Quantum Monte Calo studies of spinons in one-dimensional spin systems Ying Tang and Andes W. Sandvik Depatment of Physics, Boston Univesity, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA (Dated: July 13, 2018) whee the neaest-neighbo coupling J > 0, N is the total numbe of spins, and we apply peiodic bounday conditions. We will add othe inteactions to this model late, in ode to bing the system to the diffeent types of gound states mentioned above. The gound state of the plain Heisenbeg model (1) can in pinciple be solved exactly by the Bethe-Ansatz appoach 14, but in pactice many of its salient featues, such as the powe-law decaying spin-spin coelations, wee found using the bosonization method 15. Reflecting the deconfined spinons, the lowest excited states of the Heisenbeg model fom bands of degeneate singlets and tiplets 5,16,17 with the enegy ɛ 1 (q) as a function of the total momentum q of the state being ɛ 1 (q) = (π/2)j sin(q), which was fist calculated by des Cloiseaux and Peason using the Bethe ansatz. 16 FomaXiv:1404.6697v2 [cond-mat.st-el] 3 Jan 2016 Obseving constituent paticles with factional quantum numbes in confined and deconfined states is an inteesting and challenging poblem in quantum many-body physics. Hee we futhe exploe a computational scheme [Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011)] based on valence-bond quantum Monte Calo simulations of quantum spin systems. Using seveal diffeent one-dimensional models, we chaacteize S = 1/2 spinon excitations using the intinsic spinon size λ and confinement length Λ (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the citical egion (with ovelaps chaacteized by powe laws), and become ill defined (completely unlocalizable) in the Néel state (which we stabilize in one dimension by intoducing long-ange inteactions). We also veify that pais of spinons ae deconfined in unifom spin chains but become confined upon intoducing a patten of altenating coupling stengths (dimeization) o coupling two chains (foming a ladde). In the dimeized system, an individual spinon can be small when the confinement length is lage; this is the case when the imposed dimeization is weak but the gound state of the coesponding unifom chain is a spontaneously fomed valence-bond-solid (whee the spinons ae deconfined). Based on ou numeical esults, we ague that a system with λ Λ is associated with weak epulsive shot-ange spinon-spinon inteactions. In pinciple, both the length scales λ and Λ can still be individually tuned fom small to infinite (with λ Λ) by vaying model paametes. In contast, in the ladde system the two lengths ae always simila, and this is the case also in the weakly dimeized systems when the coesponding unifom chain is in the citical phase. In these systems, the effective spinon-spinon inteactions ae puely attactive and thee is only a single lage length scale close to citicality, which is eflected in the standad spin coelations as well as in the spinon chaacteistics. PACS numbes: 75.10.Jm, 75.10.Pq, 75.40.Mg, 75.40.Cx I. INTRODUCTION In one-dimensional (1D) stongly coelated systems, the emegence of factional quantum numbes is a geneic consequence of collective behavios 1. In the exactly solvable citical S = 1/2 antifeomagnetic (AFM) spin chain, the fundamental excitations ae soliton-like quasipaticles (kinks and anti-kinks), called spinons, which cay spin 1/2 2,3. Simila objects exist also in the valence-bond-solid (VBS) state stabilized by fustated inteactions 4. A bound state of spinons can be induced in the Heisenbeg chain by an extenal magnetic field 5. In highe dimensions, in systems with long-ange AFM ode, the fundamental excitations ae magnons with spin 1, as explained successfully by spin-wave theoy 6. Spinon excitations ae associated with spin-liquid gound states, which have no boken symmeties descibed by conventional local ode paametes (but do have non-local, topological ode) 7. In two-dimensional (2D) AFM systems, deconfined spinons should emege when a tansition into a VBS state is appoached, accoding to the theoy of deconfined quantum-citical points 8 10. The seach fo spinons has been a quest in expeimental and theoetical condensed matte physics fo decades, pimaily because the factionalization of excitations is a chaacteistic of exotic collective quantum many-body states, such as the spin liquids 7,10,11. Moeove, in some cases the mechanism of confinement of spinons is a condensed-matte analog of the confinement of quaks in quantum chomodynamics. In this pape, building on a pevious bief pesentation 12, we will exploe systems whee confinement and deconfinement of spinons can be detected and chaacteized using lagescale quantum Monte Calo (QMC) simulations in the valence-bond (VB) basis. We hee focus on a ange of diffeent 1D systems but note that the same ideas have also aleady been applied to 2D systems in the context of deconfined quantum-citicality 13. The stating point of ou studies is the S = 1/2 AFM Heisenbeg chain, defined by the Hamiltonian H = J N S i S i+1, (1)
2 ing all possible combinations of two spinons popagating independently with fixed momenta, q 1 and q 2 with q = q 1 + q 2 gives a continuum above the lowe bound and an uppe bound given by ɛ 2 (q) = πj sin(q/2). A lage spectal weight between these bounds (concentated close to the lowe bound because of matix elements 18 ), which is detectable in inelastic neuton scatteing expeiments 19, is consideed a good indicato of spinons in one dimension. The continuum spectum of spinons has been obseved in weakly coupled-chain compounds such as coppe pyazine dinitate and KCuF 3 at zeo magnetic field 19,20, while in none-zeo magnetic fields incommensuate modes have been obseved 20,21. In anothe chain compound, CuCl 2(dimethylsulfoxide), thee is an effective intenal staggeed magnetic field pesent, and spinon bound states have been obseved 22. In addition, in the spin ladde system (C 5 H 12 N) 2 CuB 4, it was epoted that the magnon could be factionalized into spinons by tuning the extenal magnetic field 23. The above expeimental esults can be modeled using the Heisenbeg Hamiltonian (1) including the othe effects mentioned above (extenal fields, inte-chain couplings). In addition to neuton scatteing, othe expeimental signals of spinons have also been poposed 24. So fa, howeve, all the expeimental pobes give indiect infomation on the existence of spinons, and not much infomation on the popeties of spinons othe than thei dispesion and excitation continuum. Motivated by the on-going inteest in the quantum physics of factionalization, in this pape we ae inteested in exploing othe aspects of spinons and thei confinement-deconfinement tansitions. Using the QMC appoach intoduced in Refs. 12,25 and used in Ref. 13 to study 2D systems, we hee exploe a wide ange of 1D systems whee confinement and deconfinement can be studied systematically unde vaious conditions. The method opeates in a basis of VBs (two-spin singlets) and unpaied spins and allows us to compute quantities defining the size of an isolated spinon as well as the size of an S = 1 bound state. We also show that the same length scales appea in standad spin coelation functions, but ae hade to access thee in pactice because the signal only appeas in the diffeences between coelations in diffeent spin sectos (and is theefoe vey noisy in QMC calculations of lage systems). The stuctue of the est of the pape is as follows: In Sec. II, we intoduce the pojecto QMC method and calculate obsevables used to chaacteize spinons. in Sec. III, we pesent esults fo the J-Q chain model 12,25, which undegoes a quantum phase tansition fom the Heisenbeg citical phase to a spontaneously symmetyboken valence-bond solid (VBS). This system has deconfined spinon excitation in the entie ange of the atio Q/J of the Heisenbeg exchange J and a multi-spin coupling Q. To achieve confinement, in Sec. IV we intoduce a staggeed patten of J-inteactions, as ecently done also in an investigation of spinons binding to a static impuity 26. In Sec. V we study spinon confinement when two Heisenbeg chains ae coupled to fom a ladde. In Sec. VI, we discuss the fact that the same length scales that appea in ou VB-based definition of spinons can also be identified in the fine-stuctue of the spin-spin coelations in the highe-spin states, thus confiming that these length scales ae not basis dependent and can be investigated using othe methods as well. We summaize ou wok and discuss futue pospects in Sec. VII. II. METHODS AND CALCULATED OBSERVABLES We use VB pojecto QMC (VBPQMC) algoithm, which has been descibed in detail in Refs. 12,27,28. Hee we fist biefly eview the essential ideas undelying simulations of spin systems with this algoithm, and then focus on the definitions of spinon quantities and how to evaluate them. A. VB basis and pojecto QMC method Seaching fo the gound state of a Hamiltonian H, we stat with a tial wave function and wite it as the linea supeposition of all eigenstates of H as Ψ t = n c n Ψ n. (2) We then opeate with H a numbe m times on this tial state to poject out the gound state Ψ 0 ; [ ( H) m Ψ t = c 0 ( E 0 ) m Ψ 0 + ( ) m c n En Ψ n ], c n>0 0 E 0 (3) whee, since nomally E 0 < 0, we have added a minus sign in font of H. Povided that E n /E 0 < 1 fo all n > 0, which can always be accomplished by adding some negative constant to H, the gound state is pojected out when m. While the gound-state pojection appoach fomulated above is completely geneal, the use of the VB basis has distinct advantages 29,30, as the spin of the tial state can be chosen to match that of the gound state unde investigation. Fo the bipatite spin models we ae inteested in hee, if the numbe of spins N is even, then the gound state is a singlet and a VB basis state can be witten as V α = N/2 whee a, b i is the ith VB (singlet), a, b i, (4) a, b i = 1 2 ( a(i) b(i) a(i) b(i) ), (5)
3 with a(i) and b(i) sites on sublattices A and B, espectively. The tial state can be expanded in these VB basis states as Ψ t = α f α V α, (6) whee the coefficients f α 0, eflecting Mashall s sign ule fo the gound state of a bipatite system 31,32. It should be noted that the VB basis is ovecomplete and, theefoe, the expansion coefficients f α ae in pinciple not unique, which, howeve, is not explicitly of impotance in the wok discussed hee. What is impotant is that the basis is non-othogonal, with the ovelap between two states given by 31,32 V α V β 2 n loop N/2, (7) whee n loop is the numbe of loops in the tansition gaph fomed when supeimposing the bond configuations of V α and V β. An example with n loop = 2 is shown in Fig. 1. Expectation values of inteest can nomally also be expessed using tansition gaphs, e.g., fo studying the spin-spin coelation opeato Ĉ = 1 N N Ŝ i Ŝi+, (8) we need matix elements of the fom, V α Ŝi Ŝj V β V α V β = { ±3/4, i, j in same loop, 0, i, j in diffeent loops. (9) whee the + and sign in font of 3/4 applies fo sites on the same and diffeent sublattices, espectively. Othe examples of tansition-gaph estimatos, e.g., dimedime coelations of the fom ˆD xx = 1 N N (Ŝi Ŝi+ˆx)(Ŝi+ Ŝi++ˆx), (10) have been discussed in Refs. 30 and 33. In the double pojection vesion of the VBPQMC method 29 that we use hee, ba and ket VB states ae geneated stochastically by opeating on the ba and ket vesions of the tial state with stings of m Hamiltonian tems (opeatos defined on bonds o goups of bonds fo J and Q inteactions, espectively). The pobability of the ba V α and ket V β appeaing togethe is given by P α,β = g α g β V α V β, (11) whee the unknown coefficients ae such that α g α V α appoaches the gound state of H when m and expectation values in this gound state ae obtained using the stochastically geneated tansition gaphs V α V β. Fo details of the computational pocedues, which make use of vey efficient loop updates, we efe to Ref. 27. (c) FIG. 1: (Colo online) Tansition gaph fomed by ba (uppe, black) and ket (lowe, geen) valence bond states on a spin chain. Pat shows an S = 0 state on an even numbe of sites. In the numbe of sites is odd and thee is an unpaied spin in both the ba and the ket states. Pat (c) shows an S = 1 configuation, whee thee ae two unpaied spins. In VBPQMC simulations, the distance distibution of the unpaied spins in gives infomation on the size of an individual spinon, while the size of an S = 1 bound state of two spinons is eflected in the distance distibution of unpaied spins on diffeent sublattices in (c). Fo the tial state, we nomally choose an amplitudepoduct state 31, whee the coefficients f α in (6) ae simple poducts of amplitudes h α coesponding to bondlengths ; f α = N/2 h nα α, (12) whee n α is the numbe of bonds of length in VB configuation α. These amplitudes can in pinciple be detemined vaiationally 27,31,34 to optimize the tial state, but in pactice such optimization is not cucial and the simulations convege well egadless of the details of the tial states. We typically choose a powe-law fom, e.g., h α = 2. The bonds configuations of the tial state ae sampled stochastically as well 27. Ou VBPQMC calculation pojects out the lowest state with given total spin, S = 0 as discussed above o highe spins, as will be discussed futhe in the following. With peiodic systems, the momentum is also a good quantum numbe and is detemined by the tial state. With the simple amplitude-poduct tial states we ae using, the momentum can be obtained vey easily by tanslating the bonds by one lattice spacing. If the numbe of bonds is odd, i.e., the numbe of sites is of the fom N = 4n + 2 fo some intege n, this esults in a negative phase, and, thus, the momentum k = π. Othewise, fo N = 4n, thee is no phase and k = 0. These ae exactly the momenta of the gound states of bipatite spin chains.
4 B. Genealized VB basis fo S > 0 In addition to the use of the VB basis fo singlet gound states, extensions of the VB basis with unpaied spins also povide a natual and convenient way to descibe excitations with highe spin 12,25,28. In ou study of spinons, we will study systems with one o two unpaied spins. In the fome case, the total numbe of sites N is odd, and a genealized VB state can be witten as V α = [ (N 1)/2 a, b α i ], (13) whee the notation explicitly indicates the location in the chain of the unpaied spin and α labels the possible (N 1)/2-bond configuations with this site excluded. Fo system with even N and two unpaied spins, analogously an extended VB basis state is witten as V α ( a, b ) = [ N/2 1 a, b α i ] a b, (14) with N/2 1 singlet pais and two unpaied spins on diffeent sublattices. These extended VB bases ae also ovecomplete and non-othogonal in thei espective total-spin sectos S, and, if we choose (as we do hee) the unpaied spins to have Si z = 1/2, the z-pojection of the total spin is S z = S. The tansition gaphs shown in Figs. 1 and 1(c) have open stings [with an open sting of length zeo being a special case coesponding to a ba and ket spinon esiding on the same site, an example of which is seen in Fig.1 (c)] in addition to loops. If we fix the spin-z oientation of the unpaied spins, as we do hee, the stings do not contibute to the weight (since they only have one allowed state, in contast to the two allowed states of each loop) and the ovelap of two states is still given by Eq. (7). Note, in paticula, that the unpaied spins can be at diffeent lattice locations and the states still always have non-zeo ovelap. The stings do contibute to expectation values. It should be pointed out that, in peiodic chains of odd size N, which we use hee to study a single unpaied spin in S = 1/2 states, thee is magnetic fustation caused by the bounday condition and the lattice is no longe stictly bipatite. Thus, maintaining the updating ules in the simulations 27,35 the VB singlets hee can some times be fomed between sites on the same sublattices if we continue to label the sites as altenating A and B, except fo one instance of adjacent AA o BB sites. (in the simulation we do not explicitly label the sites and thee is no beaking of tanslational symmety as we just use the same updating ules fo the bonds and unpaied spins as fo the even-n chains). The distance between the unpaied spin in the ba and ket can then be an odd numbe of lattice spacings (while it is always even in a tue bipatite chain). In many cases (which we will discuss in detail in Sec. III) the system is completely dominated by shot bonds and the distance between the ba and ket spinon is then always even in pactice. The tial states used fo S > 0 calculations ae simple genealizations of the amplitude-poduct states discussed in Sec. II A, with the wave-function coefficient given by Eq. (12) with no dependence on the unpaied spins. In pinciple one could impove the tial states by factos depending on the unpaied spins and spin-bond coelations as well (as ecently investigated in detail in Ref. 37), but this is not necessay hee. Following the easoning in Sec. II A, fo S = 1, k = π fo N = 4n and k = 0 fo N = 4n + 2, i.e., the momentum diffeence with espect to the S = 0 gound state is π in both cases, as it should be fo the lowest tiplet excitation. Fo the S = 1/2 states, if we stictly label the sites with sublattice labels A and B, thee is a defect in the odd-n system, as discussed above. Howeve, in the simulations thee ae no explicit efeences to sublattices and in effect the system is then tanslationally invaiant. Then, unde the futhe assumption that no bonds with length as lage as N/4 ae pesent (such configuations having ill-defined signs) 36, the momentum is k = 0 o π, fo N of the foms 4n + 1 and 4n + 3, espectively. C. Chaacteization of spinons in the VB basis In ode to study spinon sizes and confinement lengths, we conside ovelaps witten in the fom 1 Ψ 0 Ψ 2 0 1 = g 2 α g β ( ) V α V β ( ), (15), α,β genealizing Eq. (11) to S = 1/2 (single-spinon) systems and witten explicitly using sums of tems with all possible locations of the unpaied spins. We have an analogous fom 1 Ψ 0 Ψ 0 1 = g α ( a, b )g β ( a, b) a, b α,β (16) a, b V α ( a, b ) V β ( a, b), fo S = 1 (spinon-pai) systems. The ovelaps ae not computed explicitly in the simulations but seve as nomalization factos and weights in the sampling pocedues, such that the diffeent contibutions to the above sums appea accoding to thei elative weights. The pactical simulation pocedues fo S > 0 ae elatively staight-fowad genealizations of the method with loop updates fo S = 0. We efe to Refs. 25,28,33 fo technical details. In the following, we discuss distibution functions used to chaacteize spinons. We will hee make us of the unpaied spins, although in pinciple one can also define spinon quantities using the entie stings, of which the unpaied spins ae the end points.
5 1. Single-spinon distibution function As discussed above, in the VBPQMC method the ba and ket states ae geneated stochastically, and fo S = 1/2 we can use Eq. (15) to define a distibution of the sepaation of the unpaied spins in the ba and ket states. Resticting ouselves to a tanslationally invaiant system we have the pobability of sepaation (up to an ielevant nomalization facto which is easily computed at the end): P AA ( ) = α,β g α g β ( ) V α V β ( ), (17) whee the subscipt AA seves to indicate that the unpaied spins should be on the same sublattice (because thee is an excess of one site on one of the sublattices, which is the sublattice with the unpaied spin), which we can take as the A sublattice. Thus, P AA should vanish when the sepaation is an odd numbe of lattice spacings. Ou basic assetion is that, if spinons ae well-defined quasipaticles of the system, then we expect P AA to eflect the size and shape of an intinsic wave packet within which the net magnetization S z = 1/2 caied by the spinon is concentated. We will show in the following that 1D VBS states ae chaacteized by an exponentially decaying ovelap, P AA e /λ, and it is then natual to take λ as a definition of the intinsic spinon size. We should hee note again that, fo a peiodic system with an odd numbe of sites, thee is, stictly speaking, no absolute distinction between the sublattices (i.e., the system is stictly speaking not bipatite). Howeve, when the system size N we in geneal expect the ole of the bounday condition to diminish and P AA to tend to zeo fo any given odd. In Sec. III, we will discuss in detail how this limit is appoached, and we will also see an example (one whee spinons ae not well-defined quasi-paticles) whee the boundaies continue to play a ole even fo infinite size. 2. Two-spinon distance distibution function In the case of S = 1 states (two spinons), we can define seveal diffeent distibutions. Hee, we will focus on the sepaation of spinons on diffeent sublattices in the ba and ket; P AB ( a b) = g α ( a, b )g β ( a, b) α,β b, a V α ( a, b ) V β ( a, b). (18) In the case whee a single spinon is a well-defined quasipaticle, i.e., λ <, we expect this quantity to give us infomation on the confinement o deconfinement of two spinons. In the fome case, we will see that asymptotically P AB e /Λ and, thus, we conside Λ as a definition of the confinement length-scale (i.e., the size of the S = 1 spinon bound state). We will see that deconfined spinons give ise to chaacteistic boad distibutions. We could also have defined the above distance distibution with the two unpaied spins both in the ba o in the ket, and we have also investigated it. This distibution typically does not diffe significantly fom the one defined in Eq. (18). 3. Same-sublattice distibution in two-spinon states We will also study the analog of the S = 1/2 quantity P AA [Eq. (17)] in the tiplet state, defined as P AA( a a) = α,β g α ( a, b )g β ( a, b) b, b V α ( a, b ) V β ( a, b), (19) whee we use the supescipt to distinguish this distibution fom the single-spinon distibution (17). We can define PBB in the same way, and use P AA = PBB to impove the statistics. We will see that, unde cetain conditions, PAA of the tiplet state contains the same infomation fo the spinon size λ as the S = 1/2 quantity P AA, and we can use this popety of the S = 1 state to chaacteize the intinsic spinon size also in cases whee the S = 1/2 state beaks tanslational invaiance and is not appopiate fo use with ou calculations pesuming tanslational invaiance (the 2-leg ladde system being such an example, which will be studied below in Sec. V). III. DECONFINED SPINONS IN UNIFORM SPIN CHAINS We hee fist test the concepts and methods fo a class of spin chains, the J-Q 3 model, which can be tuned between a gound-state phase with popeties simila to the standad citical Heisenbeg chain and a VBS phase with VBs cystallizing on altenating neaest-neighbo bonds. In the citical state, spinons ae igoously known to be elementay excitations based on the exact Bethe-ansatz wave function of the plain Heisenbeg chain, and in a VBS state thee ae also stong aguments fo spinons 4. In eithe case, a pai of spinons can be egaded as a kink and an anti-kink of an odeed (in the case of the VBS) o quasi-odeed (in the citical state) medium. Thee is no appaent confining potential between these defects in one dimension (and clealy any effectively attactive potential would lead to a bound state and confinement of the spinons in the gound state, although deconfinement could still take place at highe enegy). Ou calculations show explicitly that thee ae instead weak epulsive inteactions, the effects of which diminish with the system size, thus leading to independently popagating spinons in the themodynamic limit down to the lowest enegies. We will also investigate a modified J-Q 3 model
6 with long-ange inteactions, which hosts a Néel odeed gound state. Hee, spinons ae not expected to be deconfined and we investigate the beak-down of the spinon as well-defined quasi-paticle in this case. A. Results fo the J-Q 3 chain We hee conside the 1D J-Q 3 chain Hamiltonian 12, H = N (JC i,i+1 + Q 3 C i,i+1 C i+2,i+3 C i+4,i+5 ), (20) i whee C ij is a singlet-pojection opeato on two sites, C i,j = 1/4 S i S j, (21) and the J tem is simply the standad antifeomagnetic Heisenbeg inteaction. We hee use the Q 3 tem with thee pojectos, as its gound state at the exteme point J = 0 is moe stongly VBS-odeed than that of the Q 2 model with only two pojectos. When the coupling atio g = Q 3 /J is small, the system emains in the Heisenbeg-type citical state, whee the spin-spin coelation function C, i.e., the expectation value of Eq. (9), has the asymptotic fom C ln 1/2 / 15,38,39. When g is lage, the Q 3 tem enfoces VBS odeing and C is exponentially decaying. The VBS state is two-fold degeneate. The physics of this phase tansition is identical (in the sense of univesality) 12,35 to that in the fustated J 1 - chain, whee spinons in the VBS state wee discussed on the basis of a vaiational state by Shasty and Sutheland 4,40. In field-theoy language, the phase tansition is diven by the sign-change of a maginal opeato, and this opeato is also the oot cause of the logaithmic coection to C in the citical phase. Exactly at the citical VBS tansition point the coelations decay as 1/ with only vey small coections. The tansition point of the J-Q 3 model is at g c = (Q 3 /J) c 0.1645, as detemined fom level spectoscopy 12 (excited-state singlettiplet cossing 41 ) and VBPQMC calculations of coelation functions 35. Although we do not expect the Hamiltonian (20) to be natually ealizable in any specific mateial, the fact that it has the same kind of gound state phases as the moe ealistic fustated J 1 - chain still makes its physics inteesting, and not being fustated in the standad sense it is not associated with sign poblems in QMC simulations. The same physics of spontaneous dimeization also occus in spin chains with phonons (often called spin- Peiels systems) 42. We expect the popeties of spinons to be discussed below to apply also to fustated chains and spin-peiels systems. P AA P AA J=0 g=4 g=1 0 50 100 150 200 250 10 1 10 2 N=1025 N=257 N=513 N=1025 FIG. 2: (Colo online) Single spinon ovelap distibution in the J-Q 3 chain. Exponential decays indicating welldefined quasi-paticles in VBS states at diffeent values of g = Q 3/J. The size λ of the spinon (the invese of the slopes of the lines on the lin-log plot) diveges as the citical point is appoached. Panel shows that the spinon is maginally defined at the citical point, with the ovelap decaying as a powe-law with exponent α = 0.500(2) (with a fitted line to the even- points shown fo N = 1025). The even-odd oscillations ae due to the fustation caused by the single-spinon defect in a peiodic chain (with the odd- contibutions only possible in a non-bipatite system). The effects of fustation fo less than N/2 diminish as the chain size inceases. 1. Single spinons in states with total-spin 1/2 We hee fist investigate P AA as defined in Eq. (17) to study the size of spinons in the VBS phase at diffeent coupling atios g = Q 3 /J. In Fig. 2, we see that the intinsic spinon wave packet has a ponounced exponential decaying fom, P AA e /λ, showing that spinons indeed ae well-defined quasi-paticles of the VBS, with a chaacteistic size λ. The spinon size deceases with inceasing g (going deepe into the VBS phase), with λ = 30.0(1) when g = 1 and λ = 9.2(1) when g (the pue Q 3 model). When λ is lage, thee ae also significant deviations fom the pue exponential fom fo a ange of small, indicating coss-ove behavios to a diffeent fom obtaining when g g c. As shown in Fig. 2, exactly at the tansition point g c the decaying fom is indeed no longe exponential, instead it is vey well descibed by α with the powe α = 0.500(2). Ou physical intepetation of this esult g c
7 10 1 P AA g=0 g=0.05 g=0.1 g c =0.1645 10 1 10 2 NP AB 10 1 g=4 N=128 N=256 N=512 NP AA 10 1 N=257 N=513 N=1025 /N FIG. 3: (Colo online) Single-spinon distibution function at the VBS tansition point and inside the citical phase (g g c = 0.1645) computed using chains of length N = 513. The data at g c fo seveal system sizes, escaled such that data collapse is achieved. The lines in both and coespond to the 1/2 fom. NP AB N=256 g=8 g=4 g=2 g=0.5 0 0.1 0.2 0.3 0.4 0.5 /N FIG. 4: (Colo online) Two-spinon distance distibution in VBS states of the J-Q 3 chain at fixed g = 4 and diffeent chain lengths, and fixed chain length N = 256 and diffeent coupling atios. The y and x axes have been escaled with N and 1/N, espectively, in ode to achieve data collapse fo lage in. The incease in the small- distibution fo the lowest g-value in shows that the effective shot-distance spinon-spinon epulsion becomes weake as the system appoaches the the tansition point (g c = 0.1645). is that, the spinon at the tansition point can be consideed only as a maginally well-defined quasi-paticle in eal space. As we discussed in Sec. II B, fo N odd thee is a complication with the peiodic boundaies, which endes the system non-bipatite in pinciple. The distance between the unpaied spin in the ba and ket can then be odd. Howeve, the pobability of these odd distances is exceedingly small in the VBS state of the N = 1025 chains used in Fig. 2, but in the citical-chain esults in Fig. 2 we clealy can see non-zeo odd- pobabilities. Relative to the even- pobabilities, fo fixed they decease apidly as N gows, while appoaching the even- pobabilities as N/2 (and, inteestingly, the odd banch follows almost an invese of the behavio of the even banch, inceasing as 0.5 in the elevant ange of ). In ou simulations we neglect the non-tivial (non-mashall) signs in the wave function aising fom the even-length bonds (whee we define the length as the shotest of the two possible distances between the two paied spins unde the peiodic bounday conditions), but we find it unlikely that this appoximation would affect ou conclusions on the natue of the spinon as these signs also ae due to boundaies and we ae inteested in the themodynamic limit. We will also see futhe in what follows that we ob- tain the same exponential (fo g < g c ) o powe-law (fo g = g c ) decay also in PAA [Eq. (19)], in the chains with two unpaied spins, whee the lattice emains bipatite and thee ae no fustation effects. Given the fact that the exponent α of the citical spinon ovelap in Fig. 2 is vey close to 1/2, and the behavio is seen to emakable consistency ove two odes of magnitude of, we conjectue that the exponent should in fact be exactly 1/2. It is tempting to associate it with the squae-oot of the spin coelation function C = 1/, although we have not tied to fomally compute this quantity within the bosonization appoach (which in pinciple should be possible 43 ). Anothe inteesting question to ask is as follows: How is the citical 1/2 fom of the single-spinon distibution P AA at g c changed when going futhe into the citical egion (g < g c )? The logaithmic coection to the coelation function 1/ is a well known consequence of the pesence of a maginal opeato, as mentioned above. One would then expect coections to P AA as well. As seen in Fig. 3, P AA indeed changes noticeably when moving away fom the tansition point into the g < g c citical phase. The behavio can be fitted to a powe-law with exponent depending on g, but most
8 J=0 g=4 g=1 P AB g=0.0 g=0.05 g=0.1 g c =1.645 P * AA 10 1 10 2 0 50 100 150 NP AB 5 1 0 L=128 L=256 L=512 L=1024 FIG. 6: (Colo online) The same-sublattice distibution function fo S = 1 states at thee diffeent values of the coupling atio. The coesponding distibutions P AA fo the S = 1/2 states at the same couplings ae shown in lighte (bown) colo and they coincide vey closely with the S = 1 functions (thus, demonstating that the single-spinon size can be obtained also fom the S = 1 simulations). The system size hee is N = 1024 fo S = 1 and 1025 fo S = 1/2. /N FIG. 5: (Colo online) Distibution of spinon sepaations in S = 1 states at and below the VBS tansition point g c; in fo fixed chain-length N = 512 and vaying g, and in at g c fo diffeent chain lengths. The lines going though the g c points have slope 0.7. likely the 1/2 behavio pesists fo all 0 g g c and it is only the stength of a logaithmic coection that changes. While the data can be fitted to the 1/2 with a multiplicative logaithmic coection, the powe of the logaithm is not clea, and futhe quantitative studies of this behavio would equie much longe chains. In Fig. 3, we futhe analyze the behavio at g c fo diffeent system sizes, e-gaphing the even banch of Fig. 2 such that data collapse is achieved: NP AA vesus /N. An inteesting aspect of these esults is that thee ae no noticeable enhancements due to the peiodic boundaies at the longest distances, N/2 (which ae typically seen pominently in coelation functions), with the powe law descibing the data vey well fom the smallest to lagest distances fo all system sizes. 2. Two spinons in states with total spin 1 Next, we conside chains with even N and two unpaied spins. The distibution function P AB hee eflects the effective mutual inteaction between two spinons, mediated by the backgound of singlets. Fo a confining case, we would expect to obseve P AB e /Λ, with a finite confinement length Λ. Deconfinement should be signaled by a divegence of Λ. Results fo the J-Q 3 chain in the VBS phase, gaphed in Fig. 4, show distibution functions with no decay at long distances. Instead, P AB exhibits a vey boad maximum at the lagest distance, which we natually intepet as esulting fom a weak epulsion between two spinons. As shown in Fig. 4, the epulsion diminishes somewhat when tuning down the coupling atio towad the citical point, whee, appaently, inceasing quantum fluctuations (including an inceasing faction of long VBs) educe the epulsive potential. The ange of ove which the distibution is almost flat inceases essentially popotionally with N. In Fig. 4, we have multiplied the distibution function with N fo seveal N at a fixed g inside the VBS phase, and find that the cuves collapse well on top of each othe fo /N oughly in the ange 0.1 to 0.5. This indicates that the effective inteactions ae shot-ange in natue, with spinons fa away fom each othe behaving as fee paticles. Clealy, all these esults point to deconfined spinons, as expected. While the details of the cause of the epulsive potential ae uncetain, it is clea that the sign of the effective inteaction is cucial fo deconfinement (at the lowest enegies studied hee); any weak attactive potential would bind the spinons, while shot-ange epulsive inteactions aid deconfinement. Results fo P AB at the VBS tansition and inside the citical phase ae shown in Fig. 5, while esults fo seveal chain lengths at the citical point ae shown with escaled axis to achieve data collapse in 5. The citical distibution is also hee consistent with a powe-law, P AB γ, with γ 0.7 (and with a pefacto deceasing with the system size). Based on these esults one may
9 ague that the effective spinon-spinon inteactions become inceasingly long-anged as g c is appoached fom the VBS side, although the shot-ange pat is deceasing, based on the fact that distibution at shot distances gows upon deceasing g. Inside the citical phase thee ae again likely logaithmic coections, and the tend of deceasing effective shot-distance spinon-spinon inteactions continues as g deceases. Next, we conside the same-sublattice distibution function PAA, defined in Eq. (19). Since the spinons ae deconfined and typically ae futhe away fom each othe than the single-spinon length-scale λ, one would expect that PAA contains essentially the same infomation as the single-spinon function P AA fo the S = 1/2 state, defined in Eq. (17). This is indeed the case in the VBS phase, as demonstated in Fig. 6. Clea exponential decays ae obseved, and the esults coincide almost pefectly with the pevious esults fo P AA in Fig. 2. To eiteate what is going on hee, the two spinons in the S = 1 state ae on diffeent sublattices, and the unpaied spin on sublattice A in the ket state is coelated to the one on the same sublattice in the ba state, to within the length-scale λ that we have agued descibes the intenal spinon size. The same holds fo the unpaied ba and ket spins on sublattice B. Due to spinon deconfinement the A and B spinons ae not bound to each othe, howeve, and typically ae fa away fom each othe. Unde these conditions, the distibution functions P AA and PAA ae essentially the same. To illustate this point moe explicitly, in Fig. 7 we plot esults in the VBS state and appoaching the citical point fo the spinon-size estimates λ and λ [extacted fom the distibution functions P AA and PAA ], togethe with the standad spin coelation length ξ c [obtained fom the spin-spin coelation function (9)] and the VBS coelation length ξ d [extacted fom dimedime coelation function (10)]. It can be seen that λ and λ ae almost identical to each othe, as expected. The fou lengths: ξ c, ξ d, λ, λ, divege at a simila ate upon appoaching the citical point g c = 0.1645. Since the phase tansition fom the odeed VBS state to the citical state in the 1D J-Q 3 model is simila to a 2D classical Kostelitz-Thouless (KT) tansition, we fit these fou lengths with functions to the fom of the coelation length in that case, ξ ae b/ g g c, whee a, b ae fitting paametes. Due to the statistical eos and the small numbe of data points, we cannot detemine these fitting paametes vey pecisely. Repesentative cuves fom these fits ae shown in Fig. 7. We also notice in Fig. 7 that the spinon size λ extacted this way is much lage than the coelation lengths ξ c and ξ d, which we will discuss again late in Sec. VI, in connection with the coelation functions in S = 1/2 o 1 states (which, we ague, should also contain the spinon size). As shown in Fig. 8, the S = 1 function PAA inside the citical phase exhibits an inteesting coss-ove behavio, most clealy visible at g = g c. The behavio at shot distances is well descibed by the same 1/2 10 2 ξ c ξ d λ λ * 10 1 0 2 4 6 8 10 g-g c FIG. 7: (Colo online) Spin and dime coelation lengths, ξ c and ξ d, along with the spinon size measued in the S = 1/2 and 1 states, λ and λ, upon appoaching the citical point g c = 0.1645 fom the VBS phase in 1D J-Q 3 model. Since this tansition is of the KT type, we fit the data to the fom ae b/ g g c (solid lines). behavio as the coesponding single-spinon function in Fig. 3. Howeve, at lage distances the behavio changes to 1/. We do not have any explanation fo this behavio and it would be inteesting to investigate it within bosonization. B. Beak-down of spinons as quasi-paticles of a Néel state in one dimension In a long-ange odeed Néel AFM state, the elementay excitations ae spin waves (magnons) caying spin S = 1. It is then inteesting to ask how the change in the natue of the excitations is manifested in ou spinon distibution functions if the system can be diven to a Néel state. The continuous spin-otational symmety of the gound state of the Heisenbeg o J-Q chains cannot be spontaneously boken, howeve, accoding to the Memin-Wagne theoem 44. We can cicumvent this limitation on 1D gound states by including long-ange inteactions, in which case the theoem does not apply. We hee conside unfustated powe-law decaying inteactions defined by the Hamiltonian H = N N/2 odd J S i S i+, J > 0, (22) whee thee ae no couplings fo even sepaations of spins, while fo odd sepaations the coupling is J = 1/ α. A simila Hamiltonian was studied befoe in Ref. 45, whee it was found that by tuning the decay exponent α the system undegoes a continuous phase tansition fom citical states when α > α c to a long-ange odeed Néel states when α < α c. The citical powe depends on details, e.g., on the stength of the neaest-neighbo coupling, and in
10 P * AA NP * AA g=0 g=0.05 g=0.1 g c =0.1645 10 1 10 2 10 1 N=128 N=256 N=512 N=1024 /N FIG. 8: (Colo online) Same-sublattice distibution functions fo S = 1 states in the citical phase. Shows esults fo diffeent coupling atios fo fixed system size N = 512, while in esults at g c ae e-scaled to achieve data collapse fo seveal system sizes. The lines have slope 1/2 and 1 fo small and lage, espectively. NP 2.5 2 N=257 N=513 N=1025 NP 2.2 2.1 2 1.9 1.8 10 2 10 1 10 2 10 3 FIG. 9: (Colo online) Size-scaled spinon ovelap function in a Néel-odeed chain with total S = 1/2, computed fo chain lengths N = 257, 513, and 1025. The asymptotically flat (with even and odd- banches) distibution shows that the spinon is not a well-defined quasi-paticle in the Néel state, as expected. The inset shows the tail of the spinon ovelap function of Néel-odeed chains with a cleae view of the N = 217 and 513 data. the cases studied in Ref. 45 α c 2.2. In Ref. 46 fustation was added to the model in ode to dive it to a VBS phase. In ou study we ae just inteested in studying an example of a 1D Néel state and choose J = 3/2 (odd ) in Eq. (22), fo which we veified that indeed the system is AFM odeed. We investigate the single-spinon distibution function P AA in an S = 1/2 state fo odd N. In Fig. 9, we plot P AA scaled by N vesus fo diffeent system sizes and find good convegence as a function of the system sizes, although the eo bas ae lage at the lagest distances. The behavio hee is quite diffeent fom the pevious cases, Figs. 2 and 3, with (i) no vanishing of the pobability of odd- sepaation and (ii) no decay of the escaled function. The latte behavio indicates that the spinon hee is not a well-defined paticle, with no concentation of the net magnetization to within an intinsic wave packet. This is of couse not supising, in the sense that spinons ae not expected to be the elementay quasi-paticle excitations of the Néel state. We had also aleady found above that in the citical state the quasi-paticles ae only maginal, chaacteized by powe-law ovelaps (and hence any futhe enhancement of antifeomagnetic coelations should completely destoy the spinons). It is still inteesting to see that the beak-down of the spinons as quasi-paticles can be explicitly obseved in the distibution function P AA. IV. SPINON CONFINEMENT ARISING FROM MODULATED COUPLINGS In ode to obseve confinement of spinons, we hee use a genealized vesion of the J-Q 3 model with diffeent neaest-neighbo coupling constants on even and odd bonds. The Hamiltonian is H = (J 1 C i,i+1 + C i+1,i+2 ) even i Q 3 C i,i+1 C i+1,i+2 C i+2,i+3. (23) i When the modulation paamete ρ = /J 1 1, the Hamiltonian itself beaks tanslational invaiance and thee is no longe a VBS phase tansition with spontaneously boken symmety. If we stat in a spontaneously fomed VBS (Q 3 /J 1 > g c ) fo ρ = 1, the gound state is doubly degeneate, but once ρ > 1 the degeneacy is boken and the gound state is unique. This is expected to confine the spinons, as the sting of out-of-phase bonds fomed between two sepaated spinons is now associated with an enegy cost inceasing linealy with the sepaation, instead of the enegy only being associated with the domain walls when ρ = 1. This model was also studied in the pesence of an impuity in Ref. 26, and it was found that the localization length of the magnetization distibution foming aound the impuity could be tuned by ρ. It was agued that two egions of confinement could be defined; (i) stong confinement, whee the size of the
11 P AA ρ = 8 ρ = 2 ρ = 1.1 ρ = 1 0 20 40 60 80 100 L=512 P AA L=512 ρ = 8 ρ = 2 ρ = 1.1 ρ = 1 P AB ρ = 8 ρ = 2 ρ = 1.1 ρ = 1 L=512 10-6 0 20 40 60 80 100 120 140 FIG. 10: (Colo online) Spinon distibution functions in the J 1--Q 3 chain with Q 3/J 1 = 4 and seveal values of the modulation paamete ρ = /J 1. Shows exponential decays, P AA e /λ, of the single-spinon distibution function of the S = 1/2 state, demonstating well-defined spinons with finite intinsic size λ. In, spinon confinement fo ρ 1 is demonstated in the spinon-distance distibution function; P AB e /Λ. The size of the bound state (the confinement length scale) deceases as the coupling modulation is inceased. Data fo ρ = 1 ae gaphed fo compaison; in this case, the spinons ae deconfined and the distibution function does not decay with the sepaation. bound state is simila to the standad spin coelation length, and (ii) weak deconfinement, whee the bound state is much lage than the coelation length. Hee we find simila behavio fo two spinons binding to each othe instead of a static impuity. A pioi it is not clea that the situations ae identical, as the impuity-spinon and spinon-spinon potentials ae not identical (since a dynamic spinon petubs its singlet envionment diffeently than a static impuity). We fist test fo confinement deep inside the VBS phase at g = Q 3 /J 1 = 4. As shown in Fig. 10, the spinon size λ computed fom P AA in the S = 1/2 gound state becomes smalle when the confining potential inceases (tuning ρ fom 1 to 8). Figue 10 shows that the confinement length Λ indeed becomes finite once we tune ρ off 1. Fo ρ vey close to 1 it is difficult to extact Λ because we also need to satisfy L Λ and the calculations become vey demanding. Upon inceasing ρ we P AB 10-6 0 50 100 150 ρ = 8 ρ = 2 ρ = 1.1 ρ = 1 L=512 FIG. 11: (Colo online) The same quantities as in Fig. 10 but with the atio Q 3/J 1 = g c = 0.1645. Hee, the tuning of the modulation paamete ρ towad 1 coesponds to appoaching a citical point. find that Λ appoaches λ. An inteesting obsevation in Fig. 10 is the maximum developing in P AB, seen aound = 20 fo ρ = 1.1 and moving to R = N/2 at the unifom point ρ = 1. In Sec. III, we aleady agued that thee is an effective shot-ange epulsive inteaction between the spinons in the unifom chains, and it is natual that these inteactions should pesist also fo some ange of ρ away fom 1, although thee is also an attactive pat binding the spinons. Thus, we aive at the conclusion that when ρ is close to 1 thee is a shot-ange epulsion followed by the linea confining attactive potential at longe distances. Judging fom the fact that the maximum pobability moves towad = 0 fo lage modulation paametes, ρ = 2, 8 in Fig. 10, the ole of the shot-ange epulsion diminishes (leading to the spinon coe being cushed ) elative to the linea attactive confinement potential, which gows with ρ. The cases of λ Λ and maximum pobability at = 0 seems vey simila to the case of stong confinement by an impuity in Ref. 26, while the case of emaining effects of epulsions pushing the maximum pobability away fom = 0 is like the weak confinement case. It would be inteesting to compae the two cases moe quantitatively, but we leave this fo futue studies. We also obseve simila behavios in the dimeized model at the citical Q 3 /J 1 value, as shown in Fig. 11.
12 The main diffeence is that now the spinon size λ diveges as ρ 1, instead of tending to a finite value in the VBS phase. Both length scales ae actually smalle than in the VBS phase fo lage ρ, e.g., fo ρ = 2, Λ 2.42(1) at g c while Λ 3.78(4) at g = 4. This implies that the imposed dimeization in the citical egion has a stonge effect than in the odeed VBS phase. In the citical egion all lengths divege, and, theefoe, once we add the explicit dimeization ρ 1 it dominates the physics immediately. In contast, in the VBS phase thee ae competition effects between the spontaneous VBS and the explicit dimeization, which appaently educe the effects on the spinon size and confinement length. Also hee we can see a maximum in P AB away fom = 0, and Λ hee is somewhat lage than λ. It would be inteesting to study in detail the divegence of these lengths as ρ 1 and compae them with both the spin and VBS coelation lengths (and also to compae with the impuity-binding case), but we also have to leave this fo futue studies. V. HEISENBERG LADDERS Anothe way to confine the spinons of the Heisenbeg chain is to couple two chains into a ladde, descibed by the Hamiltonian H = J 1 L (S 1 i S 1 i+1 + S 2 i S 2 i+1) + S 1 i S 2 i, (24) whee the supescipts 1 and 2 label the two chains, J 1 is the neaest-neighbo coupling within the chains, and is the inte-chain (ung) coupling. It is known that any inte-chain coupling opens a gap in the excitation spectum and changes the citical coelations to an exponentially decaying fom 47. This is tue fo laddes with any even numbe of legs, while odd-leg laddes ae citical and exhibit the univesality of the single chain 48. The situation hee is simila to single chains of Heisenbegcoupled intege o half-odd-intege spins, with the fome always being gapped accoding to the now well confimed Haldane conjectue 49. The intege-s chains have localized spinons at the ends of open chains, and this is also the case (pehaps less supisingly) in open laddes whee a spin is emoved fom each end. We hee investigate the spinon confinement mechanism in the peiodic, tanslationally invaiant ladde. Gapped tiplons (S = 1), which ae the low-lying excitations of ladde systems, have aleady been obseved in the excitation spectum of eal mateials by inelastic neuton scatteing 23. It has been agued that this obsevation makes the ladde system the simplest condensed matte system whee one can in pactice ealize a phenomenon simila to quak confinement in paticle physics 50. The enegy gap, spin-tiplet dispesion elation and the dynamic spin stuctual facto of the C(x,0) 10-6 10-7 L=128 10 1 =0 =0.1 =0.5 =1 =2 0 10 20 30 40 50 60 x =0 =0.1 FIG. 12: (Colo online) Spin coelation function in Heisenbeg ladde systems. Hee the inta-chain coupling J 1 = 1 and esults ae shown fo seveal values of inte-chain couplings. C(x, 0) decays exponentially when 0 and exhibits the powe-law decay of the isolated chain when = 0. In the inset, the coelations ae lage distances on a log-log scale at = 0 and 0.1. Because hee the system length L is smalle than the coelation length it is not yet possible to obseve the exponential decay. Heisenbeg two-leg ladde model have also been extensively studied by numeical methods 47. We begin by discussing the standad spin-spin coelation function in the S = 0 gound state. We fit it to the fom C e /ξ when g = /J 1 > 0, and will late compae the spinon-elated length-scales with the coelation length ξ. Results ae shown in Fig. 12. Note that it is vey difficult to extact ξ when g is small, as ξ then becomes lage and the system size has to be even lage, L ξ. The inset of Fig. 12 illustates this poblem fo g = 0.1. We hee focus on ung couplings sufficiently lage fo extacting ξ eliably based on ou available ladde sizes. We now tun to the chaacteization of the spinons. In the two-leg ladde it is not possible to study a system with an odd numbe of spins N (N = 2L) without beaking the tanslational symmety of the system (which is a much moe sevee issue than the bounday subtleties in the single chain, discussed in Sec. II B, which do not uin the tanslational symmety). We hee only discuss calculations in the S = 1 state fo even N and pesent esults fo the distibutions PAA and P AB in Fig. 13. As we discussed in Sec. III, PAA can eliably give the intinsic spinon size λ if this length-scale is smalle than the size Λ of the bound state in pinciple one would expect to need Λ λ but in pactice, as shown in Figs. 6 and 10, it seems to wok also othewise. In the ladde, the length λ as extacted fom PAA is always vey simila to Λ fom P AB, howeve, and, theefoe, it is not clea whethe λ can be intepeted stictly as the size of an individual spinon, although based on the pevious compaisons one may well ague that it is the case. In the ladde systems, λ is even somewhat lage than