arXiv:1704.05085v1 [cond-mat.str-el] 17 Apr 2017 oeso h Dbprieltie hssaecnbe can state This lattice. pair-singlet bipartite all 1D of the out of made covers state superposition metric admes aeyaciia mltd-rdc state -1 a amplitude-product without we on critical system Here (APS) a a invariant. namely in same conformally wouldrandomness, the which be chains numercially to spin demonstrate expected random be shows of entanglement not Moore case the of and the scaling in Rafael universal by func- a [3] of correlation the possibility work prove spin the previous we four-point CFT; A a and by tions. two- described by not using predicted is as latter yet region entangled and the CFT of a size the depen- the on shows dence entropy entanglement whose chain spin CFT. underlying serve an are can for which These tests CFT. form litmus the as functional by controlled and functions completely behavior correlation both scaling two-point the a for contrast, have only meaning In that CFT. with has a course associated of on number constraining charge rely the conformal less the though not the [1], does on invariance form only scale fact, scaling but In EE invariance, the some conformal well. in of as derivation seen invariance the be conformal could exclu- without and not was invariance systems EE conformal the that for been to [3–5] seen sive behavior often analytically Later the and cases has invariance. some [2] it in conformal numerically of shown such was indicator as it an and be conformal to [1], corresponding taken (CFT) the of theory charge field re- central would which the behaviors (EE) flect distinct entropy have entanglement to methods The predicted reliable was have same. we quite the that determine is quan- important to invariance is a it conformal that and has knowledge useful transition Thus, phase tum of exponents. as- determination many critical including behavior, understanding the critical for the of helpful pects is transition phase eew ilpeetasml xml faquantum a of example simple a present will we Here quantum a at invariance conformal of emergence The eateto hsc,Bso nvriy 9 Commonwealt 590 University, Boston Physics, of Department nesod htbhvo sntrpoue nteamplitud the in reproduced not is behavior That understood. epoienmrcltsso h eairpeitdb F f CFT the by and predicted chain behavior a Ising the there of transverse-field behavior, of product critical CFT tests the expected numerical is the provide function exhibits wave We EE the the and though length even d bond the 1+1 bas of in valence-bond law the models w in spin tests expressed quantum state alternative amplitude-product provide descrip for to cons CFT examples symmetry showing a use SU(2) functions, We when with even CFT scale-invariant). behavior the asymptotically by same out is the point system shows we often the Here EE (CFT). the theory as field conformal by scriptions h nageetetoy(E fqatmssesi fe us often is systems quantum of (EE) entropy entanglement The .INTRODUCTION I. / hi.TeASi nS()sym- SU(2) an is APS The chain. 2 nageetetoyadmlil-on correlators multiple-point and entropy entanglement rnyPtl igTn,EaulKt,adAdr .Sandvik W. Anders and Katz, Emanuel Tang, Ying Patil, Pranay niaoso ofra il Theory: Field Conformal of Indicators J - Q pncan hr h ofra tutr swell is structure conformal the where chain, spin FMcan nSc Vw oeo oQCrslsfor results QMC to on move symmetric we the SU(2) IV as the well Sec. as In Monte model chain. cor- 2D using TFIM classical for the model, results of (QMC) Ising functions Carlo relation the CFT Monte quantum of of and tests setting Carlo numerical simpler develop the we III in in- Sec. SU(2) In by variance. restricted CFTs two-dimensional for results interactions. frustrated to with similar chain phase that Heisenberg dimerized the to a in and that similar chain state Heisenberg the be- (critical) of transition quasi-ordered phase a the hosts tween latter The interaction). spin rnvrefil sn oe TI)o hi n the and chain the a Hamiltonians; on two J (TFIM) of model points APS Ising critical transverse-field the study for we also tests systems, conformally-invariant will CFT actual negative for results the with should contrast are Hamiltonians To states parent the know such exist. APS not though the do states, we which here ground for that noted Hamiltonians be functions underlying should correlation It the not. exhibits behavior, do EE the scaling criti- While expected such tests. the at CFT conduct functions and correlation points cal and here EE We Heisen- way. the the this compute constructed of ask be state could ground then Hamiltonian the may berg to one sin- similar and state the it, a of to whether contribute weightings which particular covers for glet criticality to tuned igo u td n nrdc entos ehr give here we definitions, introduce set- and study the the our establish to To of ting due [6]. constrained condition invariance are conformal functions correlation whose ihohrCTpeitos egv re summary brief a VI. inconsistencies give Sec. and We in concerned, conclusions predictions. with is and CFT cosistency EE its other the show as with and far detail as in CFT APS the troduce - Q nSc Iw ilbifl eiwtekonanalytical known the review briefly will we II Sec. In ofra edter otispiayoperators primary contains theory field conformal A hi teHiebr hi upeetdb four- a by supplemented chain Heisenberg (the chain vne otn ascuet 21,USA 02215, Massachusetts Boston, Avenue, h s(hr h mltdsdcya power a as decay amplitudes the (where is rit ntesaigdmningiven scaling the on traints -rdc state. e-product htti sntarlal indicator, reliable a not is this that sn F ecito fti state. this of description CFT no is mnin.I h aeo critical a of case the In imensions. rtecreainfntosi the in functions correlation the or t w-adfu-on correlation four-point and two- ith lbn mltds,w hwthat show we amplitudes), bond ll I EUT RMCFT FROM RESULTS II. da eto o-nryde- low-energy of test a as ed ini o orc a ogas long (as correct not is tion J - Q hi.I e.V ewl in- will we V, Sec. In chain. 2 a brief summary of correlation functions within CFT. point vanish, thus removing the small but visible viola- It has been shown [7, 8] that quantum spin chains with tions of conformal invariance. For simplicity we here dis- a certain set of interactions allowed by spin and trans- cuss the theoretical results within the Heisenberg Hamil- lation symmetries have conserved currents which can be tonian, given by recreated by a scalar field theory; in this case the k = 1 N Wess-Zumino-Witten (WZW) model. The WZW model H = S · S +1, (4) has a conformal symmetry as well as a SU(2) symme- i i i=1 try. Using the conformal structure, one can predict that X the correlation functions have functional forms controlled where Si are the standard spin-1/2 operators, and later by the operator product expansion in the CFT. The two carry them over into the J-Q chain. The spin operators symmetries lead to two currents that follow a combined can be written in terms of standard fermion creation and Virasoro Kac-Moody algebra [9] that controls the prop- annihilation operators [7], which can then be expanded erties of the primary fields making up the CFT. in the Fourier modes close to the Fermi surface. Follow- The k=1 WZW Lagrangian is written in terms of a ing the work by Affleck [7], the Fourier-mode fermionic spinless primary field g, which has scaling dimension operators can be written in terms of currents and g field ∆=1/2, as can be seen by using the algebra and the operators of the WZW theory. The identification of the operator product expansion of g with the SU(2) current fermionic operators with g is done by noting that the [10]. These operator product expansions lead to Ward currents in both models have the same commutation re- identities [9], which are differential equations obeyed by lations [9], thus implying that they can be used inter- the correlation functions of primary operators. The equa- changeably. Coarse graining the spin chain so that the tions force correlation functions to follow power-law be- current J and field g can be treated as smooth objects, havior where the exponents are controlled by the scaling we can write the spin operator as of the primary operators it is made up of. b b ¯b n † b Recalling that g itself is a 2×2 matrix, we expect the cor- Sn = a[J + J + (−1) tr[(g + g )τ )], (5) relation functions to be matrices themselves. In our nu- where b is one of {x,y,z}, n is the lattice site number merical investigations we will make use of the two-point and τ b is the correspoding Pauli matrix here. The scal- , which has the expected asymptotic ing dimension of J is unity as it is a conserved current form given by and that of the g field is 1/2 as mentioned earlier. Us- ∆ ∆ ing these observations one can predict that the two-point hg(z, z¯)g(0, 0)i∼ z z¯ M4, (1) spin correlation should scale with exponent 1 to leading and the four-point function order in 1/L, the three-point one should scale with the exponent of 3/2, etc. hg(z1, z¯1)g(z2, z¯2)g(z3, z¯3)g(z4, z¯4)i (2) In our case we will only use equal-time correlation func- −∆ ∼ [(z1 − z4)(z2 − z3)(z ¯1 − z¯4)(z ¯2 − z¯3)] F (p, p¯)M16, tions, motivated primarily by the fact that the APS is only given as a wave-function, with no Hamiltonian to where z lives on the infinite complex plane, M is a short- n generate time correlations. This is not an uncommon hand for an n × n matrix and we define the set of anhar- situation when exploring various quantum states. Never- monic coefficients (p, p¯) as [9] theless, as mentioned above, in principle there still exists

(z1 − z2)(z3 − z4) (¯z1 − z¯2)(z ¯3 − z¯4) a parent Hamiltonian for a given wave function, and the p = , p¯ = . (3) constraints of the CFT allow for tests for a CFT by con- (z1 − z4)(z3 − z2) (¯z1 − z¯4)(z ¯3 − z¯2) sidering solely equal-time correlations. Through the numerics, we will observe the behavior of Possible equal-time correlation functions for two and the four-point correlation function for fixed anharmonic three spins at different sites on a chain are given by ratios. The structure of the matrix M is controlled by Oab = hSaSbi, Oabc = hSaSbSci, (6) selection rules given by the SU(2)×SU(2) decomposition ij i j ijk i j k that it follows [9]. This will not be important in our case, where a,b,c each are one of {x,y,z}. These correla- as we will be dealing with correlators of tr(g), and the tion functions are similar to the ones mentioned above matrix will be traced over with some constraints, but this and can be written in terms of the Wess-Zumino g field. will not affect the scaling behavior. Note, however, that unless a = b the two-point correla- The k=1 WZW model can be connected to spin chains tor vanishes in the ground states of the SU(2) symmetric by looking at the Heisenberg model at criticality and Hamiltonians we will consider, while the three-point cor- the additional interactions allowed [7]. From a numer- relator vanishes for any choise of the components a,b,c. ical standpoint, we study the J-Q chain, which has the For models and combinations of the components where same critical behavior as the Heisenberg chain [11] and the correlators do not vanish, the general conformal con- can be tuned through a quantum into traints discussed above forces them to be of the form a dimerized state. Similar to the case of the frustrated 1 1 Heisenberg chain [12], at the dimerization point the log- hOiOj i∼ , hOiOj Oki∼ , (7) 2Λi ∆ij ∆jk ∆ik arithmic corrections existing in the system up until this zij zij zjk zik 3 where zij = |zi − zj| is the spacetime separation (in our states being made out of singlets, would have the same case purely spatial separation) between the operators on scaling dimensions as predicted for the J-Q model if it site i and j, and ∆12 =Λ1 +Λ2 − Λ3 and so on. The lat- had the correct interaction terms. It is known [8] that tice spin correlation functions approximate to the contin- given a quantum spin chain which is translationally in- uum functions constructed out of the fields of the CFT; variant against a shift of 1 site, the only local interaction in the case of the spin correlations we will denote the which can push the system away from a SU(2)×SU(2) corresponding field by σ. symmetry is a negative coupling in the term coupling Averaging over the spin components, we will study the left and right currents (JL · JR). This does not hap- two-point correlator pen in the Heisenberg chain [8] as the coupling starts off positive and is thus irrelevant, and we also would not S S C2(x1)= h 0 · x1 i, (8) expect it in the J-Q chain before its dimerization transi- tion. The situation is less clear for the APS, where we do where 0 is a reference site and x1 the distance to it. Since not know the parent Hamiltonian, which, apart from the the three-point correlator vanishes for the SU(2) Hamil- potential attractive local interactions mentioned above, tonians, when considering those we will instead study the may contain long-range interactions that ruin the CFT rotationally-averaged four-point correlator description. S S S S C4(x1, x2, x3)= h( 0 · x1 )( x2 · x3 )i, (9) III. CLASSICAL AND QUANTUM ISING for which scaling forms generalizing Eqs. (7) will apply. MODELS For convenience in finite-size scaling, we will often choose all distances to be fractions of the system size L. Though the main focus of this paper is on systems with The global group of a CFT, which consists of transla- SU(2) symmetry, we make a brief detour to systems with- tions, rotations, scaling and conformal transformations, out this symmetry, namely the classical 2D restricts the scaling forms of correlation functions to and the transverse-field Ising chain. We develop and test Eq. (7) and the full Virasoro algebra forces these func- practical procedures to detect the signatures of the two- tions to depend only on the conformal distance [10]. For and three-point correlators predicted by a CFT. all the 1D systems we study in this work, we will use peri- The critical-point TFIM Hamiltonian is odic boundary conditions, leading to a cylindrical space- time geometry that can be mapped to the complex plane N N through a conformal transformation (x → θ,t → r) [1]. z z x H = − σi σi+1 − σi . (12) The conformal equal-time distances on the complex plane i=1 i=1 correspond to spatial distances on the lattice through [10] X X This Hamiltonian can be written in terms of Majorana ∆x fermions in the continuum as follows [13]: ∆z = L sin π . (10) L   1 ∂ψ (x) 1 ∂ψ (x) H = −i R ψ (x) − −i L ψ (x) . (13) Thus, the two-point function on periodic chains, which 2 ∂x R 2 ∂x L scales with twice the spin dimension Λσ [which is 1/2 for     the SU(2) case], should take the form In general this Hamiltonian should also have a mass term, but it disappears at criticality, leaving H invariant under 2Λ 1 σ scaling. From the Hamiltonian we see that the equations C2(x1) ∝ , (11) of motion imply that the two fermions are completely L sin(πx1/L)   disconnected (due to the absence of a mass term) and thus it is sufficient to study just one of them. for sufficiently large distance x1. The four-point function The spin operators in which the Hamiltonian of should also take a similar form where all the zij ’s are replaced by the conformal distance. Eq. (12) is expressed are found to be nonlocal in the ma- The global group of a CFT is invariant in all dimen- jorana fermions [13]. With periodic boundary conditions sions and the scaling constraints it sets on correlation on the majorana fermions, it was found that a possible functions can be easily seen in finite-size scaling. For set of non-trivial primary operators of the CFT could be the SU(2) symmetry, we were not able to construct a σ,µ,ǫ which are the order, disorder, and energy density three-point correlator as the only field operator we could operators, respectively [13]. As discussed in the previous test was σ. We would expect any quantum spin chain section, the correlation functions of these primary oper- with a Hamiltonian which permits the full SU(2)×SU(2) ators are constrained by the operator product expansion symmetry (coming from decoupling of left and right mov- in a CFT [6] to power laws which are controlled by the ing currents) in its interaction terms to be described by scaling dimensions of the operators as in Eq. (7). The the k = 1 WZW theory. This would imply that the scaling dimensions for the primary operators are [6] correlation functions in the APS, whose unknown par- 1 1 ∆ = , ∆ = , ∆ =1. (14) ent Hamiltonian must have SU(2) symmetry due to the σ 8 µ 8 ǫ 4

3.5 102 C3(L/4,L/2) C3(L/8,L/4) 3.0 C (L/2) C (L/2) e 0 s Cs(L/2) 10 Ce(L/4) 2.5 -2 2.0 10

1.5 10-4 1.0 10-6 0.5

0.0 10-8 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 10 50 100 200 500 1/L L

FIG. 1. Two- and three-point functions for the classical crit- FIG. 2. Power-law behavior of correlation functions in the 1D ical 2D Ising model. The exponents for hσσi, hǫǫi, and hσσǫi TFIM chain at criticality. The exponents for the spin corre- are found to be 0.252(4), 2.01(3), and 1.263(5) respectively, lation, energy density correlation and three-point correlation where the numbers within parenthesis indicate the statistical are 0.2498(2), 1.9(4) and 1.29(1) respectively. error (one standard deviation) of the preceding digit. The curves show fits to Eq. (18) using b = 2. case the conformal distance assumes a more complicated form [1]. For the 1D TFIM this is not an issue as we use Using these and the forms predicted by the CFT, Eq. (7), projector QMC [14, 15] to obtain results at temperature we expect the following connected two-point and three- T → 0, which corresponds to the time dimension being point functions to scale as much larger (effectively infinite) than the space dimen- 1 sion and the mapping leading to the conformal distance − 4 Cs(aL)= hσ(0)σ(aL)ic ∼ L , (15) of Eq. (10) is applicable. In principle this form could also be used for the classical model if Lx ×Ly lattices are used −2 Ce(aL)= hǫ(0)ǫ(aL)ic ∼ L , (16) with a very large aspect ratio Ly/Lx. We begin with results for the classical 2D model, for

3 which we have used the Swendsen-Wang cluster Monte 4 −1 −1 C3(aL,bL)= hσ(0)ǫ(aL)σ(bL)ic ∼ L L L , (17) Carlo algorithm [16]. Though the critical temperature Tc is known exactly in this case, in order to test procedures where a,b ∈ (0, 1) and the functional dependence on a,b where this is not the case we here test a method that depends on the boundary conditions being used; an ex- works also if Tc is not known. We employ the method of ample of which is provided in Eq. (11). It is worth noting flowing Binder cumulant crossing points [17] to extract that observation of scaling forms such as (15)-(17) do not exponents for the critical correlation functions, collecting in themselves guarantee a CFT as fields outside of a CFT data for the Binder cumulant and all the correlations we can also have well defined scaling dimensions, while a de- are interested in over a range of system sizes and tem- tailed form of the dependence on the conformal distance peratures close to the known critical temperature. For similar to Eq. (11) is more specific to a CFT. a pair of system sizes (L, 2L), we find the crossing point We would expect to see these CFT behaviors in both of the Binder cumulants, and we can regard this as a the 2D classical Ising model and the transverse-field definition of the critical temperature for size L. (This quantum Ising chain, as they are connected through the procedure is illustrated in the case of the APS further transfer-matrix approach. On the lattice, σ(i) corre- below in Sec. V.) sponds to the spin at location i and ǫ(i) corresponds to We extract the correlators at the crossing tempera- the Ising part of the energy density, σ(i)σ(i + 1). For the ture by interpolating within the data set, and extract TFIM, the points which are used to observe the corre- the floating exponents from the two so obtained values lation functions are picked along the spin chain at equal by assuming a power-law dependence in L. This process time, whereas in the 2D classical case we expect emer- gives us an estimate of the exponents at size L, and we gent rotational invariance for large separation and see the repeat the same for different size pairs to obtain a series same correlation functions should be observed provided of exponent values that should converge to the correct the distances are the same. exponents in the thermodynamic limit. The finite-size Note that the conformal-distance form (11) of the two- behavior for an exponent γ is expected to be of the form point function is not applicable to the 2D classical Ising model, as our simulations use L × L lattices with peri- 1 b γ(L)= γ0 + a , (18) odic boundary conditions in both dimensions, in which L   5

2.0 dimer 0.646 C (L=384) L=64 Cdimer(L=192) L=96 spin 1.5 C (L=384) L=128 Cspin(L=192) L=192 0.645

(x) 1.0 sc C 0.644

0.5

0.643

0.0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 x/L x/L

FIG. 3. Scaled spin correlation function, Eq. (19), for the FIG. 4. The spin and dimer two-point correlation functions critical TFIM on chains of different lengths (x/L = 0.5 is the of the critical J-Q chain scaled with the conformal distance largest separation on a periodic chain). Here we have used according to Eq. (19). ∆σ = 1/2 as that is the scaling dimension of the Ising spin operator [9]. distance as

x 2Λσ C (x)= L sin π C2(x), (19) where γ0 is the value that we conclude to be the final sc L exponent value and b is an exponent originating from an    irrelevant field (or, in some cases, from the size correc- where C2(x) is the original correlation function measured tions to the non-singular part of the free energy). Our at lattice distance x. If C2(x) follows Eq. (11), the scaled results are presented in Fig. 1, and the extracted expo- correlation should be a constant with respect to the dis- nents (listed in the figure caption) are seen to agree very tance x. As shown in Fig. 3, we find that the scaled spin well with the expected values. We have used distances correlation function indeed converges to a constant, even proportional to the system size L and fit the flow of the for relatively small system sizes (indicating small scaling exponents to Eq. (18) using the leading correction expo- corrections in this case), and to good precision except for nent b = 2 for the Ising . x/L ≈ 0 (and signs of convergence even when x → 0 are Turning now to the quantum TFIM, we run projector seen when the system size grows). Thus, as expected the QMC simulations [14, 15] to study the ground state of the TFIM passes the CFT test. critical Hamiltonian from Eq. (12) and extract exponents using different sizes, as illustrated in Fig. 2. Here we have IV. J-Q CHAIN only carried out simulations exactly at the critical point and we test fitting without corrections in this case. Since we expect the larger sizes to show better agreement to To test numerical signatures of the k = 1 WZW model, the asymptotic power-law forms than smaller sizes, due we study the J-Q chain [11] which, in the simplest incar- to corrections, we fit to a power-law starting with the nation, is the Heisenberg chain with a particular four-spin largest available sizes and add smaller sizes as long as interaction added. The Hamiltonian is the χ2 value of the fit is acceptable. Once we find the largest set of points with a statistically acceptable χ2 H = −J Pi,i+1 − Q Pi,i+1Pi+2,i+3, (20) i i value, we drop the smallest size from this set and use X X the remaining set for our fits. These fits are presented in where Pi,j =1/4−Si·Sj is the singlet projection operator Fig. 2, and the so obtained exponent values (listed in the between the sites i and j. For Q/J → 0 we recover the figure caption) match well the predictions from Eq. (15). Heisenberg chain, whereas for Q/J ≫ 1 the four-spin in- It is important to note here again that through Fig. 1 teraction term forces the chain into a valence-bond solid and Fig. 2 we have only proved the scaling behavior of state with spontaneously broken translational symmetry. the two- and three-point correlator, not the conformal be- The transition point Q/J ≈ 0.8483 is known from pre- havior. To study the exact functional form and establish vious studies [11, 18], and at this point the logarithmic the CFT form Eq. (7), we need to investigate their de- corrections present for smaller coupling ratios vanish. All pendence on both the point separation and system size our simulations are done at this critical point, using a and analyze according to the form (11). To make the projector QMC method formulated in the valence-bond functional dependence expressed there more explicit in basis to extract the properties in the ground state [19]. graphs, we define the correlator scaled by the conformal The two-point function, C2(x), has already been found 6

1.8 C4(L/8,L/4,3L/8) C (L/8,3L/8,L/2) 10-2 4 1.7 VBS

1.6

κ 1.5 10-3 1.4

1.3 Néel

10-4 1.2 50 100 200 0 0.5 1 1.5 2 2.5 3 3.5 4 L λ

FIG. 5. Four-point correlators of the critical J-Q chain along FIG. 6. Phase diagram of the APS with a critical curve sep- with a fit to the form expected for scaling dimension 2, shown arating the N´eel phase and the VBS phase; this matches well here for two different combinations of cross-ratios and system the phase diagram previously reported in Ref. [20]. sizes. The prefactor for the two fits is almost the same.

0.08 0.07 L=384 to have a scaling exponent of unity [11] as expected of L=768 the underlying CFT. 0.06 L=1536 L=3072 In the J-Q chain, we have used the two-point correla- 0.05 tors for the spin and dimer operators, which act as order 0.04 parameters for the critical and dimerized phases, respec- c B tively. On the lattice, the dimer operator is defined as 0.03 Di = Si · Si+1 and its two-point scaled connected cor- 0.02 relator approaches a constant when the system size is 0.01 sufficiently large, as does the spin correlator; see Fig. 4. 0.00 To check the scaling behavior of the four-point func- tion, as defined in Eq. (9), we plot its finite-size scaled -0.01 form and fit it assuming the scaling dimension 2, as dis- 1.265 1.275 1.285 1.295 1.305 1.315 1.325 κ cussed in Sec. II. The prediction for the four-point func- tion also tells us that using different cross ratios should give us the same scaling. Examples of this are shown in FIG. 7. Flow of the Binder cumulant of the APS with κ for Fig. 5. In cases where we do not know the scaling dimen- different system sizes at λ = 3. We see here the crossing points for three sets of (L, 2L) (example of extrapolation to critical sion of the operator making up the correlator, we can use point shown in inset of Fig. 8) and also that Bc becomes the four-point function to determine it, as we will do for negative for large sizes after the crossing point. the APS in Sec. V.

on the bond-lengths present in |Vii;

V. AMPLITUDE-PRODUCT STATE N b λ, d =1, Ai = h(dj ), h(d)= (22) d−κ, d> 1. j=1 ( The APS is a wavefunction of the resonating valence Y bond type on a bipartite lattice [20, 21]. Here we define It should be noted that the asymptotic properties of the it in 1D, using parameters λ and κ, as a superposition of state are not controlled only by the asymptotic power valence-bond states as, law, as might be naively expected, but also by the short- length weights which are important in this regard [22]. |Ψ(λ, κ)i = Ai(λ, κ)|Vii, (21) We have here chosen a very simple parametrization i that allows us to study the effects of both asymptotic and X short-length amplitudes, as in Refs. [20, 22]. For κ → 0 where |Vii is a tiling of Nb = L/2 two-spin singlets be- and λ ≈ 1 all valence-bond coverings contribute equally tween the A-sites and the B-sites on the 1D periodic and we obtain a long-range ordered N´eel state, whereas chain, and it contributes with a weight Ai to the APS. for κ → ∞ we have only two contributing components, The weight is given by an amplitude-product depending each with short bonds on alternating spin pairs, i.e., two 7

1.5 1.4 L=32 1.4 L=64 L=100 1.2 1.3

) 1.52 A 1.2 (L

1.0 c 2 1.50 S c

κ 1.1 1.48 0.8 1.0 1.46

0⋅100 1⋅10-3 2⋅10-3 3⋅10-3 4⋅10-3 0.9 0.6 1/L 0.8 0 5 10 15 20 25 30 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 L A λ

FIG. 8. EE of the 1D critical APS at λ = 1 for a periodic FIG. 9. EE for the 1D critical APS versus the parameter λ, chain of L = 64 sites. A fit to the form predicted by Eq. (23), extracted as illustrated in Fig. 8. along with finite size corrections [25] causing the visible even- odd oscillations, gave c = 1.08(4). Error bars are of the order of the symbol size. The inset shows the extrapolation of the a periodic chain in Fig. 8), i.e., the EE depends on the Binder cumulant crossings points (examples of which are seen size of the entangled region (lA) as follows: in Fig. 7), which we have used to extract the critical κ for the infinite system. Here κcrit is 1.519(5). c L πl S2(l )= log sin A + d, (23) A 4 πa L    degenerate dimerized states. In (λ, κ) space we have a where L is the total system size, and a is the nearest- critical curve, κcrit(λ), separating these two phases, as neighbour distance (set to 1 in our case). This form can shown in Fig. 6. The critical spin and dimer exponents be derived from the general form for the nth Renyi en- vary continuously on this curve [20]. All our scaling re- tropy [1]. The constant d is different for odd and even sults for the correlators were obtained for parameter val- sizes and the constant c corresponds to the central charge ues falling on this critical curve, and we have improved of the CFT. the estimation of this curve over Ref. [20] by going to Using this formulation, we can extract the effective larger chain lengths. Measurements in the valence-bond central charge of the apparent CFT. We have done this basis are done by following the mapping to loop esti- for a set of different λ (shown in Fig. 9) for different sizes mators developed in [23]. To avoid getting locked into and find the effective central charge to be close to unity winding sectors at large λ, we have also introduced si- for λ 6= 0. At λ = 0, the VBS phase shifts from being multaneous updates for identical singlets in the bra and composed of bonds of unit length to one made out of ket confgurations [24] which considerably improves the bonds of length 3 (as unit-length bonds have null weight), sampling efficiency. and we find the extracted c to be closer to 3/2 in this We extract crossing points using the Binder cumulant case. The values 1 and 3/2 are both possible CFT central at a particular value of λ, as shown in Fig. 7, and estimate charges in the case of SU(2) symmetry, and c = 1 is the of the spin operator by looking at associated with continuously varying exponents. Here the flow of the exponent with chain length, as described we have not studied finite-size effects, due to difficulties in Sec. III in the case of the classical Ising model. We in accurately computing the EE for larger system sizes, also see that the Binder cumulant becomes negative for and it is not clear whether c will flow with increasing size κ & κcrit and large system sizes (it approaches 0 for larger to 1 (in general) and 3/2 (in the special case of λ = 0), κ). A negative Binder cumulant is often taken as an or whether the value changes continuously as we vary λ. indicator of a first-order transition, but for that purpose In any case, as we will see, the system should not have a one also has to check that the negative peak value grows CFT description and it is interesting that the “effective as the system volume. Rigorous examples exist of more c” nevertheless is close to common CFT values. slowly divergent negative peaks at continuous transitions Next we look at the critical exponent for the spin corre- Ref. [26], and, as indicated by our results, the APS may lation using the technique of Binder crossings and flowing provide yet another example where this may be the case. exponents at different λ. The spin dimension calculated Next we consider the EE of the APS, using the second from the two- and four-point functions in Fig. 10 disagree 2 Renyi variant S (lA), which is accessible in simulations slightly when λ & 1, but considering the relatively large using the swap operation and the ratio trick [27]. We error bars we consider the agreement satisfactory nev- find that the EE follows a profile identical to what we ertheless. The exponent values differ substantially from would expect from a CFT [1] on a circle (shown for half the expected CFT value 1/2 for the entire range of λ 8

0.6 C (L/8,L/4,3L/8) 4 0.65 C2(L/4) R(0.5,0.2) 1.40 0.5 0.55 1.30 0.4 1.20 S (x/L) ∆ 0.45 0⋅100 1⋅10-4 2⋅10-4 3⋅10-4 4⋅10-4 0.3 sc

C 1/L 0.2 0.35 12288 9216 6144 0.1 4608 0.25 3072 2304 0.0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 λ x/L

FIG. 10. Dimension of the spin operator in the 1D critical FIG. 11. Scaled spin correlation functions, defined in Eq.(19), APS graphed vs the short-bond parameter λ. The values plotted against the separation as a fraction of the system size were deduced using two- and four-point functions calculated for the critical APS at λ = 1 (where κcrit = 1.519). The inset on the critical curve. shows the ratio R(0.5, 0.2) Eq. (24), of the correlations at two distances. studied. This rules out a k=1 WZW description of the critical APS. cality. We have determined the inapplicability of a CFT We can also study the scaled r-dependent correlation description by extracting the dimension of the spin oper- function as defined in Eq. (19), and we expect that it ator and by investigating the behavior of the two-point would not flow to a constant with respect to point sep- spin correlator. As the APS is capable of harbouring a aration with increasing size as there does not appear to long-range ordered N´eel state in 1D, it is apparent that be a CFT description for this system. In Fig. 11 we show the Hamiltonian must have long-range interactions on results for λ = 1 using the scaling dimension Λσ = 0.38 this side of the phase transition, and it is likely that the from Fig. 10 and tuning slightly to 0.40 to get data col- effects of these interactions carry on to the critical point x lapse for a large range of L , and find that it does not as well, thus perhaps preventing a CFT description (as follow the CFT functional form even for a large system that requires sufficient locality of the interactions). Note of 12288 sites, whereas the J-Q chain showed this (Fig. 4) that long-range interactions do not automatically pre- already for a system of 192 sites. To check the deviations clude a CFT description and there are indeed examples more explicitly, we define the ratio of the scaled correla- of this, e.g., SU(N) spin chains with exchange interac- tions at two different distances, tions decaying with distance r as 1/r2 [28]. As the rate of decay is decreased, one would at some point expect Csc(r1) R(r1, r2)= , (24) the CFT description to break down. C (r2) sc Examples of direct transitions between antiferromag- which should approach unity for all r1, r2 when L → ∞ netic and VBS phases have already been seen in both if the CFT description is valid. In the inset of Fig. 11 frustrated and [29] and unfrustrated [30] spin chains with we plot R(0.2, 0.5) as a function of 1/L at criticality and tunable power-law decaying long-range interactions, and extrapolate it to infinite size using a second-order poly- it was found that the dynamic exponent z < 1. Thus, nomial. We see that the result deviates substantially space and time do not scale in the same manner and we from unity, suggesting that this indeed is a good CFT cannot have a 1 + 1-dimensional CFT description of such test (here with a negative result). We were not able to a system (though still holds). As the look at the behavior of dimer correlators in the APS, as Hamiltonian of the APS is unknown, it is not possible to we have done with the J-Q chain, as the dimer order is determine z, but given that there is a long-range ordered quite weak at criticality and not substantial enough to N´eel state terminating at the critical curve it seems plau- perform a careful finite-size scaling analysis with large sible that the Hamiltonian could have z 6= 1. It might system sizes. indeed be worthwhile to find the parent Hamiltonian and to test this conjecture. Our numerics have also shown that the critical APS VI. CONCLUSION looks like a CFT when examined only through the lenses of entanglement, following the predicted scaling with the . size of the subsystem of the bipartition and even deliv- We have found that the EE of the APS falsely points ering values of c in the range of common CFT values towards the existence of a conformal description at criti- (the effective c being close to 1 and 3/2). Thus, our 9 study shows that care has to be taken when using the VII. ACKNOWLEDGEMENTS EE to establish the proper field-theory description, and it is important to also tests the constraints of CFTs in We would like to thank Ian Affleck, Roger Melko, the details of correlation functions. and Ronny Thomale for useful discussions. The com- putational work reported in this paper was performed in part on the Shared Computing Cluster administered by Boston University’s Research Computing Services. AWS is supported by the NSF under Grant No. DMR-1410126.

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