible systems. The gap of the infinite chain is slightly larger than . The competition between edge and bulk excitations in short HAFs with is discussed elsewhere. Qin et al. (1995); Fáth et al. (2006); Machens et al. (2013); Lou et al. (2002)

N , , ,
64 0.01583 0.05578 0.01276
100 0.00721 0.02452 0.00725
150 0.00255 0.00835 0.00388
200 0.00081 0.00285 0.00268
300 0.00194
400 0.00090
450 0.00065
Table 3: Edge-state energy gaps , Eq. 2, of HAFs with spins and in Eq. 1.

Figure 4 shows and for and even . The gaps are exponential in , as expected for integer . The correlation length, , is the same within our numerical accuracy. The ratio is based on the fitted lines and it varies between 3.27 and 3.56 for individual points. Although is approximate, the ratio is larger than the NLM value of 3 based on Eq. 4. We return to gaps after presenting results for spin densities.

Figure 4: Edge-state gaps and of chains of spins in Eq. 1.

chains with odd have a quintet GS and excitations to the triplet and singlet. We again use and the BI-SDWs analysis. Figure 5 shows in the sector up to the middle of the chains. For the sake of clarity, not all points are shown. Even-odd effects now extend to about the first 25 sites and become size independent in long chains. The thermodynamic limit is with . The magnitudes of spin-densities in Fig. 6 are fit as a function of with the same and in Eq. 7. Two parameters are nearly quantitative aside from sites . The triplet is an excited state for either even or odd . It is the lowest state in the sector for even and the first excited state in that sector for odd . DMRG calculations for even converge slowly for reasons we do not understand in detail. The triplet spin densities return the same as the quintets in Fig. 5.

Figure 5: Open symbols are DMRG spin densities in the sector for , Eq. 5, to the middle of chains of spins. Solid lines are Eq. 8 with correlation length . The horizontal dashed line in the thermodynamic limit.
Figure 6: Open symbols are DMRG results for in the sector to the middle of chains. Lines are Eq. 8 with and . Even and odd deviate from in the middle.

The correlation length based on spin densities is more accurate than from energy gaps. The fits account for of even and odd chains that extend to 500 spins, whereas numerical accuracy limits to . Schollwöck et al., Schollwöck et al. (1996) argued that the thermodynamic limit requires and obtained (Fig. 6 of [Schollwöck et al., 1996]) for with an auxiliary spin-1 at the other end using the local correlation length . Qin et al., Qin et al. (1995) estimated that for S = 2 chains up to and remarked that the accuracy was much worse than for chains. Indeed, spin densities for return . As seen in Figs. 3 and 6 for and 2, respectively, the thermodynamic limit requires even when the contribution of the BI-SDW at the other end is included. The present results for chains offer more stringent comparisons of the NLM. The model is semi quantitative: The ratio is greater than 3. We note that BI-SDWs with exponentially decreasing would assumed on general grounds and follows directly from , but not the same for gaps and spin densities.

Iv Half integer spin,

HAF chains with half integer are gapless and their edge states are fundamentally different. Even chains have a singlet GS and BI-SDWs with integer ; odd chains have and BI-SDWs with half integer . The even chain has a singlet-triplet gap that decreases faster than and has been studied by Qin et al., Qin et al. (1995) and in greater detail by Fáth et al. Fáth et al. (2006) The NLM gap goes as Fáth et al. (2006)


Fath et al. Fáth et al. (2006) used DMRG to compute for chains from to in steps of 12 spins. The first term of Eq. 9 leads to parameters and whose size dependence was obtained from successive gaps and . Extrapolation in gave the thermodynamic values of and with uncertainties. In the present study, we are characterizing BI-SDWs in spin chains and take the first term with constant , as a two-parameter approximation.

Figure 7 shows the calculated gaps of HAFs as . The gaps decrease faster than as expected for edge states. The NLM size dependence for even is Eq. 4 with . The dashed line has and as inferred by Fáth et al. Fáth et al. (2006) The solid line for even is a power law with two parameters, . Either fit is adequate over this range of system sizes, and neither accounts the decrease at . The shortest chains in which edge and bulk excitations are decoupled are probably in the range to 60, and the desired fits are for long chains. The gaps for odd are several times larger and their size dependence is weaker. They can be approximation by a different logarithm or power law. The gaps and of the chain are in marked contrast to equal in Fig. 1 for chains with even and odd .

Figure 7: Open symbols are DMRG results for the gaps and for chains with spins in Eq. 1. The solid and dashed lines are power law and logarithmic fits, respectively, with two parameters. For even , the NLM parameters in Eq. 9 are , .

The BI-SDWs of even chains are triplets. The ratios in Eq. 5 are quite different for the chain, either even or odd, and are not shown. The upper panel of Fig. 8 shows the magnitude of spin densities up to the middle of chains. The SDWs converge at small but are not localized in the chain. The spin densities add up to for even . They decrease slowly and the sum over diverges in the thermodynamic limit. The lines are fits that are discussed below. The lower panel of Fig. 8 shows the cumulative spin density to site that we define as


The total spin density is . increases rapidly to 0.5 around , reaches a broad maximum that depends on system size and decreases as required by symmetry to in the middle of the chain. The VBS valence bond diagram in the inset has unpaired spins at each end that correspond Ng (1994) to . Each site forms three singlet-paired spins to a neighbor. The middle and either the top or bottom line corresponds to the VBS diagram of the chain with a localized spin at the ends. The remaining line with paired spins is a singlet valence bond diagram of the chain. The slow variation of in the middle and no net spin between and is consistent with singlet-paired spins.

Figure 8: Upper panel: Open symbols are DMRG spin densities to the middle of HAFs with even in Eq. 1. The lines are fits based on Eq. 11 with and the indicated scale factors . Lower panel: Cumulative spin densities, Eq. 10, up to site ; inset: VBS valence bond diagram for even .

The GS of odd chains is a quartet, . Figure 9, upper panel, shows to the middle of chains. The large amplitude of in-phase BI-SDWs in the middle decreases slowly with system size. The cumulative spin density in the lower panel is again given by Eq. 11 except that the spin density is shared equally between the two halves. The total is for the entire chain, or 0.75 for the half chain. The rapid initial increase to by suggests a spin-1/2, as does the gradual increase to in the middle. The VBS valence bond diagram in the lower panel has three unpaired spins, two at one end, one at the other end; the diagram with reversed unpaired spins at the ends contributes equally by symmetry. The middle and either top or bottom line is again the VBS diagram. The remaining line is an valence bond diagram with an unpaired spin at either end. Although the diagram correctly has three unpaired spins, the DMRG spin densities clearly show one spin in the central region rather than at the ends.

The HAF with odd does not support edge states. The spin density is delocalized over the entire chain. Soos and Ramasesha (1983) Even more simply, a half-filled tight binding or Hückel band of sites has spin density at odd numbered sites and at even numbered sites; in that case, goes as and immediately rationalizes the linear increase in Fig. 9, lower panel. We attribute the larger gap in Fig. 7 and its weaker dependence of system size to enhanced coupling between the ends by the delocalized spin in the middle.

Figure 9: Upper panel: Open symbols are DMRG spin densities to the middle of HAFs with odd in Eq. 1. The lines are fits based on Eq. 11 with and the indicated scale factors . Lower panel: Cumulative spin densities, Eq. 10, up to site ; inset: one of two equivalent VBS valence bond diagrams for odd ).

The BI-SDW amplitude at the middle in the upper panel of Fig. 9 decreases slightly faster than . The size dependence of the amplitude suggests modeling the spin densities as


The amplitude depends on system size because the SDWs are not localized. We took with and the indicated to generate the lines in the upper panels of Figs. 8 and 9. The spin densities are adequately fit in the central region in either case. Deviations are seen for when is even and for when is odd.

To some extent, Eq. 11 can be understood in terms of the NLM. In the thermodynamic limit, the GS spin correlation functions depend only on the separation between spins. The NLM result is Sørensen and Affleck (1994)


for integer spin HAFs and . Several authors Nomura (1989); White and Huse (1993); Sørensen and Affleck (1994) have remarked that DMRG results for are noticeably closer to exponential in chains of 60 or 100 spins. Since converged are limited to about , such agreement is promising but not forced. White and Huse discuss White and Huse (1993) the point explicitly and show (Fig. 4 of [White and Huse, 1993]) that the ratio to the NLM correlation function becomes constant at . The first few sites where can be computed most accurately are inevitably excluded from direct comparison since the NLM describes a continuous rather than a discrete system. The factor in does not appear in the spin densities of integer chains, Sørensen and Affleck (1994) whose exponential decrease with is shown in Figs. 3 and 6. The spin correlations of the HAF go as according to field theory Affleck et al. (1989) and Monte Carlo calculations Sandvik (2010) up to return . But exact results for in finite PBC systems Sandvik (2010) still show significant deviations at .

Hallberg et al. Hallberg et al. (1996) applied the NLM and DMRG to the chain and confirmed that it belongs to the same universality class as the chain. They report and estimate from to in a 60-spin chain. We find in similar calculations for = 200. Fáth et al. Fáth et al. (2006) extrapolate to for in the thermodynamic limit. The differences are negligible in the context of spin densities. Then gives Eq. 11 when contributions from both ends are taken into account. DMRG results for deviate from Eq. 11 near the ends of chains and from Eq. 7 in integer chains. The choice of changes the fits at small . Since small is not modeled quantitatively in either case and does not concern us here, we took in Eq. 11 for the spin densities of chains.

Three effective spins are needed for the spin densities when is odd, a spin in the middle in addition to spins at the ends. The generalization of Eq. 4 to half integer and odd is


The eight microstates of correspond to the GS quartet and two doublets. Both the total effective spin and are conserved, with , in the GS. The doublets have and or 1. The spectrum is


The gap is to the doublet with singlet-paired spins at the ends; the gap to parallel spins is . The effective exchanges in Eq. 13 can be fit to DMRG results for the doublets with the lowest and second lowest energy in the sector. We find , and , . Large in Fig. 7 for odd is due to and coupling through the delocalized effective spin . The small effective exchange is antiferromagnetic.

To conclude the discussion of chains, we recall that the GS for PBC and odd has . Since is between sites in the same sublattice, the system is not bipartite, and the GS has a domain wall or topological soliton. The OBC system is bipartite. The doublet with the lowest energy has positive spin densities at odd-numbered sites and negative spin densities at even-numbered sites, respectively, with singlet paired and in Eq. 13. Figure 10 shows for to the middle of chains in the upper panel and the cumulative spin density in the lower panel. The magnitude of the spin density at the middle decreases roughly as . There are no boundary-induced edge states. The spin is delocalized as expected on general grounds and becomes the effective spin in Eq. 13. By contrast, the spin densities are entirely associated with BI-SDWs in OBC systems with even or integer since singlet states have at all sites.

Figure 10: Upper panel: DMRG spin densities to the middle of HAFs with odd in Eq. 1 for the lowest energy doublet state, . Lower panel: Cumulative spin densities, Eq. 10, up to site .

V Discussion

We have applied different DMRG algorithms to spin- HAFs, Eq. 1, with even and odd number sites in order to obtain accurate edge states in chains of several hundred spins. The principal results are the energy gaps , Eq. 2, and the spin densities , Eq. 3, that are modeled as boundary-induced spin density waves (BI-SDWs) at both ends. For the HAF, we reproduce and refine previous studies on even chains of 60 or 100 spins that exceed the correlation length by an order of magnitude. We confirm that the gap goes as in chains with odd . Two parameters, and the SDW amplitude, account quantitatively for and for chains from to at least 120. The BI-SDWs are in phase for odd , out of phase for even .

The smaller Haldane gap of the HAF or the gapless HAF requires substantially longer chains, here up to 500 spins, whose edge states have previously been studied in shorter chains . The spin densities of HAFs beteen and 502 are modeled by BI-SDWs with correlation length and amplitude . There are now two gaps, and , that decrease exponentially as up to the limit of our numerical accuracy. The gap ratio is . The gap of the HAF with even decreases faster than , roughly as or as . The gap for odd has larger amplitude and weaker size dependence. The BI-SDWs of the chain have maximum spin density at the ends but are not localized. The spin densities in chains of more than 100 spins have not been previously reported to the best of our knowledge. The ground state for odd can be modeled as a spin-1/2 at each end and a spin- in between.

DMRG calculations can be performed on longer chains of and/or larger . But the condition for integer is increasingly difficult to satisfy for small Haldane gaps whose rapid decrease has been reported Nakano and Terai (2009) to . Moreover, the gaps will require extraordinary accuracy since, as shown in Table 3, is reached at for or at for . Spin densities are more promising probes of long chains in terms of the sectors of and spins. But the required system size for half integer is poorly known and may not have been reached in the present work.

The nonlinear sigma model (NLM) and valence bond solid (VBS) have been applied to spin chains, primarily to the HAF in the thermodynamic limit. Machens et al. Machens et al. (2013) summarize and critically evaluate both the NLM and VBS in connection with short chains of less than 20 spins. In partial disagreement with earlier works, they find that the effective coupling between edge states in Eq. 4 in short chains is influenced by the comparably small finite-size gaps of bulk excitations. We have characterized long chains whose prior modeling has mainly been for .

Accurate DMRG results for or HAFs are a prerequisite for comparisons, mainly via in Eq. 4, with either the NLM or VBS. Good agreement in chains carries over to some extent to chains and less so to chains. Spin densities to yield for the correlation length of the chain. The gaps in shorter chains return the same , but the ratio is 3.45 instead of 3. The deviation is real. The 3/2 chain does not follow the Lou et al. (2002) pattern of in Eq. 4. The BI-DWIs are not localized. Two effective spins at the ends account for when is even. A third in the middle leads to and in Eq. 13 for odd .

In other ways, however, comparisons are simply not possible. Since field theory starts with a continuous system rather than a discrete chain, the ends can be distinguished from the bulk but not sites at a finite distance from the ends. Similarly, VBS deals with special Hamiltonians, Affleck et al. (1988); Machens et al. (2013); Schollwöck et al. (1996); Totsuka and Suzuki (1995) that contain, in addition to Eq. 1, terms that go as with and coefficients . Exact GS are obtained in the thermodynamic limit of these models. The relevant valence bond diagrams have paired spins, as shown, except at the first and last sites. Either the NLM or VBS correctly places localized states or unpaired spins for integer , but neither describes the BI-SDWs found in DMRG calculations spin- HAFs. The BI-SDWs are not localized in half integer chains and have different effective coupling between ends. Direct solution of Eq. 1 for chains inevitably leads to edge states whose features are blurred or lost in the NLM or VBS. Comparisons may well be limited to effective spins and exchange at the ends.

The occurrence of edge states in HAFs with follows directly from Eq. 1, as shown in the Introduction. PBC systems with have except for half integer and odd , when . OBC systems with have for even and for odd . The energy per site in the thermodynamic limit cannot depend on boundary conditions for short-range interactions. Different under OBC and PBC implies edge states, or BI-SDWs, in HAF with and gaps relative to for even or integer S or to for odd and half integer S. The size dependence and interpretation of gaps or spin densities are standard for integer . The spin densities and gaps of the chain lead to different BI-SDWs for even and odd . The NLM or VBS provides useful guidance for quantitative modeling of BI-SDWs obtained by DMRG for HAFs with .

We thank S. Ramasesha and D. Huse for discussions and the NSF for partial support of this work through the Princeton MRSEC (DMR-0819860). MK thanks DST for a Ramanujan Fellowship SR/S2/RJN-69/2012 and DST for funding computation facility through SNB/MK/14-15/137.


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