Floating, critical and dimerized phases in a frustrated spin-3/2 chain
FFloating, critical and dimerized phases in a frustrated spin-3/2 chain
Natalia Chepiga, Ian Affleck, and Fr´ed´eric Mila Institute for Theoretical Physics, University of Amsterdam,Science Park 904 Postbus 94485, 1090 GL Amsterdam, The Netherlands Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (Dated: February 24, 2020)We study spontaneous dimerization and emergent criticality in a spin-3/2 chain with antiferromag-netic nearest-neighbor J , next-nearest-neighbor J and three-site J interactions. In the absenceof three-site interaction J , we provide evidence that the model undergoes a remarkable sequenceof three phase transitions as a function of J /J , going successively through a critical commesuratephase, a partially dimerized gapped phase, a critical floating phase with quasi-long-range incommen-surate order, to end up in a fully dimerized phase at very large J /J . In the field theory language,this implies that the coupling constant of the marginal operator responsible for dimerization changessign three times. For large enough J , the fully dimerized phase is stabilized for all J , and the phasetransitions between the critical phases and this phase are both Wess-Zumino-Witten (WZW) SU(2) along part of the boundary and turn first order at some point due to the presence of a marginaloperator in the WZW SU(2) model. By contrast, the transition between the two dimerized phase isalways first order, and the phase transitions between the partially dimerized phase and the criticalphases are Kosterlitz-Thouless. Finally, we discuss the intriguing spin-1/2 edge states that emergein the partially dimerized phase for even chains. Unlike their counterparts in the spin-1 chain, theyare not confined and disappear upon increasing J in favour of a reorganization of the dimerizationpattern. I. INTRODUCTION
Antiferromagnetic Heisenberg spin chains have at-tracted a lot of attention over the years. Competing in-teractions induce frustration and are known to lead tonew phases and quantum phase transitions. For exam-ple, the J − J spin-1/2 chain undergoes spontaneousdimerization when the ratio of the next-nearest neigh-bor interaction to the nearest neighbor one J /J > . . In the spin-1 chain, spontaneous dimerizationis known to be induced by a negative biquadratic inter-action J biq /J < −
1. Recently, it has been shown thatthe three-site interaction J [( S i − · S i )( S i · S i +1 ) + h . c]induces a fully dimerized state in spin-S chains and re-duces to the J − J model for spin-1/2 . Each of thesetwo terms leads, when combined with nearest and next-nearest-neighbor interaction, to a rich phase diagram forthe spin-1 chain. In particular, the quantum phase tran-sition between the next-nearest-neighbor (NNN-) Hal-dane phase that appears at large J coupling and thedimerized phase has been shown to be in the Ising uni-versality class with a singlet-triplet gap that remainsopen . Besides, the Wess-Zumino-Witten (WZW)SU(2) critical line between Haldane and dimerizedphases turns into a first order transition due to the pres-ence of a marginal operator in the underlying critical the-ory.The Heisenberg spin-1 chain has a bulk gap , but spin-1/2 edge states. In finite systems, the coupling betweenthese two edge spins decay exponentially fast with thelength of the chain. This causes quasi-degenerate low-lying in-gap states, a singlet and the so-called Kennedytriplet . By contrast, half-integer-spin chains with isotropic nearest-neighbor Heisenberg interaction ( J = J = 0) are known to be gappless with algebraically de-caying spin-spin correlations. Edge spins do not appearin the critical spin-1/2 chain, although they emerge forhigher spins. Edge states in critical systems are funda-mentally different from those in the gapped phase: inthe critical spin-3/2 chain the spin-1/2 edge states arelogarithmically delocalized over the entire chain, more-over the energy splitting between the singlet and tripletlow-lying states scales with the length of the chain Las ∆ e dge ∝ / ( L ln( BL )), where B is a non-universalconstant .A previous investigation of the spin-3/2 J − J chainhas shown that the system undergoes spontaneous dimer-ization when J /J > .
29. The transition between thecritical and the gapped dimerized phases is expected tobe in the WZW SU(2) universality class in analogywith the spin-1/2 J − J model. In this gapped phase,emergent spin-1/2 edge states are localized at the openends of a chain. The appearance of these edge states inthe gapped dimerized phase suggests that it is ratherpartially dimerized with alternating single and doublevalence bond singlets (VBS) between nearest-neighborsites. The edge states disappear around J /J ≈ . .Since the correlation length remains finite, it has beenproposed that the system undergoes a first order phasetransition at that ratio by analogy with the spin-1 J − J model .In the spin-3/2 J − J model the transition from thecritical WZW SU(2) phase to a spontaneously dimer-ized one occurs at J /J ≈ . . This phase transitionis continuous and belongs to the SU(2) k =3 WZW univer-sality class characterized by the central charge c = 2. a r X i v : . [ c ond - m a t . s t r- e l ] F e b This is in agreement with a recent prediction on symme-try protection of critical phases that a renormalization-group flow from WZW SU(2) k to SU(2) k theory is pos-sible only if the parity from k to k is preserved.The dimerized phase induced by the three-site inter-action corresponds to a fully dimerized phase with threeVBS singlets on every other nearest-neighbor bond. Tosee this we refer to a special point in J − J model wherethe ground state is known exactly. Michaud et al. have shown that at J /J = 1 / (4 S ( S +1) −
2) the ground-state is an exactly dimerized state for all spin-S chains.Further investigations have shown that this exact resultcan be extended to the case where a next-nearest neigh-bor exchange J is included . The two fully dimerizedstates are eigenstates along the line J J − J = 14 S ( S + 1) − , (1)and they are ground states for J not too large. Forspin-3/2, eq.1 implies J / ( J − J ) = 1 / J − J − J Hamiltonian: H = J (cid:88) i S i · S i +1 + J (cid:88) i S i − · S i +1 + J (cid:88) i [( S i − · S i )( S i · S i +1 ) + h .c. ] . (2)In the following, we focus on J , J , J >
0, and withoutloss of generality we fix J = 1. The numerical simula-tions have been performed with a state-of-the art densitymatrix renormalization group (DMRG) algorithm .Throughout the paper we consider chains of even lengthwith open boundary conditions.The paper is organized as follows. In Sec. II we pro-vide an overview of the phase diagram and discuss itsmain features. In Sec. III we focus on small values of J and study the nature of the phase transition between thecommensurate-critical phase and the fully dimerized one.In Sec. IV we provide numerical evidence for a Kosterlitz-Thouless transition between the commensurate-criticaland partially dimerized phases. In the following Sec. Vwe provide numerical evidence in favor of a first ordertransition between the two dimerized phases. We studythe critical incommensurate phase that emerges at largevalues of J and discuss the nature of the phase transi-tions between this floating phase and the two dimerizedphases in Sec. VI. In Sec. VII we discuss the behavior ofthe edge states in the partially dimerized phase. Sec. VIIIcontains our final discussion and conclusions. II. PHASE DIAGRAM
Our numerical results are summarized in the phase di-agram of Fig. 1. It contains two dimerized phases - par-tially and fully dimerized; and two critical phases with commensurate and incommensurate correlations. Thedimerized phases can be schematically illustrated usinga valence bond singlet (VBS) representation as shown insketches of Fig. 1. The fully dimerized phase correspondsto three valence-bonds on every other J bond, while thepartially dimerized phase corresponds to alternating oneand two valence bonds.The conventional or commensurate (c-) critical phaseis similar to that of the Heisenberg spin-1/2 chain. Itis stabilized when both J and J couplings are small.The dominant wave-vector of the spin-spin correlations is q = π . This critical phase can be described by the WZWSU(2) conformal field theory . In terms of VBS sin-glets, this phase can be visualized as one valence bond per J bonds, and on top of that one valence bond that res-onates between two neighboring bonds (shown schemat-ically with a dashed line).By contrast, inside the critical phase stabilized atlarger values of J , the wave-vector q is not locked andchanges within the phase. Following the classification byBak we will refer to this critical phase as a floatingphase. As in the previous case, the underlying criticaltheory is WZW SU(2) topped with incommensurate os-cillations, that, in particular, affect the boundary condi-tions. In terms of VBS singlets, the floating phase can beviewed as a sequence of different domains. The size of thedomains, i.e. the period, changes with the wave-vector q .The transition between the c-critical and partiallydimerized phases is in the Kosterlitz-Thouless (KT)universality class, in agreement with the previous studyof the J − J model . Both the c-critical phase andthe KT critical line are described by WZW SU(2) k =1 theory, however, in complete analogy with the critical J − J spin-1/2 chain, they can be distinguished by log-arithmic corrections. Due to the presence of marginaloperators the logarithmic corrections appear on top ofall finite-size scaling inside the critical phase. By con-trast, at the KT critical line the coupling constant of themarginal operator vanishes, and the log-corrections dis-appear. In Sec. IV we will explain how one can exploitthe log-corrections to determine the location of the KTcritical line.The transition between the critical and fully dimer-ized phases is continuous in the WZW SU(2) k =3 uni-versality below and up to the end point and first orderbeyond it. The end point is located around J ≈ . J ≈ . the switch from continuous WZW SU(2) k to firstorder transition is induced by the change of sign of amarginal operator. The logarithmic corrections that arenon-zero all along the continuous transition vanish at theend point, where the marginal coupling vanishes. Inter-estingly, in the spin-3/2 chain the end point is locatedat J ≈ .
1, which changes very little compared to thecorresponding value for the spin-1 chain J = 0 . .The first order transition line continues towards small J and separates the fully dimerized from the partiallydimerized phase. Then the line continues as a first order Fully dimerizedor c=9/5 c=9/5 E x a c t l y d i m e r i z e d l i n e P a r t i a ll y d i m e r i z e d W Z W S U ( ) st o r de r Floating W Z W S U ( ) K o s t e r l i t z - T h ou l e ss K o s t e r l i t z - T h o u l e ss C-critical or or
Figure 1. Phase diagram of the S = 3 / J and three-site interactions J . Both par-tially and fully dimerized phases are gapped and sponta-neously break the translation symmetry. The fully dimerizedstate is an exact ground-state along the dotted line. Both thec-critical and the floating phases are characterized by a gap-less spectrum and algebraically decaying correlations, how-ever, by contrast to the commensurate-critical phase, the cor-relations in the floating phase are incommensurate with thelattice. The fully dimerized phase is separated from both thecritical and the floating phases by a continuous WZW SU(2) transition along the solid line and by a first order transitionalong the dashed one. The partially dimerized phase is sepa-rated from both the floating and critical phases by Kosterlitz-Thouless critical lines. The transition between the partiallyand fully dimerized phases is always first order. The preciselocation of the boundaries of the floating phase is not known,thin black lines are just indicative. transition between the floating and the fully dimerizedphases until around J ≈ .
42 where it eventually turnsagain into a continuous WZW SU(2) critical line. Thisis remarkable since, to the best of our knowledge, neithera first order transition nor a higher level k > D ( j, N ) = |(cid:104) (cid:126)S j · (cid:126)S j +1 (cid:105) −(cid:104) (cid:126)S j − · (cid:126)S j (cid:105)| as an order parameter to probe numericallythe phase diagram. Fig. 2 shows examples of the mid-dle chain dimerization D ( N/ , N ) as a function of J forthree different values of J . The dimerization changescontinuously for J = 0 (Fig. 2 (a)), in agreement witha continuous WZW SU(2) transition. A finite jump indimerization as in Fig. 2(b) for J = 0 . J = 0 .
3, the dimerization is very small up tothe transition from the critical to the partially dimerizedphase beyond which it increases up to a value approx-imately equal to 1. The Kosterlitz-Thouless transitionbetween the two occurs around J ≈ . a) c) b) Figure 2. Middle chain dimerization for N = 90 (blue) and N = 150 (red) across different transitions. (a) Continuousgrowth of dimerization across the WZW SU(2) critical line.(b) Finite jump in dimerization across the first order phasetransition from the c-critical phase to the fully dimerizedphase. (c) Continuous change of the finite-size dimerizationfrom the non-dimerized critical phase to the partially dimer-ized phase across a Kosterlitz-Thouless transition (the criti-cal line goes through J = 0 . J ≈ . J ≈ . ther increase of the J coupling, the dimerization jumpsabruptly to approximately D ( N/ , N ) ≈
4, indicating afirst order phase transition to the fully dimerized phase.On top of long-range dimerization one can distinguishregions of fully and partially dimerized phases by short-range order as shown in Fig. 3. Besides, in the partiallydimerized phase certain regions can be distinguished byemergent spin-1/2 edge states. Below we provide a shortdescription of each sub-phase.
Fully dimerized phase: • FDC1:
Real-space correlations are commensuratewith wave-vector q = π . • FDIC:
Real-space correlations are incommensu-rate with wave-vector π/ < q < π • FDC2:
Real-space correlations are commensuratewith wave-vector q = π/ , the disorder line thatseparates FDC1 from FDIC coincides with the exact linegiven by Eq.1. Partially dimerized phase: • PDC:
Real-space correlations are commensuratewith wave-vector q = π . On finite-size chains thereare two spin-1/2 edge states that couple to eachother and form a singlet (triplet) ground-state andtriplet (singlet) excited state when the entire chaincontains an even (odd) number of sites. The en-ergy splitting between these in-gap states is expo-nentially small with the system size. • PDIC:
Real-space correlations are commensuratewith wave-vector q < π . Spin-1/2 edge states arepresent, but depending on the wave-vector q andthe distance between the edges L they can eventu-ally become completely decoupled from each other, FDC2FloatingPDICRPDPDCC-critical FDC1FDIC
Figure 3. Extended phase diagram of the spin-3/2 J − J − J model. The dotted lines correspond to three disorder lines,one of which coincides with the exactly solvable line givenby Eq.1. There are three regions in the fully dimerized (FD)phase that can be distinguished by short-range order: com-mensurate (C) with wave-vector q = π below the exactlydimerized line; incommensurate with π/ < q < π (FDIC)above the exact line and not too far from the phase bound-ary and commensurate with q = π/ q = π . The floating phase is always incom-mensurate. which leads to exact zero modes - level crossing be-tween singlet and triplet in-gap states. • RPD:
As in the PDIC, real-space correlations arecommensurate with wave-vector q < π . Edge statesdisappear, which leads to the re-orientation of thedimers in finite-size chains: the pattern with (2,1)VBS singlets changes to pattern (1,2).
III. TRANSITION BETWEEN THEC-CRITICAL AND FULLY DIMERIZED PHASES
In order to determine the precise location of the criticalline between the non-dimerized c-critical phase and thefully dimerized one, we looked at the finite-size scaling ofthe middle-chain dimerization D ( N/ , N ). For each fixedvalue of J we compute the dimerization as a function ofsystem size for several values of J . Then the critical lineis associated with a straight line (the separatrix) in a log-log plot of the dimerization D ( N/ , N ) as a function ofthe chain length N (see Fig. 4(a)).Based on numerical calculations of the central charge,Michaud et al. have shown that for J = 0 the corre-sponding transition is in the WZW SU(2) universality class. Conformal field theory (CFT) predicts for WZWSU(2) k =3 the scaling dimension of the dimerization op-erator to be d = 3 / [2(2 + k )] = 3 /
10. The slope of theseparatrix gives an ‘apparent’ critical exponent, which isdifferent from d = 3 /
10 due to logarithmic corrections.At the end point however, the coupling constant of themarginal operator vanishes, and the logarithmic correc-tions disappear. So this is the only point along the tran-sition where the critical exponents can be accurately ex-tracted from finite sizes. By keeping track of the apparentcritical exponent along the critical line, we find that itcrosses the line d = 3 /
10 at J ≈ .
10 and J ≈ . Fit
DMRG , a)
50 100 150 2000.511.5 b)c) d)
Figure 4. (a) Log-log plot of the middle-chain dimerization D ( N/ , N ) as a function of the number of sites N for J = 0 . J around the critical value. Thelinear curve corresponds to the critical point and the slopegives the critical exponent d ≈ .
307 in good agreement withthe CFT prediction 3 /
10 for WZW SU(2) k =3 critical theory.(b) Apparent critical exponent along the SU (2) critical lineas a function of J . Red circles: from the slope of the log-log plot D ( N/ , N ) as a function of N for the value of J forwhich it is linear. Blue circles: from fitting D ( j, /
10. Thusthe end point is located at J = 0 . J = 0 . D ( j, N ) along the chainwith N = 200 sites at the SU (2) critical end point fitted toEq.3. The extracted exponent is in excellent agreement with d = 3 /
10. (d) Entanglement entropy at the end point for N =200 after removing the Friedel oscillations with weight ζ ≈− .
3. The central charge obtained from the fit to Calabrese-Cardy formula c ≈ .
781 agrees within 2% with the CFTpredictions 9 / It turns out that the fully dimerized phase first appearsat the edges of an open chain and therefore free bound-ary conditions in the spin-3/2 chain correspond to fixedboundary conditions in CFT. A similar effect has beenpreviously reported for the spin-1 J − J − J chain .Then, according to the boundary CFT, the dimerizationin the finite-size chain at the critical line scales (up tologarithmic corrections) as: D ( j, N ) ∝ / [( N/π ) sin( πj/N )] d (3)where j is the position index, and the critical exponentis d = 3 /
10 for WZW SU(2) k =3 . This effect is knownas Friedel oscillations. An example of the scaling of thedimerization along a finite chain is shown in Fig. 4(c).The critical exponents extracted along the transition lineare summarized in Fig. 4(b) and are in a perfect agree-ment with those extracted from the finite-size scaling ofthe middle-chain dimerization D ( N/ , N ).We extract the central charge numerically from thefinite-size scaling of the entanglement entropy in an openchain: ˜ S N ( n ) = c d ( n ) + ζ (cid:104) S n S n +1 (cid:105) + s + ln g, (4)where d = Nπ sin (cid:0) πnN (cid:1) is the conformal distance and ζ isa non-universal constant introduced in order to suppressFriedel oscillations. Fig. 4(d) provides an example of a fitof the reduced entanglement entropy ˜ S N ( n ) with Eq.4.The values of the central charge along the continuouspart of the transition always agree within 3% with theCFT prediction c = 9 / theory.For any conformally invariant boundary condition, theground state scales with the system size as E = ε N + ε + πvN (cid:16) − c
24 + x (cid:17) , (5)where ε and ε are non-universal constants, c is the cen-tral charge and x is the scaling dimension of the corre-sponding primary field. For the SU (2) k =3 WZW modelthere are 4 conformal towers labeled by the spin of thelowest energy states, j = 0, 1 /
2, 1 and 3 /
2. The scal-ing dimension of the corresponding operator is given by x = j ( j + 1) / (2 + k ). Chains with an even number ofsites have a singlet ground-state and are thus describedby the conformal tower j = 0 with scaling dimension x = 0. By contrast, the ground state of a chain with anodd number of sites belongs to the conformal tower withthe largest j = 3 / x = 3 /
4. Thusthe ground state energies of an open chain with even orodd numbers of sites scale as: E even = ε N + ε − πv N , (6) E odd = ε N + ε + 27 πv N . (7)Examples of finite-size scaling of the ground-state energyfor even and odd numbers of sites are shown in Fig. 5(a)and (b).We have extracted several excited states by comput-ing the lowest states within different symmetry sectorsof total magnetization 0 ≤ S z tot ≤
5. In order to con-struct WZW SU(2) k =3 conformal towers we have closelyfollowed Ref. 25. Since we are interested only in the a) OBC-even b) OBC-odd d) OBC-odd c) OBC-even
Figure 5. Ground state and excitation energy at J = 0 . J = 0 . /N after subtracting ε and ε in open chains with (a) even and (b) odd numbers ofsites N . (c) and (d): Energy gap between the ground stateand the lowest energies in different sectors of S tot z = 1 , ..., S tot z = 5 / , ..., / /N for even and odd numbers of sites. Red linesare CFT predictions for j = 0 and j = 3 / lowest state for different values of the total spin s , theenergy level that corresponds to this state is defined byan integer n that satisfies: j − S k ≤ n < j − S + kk (8)The results for j = 0 and j = 3 / confor-mal towers are summarized in Table I. For the j = 3 / S tot = 3 / S z tot = 1 / / J = 0 . J = 0 . j=0 s E − E ) N/πv s E − E ) N/πv , 2 0 2 4 6 10Table I. Lowest excitation energy with spin s for both j = 0and j = 3 / conformal towers.DMRG: J = 0 . J = 0 . S tot z = 0 -3/40 -3/40OBC, Even, GS S tot z = 1 1 1.0065OBC, Even, GS S tot z = 2 2 2.0003OBC, Even, GS S tot z = 3 3 2.9999OBC, Even, GS S tot z = 4 6 6.057OBC, Even, g GS S tot z = 5 9 9.12OBC, Odd, GS S tot z = 3 / S tot z = 5 / S tot z = 7 / S tot z = 9 / S tot z = 11 / SU (2) critical point. Theground state for N even S tot z = 0 and odd S tot z = 3 / /N term in the ground state energy. For the rest, thegap above the ground state is given. The results are in unitsof πv/N with v = 1 . plotted the velocities extracted from three different ex-citation levels n according to v n = ( E n − E ) N/ ( πn )(Fig. 6). At the end point, all velocities are the same,implying that the conformal tower is restored. This oc-curs around J = 0 .
1, in agreement with the value de-termined from the critical exponent. Due to logarithmiccorrections the velocities split but remain relatively closeto each other along the continuous transition. Above theend point however the spitting of ( E n − E ) N/ ( πn ) ismuch faster, in agreement with the first order transitionwith a very different structure of the spectrum.In order to characterize the phase transition beyondthe end point we have looked at the dimerization andthe ground-state energy. Both quantities were computedin the middle of fairly long chains with N = 200 and N = 400 sites to reduce the impact from the finite-sizeeffects and provide an estimate of their values in the ther-modynamic limit. The energy per bond ε N is defined by: ε N = ε + ε + ε , where ε = J (cid:104) S i − · S i + S i · S i +1 (cid:105) , a) b) Figure 6. Velocity along the critical line between the crit-ical and the fully dimerized phases extracted from the gapbetween the n th energy level and the ground state for (a) N = 50 and (b) N = 51 ε = J (cid:104) S i − · S i +1 (cid:105) ,ε = J (cid:104) ( S i − · S i )( S i · S i +1 ) + h .c. (cid:105) , where ( i, i + 1) is the central bond.Beyond the end point, for J = 0 .
18, we detect a kinkin the ground-state energy ε N as shown in Fig. 7(a).In the vicinity of the transition the energy increasesmonotonously with J ; thus in order to see the changeof the slope we have to look at the narrow windowaround the phase transition. By extrapolating the nu-merical data with a second order polynomial (black linesin Fig. 7(a)) we find the crossing point of the two fitsaround J = 0 . N = 200 to N = 400, so that the edgeeffects are negligibly small in the middle of the chain forthe chosen system sizes. (a) (b) Figure 7. (a) Kink in the energy per site plotted as a functionof J for J = 0 .
18 above the end point. (b) Finite jump in thedimerization as a function of J for J = 0 .
18 in agreementwith the first order phase transition.
We also detect a finite-jump in the dimerization D ( N/ , N ) between J = 0 . J = 0 . universality class below and including at the end pointand it is first order beyond it. This result is rather sur-prising since the critical phase is connected to a gappedphase with spontaneously broken symmetry via a firstorder transition. To the best of our knowledge such ascenario has not yet been observed in the context of one-dimensional spin systems. The conformal field theoryexplains the appearance of the first order transition bythe change of the sign of the marginal coupling constant. IV. C-CRITICAL PHASE AND FIRSTKOSTERLITZ-THOULESS TRANSITION
As pointed out above, the commensurate critical phasethat appears at small values of J and J and theKosterlitz-Thouless transition to the partially dimerizedphase are both characterized by the WZW SU(2) criticaltheory and can be distinguished only by the logarithmiccorrections.We have extracted the central charge numerically byfitting the reduced entanglement entropy to Eq.4. Ex-amples of fits of finite-size results are provided in Fig. 8.CFT predicts the central charge c = 1 for WZW SU(2) critical theory. Due to large logarithmic corrections thecentral charge extracted from the entanglement entropyin finite-size clusters differs significantly from this predic-tion deep inside the critical phase, as can be observed inFig. 8(a). By contrast, close to the Kosterlitz-Thoulesscritical line, the logarithmic corrections are suppressed,and the central charge can be extracted with sufficientaccuracy even from relatively small chains Fig. 8(b).
20 40 60 80 1003.23.43.6 20 40 60 80 1003.23.33.43.5 a) b)
Figure 8. Extraction of the central charge for open chainswith N = 90 (green) and N = 150 (red) by fitting the re-duced entanglement entropy ˜ S N ( n ) with the Calabrese-Cardyformula of Eq.4 inside the critical phase (a) far from and (b)close to the Kosterlitz-Thouless transition Close to the Kosterlitz-Thouless transition the dimer-ization decreases almost linearly on a log-log scale onboth sides of the transition, and locating the critical lineby identifying the separatrix becomes extremely chal-lenging (see Fig. 9(a)). To locate the phase transition,one can extract the central charge, that differs slightlyfrom c = 1 inside critical phase close to the transition,but rapidly decreases in the gapped partially dimerized C e n t r a l c h a r g e a) b) Figure 9. (a) Middle-chain dimerization D ( N/ , N ) as afunction of the system size N for J = 0 .
01. For finite-sizesystems the apparent finite-size scaling is linear even abovethe transition that occur at 0 . ≥ J ≥ .
30. (b) Centralcharge as a function of J for J = 0 .
01. It is given by c = 1inside the critical phase close to the transition and decreasestowards zero in the gapped partially dimerized phase. Theresults are for an open chain with N = 150 sites. phase, as shown in Fig. 9.An alternative way to locate the KT phase transitionis based on the effective velocities that can be extractedfrom the excitation spectrum. For the SU (2) k =1 WZWmodel, there are only two conformal towers labeled by thetotal spin: j = 0 and 1 /
2. The levels corresponding tothe lowest states with different magnetization sectors canbe extracted from Eq. 8. They are summarized in TableIII. Due to presence of the low-lying edge states aroundthe Kosterlitz-Thouless transition, the listed states canbe approximately found as ground states in the symmetrysector S z tot = s + 1. The numerical results obtained for J = 0 .
01 are summarized in Fig. 10. The crossing pointwhere all velocities are almost the same and thereforethe conformal towers are restored is around J ≈ . J ≈ . J = 0 .
01. In generalone can improve the results by removing or fixing theedge states, which themselves contribute logarithmicallyto the energy. Here, since we are interested only in thelocation of the critical line, shifting the sectors by S z tot =+1 seems sufficient. j=0 s E − E ) N/πv s E − E ) N/πv , 0 2 6 12 20 30Table III. Lowest excitation energy with spin s for both j = 0and j = 1 / conformal towers. a) b)c) d) Figure 10. Velocities across the Kosterlitz-Thouless transi-tion between the critical and partially dimerized phases ex-tracted from the gap between various energy levels and theground state as a function of J for fixed value of J = 0 . V. TRANSITION BETWEEN THE TWODIMERIZED PHASES
A phase transition between the partially and the fullydimerized phases occurs for 0 . ≤ J ≤ .
35, and it is offirst order. This transition can be seen as a pronouncedkink in the energy per site (cid:15) mid calculated in the middleof the chain. A small hysteresis behavior appears be-cause the dimerization is favored at the open edges (seeFig. 11). It decreases with increasing system size. Apartfrom that, the finite-size effects are very small, and thelocation of the critical point can be extracted accuratelyfrom relatively small clusters.
Figure 11. Kink in the energy across the first order phasetransition between partially and fully dimerized phases.
The simplest domain wall between the fully and par-tially dimerized domains carries spin-1/2. One can detectit as a pair of solitons (see Fig. 12(a)) in the magnetiza-tion profile of a chain with S z tot = 1 at the transition linebetween the two phases. From Fig. 12(b) one can alsoconclude that the domains of fully dimerized states are located close to the open edges of the chain, while thedomain in the middle is in the partially dimerized state. -0.100.10.20 20 40 60 80 100 120-1-0.50 (a)(b) Figure 12. (a) Local magnetization and (b) nearest-neighborcorrelation profiles for N = 120 at J = 0 .
28 and J = 0 . VI. FLOATING PHASE
The numerical investigation of floating phases is alwayschallenging. Since the floating phase is critical and char-acterized by a divergent correlation length a proper con-vergence in DMRG can be achieved for relatively smallsystem sizes only. At the same time, the incommensu-rate wave-vector q has to be compatible with the bound-ary conditions, either open (and usually spontaneouslyfixed) or periodic. Therefore the system size should besufficiently large to resolve the true wave-vector. Thecloser q is to a commensurate value, the longer the systemshould be to resolve the difference. Moreover, an increas-ing next-nearest-neighbor interaction naturally increasesthe amount of entanglement carried by the J bonds,that are superimposed when the system is bipartite inthe DMRG. We keep up to 1500 states and perform upto 7 DMRG sweeps in the two-site routine. A. Incommensurate correlations
In order to extract the wave-vector q we fit the real-space spin-spin correlations C i,j = (cid:104) S i · S j (cid:105) − (cid:104) S i (cid:105) · (cid:104) S j (cid:105) to the Ornstein-Zernicke form: C OZ i,j ∝ e −| i − j | /ξ (cid:112) | i − j | cos( q | i − j | + ϕ ) , (9)where the correlation length ξ , the wave-vector q and thephase shift ϕ are fitting parameters. We equally use thesame form to fit the correlations inside the critical phase,because the finite length of the chain and the finite MPSbond dimension both induce an effective finite correlationlength. We find that the quality of the fit is improved ifit is done in two steps. First, we discard the oscillationsand fit the main slope of the exponential decay. Thisallows us to perform a fit in a semi-log scale log C ( x = | i − j | ) ≈ c − x/ξ − log( x ) /
2. Second, we define a reducedcorrelation function˜ C i,j = (cid:112) | i − j | e −| i − j | /ξ + c C i,j (10)and fit it with a cosine ˜ C i,j ≈ a cos( q | i − j | + ϕ ).Fig. 13(a) summarizes our results and shows three dis-order lines and a set of equal-q lines. Close to the ex-act line in the fully dimerized states the wave-vectorchanges very fast and the equal-q lines are quite con-densed. In the partially dimerized phase, we observe anabrupt change from q = π to q ≈ . π at the disorderline. The wave-vector q seems to be locked at this valuefor a short parameter range not too far from the disorderline. We believe that this plateau is purely a finite-sizeeffect, however a deeper understanding of its nature isbeyond the scope of this work. Apart from that, the q -vector changes with the parameters J and J in a con-tinuous and smooth way, in particular inside the floatingphase. Fig. 13(b) shows q as a function of J for a fixedvalue of J . One can clearly see that inside the float-ing phase the wave-vector is not locked to a particularvalue, but indeed changes very smoothly with couplingconstant. Floating Fullydimerized (a) (b)
Figure 13. (a) Phase diagram with the lines of constant wave-vector q extracted by fitting real-space correlations to theOrnstein-Zernicke form. (b) Dependence of the wave-vector q as a function of J for J = 0 . q reaches its commensurate value π/ J ≈ . In order to understand the nature of the floating phase,let us think of the spin-3/2 chain as a composite sys-tem made of a spin-1 chain and a spin-1/2 chain. Atsufficiently large value of the next-nearest-neighbor cou-pling J , the spin-1 chain is in the next-nearest-neighbor(NNN) Haldane phase with one VBS singlet per J bond . It has been shown however that this simple pic-ture is only true at infinitely large J . At finite val-ues, the ground state corresponds to what was called theintertwined Haldane chain , the periodic twist betweenthe two chains being responsible for the incommensurateshort-range correlations. The superposition of this spin-1 state with the remaining critical spin-1/2 sketched inFig. 14 then naturally leads to an intuitive picture of thefloating phase. ++ +... Figure 14. VBS sketch of the state in the floating phase thatconsists of the intertwined Haldane chains (balck solid lines)and a critical spin-1/2 chain (red dashed line).
B. Dimerization
Inside the floating phase, the incommensurability af-fects also local quantities such as the nearest-neighborcorrelations at the center of the chain because the cor-relation length diverges; therefore the definition of thedimerization D ( N/ , N ) that we used before is no longerapplicable. Instead, we compute the amplitude of thenearest-neighbor correlations: D ampl = max( (cid:104) S i S i +1 (cid:105) ) − min( (cid:104) S i S i +1 (cid:105) ) (11)and the minimum of the local dimerization: D min = min( (cid:104) S i − S i (cid:105) − (cid:104) S i S i +1 (cid:105) ) , (12)where N/ − < i < N/ D ( N/ , N ) used before.In Fig. 15 we show D ampl and D min as a function ofthe next-nearest-neighbor interaction J for three valuesof J . In each of the three cases the curve starts witha small finite-size dimerization inside the commensuratecritical phase, followed by a pronounced peak with finitedimerization over an extended region that correspondsto the partially dimerized phase. Upon further increaseof J the dimerization is non monotonous - it decreasesand remains extremely small over an extended param-eter range; eventually it increases again indicating theentrance to the fully dimerized phase. By analogy withspin-1/2 we expect the dimerization to decrease at large J ; our results for J = 0 .
02 support this scenario.0 D i m e r i z a t i o n D i m e r i z a t i o n D i m e r i z a t i o n = 0= 0.01= 0.02 (a)(b)(c) D min D ampl N = 30 N = 60 N = 90 N = 150 = 1.5 D m i n Figure 15. Dimerization as a function of J for (a) J = 0; (b) J = 0 .
01; and (c) J = 0 .
02 and for four different system sizes N = 30 , , , J ≈ . J the finite dimerization corresponds to the fullydimerized phase. Between these two regimes, the vanishinglysmall dimerization corresponds to the floating phase. The in-set in panel (a) shows the finite-size scaling of the dimerizationat J = 0 and J = 1 . C. Second Kosterlitz-Thouless transition
Finding the location of the Kosterlitz-Thouless phasetransition between the partially dimerized and the float-ing phase is extremely challenging because of the incom-mensurate correlations on both sides of the transition.In particular, for any finite size chain, open edges withstronger dimerization correspond to conformally invari-ant boundary conditions only at particular values of thewave-vector q , and therefore only at specific points in thephase diagram.In Fig. 16(a) we plot the effective velocities extractedfrom the finite-size spectrum along the line J = 0. Asin the previous case, we expect the velocities to crossat the points where the conformal towers are restored.Luckily enough, the logarithmic corrections grow slowlywith J , so for every system size N we observe a veryclear crossing of all three lines.In order to show that the observed crossings are not acoincidence, but indeed signal the WZW SU(2) criticaltheory, we look at the higher excited states. In Fig. 16(b)we compare the structure of the excitation spectrum atthe crossing point with the CFT prediction for WZWSU(2) . Since the position of the crossing changes with
30 40 50 700510152025 (a) (b)
Figure 16. (a) Effective velocities across the transition be-tween partially dimerized and floating phases extracted fromthe gap between various energy levels and the ground stateas a function of J for fixed value of J = 0 and differ-ent system sizes. Only first crossing is shown. (b) Confor-mal towers of states extracted at the crossing points in (a).The DMRG data (circles) agrees with the CFT prediction forWZW SU(2) (gray lines) within 2%. the size of the chain, one cannot expect the effective ve-locity to remain the same. Therefore, we extract an effec-tive conformal level with respect to the singlet-triplet gapfor each system size N as n eff = ( E n − E ) / ( E − E ).This mean that the lowest level in Fig. 16(b) that cor-responds to n = 1 is trivial and shown only for com-pleteness. The next two levels with n = 4 and n = 9show how well we identified the location of the crossingin Fig. 16(a). Finally, the two highest levels provide theresults for S z tot = 4 and 5 and show how close the spec-trum is to the WZW SU(2) conformal tower. All n eff extracted here agrees with the CFT prediction within2%. This agreement is surprisingly good and suggests ei-ther that logarithmic corrections grow inside the floatingphase very slow, so that none of the observed crossingsare essentially affected, or that there is a process asso-ciated with incommensurability that compensates loga-rithmic corrections at the crossing points.The position of the crossing point scales with the sys-tem size in a non-monotonous way, but oscillates within awide range of parameters. This makes it impossible withthe available numerical method to identify accurately thelocation of the Kosterlitz-Thouless transition in the ther-modynamic limit. D. Phase transition between the floating and fullydimerized phases
In order to locate the phase transition between thefloating and the fully dimerized phases, we look at thefinite-size scaling of the dimerization. Despite the pres-ence of algebraic incommensurate order, this methodworks reasonably well when J is not too large. An exam-ple of such a finite-size scaling is shown in Fig. 17(b). Theslope gives an apparent critical exponent that changes1along the transition due to the presence of logarithmiccorrections. By analogy with the lower part of the phasediagram we expect the transition between the critical andthe fully dimerized phase to be in the WZW SU(2) uni-versality class. If this is so, the expected critical exponent(in the absence of log-corrections) take the value d = 0 . J ≈ .
42 and J ≈ . N = 90 sites; and it is hard, if actuallypossible, to estimate finite-size effects due to the presenceof quasi-long-range incommensurability. We therefore ex-pect that the location of the end point can be slightlydifferent in the thermodynamic limit, however, the veryexistence of the end point at which the transition switchesfrom first order to continuous is solid. (a) (b) Figure 17. (a) Apparent critical exponent along the transi-tion between the floating and fully dimerized phases extractedfrom a finite-size extrapolation of the middle-chain dimeriza-tion (blue circles) and from the fit of the Friedel oscillationprofile for two different chain lengths. (b) Example of finite-size scaling of the middle-chain dimerization for J = 0 .
42 andvarious values of J in the vicinity of the critical point. Theslope of the separatrix gives an apparent critical exponent. At the end point, where logarithmic corrections are ex-pected to vanish, we extract the central charge from thefinite-size scaling of the reduced entanglement entropy aspresented in Fig. 18. The extracted values of the centralcharge are in very good agreement with the CFT predic-tion c = 9 / . VII. EDGE STATES IN PARTIALLYDIMERIZED PHASEA. Disappearance of the edge states
One of the most intriguing and potentially mislead-ing features of the phase diagram is the line where thespin-1/2 edge states disappear. This happens inside thegapped partially dimerized phase and well away fromeach of the phase boundaries. This is very uncommon;in the original study of J − J model, the disappearanceof the edge states has been interpreted as the indicationof a phase transition . And since the correlation length Figure 18. Extraction of the central charge for open chainsby fitting the reduced entanglement entropy ˜ S N ( n ) with theCalabrese-Cardy formula of Eq.4 at the end point of the WZWline between the floating and the fully dimerized phases at J ≈ .
42 and J ≈ . remains finite at the point where the edge states disap-pear, it has been suggested that the transition is firstorder.We would like to propose a different explanation of thisphenomenon. Starting from a certain value of the next-nearest-neighbor coupling, open edges favor domains ina ‘ladder’ state, in which some VBS singlets are locatedon a few J bonds not too far from the edges, as shown inthe sketch in Fig. 19(b). It is easy to see that such edgedomains can be connected to a domain in the partiallydimerized state via non-magnetic domain walls, so thatthe localized spin-1/2 disappear at both edges. However,this requires a reorientation of dimers: if in the regionwhere the edge states are present, strong dimers are lo-cated on every odd bond, then, in the absence of edgestates, even bonds becomes stronger. This is well illus-trated by Fig. 19(c) and Fig. 19(d). The correspondingregions with and without edge state are marked in thephase diagram of Fig. 20.Note also that if one keeps track of the sign of thedimerization D s ( j, N ) = (cid:104) (cid:126)S j · (cid:126)S j +1 (cid:105) − (cid:104) (cid:126)S j − · (cid:126)S j (cid:105) , there isa small region before the first order transition where thesign of the dimerization changes.We find it instructive to illustrate the deconfinement ofthe edge states in the partially dimerized phase using theVBS sketches shown in Fig. 21. Note that the VBS pic-ture changes only in the vicinity of the edges when thespin-1/2 is moved along the chain. In the bulk, it canmove at no energy cost as a domain wall between par-tially dimerized state with different dimer orientations. B. Exact zero modes
Recently it has been shown that the effective couplingbetween the spin-1/2 edge states of a spin-1 chain of fi-nite length can be continuously tuned by frustration ifit also induces short-range incommensurability . It im-plies the existence of several level crossings between thesinglet and triplet in-gap states, i.e. points where theedge states are completely decoupled from each other.2 edge states disappear (a) (b) (c)(d) Figure 19. (a) Difference between strong and weak consecu-tive nearest-neighbor correlations computed in the middle of afinite-size chain in the vicinity of the first order transition be-tween the partially and fully dimerized phases. Shortly beforethe transition, the dimerization switches to a negative value,while its absolute value remains continuous (see Fig. 2(c)). (b)Sketch that show the mechanism of disappearance of the edgestates upon increasing the J coupling. (c) and (d) Nearestneighbor correlations along a finite-size chain with N = 150sites below(c) and above (d) the line where the edge statesdisappear. Blue (red) dots correspond to odd (even) bonds.One can clearly see an abrupt reorientation of the dimers. Later,this conclusion has been generalized to various spinladders and to a spin-2 chain . In all these cases, how-ever, the translation invariance has been preserved.In the present case, localized spin-1/2 edges statesemerge in the partially dimerized phase, inside which thetranslation symmetry is spontaneously broken. However,it turns out that the broken symmetry does not preventthe appearance of exact zero modes. In the region labeledPDIC and located between the disorder line and a linealong which the edge states vanish, we observe severallevel crossings between singlet and triplet in-gap statesas shown in Fig. 22 and Fig. 23.We have used the average energy as a reference toplot the relative energy of singlet and triplet states: ε S,T = E S,T − ( E S + E T ) /
2. So, the level with negativerelative energy corresponds to the ground state. Note
FloatingPDICRPDPDCC-critical Fully dimerized
Figure 20. Phase diagram with sketches of partially dimerizedstates with and without edge states.Figure 21. Deconfinment of the edge states in the partiallydimerized states that below the disorder line the ground-state is singletif N is even and it is a triplet if N is odd, while abovethe line, at which the edge states disappear, the ground-state is always a singlet. It implies that the number ofcrossings are even for N even and are odd when N is odd.These results have been obtained by targeting twostates in the sector of S z tot = 0 as explained in Ref. 28.Several data points for triplets have been cross-checkedby computing the energy of the lowest-energy state in thesector S z tot = 1.3 -4 a)b) SingletTriplet
Figure 22. Two crossings between singlet and triplet low-lyingenergy levels for N = 16 as a function of the next-nearest-neighbor coupling constant. Panel (b) are enlarged parts of(a) -4 SingletTriplet a)b)
Figure 23. Four crossings between singlet and triplet low-lyingenergy levels for N = 24 as a function of the next-nearest-neighbor coupling constant. Panel (b) are enlarged parts of(a). VIII. CONCLUSIONS
To summarize, the J − J − J model leads to a veryrich phase diagram that contains two dimerized phasesand two critical phases, to be compared with three phasesin the spin-1 case and only two phases in the spin-1/2 case. The combination of both J and J terms was in-strumental to understand the differences between thesephases and the transition between them. For instance,the presence of an exactly dimerized line for J largeenough is very important to support the presence of afirst-order transition between the fully dimerized phaseand a partially dimerized phase. Still, let us emphasizethat all these phases appear in the simple J − J model, arealistic model that is naturally realized in zig-zag chains.This phase diagram reveals a number of unexpectedfeatures. The change in the nature of the phase transi-tion between the c-critical and fully dimerized phase fromcontinuous to first order agrees with our previous resultson spin-1 and confirms our prediction that the realizedscenario is generic for theories with a marginal operator .It is nevertheless remarkable because the first order tran-sition appears at the boundary of a critical gapless phase.Surprisingly enough the upper part of the phase dia-gram contains a reflected version of this transition - thefirst order line turns into a continuous WZW SU(2) crit-ical line upon increasing the next-nearest-neighbor inter-action. We expect only one marginal operator in thetheory, so the double change of nature of the critical linesuggests that the coupling constant of this marginal op-erator has a minimum as a function of J and thereforecrosses zero at the end points.The appearance of a floating phase over such a wideparameter range is also quite uncommon. Note that anaccurate determination of the phase boundaries of thefloating phase would require further advances in numer-ical algorithms.Finally, we have clarified the origin of the behaviorof the edge states in the partially dimerized phase. Inparticular, we have shown that the disappearance of theedge states does not necessarily imply a phase transition,but can signal local changes of the edges that do not affectthe bulk.Altogether the physics of the frustrated spin-3/2 chainturns out not to be a simple extension of that of the spin-1/2 chain, even if in the absence of frustration they aregapless and described by the same field theory. We hopethat the present results will stimulate further experimen-tal investigation in this direction. IX. ACKNOWLEDGMENTS
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