Detection and characterization of symmetry-broken long-range orders in the spin-\frac{1}{2} triangular Heisenberg model
DDetection and characterization of symmetry-broken long-range orders in the spin- triangular Heisenberg model S. N. Saadatmand ∗ and I. P. McCulloch ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics,The University of Queensland, St Lucia, QLD 4072, Australia (Dated: April 4, 2019)We present new numerical tools to analyze symmetry-broken phases in the context of SU (2)-symmetric translation-invariant matrix product states (MPS) and density-matrix renormalization-group (DMRG) methods for infinite cylinders, and determine the phase diagram of the geometrically-frustrated triangular Heisenberg model with nearest and next-nearest neighbor (NN and NNN)interactions. The appearance of Nambu-Goldstone modes in the excitation spectrum is characterizedby “tower of states” levels in the momentum-resolved entanglement spectrum. Symmetry-breakingphase transitions are detected by a combination of the correlation lengths and second and fourthcumulants of the magnetic order parameters (which we call the Binder ratio), even though symmetryimplies that the order parameter itself is strictly zero. Using this approach, we have identified 120 ◦ order, a columnar order, and an algebraic spin liquid (specific to width-6 systems), alongside thepreviously studied topological spin liquid phase. For the latter, we also demonstrate robustnessagainst chiral perturbations. PACS numbers: 03.65.Vf, 05.30.Pr, 71.10.Pm, 75.10.Jm, 75.10.Kt, 75.10.Pq, 75.40.Mg
I. INTRODUCTION
The groundstate of the one-dimensional nearest-neighbor Heisenberg model (originally determined in animportant work by Heisenberg ), H NN = J (cid:80) i S i · S i +1 ,for J <
0, exhibits long-range ferromagnetic (FM) or-der, which breaks the spins’ rotational symmetry (the SU (2) group) and elementary excitations are spin-waves(also known as Nambu-Goldstone bosons or Magnons;see for example 2–4). The Bethe Ansatz can be em-ployed (e.g. see 6) to study the antiferromagnet (AFM)spin- Heisenberg model,
J >
0, which demonstratesthe absence of magnetic ordering in clear contrast to theFM case. Today, we know there exist no continuous-symmetry-broken long-range order (LRO) in any one-dimensional system. In fact, magnetism in 1D andfew-leg ladders is peculiarly different to higher dimen-sions (where LROs exist; see below), since the mag-netic ordering at zero temperature is suppressed byquantum fluctuations due to the same mechanism asdescribed by Mermin-Wagner-Hohenberg theorem forfinite-temperatures (i.e. due to the low cost of cre-ating quantum long-range fluctuations, which increasesthe entropy). In contrast to 1D, long range magneticordering is possible in 2D Heisenberg-type Hamiltoni-ans; early examples arose from studying anisotropy , theAFM Heisenberg model with S =
12 10–15 , S ≥
32 16 , large- S values and for the antiferromagnetic XY andXXZ models for all spin magnitudes. For the major-ity of two-dimensional magnetic materials, if there exist no frustration, the groundstate exhibit either fer-romagnetism or antiferromagnetism (i.e. the well-knownbipartite N´eel order ). It is widely believed that thecelebrated Landau symmetry-breaking theorem ex-plains the physics behind all such conventional magneticordering: Hamiltonians such as H NN contain a set of symmetries which are absent in the groundstate, a fea-ture known as spontaneous symmetry breaking (SSB).As a result of symmetry-breaking, a well-defined or-der parameter exists in the model that can be used tocharacterize the magnetic ordering, unambiguously. Af-ter uncovering the key mechanisms behind the conven-tional ordering (in particular, ferromagnetism and bi-partite antiferromagnetism), the field of low-dimensionalquantum magnetism enjoyed a new boost of attentionaimed at understanding exotic phases of quantum matterthat appear in frustrated one-dimensional and two-dimensional systems. This happened partly dueto the rise of the geometrically-frustrated antiferromag-nets on non-bipartite Archimedean lattices .Interestingly, the existence of geometrical frustration isenough by itself to often lead to the ‘melting’ of themagnetic ordering, stabilizing a family of nonmagneticphases, collectively classified as spin liquids (also knownas paramagnetic states) . Such quantum liq-uids preserve all Hamiltonian symmetries and, conse-quently, their existence cannot be understood throughLandau’s symmetry-breaking paradigm. The search fornew, hidden order parameters has been challenging the-orists for the last 20 years, and has led to the discov-ery of even more intriguing phases of the quantum mat-ter. A canonical example is the discovery of the topo-logical order , such as symmetry-protected topo-logical (SPT) ordering (including the Haldane phase andthe closely relevant Affleck-Kennedy-Lieb-Tasaki ground-state ) and the intrinsic topological states (in-cluding the Z -gauge groundstate of the toric code ),which can only exist in D ≥ a r X i v : . [ c ond - m a t . s t r- e l ] A p r length, ξ , as | G ( i, i (cid:48) ) | = (cid:104) S i · S i (cid:48) (cid:105) ∼ C + e − rii (cid:48) ξ + · · · ,where C stands for a constant (which can be zero), r ii (cid:48) isthe distance between sites, and ellipses represent fasterdecaying terms. Different type of ordering can be definedas follows. For magnetically disordered states, with noconventional order parameter (i.e. no broken symmetry),the correlation function decays to zero exponentially fast, C is zero, ξ is finite, and there is a bulk gap in the excita-tion spectrum. In this case, instead of symmetry break-ing, we have symmetry protection , giving rise to the SPTorder. Such an exponential drop is observed in the Hal-dane phase (as an example, see the original calculationsby White and Huse ). For true LROs, such as N´eel-type AFMs and the FM state with a conventional orderparameter, the correlation function tends to a constantat large distances, ξ → ∞ . There exists another dis-tinct long-range phase of the quantum matter, which isreferred to as a quantum critical state (or a quasi-LRO).In such phases, the correlation function decays with a power-law with distance. Power-law decaying correlationfunctions can be approximated as the sum of many expo-nential functions, as occurs in the MPS ansatz , whichagain translates to having diverging ξ , consistent withthe Bethe Ansatz’ prediction for the spin- Heisenbergchain. Critical states are common in 1D quantum mag-netism and appear at a transition between two gappeddisordered phases with different symmetries, when thegap necessarily closes; however, they can also stabilize inan extended region, as in the
XY-phase of the anisotropicHeisenberg chain .High-accuracy numerical methods, such as exact di-agonalization (ED), quantum Monte Carlo (QMC),(see 50 for a review), and coupled cluster meth-ods, are often used for low-temperature frustrated mag-nets, modeled as strongly-interacting spin Hamiltoni-ans exhibiting many-sublattice groundstates. In thispaper, we employ and expand the functionality of thefinite DMRG (fDMRG) , and the state-of-the-artinfinite DMRG (iDMRG) methods to characterizeLROs of a geometrically-frustrated system, when themany-body states are constructed through the SU (2)-symmetric (non-Abelian) MPS and infinite MPS (iMPS)ans¨atze , respectively. The latter is a translation-invariant MPS that allows the calculation of many use-ful quantities directly in the thermodynamic limit viatransfer matrix methods. Currently, there exist fewwell-established numerical tools, in the context of non-symmetric DMRG, to identify LROs. In finite-systemMPS studies, SSB needs to be treated carefully becausein exact calculations SSB does not occur at all , asfinite size effects induce a gap between states that wouldbe degenerate in the thermodynamic limit. In practice,with finite-precision arithmetic symmetry breaking canoccur when the finite-size gap is smaller than the char-acteristic energy scale set by the accuracy of the numer-ics (in MPS calculations, this is set by the energy scaleassociated with the basis truncation). This can be dif-ficult to control, as symmetry breaking might occur as a side-effect of the numerical algorithm or it might re-quire an additional perturbation. Infinite-size MPS (orvery large finite MPS) are better behaved in this respect,where there are a variety of techniques; one can lookat the scaling of the correlation length of the ground-state against MPS number of states, m , which distin-guishes gapped and gapless states , direct measure-ment of local magnetization order parameters, the en-tanglement entropy , and the static spin structure fac-tor (SSF – see below). However, when the Hamiltoniansymmetries are preserved explicitly, the order parameteris zero by construction and a robust set of numerical toolsfor characterizing magnetic ordering is not readily avail-able. Here, we introduce and verify the accuracy of twonew numerical tools, in the context of SU (2)-symmetriciMPS/iDMRG, to characterize and locate phase transi-tions incorporating LROs in the triangular Heisenbergmodel (THM) on infinite cylinders. New tools includestudy of the cumulants (cf. 61 for the definitions and rel-evant discussions on the non-central moments and thecumulants in the context of the probability theory) anda Binder ratio of magnetization order parameters, andfurther developments on tower-of-states (TOS) level pat-terns in the momentum-resolved entanglement spectra(ES) .The triangular lattice has the highest geometrical frus-tration in the Archimedean crystal family with a coordi-nation number of Z c = 6. Anderson and Fazekas argued that the high frustration of the triangular latticemight be enough to melt the long-range magnetic or-dering observed for the Heisenberg model on the squarelattice (e.g. see 4, 20, and 21). In the first work, Ander-son conjectured that the spin- THM with antiferromag-netic NN bonds should stabilize a resonating-valance-bond (RVB) groundstate (i.e. the equally-weighted su-perposition of all possible arrangements of the singletdimers on the lattice; RVBs are the building blocks of thequantum liquids). The failure of robust analytical andnumerical studies to find an RVB groundstate motivatesthe search for a minimal extension to H NN that increasesthe frustration. The obvious choice is frustration throughthe addition of a NNN coupling term, which frustrates aLRO 120 ◦ -ordered arrangements of sublattices (see be-low and 12, 21, 67–78). This led to the introduction ofthe J - J THM, for which the Hamiltonian is defined as H J = J (cid:88) (cid:104) i,j (cid:105) S i · S j + J (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) S i · S j , (1)where (cid:104) i, j (cid:105) ( (cid:104)(cid:104) i, j (cid:105)(cid:105) ) indicates that the sum goes over allNN (NNN) couplings. The SU (2)-symmetry of H J canbe simply realized by noticing [ H J , S ] = 0 ( S stands forthe total spin vector), which means that eigenvalues of S are good quantum numbers and can be used to la-bel groundstate symmetry sectors. Geometrical frustra-tion forbids the bipartite N´eel order as a stable ground-state of the antiferromagnetic NN model ( J > J = 0). Consequently, one expects the groundstate, forthe majority of the phase diagram of the antiferromag-netic H J , to be a compromise, such as a 120 ◦ -orderedarrangement . By now, it is well-known that thegroundstate of the nearest-neighbor THM does not ex-hibit an RVB, but is instead a quasi-classical LRO 120 ◦ state, which is less stable than the N´eel or-der on the square lattice, since the sublattice magneti-zation of the triangular lattice is significantly reducedcompared to its classical value. Because of this reducedstability, inherent to the triangular lattice, upon perturb-ing the Hamiltonian one may expect to see a varietyof new phases. There have been some historically im-portant semi-classical spin-wave theory (SWT) and EDstudies for the model. However, such stud-ies did not cover the physics of the whole phase diagramand were not able to capture the detailed properties ofthe groundstates. Previously, we elucidated the com-plete phase diagram of the J - J THM on three-leg finite-and infinite-length cylinders to understand the crossoverof 1D and 2D physics in the model. Moreover, otherprecise numerical approaches demonstrate theexistence of a spin-liquid (SL) state that stabilizes in aregion ranging from J low2 ≈ . up to J high2 ≈ . .Some numerical studies discovered magnetic orders out-side this approximate SL region (see for example 21, 70–72, 75, and 84). However, we suggest the detailed prop-erties of the magnetic groundstates are still unclear incomparison to the well-understood counterparts in the(semi-)classical THM and the quantum model on thethree-leg cylinders. In particular, for the finite-size lat-tices, the largest system sizes for which the magnetic or-dering of the J - J THM was thoroughly studied is an18 ×
18 torus and a 30 × .It is noteworthy that the J - J THM can de-scribe some low-temperature properties of quasi-2D or-ganic lattices, such as κ -(BEDT-TTF) Cu (CN) andMe EtP[Pd(dmit) ] , and inorganic materials, such asCs CuCl , Cs CuBr , and RbFe(MoO ) (see 31, 33, and86 for details).In this work, we establish the phase diagram for the J - J THM on infinite-length cylinders with width upto 12 sites, and show that the model contains an LROcoplanar three-sublattice 120 ◦ order ( J ≤ . -gauge spin liquid(0 . ≤ J ≤ . . ≤ J ≤ . . ≤ J ≤ . . ≤ J ≤ . (ASL) groundstate (0 . ≤ J ≤ . J - J THM and the possibility of the stabilization ofa chiral LRO. We note that there were recent, indecisivediscussions on the robustness of the topological spin liquids against perturbing H J with a chiral term.Here, we confirm the non-chiral nature of such ground-states and the existence of a continuous phase transitiontoward the chiral spin liquid (CSL) phase throughthe study of a scalar chiral order parameter on width-8cylinders.The rest of the paper is organized as follows. In Sec. II,we provide the details of the employed SU (2)-symmetricMPS and DMRG methods (in particular, how we con-struct order parameter operators) and the geometry ofthe cylindrical lattices (in particular, the MPS mapping).In Sec. III, we present an overview of each of the phases,a schematic phase diagram for the model, and a morequantitative diagram in the form of short-range corre-lations. In Sec. IV, we directly measure the magne-tization order parameters on some small-width ( L y =3 , , ,
6) finite-length cylinders using MPS/fDMRG al-gorithms, to benchmark our calculations with anotheralgorithm. Afterward, we focus on presenting our moreprecise iMPS/iDMRG results on infinite cylinders (hav-ing widths up to L maxy = 12). In Sec. V, we investigatethe scaling behaviors of the correlation lengths against m ,deep in each phase region. In Sec. VI, in order to betterunderstand the entanglement entropy of the symmetry-broken LROs on cylinders, we study the entropy in thecolumnar magnetically-ordered phase. Details of our nu-merical tools are presented in Sec. VII and Sec. VIII,for cumulants and Binder ratios of the magnetization or-der parameters, and for TOS levels in the momentum-resolved ES, respectively. In Sec. IX, we test the robust-ness of a topological SL groundstate against chirality per-turbations of the Hamiltonian to investigate the forma-tion of long-range chiral orders, before some concludingremarks in Sec. X. II. METHODS
To obtain the variational groundstate of the THM fora wide range of FM and AFM J -values in Eq. (1),we set J = 1 as the unit of the energy, and employthe single-site DMRG algorithm (incorporating density-matrix mixing with subspace expansion and SU (2)symmetry ). In addition, we construct the oper-ators using the efficient formalism of matrix productoperators (MPOs), which represents an operatoranalogous to an ordinary MPS matrix. The MPO struc-ture provides a formulation of any polynomial operator(with an expectation value that scales polynomially withthe size of the lattice) in a Schur form (an upper- orlower-triangular matrix) for infinite systems (see be-low for an example and also 93 for an overview), whichallows the calculation of the asymptotic limit of the ex-pectation value per site. We keep up to m = 2 ,
000 num-ber of states (approximately equivalent to 6,000 statesof an Abelian U (1)-symmetric basis) in MPS/fDMRG,and up to m = 3 ,
000 number of states (approximately9,000 U (1)-states) in iMPS/iDMRG calculations. Due to MPS chain bipartite L-partition R-partition a -60° -directiona +60° -direction Y - d i r e c t i o n FIG. 1. (Color online) Cartoon visualization of a triangularlattice on a YC cylinder. Spins sit on spheres. An ‘efficient’mapping of the MPS chain is shown using the red spiral. Thegreen arrows represent the unit vectors on three principal lat-tice directions. The transparent gray plane corresponds tothe bipartite cut that creates partitions L and R , withoutcrossing any Y -direction bond. inherent 1D nature of the MPS, a mapping between theansatz wavefunction and the triangular lattice is neces-sary. For the mapping purposes, we wrap the lattice ina way to create a long (or infinite-length) L x × L y -sitecylinder ( L x can go to infinity; we also set L = L x × L y )as in Fig. 1. We will employ a standard notation, previ-ously presented in 79 (originally developed for single-wallcarbon nano-tubes ), to specify the wrapping vectorsof the cylinders, C , in terms of principal lattice direc-tions using a notation of (ˆ a +60 ◦ , ˆ a − ◦ ). For the majorityof the calculations, we choose the so-called YC wrap-ping, C [YC] = ( L y , − L y ) (we shall use the shorthandnotations of YC L y and YC L x × L y to specify differentYC lattice sizes). The YC structure is the only wrap-ping method with a circumference that equals to L y ( Y -axis now coincides with the lattice short-direction andthe X -axis coincides with the lattice long-direction) andis the best choice for the momentum-resolved ES (seebelow). However, in general, the choice of L y and C should respect sublattice ordering (if any) of the targetstate to avoid frustrating the groundstate. Consequently,depending on the desired width, the YC structure can-not be always used. Therefore, in finite- L x fDMRGcalculations, we use a YC6 structure in all regions (al-lowing the stabilization of up to tripartite-symmetricgroundstates), and YC3, C [ L y = 4] = (4 , − C [ L y = 5] = (5 , −
4) cylinders only in the 120 ◦ andthe SL phase regions. We also consider a YC4 structurein the columnar and the SL phase regions (occasionally,the YC3 and C [ L y = 4] = (5 , −
4) systems are employedin the columnar phase region, however they are frustrat-ing some forms of the collinear ordering – see below).For finite-length cylinders, we fix L x to a value thatafter which, an increase of the cylinder’s length wouldnot change the average bond energy in the bulk of the system up to numerical uncertainties coming from theDMRG systematic errors . In L x = ∞ iDMRG calcu-lations, we use YC6 and YC12 structures in all regions,reserving YC9 only for the 120 ◦ region, plus YC8 andYC10 in the columnar phase. We always set an effi-cient mapping for the infinite cylinders that minimizesthe one-dimensional range of NN and NNN interactions,as shown in Fig. 1. Finally to calculate bipartite quan-tities, such as reduced density matrix, ˜ ρ , and entropy ofthe DMRG wavefunctions , we make a cut that does notcross any Y -direction bond and creates partitions L and R , as shown in the figure.We now present an overview of how to calculatehigher moments of a (possibly non-local) observable ina translation-invariant infinite-size system. This is re-quired for the measurements of the cumulants and Binderratios of the magnetization order parameters (see below).For symmetry broken (or symmetry protected) states theBinder cumulant of the (string) order parameter can beevaluated directly in the thermodynamic limit . How-ever in this case, because we preserve SU (2) symmetrythe magnetic order parameter is strictly zero and theBinder cumulant is not well defined. However, as weshow below, the moments can still be used to detect thesignature of magnetic ordering. Suppose we are inter-ested in calculating the matrix elements of the momentsof an order parameter MPO, M [ k ] , of dimension ˜ m , thattransforms under SU (2) as a rank k tensor. The explicitpreservation of SU (2) symmetry leads to the vanishingof the order parameter, but the even moments can benon-zero. In this case, the measurement of the expecta-tion values of the higher-order magnetic moments, (cid:104) M n (cid:105) (of order n ), is of interest. These can be done using themethod of the transfer operator . The generalized trans-fer (super-)operator, T X , associated with some operatorof finite support (acting on a unit cell of an iMPS), ˆ X ,is defined as T X ( E a ) = (cid:88) s (cid:48) s (cid:104) s (cid:48) | ˆ X | s (cid:105) A s (cid:48) † E a A s , (2)where A s are ordinary MPS matrices and E a denote theso called E -matrices. In this context, E a is essentiallyan eigenmatrix, however, the E -matrices are more famil-iar for their role in the expectation value of an MPO (cid:104)A| ˆ M |A(cid:105) (see 53 and 55 for full details). E a is in prin-ciple extensive, and on a n -site system, can be definedrecursively as E a (cid:48) ( n ) ≡ (cid:88) s (cid:48) ,s,a A s (cid:48) n † M s (cid:48) sa (cid:48) a A s n E a ( n − . (3)In the following example, for the sake of the simplicitywe assume a one-site unit cell, although in practice fora magnetically ordered system the unit cell will be atleast as large as the number of sublattices; The gener-alization for larger unit-cell sizes is straightforward. Wegive an example here for the second moment, the highermoments can be obtained recursively . To calculate the S=0,1,0
E(L+1) A = E(L) M (a) A † E (L+1) = E (L) (b) E (L+1) = E (L) (c) X + E (L) E (L+1) = E (L) (d) X + E (L) 2X + E (L) < M > /L = E (L) (e) X + ρ E (L) 2X ρ FIG. 2. (Color online) (a) MPS diagram for the fixed-point equation of E a -matrices of the second moment of M . MPS diagramsfor the (b) first, S = 0, (c) second, S = 1, and (d) third, S = 0, columns of M , Eq. (6). (e) The MPS recipe to calculate thefinal expectation value of the second moment. asymptotic limit of (cid:104) M (cid:105) , one only needs to solve thediagrammatic fixed-point equation shown in Fig. 2(a),where the E a -matrices are connected according to E i ( L + 1) = T M ii ( E i ( L )) + (cid:88) j>i T M ij ( E j ( L )) , (4)which can be solved sequentially, from E , E , . . . , E ˜ m .In practice, Fig. 2(a) shows the fixed point at which theaddition of an extra site (or unit cell) to E a -matriceswill leave the system unchanged. The MPO form of theorder parameter on a unit cell can be written as asuper-matrix (a matrix where elements are local opera-tors acting on a single site or unit cell of the lattice): M = (cid:18) I XI (cid:19) . (5)and we can attach SU (2) quantum numbers S = 0 , M = (cid:18) I XI (cid:19) ⊗ (cid:18) I XI (cid:19) = I X X X I XI XI ⇒ I X X I XI , (6)where in the last step, we have collapsed the middle rowsto create a 3 × S = 0 , , X – for the calcula-tion of higher moments we need the other spin projectionstoo). We can now write the fixed point of the last MPOin the form of recursive equations for the E a -matrices,as shown in Fig. 2(b),(c), and (d). We note that the ob-jects that appear on the right hand sides of the figuresare nothing other than the generalized transfer opera-tors. Translating the graphical notation into equations,for example, Fig. 2(b) can be written as E ( L + 1) = T I ( E ( L )) , (7)which means E ( L ) is an eigenmatrix of the transfer op-erator, which, for a properly orthogonalized MPS is justthe identity matrix, so E ( L ) = I . As a result, equationsfor Fig. 2(c) and (d) can be written as E ( L + 1) = T X ( I ) + T I ( E ( L ))= C X + T I ( E ( L )) , (8)where C X = T X ( I ) is a constant matrix, and E ( L + 1) = T X ( I ) + 2 T X ( E ( L )) + T I ( E ( L )) . (9)The desired expectation value is encoded in the final ma-trix, i.e. (cid:104) M (cid:105) = T r ( E ˜ ρ ). However, importantly, mostof the matrix elements of E do not contribute to the ex-pectation value of the second moment per site, we needonly the component of E that has non-zero overlap with˜ ρ . Note that ˜ ρ is the right eigenmatrix of T I with theunity eigenvalue, hence the component of E that givesthe expectation value is the component in the directionof the corresponding left eigenmatrix of T I .The calculation of the matrix elements of E can bedone efficiently using a linear solver. To see how thisworks, consider the eigenmatrix expansion for the trans-fer operator, T I = (cid:80) m n =1 η n | η n )( η n | , to obtain the eigen-values η n and eigenvectors | η n ). If we write C X and E matrices in this {| η n ) } basis with expansion coefficients c (2) n and e (2) n ( L ), C X = m (cid:88) n =1 c (2) n | η n ) ,E ( L ) = m (cid:88) n =1 e (2) n ( L ) | η n ) , (10)then Eq. (8) is, for each component, e (2) n ( L + 1) = c (2) n + η n e (2) n ( L ) . (11)Following 92, we further decompose the coefficients intoa component parallel and components perpendicular tothe identity matrix, I (i.e. the left eigenmatrix of T I ,which has the largest eigenvalue of η = 1 due to theMPS orthogonalization condition), and define˜ C X = m (cid:88) n =2 c (2) n | η n ) , ˜ E ( L ) = m (cid:88) n =2 e (2) n ( L ) | η n ) , (12)so that C X = ˜ C X + c (2)1 I and E = ˜ E + e (2)1 I . Thereason for this is that the component in the direction of the identity e (2)1 diverges in the summation, whereas theother components that are perpendicular to the identitydo not. Hence, we need to find the fixed points of theseparts separately.Solving Eq. (11) for the parallel components reveal thelocal expectation value of X per site, which is an straight-forward calculation, e (2)1 ( L + 1) = e (2)1 ( L ) + c (2)1 , (13)where c (2)1 is just the expectation value of the order pa-rameter on one site. Hence e (2)1 ( L + 1) = (cid:80) Li =1 (cid:104) X i (cid:105) ,which is zero because of the SU (2) symmetry (indeed, c (2)1 = 0 by construction, since it is in the wrong quan-tum number sector for the identity eigenvector of thetransfer operator). The perpendicular components leadto ˜ E n ) ( L + 1) = ˜ C ( n ) + η n ˜ E n ) ( L ) , (14)where now n ≥
2, and the eigenvalues | η n | <
1. Thus,Eq. (14) is of the form of the sum of a convergent geo-metric series. Upon taking the limit L → ∞ and writingback the projection operators as the original matrices,Eq. (14) converges to a fixed-point:(1 − T I ) ˜ E ( ∞ ) = ˜ C X , (15)which is a rather simple system of linear equations, andis numerically stable because the condition number of1 − T I is simply related to the leading correlation length,1 / (1 − | η | ) (cid:39) ξ . In practice, generalized minimal resid-ual method (GMRES) is a good choice of linear solverfor Eq. (15). Upon obtaining the matrix elements of˜ E , we can proceed to calculate the final expectationvalue as shown in Fig. 2(e). Note that this does notrequire all of the matrix elements of E , since we onlyrequire the overlap between E and the density matrix(the right eigenvector of T I with eigenvalue 1). Thismeans that (cid:104) M (cid:105) = L × Tr(˜ ρ T X ( I ) + 2˜ ρ T X ( E ( L ))),which is demonstrated in the MPS diagrammatic equa-tion of Fig. 2(e). I.e. the only unknown is the E -matrix.This is a useful optimization and rather general – in cal-culating the expectation value of a triangular MPO ofdimension ˜ m , only the matrix elements up to E ˜ m − arerequired.For calculating the 4th moment of a magnetization or-der parameter using SU (2) symmetry, X decomposesas X = ( X · X ) + ( X ⊗ X ) · ( X ⊗ X ) , (16)where the dot product X · X = −√ X × X ] [0] and outerproduct X ⊗ X = (cid:112) / X × X ] [2] are proportional tothe S = 0 and S = 2 projections of the operator product,respectively, with an additional factor arising from the SU (2) coupling coefficients. In general, we would needto also include the cross-product term ( X × X ) · ( X × X ) (proportional to the spin-1 projection), however, thisvanishes due to antisymmetry under time reversal. III. OVERVIEW OF THE PHASE DIAGRAM
In this section, we present our findings for the phaseboundaries and properties of Eq. (1), for different J /J with J >
0, using iDMRG and some benchmark com-parisons using fDMRG.In Fig. 3, we show the summary of the phase diagram,with four distinct phases; two phases with symmetry-broken magnetic order, a Z spin liquid, and (only forthe Y C (cid:104) S i · S (cid:105) , with respect to a reference site S ,using the size and the color of some spheres, and the NNcorrelators are depicted using the thickness and the colorof some bonds. The reference site is denoted with thegray sphere. We also present the SSF up to the secondBrillouin zone. Using the discrete Fourier transform ofthe real-space correlations to switch to the momentumspace, one can writeSSF( k , N = ∞ ) = lim N →∞ N N (cid:88) i,i (cid:48) (cid:104) S i · S i (cid:48) (cid:105) e i k · ( r i − r i (cid:48) ) , (17)where r i denotes the position vector of a spin S i in the planar map of the lattice. The momentum vector, k , willsweep the extended Brillouin zones. When the momen-tum vector coincides with the lattice’s wave vector, Q ,the occurrence of the condition lim N →∞ SSF( N ) N = Const.guarantees the existence of a true LRO. Plotting the SSFin the ( k x , k y ) plane will reveal occurrence of strong FMcorrelations as Bragg peaks. However, for a fixed- L y in-finite cylinder, one can only estimate the sums appear-ing in Eq. (17) using a finite length correlation. There-fore, we consider a large enough cutoff as an upper limitfor i , namely N c . We note that it is possible to obtain SSF ( k , N = ∞ ) directly using the same method as de-scribed above for the moments (see also 92), however,this is an expensive process and for calculating the entire k space it is much faster to calculate the real-space corre-lations and perform a Fourier transform. Here, we trun-cate the real-space correlation at the first point where |(cid:104) S · S N c (cid:105)| ≤ − is met for the nonmagnetic short-range correlated states (i.e. spin liquids) and the condi-tion |(cid:104) S · S N c (cid:105)| ≤ − is met for the symmetry-brokenquasi-LROs (i.e. 120 ◦ and columnar states). The ob-tained phases are:1. J → −∞ : In this limit, one can readily show thatthe lattice decouples into three sublattices, each ofwhich is a NN triangular lattice with bond strength J . In the case of vanishing interactions betweensublattices ( J /J → −∞ ), the groundstate foreach sublattice is trivially a fully-saturated ferro-magnet (see also 79) with total spin magnetizationof S total A,B,C = L u per unit cell of each sublattice ( A , B , or C ). For a width- L y infinite-length YC struc-ture, L u = L y / S total A,B,C = L y /
6. The overall state can be any arbitrary mixture of three S total A , S total B , and S total C spin vectors, where they only haveto follow the angular momentum summation rules.This will cause a large degeneracy for the overallgroundstate, supporting total magnetization in arange of 0 ≤ S total ≤ L u . Perturbing the Hamil-tonian with a positive J would then break this de-generacy and impose a 120 ◦ -ordered groundstate.Similarly to the case of three-leg cylinders , wefind no signs of a phase transition for any J < J ≤ . ◦ order. Our investigations oninfinite YC6, YC9, and YC12 structures find athree-sublattice magnetic ordered state exhibitingSSB in the thermodynamic limit (cf. Sec. VII andSec. VIII). By imposing SU (2) symmetry, the low-lying Nambu-Goldstone modes are evident andviewing the infinite cylinder as a 1D system itappears as a 1D quantum critical gapless state(cf. Sec. V). In Fig. 4(a), we present the corre-lation function for a YC12 groundstate at J = − .
0. The appearance of L y = 4 blue (ferromag-net) spheres per ring exhibiting a roughly constantsize (for short distances) and all-AFM (red) bonds(throughout the cylinder) are characteristics of thephase. In Fig. 4(b), we present the SSF for aYC6 groundstate, deep in the 120 ◦ phase. Theformation of six strong Bragg peaks on a slightly-distorted regular hexagon is another characteris-tic for the phase. Using this data, we predicta wave vector of Q ◦ ≈ ( ± . , ± . Q theory120 ◦ = ( ± π √ , ± π ) ≈ ( ± . , ± .
09) for a 120 ◦ product state . We note that the correlationfunctions of YC6 and YC9, and SSFs of YC9 andYC12 structures in the 120 ◦ phase are essentiallyidentical to the results of Fig. 4.3. 0 . ≤ J ≤ . topologicalspin liquid (denoted by YC L y -ˆa for the anyonicsector ˆa ∈ { ˆi , ˆb , ˆf , ˆv } ; see 85 for full details). InFig. 5(a), we present the correlation function for aYC12-ˆi groundstate at J = 0 . ). In Fig. 5(b),we present the SSF for a YC10-ˆb groundstate at J = 0 . ◦ or-der. We notice that this overall pattern is virtu- J topological spin liquid columnar order columnar order columnar order FIG. 3. (Color online) Schematic phase diagram of the J - J THM, Eq. (1), on infinite cylinders. Phase transition boundariesare obtained from the Binder ratios of the magnetization order parameter (see below). (b) -6 -4 -2 0 2 4 6Kx-6-4-2 0 2 4 6Ky
SSF
FIG. 4. (Color online) Lattice visualizations for the iDMRGgroundstates of the THM on infinite cylinders at J = − . ◦ order). (a) Correlation function for a YC12 system.The size and the color of the spheres indicate the (long-range)spin-spin correlations in respect to the principal (gray) site,and the thickness and the color of the bonds indicate thestrength of the NN correlations. (b) SSF for a YC6 system.Bragg peaks are presented up to the second Brillouin zone ofthe inverse lattice. ally the same for all anyonic sectors and systemsizes. Furthermore, our qualitative studies demon-strate that the homogeneity of the SSF is growingwith increasing L y (not shown in the figures). Forthe topological SL phase, we find the lower andupper phase boundaries of J low2 = 0 . J high2 = 0 . . ≤ J ≤ . (b) -6 -4 -2 0 2 4 6Kx-6-4-2 0 2 4 6Ky SSF
FIG. 5. (Color online) Lattice visualizations for the iDMRGgroundstates of the THM on infinite cylinders at J = − . those found by other authors .4. 0 . ≤ J ≤ .
5, but excluding a region only forYC6 of 0 . ≤ J ≤ . J points on YC12structures) show that that the columnar order istwo-sublattice AFM state exhibiting SSB in thethermodynamic limit (cf. Sec. VII and Sec. VIII).Again, with SU (2) symmetry the state appears onan infinite cylinder as 1D quantum critical. The (b) -6 -4 -2 0 2 4 6Kx-6-4-2 0 2 4 6Ky SSF
FIG. 6. (Color online) Lattice visualizations for the iDMRGgroundstates of the THM on infinite cylinders at J = 0 . correlation function for a YC12 groundstate at J = 0 . a +60 ◦ -direction isclearly recognizable. In fact, the columnar orderon the triangular lattice has three possible arrange-ments of FM stripes, each aligning with one of thethree principal lattice directions, which are only de-generate in the thermodynamic limit. For the THMon three-leg (trivially) and four-leg cylinders (bothfinite and infinite-length cases), we found that thecolumnar order always has FM stripes in the latticeshort (Y) direction, while for wider-width finite-length YC structures, FM stripes will be in either of a +60 ◦ or a − ◦ -directions, producing only two de-generate groundstates. We numerically confirmedthat, upon choosing a suitable wavefunction unitcell, iDMRG states randomly converge to one ofthese two states. We present the SSF for a YC8groundstate at J = 0 . a +60 ◦ -direction FMstripes), in Fig. 6(b). The formation of four , com- (b) -6 -4 -2 0 2 4 6Kx-6-4-2 0 2 4 6Ky SSF
FIG. 7. (Color online) Lattice visualizations for the iDMRGgroundstate of the THM on an infinite YC6 system at J =0 .
185 (ASL phase). (a) Correlation function results, wherethe size and the color of the spheres indicate the (long-range)spin-spin correlations in respect to the principal (gray) site,and the thickness and the color of the bonds indicate thestrength of the NN correlations. (b) SSF results, where theBragg peaks are presented up to the second Brillouin zone ofthe inverse lattice. paratively very strong Bragg peaks on a slightly-distorted regular parallelogram (with 60 ◦ angles)is a characteristic of the phase. A wave vectorof Q striped ≈ ± (1 . , .
18) can be estimated forthe SSF, which is close to our expected theoret-ical value of Q theorystriped = ± ( π √ , π ) ≈ ± (1 . , . . We note thatthe SSFs of the columnar orders on YC6, YC10,and Y12 systems are rather similar to this result,however, the wave vector changes to Q theorystriped = ± ( π, − π √ ), when the direction of FM stripes areswitched. Our numerical calculations extend onlyto J = 0 .
5. However we expect that there willbe some additional geometry-dependent magneti-cally ordered phases for larger J before reachingthe large J limit (see below).5. 0 . ≤ J ≤ . no magneticorder. In Fig. 7, the presented correlation functionand SSF appear to be reminiscent of a columnar-like ordering, but there are subtle differences. Thesize of spheres, representing the two-point corre-lation function, Fig. 7(a), decays faster than thecolumnar phase. In addition, the size of SSF peaks,Fig. 7(b), are considerably smaller than the typicalsize of the Bragg peaks in the columnar order hav-ing the same system width. In Sec. VII and VIII,below, we show that this phase has no signatures ofmagnetic ordering, which indicates that there areno broken symmetries and hence some kind of al-gebraic spin liquid.6. J → + ∞ : Following the arguments presentedfor the J → −∞ case, in the limit of | J | (cid:29) J . For J → + ∞ , the ground-state on each new sublattice is the same as theoverall groundstate for J = 0, i.e., the 120 ◦ or-der. However for few-leg ladder systems, othersymmetry broken phases could appear due to therestricted geometry. As an example, for three-leg finite cylinders in J → ∞ , we found thatthe groundstate is three weakly-coupled copies ofa NNN Majumdar-Ghosh state. Interestingly, wefound a similar dual Majumdar-Ghosh phase forfour-leg finite cylinders ). Consistent with the ex-pected 2D limit, we did not observe any signatureof such Majumdar-Ghosh-type phases for L y > confirms that the “order from disorder” mechanismwould choose three-fold degenerate and decoupledstates for J (cid:29)
1, which are energetically favorableto arrange according to 120 ◦ ordering. Hence, weexpect that such exotic ordered phases are partic-ular features of narrow cylinders.To get a better quantitative insight on the phase di-agram of the THM, we study the short-range (NN andNNN) spin-spin correlations, (cid:104) S i · S j (cid:105) , Fig. 8. Short-distance correlators in a crystalline phase have a repeti-tive pattern reflecting the bulk properties of the ground-state. In Fig. 8, we choose six reference bonds, in-cluding three NN and three NNN correlators, to buildup a picture of the real-space correlations for differ-ent system widths. In the 120 ◦ phase region, cor-relators are very nearly isotropic, where NN (NNN)bonds are all AFM (FM). On the other hand, topolog-ical spin liquids on finite-width systems contain stronganisotropies , which is clearly seen in Fig. 8. As weshowed previously , in the thermodynamic limit anyonicsectors ˆb and ˆf are anisotropic on finite cylinders, whileˆi and ˆv are isotropic . The behavior of the correla- -0.4-0.3-0.2-0.100.10.2 A v e r a g e d S ho r t -r a ng e C o rr e l a ti on s YC10, columnar, NNYC10, columnar, NNNYC10, SL-i, NNYC10, SL-i, NNNYC10, SL-b, NNYC10, SL-b, NNNYC8, SL-f, NNYC8, SL-f, NNNYC8, SL-i, NNYC8, SL-i, NNNYC8, columnar, NNYC8, columnar, NNNYC6, columnar, NNYC6, columnar, NNNYC6, algebraic SL, NNYC6, algebraic SL, NNNYC6, 120° order, NNYC6, 120° order, NNNYC6, SL-v, NNYC6, SL-v, NNNYC6, SL-b, NNYC6, SL-b, NNN
FIG. 8. (Color online) Short-range correlation functions forthe iDMRG groundstates of the THM on YC6, YC8, andYC10 structures versus NNN coupling strength, J . Each cor-relator value is averaged over a wavefunction unit cell, thenextrapolated linearly with iDMRG truncation errors towardthe thermodynamic limit of m → ∞ . For each J , red sym-bols represent NN bonds in principal Y , a +60 ◦ , and a − ◦ directions. Similarly, blue symbols represent NNN bonds innon-principal directions of √ (1 , √ (2 , − √ (2 , tion functions is distinct in the columnar phase, wherethere are always two FM bonds (one is a NN and an-other one a NNN correlator) and four
AFM bonds (twoare NN and other two NNN correlators) out of the sixreference bonds. The FM stripes of the columnar or-der can, of course, choose either of a +60 ◦ or a − ◦ di-rections, so such data-points in this region are exchange-able. Furthermore, curiously for YC6, in the ASL phaseregion (0 . ≤ J ≤ . a ± ◦ -direction bonds. IV. DIRECT MEASUREMENT OF THE ORDERPARAMETERS ON FINITE-LENGTHCYLINDERS
To provide a verification of the phase boundaries forcomparison against our iDMRG results, we calculatedtwo magnetization order parameters on L y ≤ . Considerthe arbitrary magnetization vector order parameter of M ( m ) for a wavefunction with m number of states (thepreservation of the SU(2)-symmetry causes the structuralvanishing of all projection components). Upon a suitable1 O / O c l a ss i ca l ( L → ∞ ) FM sublattice magnetization, Eq. (19)(a 120° phase order parameter)Staggered magnetization, Eq. (20)(a columnar phase order parameter) J classical O (J =0) /O classical =%25(8) FIG. 9. (Color online) fDMRG results for the magnetiza-tion order parameters of the THM, O FM A , Eq. (19), and O stag ,Eq. (20), in the thermodynamic limit of L → ∞ . Each data-point represents a separate extrapolation (and the resultingerror) with the method of the fixed-aspect-ratio, Eq. (18). Avariety of cylindrical structures have been used for extrap-olation purposes, as listed in Sec. II. Brown (outer) stripesare predicted phase boundaries, while the middle stripe is theclassical phase transition at J = 0 . . choice of the system size and careful extrapolation to-ward the thermodynamic limit, non-zero values for the second moment of M (which is directly proportional tothe spin susceptibility) can be derived. In White andChernyshev’s method, one first extrapolates the order pa-rameter linearly with the DMRG truncation errors, ε m ,toward the thermodynamic limit of m → ∞ ( ε m → (cid:104) M ( ∞ ) (cid:105) . Then, using only fixed aspect-ratio ( L y L x = Const.) system sizes, L x and L y should besimultaneously extrapolated toward the thermodynamiclimit of L → ∞ . By employing a similar approach, plussome simple dimensional analyses and numerical exami-nation of the magnetic moments, we suggest in the MPSconstructions of the SU (2) S = 0-sector groundstates onfixed aspect-ratio cylinders ( L x > L y ), the normalizedorder parameter, M ( ∞ ) per site, scales as (cid:104) ¯ M ( ∞ ) (cid:105) = ¯ a + ¯ a L − x + ... , (18)where eclipses represents higher order terms in L x (notethat Eq. (18) is only a heuristic fit; see 99 for theoreti-cal predictions). One should note that any independent growth of L x and L y toward the L → ∞ limit, can beinterpreted as the existence of an infinitely long cylinderat some stage. This will collapse the system, essentially,to an inherently 1D state, for which the behavior of themagnetic moments is essentially different (see below).The magnetic order parameters that we selected tostudy the phase diagram on finite-size cylinders include:the FM sublattice magnetization, defined arbitrarily on sublattice A , O FM A = 2 (cid:112) N A ( N A + 2) (cid:113) (cid:104) S A (cid:105) , (19)where S A = (cid:80) i ∈ A S i is summed over all sites in sub-lattice A , and 2 / (cid:112) N A ( N A + 2) is a normalization factor( N A is the total number of sublattice- A spins on the fi-nite lattice). O FM A is a well-defined order parameter forthe 120 ◦ phase. The classical ◦ order will result inthe maximum possible value for the order parameter inthe limit of L → ∞ , i.e. O FM A [classical , L → ∞ ] = 1 .The next order parameter is the staggered magnetization, M stag , for which the second moment is a well-defined or-der parameter for the columnar phase, O stag = 1 L (cid:113) (cid:104) S (cid:105) (20)where S stag = S A − S B is the staggered magnetizationfor sublattices A and B . The classical columnar orderwill result in the maximum possible value for the orderparameter in the limit of L → ∞ , i.e. O stag [classical , L →∞ ] = 1 .Our results for O FM and O stag , in the thermodynamiclimit of L → ∞ , are presented in Fig. 9. Individual error-bars are relatively large, but the overall behavior of themagnetization curves follow the expected pattern: thereexists a small region for J , where both O FM ( L → ∞ )and O stag ( L → ∞ ) are touching the zero axis (consider-ing uncertainties), which provides SL region boundaries, J low2 = 0 . J high2 = 0 . O stag ( L → ∞ ) is touch-ing the zero axis, while O FM ( L → ∞ ) are increasing for J → −∞ (confirming the stabilization of 120 ◦ order inthis region). On the other hand, next to the SL phaseregion on the right, O stag ( L → ∞ ) increases rapidly,indicating columnar order. Interestingly, the value of O FM ( L → ∞ ) ( O stag ( L → ∞ )) is increasing (decreasing)again for large J . This is consistent with the existenceof a multi-component 120 ◦ order (three copies of a con-ventional 120 ◦ order placed on sublattices; see Sec. III)in the J → ∞ limit.It is worth noting the magnitude of the sublatticemagnetization at J = 0 (NN model). Measurementof variants of a 120 ◦ order parameter for the NNmodel has been in the center of attention to understand the degree of magnetization reduction(in comparison to their classical counterparts) in sucha frustrated model. As shown in Fig. 9, we predict O FM [ J = 0] /O FM [classical] = 25(8)%, which is con-siderably smaller than approximate results of 50% bySWT , 48% by ED , 40% by CCM , and 50% by vari-ational QMC . V. CORRELATION LENGTHS
For infinite cylinders, the gapped or gapless nature ofthe groundstate can be understood through the study2of the (principal) correlation length, ξ , since the be-havior of the magnetic ordering and the scaling be-havior of the static correlation functions are connected(cf. Sec. I). Indeed, the Hastings-Oshikawa-Lieb-Schultz-Mattis theorem relates the size of the energy gap,∆ e , to ξ , for local, translation-invariant Hamiltonians oneven-width cylinders as ξ ≤ Const . ∆ e (i.e. ξ − serves as anupper boundary for the gap size). For the (inherentlyone-dimensional) MPS ansatz, the connection betweenentanglement scaling and the correlation length is well-understood . In a critical phase, the correlationlength diverges with a signature power-law scaling withthe number of states as ξ ( m ) = ˜ κ c m ˜ κ . Furthermore, insuch states, the entanglement entropy diverges with ascaling of S EE ∼ log ξ . On the other hand, for short-range gapped states ξ saturates to a finite value as m isincreased, which in the topological spin-liquid state of theTHM is short; of the order of a few lattice spacings. In-terestingly, as we see below, the correlation length scalingfor magnetic ordering in SU (2)-symmetric MPS on infi-nite cylinders appears rather differently than the full 2Dlimit. Such cylindrical magnets exhibit some signaturesof true LRO (e.g. in the ES – see below), however, dueto the explicit preservation of SU(2) and the dominating1D physics of the MPS ansatz, the groundstates emergeas quasi-LRO critical states (note that the correlationlength can still diverge with respect to the cylinder cir-cumference). Nevertheless, in the iMPS representation ofthe wavefunction, the correlation lengths (per unit-cell)can be conveniently read from the eigenspectrum of thetransfer operator, T I (cf. Sec. II): ξ i ( m ) L u = − | η i ( m ) | , i = 2 , , , ... , (21)where η i are eigenvalues of T I (arranged as {| η | > | η | > | η | > ... } ). η i depends on the number of states, andare also labeled by an SU (2) spin sector, which is thesymmetry sector of the (block diagonal) transfer opera-tor, and corresponds to the symmetry of the associatedcorrelation function. We have discarded i = 1, as thelargest eigenvalue of T I in an orthonormalized basis al-ways corresponds to η = 1 (belonging to the identityeigenmatrix) and the principal correlation length is thesecond largest eigenvalue, ξ ,S ≡ ξ S . For a phase withmagnetic ordering, such as 120 ◦ and columnar order, theprincipal correlation length is expected to belong to the S = 1 sector, indicating that the slowest decaying cor-relations are in the spin-spin form. For the topologicaland algebraic spin liquid phases, we find that the prin-cipal correlation length is in the S = 0 sector, indicat-ing that the slowest decaying correlation is some kindof singlet-singlet correlator (we have not determined theexact form). An undesirable effect of the variational con-vergence of the groundstate using the iDMRG approachemerges from the constraint of SU (2) symmetry, wherebyspurious symmetry effects make the wavefunction non-injective (the spectrum of T I contains multiple identityeigenvalues in each S -sector). We have removed such
100 1000m110100 ξ YC6, J =0.185, algebraic SL, S=0YC6, J =-1.0, 120° order, S=1YC9, J =-1.0, 120° order, S=1YC12, J =-1.0, 120° order, S=1YC6, J =0.5, striped order, S=1YC8, J =0.5, striped order, S=1YC10, J =0.5, striped order, S=1YC12, J =0.5, striped order, S=1YC6, J =0.125, SL b-sector, S=0YC8, J =0.125, SL f-sector, S=0YC10, J =0.125, SL i-sector, S=0YC10, J =0.125, SL b-sector, S=0 FIG. 10. (Color online) iDMRG results for the principal corre-lation lengths (per unit-cell size) versus the number of states, m , in a variety of the detected phases and system sizes of theTHM on infinite cylinders. Results are labeled with SU(2)quantum numbers, S . Lines are attempted power-law fits, ξ = ˜ κ c m ˜ κ , to quasi-LROs ensuring the existence of a crit-ical phase (see also Sec. I). Green-symbol data are selectedfrom 93 to provide comparison between the magnitudes andasymptotic behaviors of ξ S ( m ) in gapped and gapless phases. wavefunctions everywhere except in the immediate vicin-ity of the J = 0 . ξ S from the topological spin liquid . We immedi-ately notice that the principal correlation length belongsto the S = 1 sector for the magnetic groundstates with120 ◦ and columnar ordering, however, it switches to the S = 0 sector for all SL states, whether they are quasi-LROs (as in ASLs) or short-range correlated (as in topo-logical spin liquids). We can see that both the ASL andthe magnetically ordered states have power-law behav-ior, reflecting their gapless and quantum critical natures.We emphasize that in the case of the ASL, this behav-ior appears to be intrinsic; however for the magneticallyordered phases the power-law correlations are a conse-quence of preserving SU (2) symmetry. In contrast, forthe topological spin liquid, the correlation length is con-siderably smaller in the size (order of few lattice spacings)and qualitatively begins to saturate in the large- m limit,although it is surprisingly difficult to do a rigorous fit.3 VI. ENTANGLEMENT ENTROPY OFQUASI-LRO MAGNETS
The entanglement entropy is a central quantity in thephysics of the many-body systems, which provides a mea-sure of how strongly conjunct subsystems are entangled.The entropy has proven to be a powerful numerical toolfor characterizing the low-energy spectrum, detection ofSSB, and topological degeneracy of the groundstate (forsome examples, see 60, 102–106). Between many dif-ferent approaches to measure entropy, we employ themethod of Jiang et al. that calculates the von Neu-mann entropy along a bipartition cut of the cylinder,as shown in Fig. 1, since it is computationally conve-nient to manipulate in the context of MPS and DMRGalgorithms. The bipartite von Neumann entropy is de-fined as S EE = − T r (˜ ρ log ˜ ρ ). In terms of the eigen-values of ˜ ρ , i.e. { λ i } , the entropy can be written as S EE = − (cid:80) i λ i log λ i . Roughly speaking, S EE countsthe number of entangled pairs on the bipartite boundary. S EE is a function of the ( D − area of the D -dimensional quantum system, i.e. the boundary size, L cut (note that L cut = L y for the YC structure). In fact,robust theoretical studies proved that for in-teracting 2D spin systems with only local couplings anda cut size significantly larger than the correlation length,the leading term in the entropy scales with the boundaryarea, S EE ∝ L cut , not the system volume, which is knownas the area-law (the area-law was originally introducedin the context of the black holes and quantum fieldtheory ). However, for strictly 1D quantum criticalstates (in the thermodynamic limit) the condition of theboundary size being considerably larger than the corre-lation length cannot be met, and the S EE behavior ismodified. In this case, the leading term in the entropyrelates to the only length scale of the system, i.e. the cor-relation length, as S EE ∼ log( L eff ) ∼ log( ξ ) , where L eff stands for the effective size of the system. For thesymmetry-broken true LROs, again, the size of the cut issignificantly smaller than the diverging correlation lengthand a logarithmic term should be added to the area-lawbehavior : S EE = β + β L cut + N G L ) , (22)where β corresponds to a non-universal constant, whichdepends on the system geometry, the topological entan-glement entropy , spin stiffness, and the number, N G , and the velocity of the Nambu-Goldstone excita-tions. In addition, β is another non-universal constant,which depends on the short-range entanglement in thevicinity of the cut and a short-distance characteristiccutoff. For the (quasi-)LRO, SU (2)-symmetric, iMPSgroundstates on the infinite cylinders, we find that theentropy scaling behavior is distinct. As discussed inSec. V, MPS-ansatz symmetry broken magnets appearas quantum critical states on the cylinder. Thus, it isexpected that the entropy exhibits a combination of the ξ )22.533.54 S EE "a (L y ) + a (L y )log( ξ )" fitsYC6, J =0.5, columnar orderYC8, J =0.5, columnar orderYC10, J =0.5, columnar orderYC12, J =0.5, columnar order y a ( L y ) y a ( L y ) critical-statebehaviorarea-lawbehavior sub-leadingcorrections FIG. 11. (Color online) iDMRG results for the entanglemententropy of the columnar order of the THM at J = 0 . S = 1 correla-tion lengths, ξ , and red lines are attempted fits accordingto Eq. (23), which is the predicted behavior for the quasi-long-range critical states on infinite cylinders. The scalingbehaviors of a and a of Eq. (23) versus the system widthare presented in the insets. area-law and the critical behaviors. Our numerical mea-surements on an SU (2)-symmetric, quasi-LRO ground-state of the J - J THM on the infinite YC structuresconfirms such a mixed scaling as of S EE (cid:39) a ( L y ) + a ( L y ) log( ξ ) , (23)where a ( L y ) = α + α L y . (24)The behavior of the non-universal constant of a provedto be more challenging to predict, but it can only containsub-leading corrections to the area-law term appearing in a (see below).In Fig. 11, we present our entropy measurements forthe groundstates of the THM deep in the columnar phaseregion. Due to exponential cost of the calculations withthe system width we only obtained a few wavefunctionsfor different L y in the columnar phase. However the re-sults shown in Fig. 11 confirm the prediction of Eq. (23)and Eq. (24). In the figure, we first fit a line to theoriginal entropy data and calculate a and a for eachsystem size. Clearly, a -values are consistent with thearea-law behavior. We measured the coefficients of a as α = − . α = 0 . a -values, but their saturat-ing nature for the large- L y limit is consistent with thisterm being a sub-leading correction to the mixed termcontaining the area-law behavior.4 VII. NUMERICAL TOOLS I: CUMULANTSAND BINDER RATIOS OF THEMAGNETIZATION ORDER PARAMETERS
In Sec. II, we constructed the theoretical frameworkfor a method to measure the non-local moments and cu-mulants of the magnetic order parameters, in the contextof SU (2)-symmetric translation-invariant MPS, where allprojection components of the magnetic order parameter, M [ k ] , vanish by construction. In this case, the highermoments can play the role of the order parameter. It isconvenient to connect the moments of the operators tothe i th cumulant per site , κ i , by employing (cid:104) M n (cid:105) = n (cid:88) i =1 B n,i ( κ L, κ L, ..., κ n − i +1 L ) , (25)where B n,i are partial Bell polynomials and L nowstands for the operator length. For some examples, weexpand Eq. (25) to write the relations for the first fewcumulants, (cid:104) M (cid:105) = κ L , (cid:104) M (cid:105) = κ L + κ L , (cid:104) M (cid:105) = κ L + 3 κ κ L + κ L , (cid:104) M (cid:105) = κ L + (4 κ κ + 3 κ ) L + 6 κ κ L + κ L . (26)The cumulants per site are obtained directly as theasymptotic large L limit obtained from the summation ofthe tensor diagrams presented in Sec. II. For the iMPSansatz, when the asymptotic limit is taken to derive atranslation-invariant infinite-size system, one should re-place the operator length with the effective system sizeas L → L eff ∝ ξ (see also Sec. VI). Below, we intro-duced the magnetic order parameters that are used tomeasure the cumulants and characterize the LROs of theTHM. We first construct the MPO forms of the highermoments of a staggered magnetization (the order param-eter for columnar order on cylinders with FM stripes in a +60 ◦ -direction), M stag = L y (cid:88) i =1 ( − i S i , (27)and a tripartite magnetization (the order parameter forthe 120 ◦ phase), M tri = L y (cid:88) i(cid:15) ( S A i + e i π S B i + e − i π S C i ) , (28)on a L y -size unit cell. Numerical computation of themoments of such order parameters is a challenging taskdue to relatively large dimensions of the resulting MPOs.Nevertheless, we succeeded to calculate the second cumu-lant, κ , and the fourth cumulants, κ , of M stag and M tri (the odd moments vanish due to the SU (2) symmetry)for a range of the groundstates. We suggest that the mostuseful choice of cumulants is κ , which is connected to the excess kurtosis , γ , of the block distribution functionassociated with the operator M [ k ] : γ = κ κ L . (29)We emphasize that the above equation is only valid forthe κ = κ = 0 case. The importance of the fourth cu-mulant was revealed by some studies on fourth magneticmoment behavior of 2D Ising antiferromagnets ,which established κ as an effective tool for pinpointingquantum critical points. In these studies, the scaling be-havior of the fourth magnetic moment is observed to varysignificantly at an Ising transition (more precisely, κ changes sign at the critical point, and changes by manyorders of magnitude nearby the critical point). Anotherrelevant and interesting (dimensionless) quantity is theBinder cumulant , U L = n H +22 (1 − n H n H +2 (cid:104) M (cid:105)(cid:104) M (cid:105) ),where n H is the number of projection spin operators usedto construct the order parameter (e.g. n H = 3 for a vec-tor magnetization). In the vicinity of a critical point,the Binder cumulant becomes independent of the systemsize (lower moments of the order parameter cancel outhigher-order finite-size effects) and can be used to pin-point the transition. Previously, we adopted U L of a(scalar) dimer order parameter to locate a critical pointin the phase diagram of the THM on three-leg cylinders.However, until now, the scaling behavior of U L was less-known for the cases where the order parameter itself isstrictly zero. In the limit of L → ∞ , as it is clear fromEq. (26), the higher-order corrections in (cid:104) M n (cid:105) vanish andthe conventional method of Binder cumulants for locat-ing the phase transitions becomes ineffective. However,the correlation length, ξ , gives us a natural length scaleand a rather precise process to scale a Binder-cumulant-type quantity in the vicinity of a critical point. As inthe case of the entropy, Sec. VI, the key to obtaining thecorrect scaling of the magnetic moments of iMPS wave-functions is to choose L eff = ˜ sξ , where ˜ s is any fixed scaling constant. For Binder cumulant, ˜ s has no qual-itative effect except to change the value of the criticalbinder cumulant, similar to the role of boundary condi-tions for the finite-size Binder cumulant. Therefore, onecan freely choose ˜ s to obtain the most numerically sta-ble fit. When the order parameter is zero by symmetry,so that κ = κ = 0, the appearance of such a con-stant is irrelevant and only the ratio of the second andfourth cumulants plays a role. By replacing the explicitrelations for (cid:104) M (cid:105) and (cid:104) M (cid:105) from Eq. (25) into U L , wepropose the ratio (which we call the “Binder ratio” – seealso Eq. (29)): U r = κ κ ξ . (30)We find that numerically this combination of the mo-ments and the correlation length removes much of the5 κ ( m → ∞ ) YC6, 120° order, M tri
YC6, striped order, M stag
YC8, striped order, M stag
YC10, striped order, M stag (a) - κ ( m → ∞ ) YC6, 120° order, M tri
YC6, striped order, M stag
YC8, striped order, M stag
YC10, striped order, M stag (b)
FIG. 12. (Color online) iDMRG results for the extrapolated(a) second cumulants and (b) fourth cumulants of the mag-netization order parameters, Eq. (27) and Eq. (28), at thethermodynamic limit of m → ∞ , on a variety of phase re-gions and system widths of the THM. Each colored data-pointrepresents a κ ( m → ∞ )-value (- κ ( m → ∞ )-value), whichis the result of an extrapolation according to a power-law fit κ = ˘ a + ˘ a e − ˘ a m for m → ∞ (see 93 for some exampleson the individual extrapolations). In part (a), brown stripesare fDMRG results for the phase transition obtained from di-rect measurements of the local magnetization, Sec. IV. Solidcircles, in part (a), and dashed-lines, in part (b), mark theborders beyond which an extrapolation was not possible dueto the magnetic disorder. numerical noise that appears in the individual moments.We present the extrapolated results of κ and | κ | for M stag and M tri , in the limit of m → ∞ , in Fig. 12. Inthe figures, each data-point is the result of a separate ex-trapolation of the cumulants versus m . Upon careful nu-merical examination of the scaling behaviors of numerousgroundstates in the various phases, we were able to estab-lish the scaling relation of | κ n | = ˘ a + ˘ a e − ˘ a m , n = 2 , , for ordered phase regions and make sense of the cumulant results in the m → ∞ limit. These results show that κ is comparatively large and negative when there is quasi-long-range magnetic ordering. Moreover, κ is large andpositive for quasi-LROs (see 120 ◦ and columnar phase re-gions in Fig. 12). This is in contrast to the behavior nearphase transitions, and within the topological and alge-braic spin liquids, where we were not able to find an ap-propriate analytical fit for the cumulants in the m → ∞ limit, as they behave irregularly or quickly decay to nu-merically vanishing values. A likely reason for this is thatfor a magnetically-ordered, SU (2) S = 0 groundstate,the moments M [ k ] acquire a set of equally-weighted non-zero values from the limited number of recovered (purely)TOS levels by iDMRG (see below). In such a case, thedistribution function would resemble a discrete uniformdistribution with very large and negative κ , and largeand positive κ . However, for disordered states with nosymmetry breaking in the thermodynamic limit, the dis-tribution function is expected to resemble the normal dis-tribution centered around zero magnetization, which hasvanishing κ . For κ ( m → ∞ ), in Fig. 12(a), we displayin bold the boundaries where we were not able to extrap-olate to m → ∞ . These are quite close to the phase tran-sitions indicated by fDMRG, Fig. 9 (except for the YC6structure, where we find an additional ASL phase), whichsupports the validity of the iDMRG cumulant method.The same behavior was observed for κ ( m → ∞ ), indi-cated by the black dashed-lines in Fig. 12(b). In addition,the extremely large (negative) values of κ ( m → ∞ ) areconsistent with our interpretation.Our attempts to pinpoint the phase transitions of theTHM on infinite cylinders, using U r , is presented inFig. 13 and Fig. 14. Based on these results, we argue that U r ( m ), as the ratio between κ ( m ) and κ ( m ), scaledwith ξ ( m ), regularly extrapolates to a finite value in the m → ∞ limit everywhere, except close to (or on) a phasetransition, or when the wavefunctions are non-injective(cf. Sec. V). Careful numerical examination suggests thatthe Binder ratios scale with a saturating behavior similarto the cumulants, U r = ˘ b + ˘ b e − ˘ b m , (31)In Fig. 13, we observe that in the topological SL phaseregion, U r ( m ) has a comparatively small value, as ex-pected for nonmagnetic phases with γ →
0. In addition,when there is magnetic ordering, U r ( m ) converges to afinite, negative value, while it appears different m -curvestend to group together. The latter should be due to thefact that the iMPS magnetic orders are quantum criticalstates with an U r independent from L eff . Furthermore,for the ASL phase of YC6 structures (see Fig. 14(a)),very close to the expected phase transition points fromthe short-range correlation data, Fig. 8, and within theentire ASL phase region, U r diverges with m (e.g. seethe inset of Fig. 14(a)), where it is impossible to extrap-olate to a finite U r ( ∞ ). In the immediate vicinity ofthe transition from the 120 ◦ to topological spin liquid(cf. Fig. 14(b)), it was not possible to employ Eq. (31)6 -40-30-20-100 B i nd e r R a ti o s o f M s t a g m=1200m=1400m=1600m=1800m →∞ columnar orderSL b-sector (a) -20-10010 B i nd e r R a ti o s o f M s t a g m=1200m=1400m=1600m=1800m →∞ columnar orderSL i-sector (b) FIG. 13. (Color online) iDMRG results for the Binder ratios, U r ( m ), Eq. (30), of M stag , Eq. (27), in the vicinity of thetopological SL and the columnar phase regions of the THMon (a) YC10 and (b) YC8 structures. Black diamonds denote U r ( m → ∞ ), i.e. extrapolations of Binder ratios according toEq. (31) to the m → ∞ limit. Brown stripes are the bestestimate for the phase transition based on the discontinuityof the dashed line, U r ( m → ∞ ), for the larger system size,part (a). due to unavoidable non-injectivity of the wavefunctions.However, we suggest that the fixed- m results are ratherreliable and can be used to estimate a phase transition.Overall, we locate critical points of the THM from thediscontinuities of U r ( m → ∞ )-lines (i.e. where there is noextrapolation possible) or when there is a significant kinkin fixed- m data. Based on this approach, we estimatethe phase transition points of J = 0 . ◦ and topological spin liquid states using YC6 resultsof Fig. 14(b), J = 0 . -20-10010203040 B i nd e r R a ti o s o f M s t a g m=800m=900m=1000m=1200m →∞
500 1000 1500 2000 m02040
SL v-sector stripedorder algebraicSL stripedorder (a) -5-4-3-2-101 B i nd e r R a ti o s o f M t r i m=800m=900m=1000m=1200 SL v-sector120° order (b)
FIG. 14. (Color online) iDMRG results for the Binder ra-tios, U r ( m ), Eq. (30), of (a) M stag , Eq. (27), and (b) M tri ,Eq. (28), of the THM on YC6 systems. In part (a), blackdiamonds denote U r ( m → ∞ ), i.e. the the extrapolation ofthe of Binder ratio according to Eq. (31) toward the m → ∞ limit. Furthermore, the inset shows the individual U r ( M stag )at J = 0 .
2. Brown stripes are the best estimate for the phasetransition based on the discontinuities or rapid changes in U r . J = 0 . , . , . U r accuracy in estimating the transitionsin case of YC6 structures, we also provide a numericalapproximation for the fidelity susceptibility , χ approx F = 1 − |(cid:104) ψ ( J ) | ψ ( J + δJ ) (cid:105)| δJ , (32)in Fig. 15, where we set δJ = 0 .
05. The fidelity suscep-tibility is known to be well-behaved and small when awayfrom a phase boundary, but can diverge at a transition. Itis clear from the figure that the diverging peaks of χ approx F (considering their tendency to lean toward the right) are7 F i d e lit y S u s ce p ti b ilit y m=800m=900m=1000m=1200 stripedorder algebraicSL striped order FIG. 15. (Color online) iDMRG results for the fidelity sus-ceptibility, χ approx F , Eq. (32), of the THM on YC6 systems.Brown stripes are the predicted phase transitions based onFig. 14(a) results. happening relatively close to the predicted phase transi-tions from the Binder ratio results of Fig. 14(a). VIII. NUMERICAL TOOLS II: ‘TOS COLUMNS’IN THE MOMENTUM-RESOLVEDENTANGLEMENT SPECTRUM
The entanglement between the partitions of a quan-tum system is encoded in the spectrum of the entan-glement Hamiltonian, H E = − log(˜ ρ ), i.e. {− log( λ i ) } ,which is known as the ES and commonly presented us-ing energy-level arrangements analogous to an energyspectrum. {− log( λ i ) } can be labeled using any global-symmetry quantum number to extract more informationon the symmetry nature of the state (as long as the cor-responding symmetry is preserved on the bipartite cut). H E maintains the symmetries of a cylindrical wavefunc-tion, however, they may exist some symmetries that are not explicitly preserved by the ansatz due to the map-ping of the 2D model onto an MPS chain. Nevertheless,one can still diagonalize such a symmetry operator in the‘auxiliary’ basis (i.e. the basis that diagonalizes H E ) tocreate a new set of good quantum numbers (see 63, 64, 85for some examples). When the SU (2)-symmetry is pre-served in the calculation, the obvious choice for the la-bels is the spin S quantum number (belonging to a sin-gle partition of the system). We refer to an H E spec-trum that is plotted against S (where no other labelexists) as the spin-resolved ES. Kolley et al. showedthat the spin-resolved ES of a magnetically ordered stateon finite-length cylinders shows signatures of symmetry-breaking at the thermodynamic limit. This emerges froma key finding: the realization that the low-energy part of the ES of magnetic orders exhibits a specific type of grouped levels, known as the entanglement-spectrumTOS (also referred to as “quasi-degenerate joint states”),closely resembling the low-lying levels in the energy spec-trum, known as the Anderson TOS levels (alsoreferred to as the “Pisa tower” structure or the “thinspectrum”), which is considered as clear-cut evidence forthe existence of true LROs on finite lattices. Kolley etal. established that, similar to the energy spectrum, fora fixed S -sector, entanglement-spectrum TOS levels arewell-separated from the denser rest of the spectrum andthe lowest energy levels of the ES, immediately abovethe TOS levels, are spin-wave states (Nambu-Goldstonemodes). In this paper, we are interested in exploitingboth the S quantum numbers ( SU (2) is explicitly pre-served in the iDMRG calculations), and the momentain the cylinder Y -direction, k , i.e. the complex phase ofthe eigenvalues of the reduced T y operator, where T y isthe translation by one site in Y -direction; we can de-compose the operator in the same way as the Schmidtdecomposition of the wavefunction , T y = T Ly ⊗ T Ry ,where T Ly and T Ry are the reduced operators and main-tain the unitary property of the original operator. T y is not preserved exactly in the calculations due to the MPSmapping on the cylinder, Fig. 1, but it can be diagonal-ized straightforwardly . We refer to an H E spectrumthat is plotted against k and additionally labeled by S ,as the momentum-resolved ES, {− log (cid:0) λ n [ k n , S n ] (cid:1) } . Fora system with PBC in Y -direction, dihedral symmetryimplies that T L y y = I . As a result, the allowed momen-tum spacing is as ∆ k n = πnL y for n = 0 , , ..., L y − k , the momentum of the lowest ES level,is not fixed due to the possibility of inserting a shift inthe expectation value of T y (one needs to first fix k ,then measure the rest of the momenta in respect to it;physically only ∆ k n matters here – see also 92 and 93).The study of momentum-resolved forms of the ES is nowfinding a place in the literature of the low-dimensionalquantum magnets. Another key breakthrough was therealization of that such ES can be used to fully clas-sify anyonic sectors of chiral and Z -gauge topolog-ical orders on infinite cylinders . Below, we argue thatthe symmetry-breaking can be recognized and character-ized using the momentum-resolved ES, which shows thesymmetry properties even more robustly than the spin-resolved ES.Upon careful examination of the momentum-resolvedES of the magnetic orders in the THM on infinite cylin-ders and noticing the underlying symmetries of the sub-lattices, we find that the spectrum contains exactly N s (number of the groundstate sublattices) column-likestructures, which are the low-lying component TOS lev-els, independent of the system width. We shall refer tothese particular patterns as ‘TOS columns’. The appear-ance of TOS columns is due to that, as previously dis-cussed, the TOS levels are clear features in the low-lyingES. These columns also have a momentum structure.Consider an ideal magnetic order that consists of N s fullyFM sublattices, represented as { ˜S , ˜S , ..., ˜S N s } ( L y = 08mod N s ) in a big-S notation of the spins. The SU (2)-symmetric groundstate is, of course, the S total = 0-singlet,constructed by adding all spins, || ˜S , ˜S , ..., ˜S N s ; 0 (cid:105) in areduced dimension basis notation (see for example 121).Importantly, this is the true groundstate of the effectiveHamiltonian of H eff ∝ √ L S describing purely theTOS levels . The only non-trivial sets of unitary sym-metry operations that are allowed to act on the S total = 0-singlet and leave a Heisenberg-type Hamiltonian betweenthe sublattices unchanged (sublattices should be still ar-ranged on the physical lattice), can be written as thecyclic translations of sublattices, T ν , where ν is the num-ber of sublattices that will be shifted (for example to theright). One can then write T ν = N s || S , S , S , ..., S N s ; 0 (cid:105) = T N s ν =1 || S , S , S , ..., S N s ; 0 (cid:105) = || S , S , S , ..., S N s ; 0 (cid:105) (33)There are obviously only N s distinct values that ν cantake, including the identity operator. Eq. (33) alreadyimplies that the TOS levels can only acquire lattice mo-menta of k TOS ν = πνN s for ν = 0 , , ..., N s −
1, betweenthe equal or greater group of general ES momenta, k n .The only complication emerges from the distribution pat-tern of n (cid:48) TOS-levels between N s momenta for a fixed S -sector. To clarify this, let us focus on the more gen-eral case of n (cid:48) > N s and choose the momentum of thelowest ES level to be k TOS0 [ S = 0] = 0, presumably, cor-responding to the action of I on the sublattices (chosendifferently in Fig. 17). Trivially, all other ( n (cid:48) − k TOS0 [ S = 0](there is no relative net momentum). So, they can ei-ther, altogether, fill the zero-momentum state on top of k TOS0 [ S = 0] or occupy ± k ν ( ν (cid:54) = 0) states around it. Theformer is not possible, due to the fact that T ν ( ν (cid:54) = 0)and I posses a distinct set of eigenvalues and thereforeproduce different momenta (this can be easily observedby writing the bipartite Schmidt decomposition of the S total = 0-singlet state and switch to the basis of fixed- S states for L or R partition to reveal distinct eigenspectraof T ν and I ). In addition, we notice that some states ap-pearing in a TOS column are not essentially TOS levels.This is partly due to the fact that the non-TOS levelsare also allowed to fill k TOS ν states, and partly becausein an MPS representation, there is always a fixed num-ber of states kept and consequently, only the first fewTOS levels of H eff will be recovered. Nevertheless, suchinitial states (having a clear gap to the higher levels)certainly follow the TOS level counting as governed bythe degree of symmetry-breaking in the thermodynamiclimit. I.e. for a state that fully breaks SU (2)-symmetry(e.g. the 120 ◦ order), there are N TOS S = (2 S + 1) levelsgrouped together, and for a state that partially breaksthe SU (2)-symmetry down to U (1) (e.g. the columnarorder), there is only N TOS S = 1 level per each fixed S -sector ( not counting the degeneracy that comes from the SU (2) quantum numbers themselves; the overall degener- -1 -0.5 0 0.5 102.557.51012.5 (a) -1 -0.5 0 0.5 102.557.51012.5 (b) -1 -0.5 0 0.5 1 k/ π - l og ( λ / λ ) S=0S=1S=2S=3S=4 (c)
FIG. 16. (Color online) iDMRG momentum-resolved ES ofthe 120 ◦ order, J = − .
0, for (a) YC6, (b) YC9, and(c) YC12 structures of the THM versus Y -direction mo-menta (the reference momentum is fixed to k TOS [ λ ] = 0).Boxes emphasize TOS columns at the unique momenta of k TOS ν = − π , , π . In part (c), dashed-lines are guides to theeyes and connect the Nambu-Goldstone modes of the ES forthe first few levels on the top of the TOS levels. acy of the ES levels is always (2 S + 1) N TOS S – see 63 and93 for more details). We discover another striking fea-ture in the momentum-resolved ES of symmetry-brokenphases, however, this time for the states between the TOScolumns: the first few Nambu-Goldstone modes exhibit sine-like dispersion patterns (as in the energy spectrum),if L y chosen to be large enough.In Fig. 16, we present the momentum-resolved ES ofthe 120 ◦ order on different width of the YC structure(for more visibility, we have limited the display of theES levels to S max = 4 in all ES figures of this sec-tion). The presence of three characterizing TOS columnsis clear for all system widths, consistent with the the-ory for a N s = 3-state. The low-lying levels inside theTOS columns (purely TOS levels) have a clear gap tothe higher levels, which qualitatively observed to con-verge to a finite value, linearly with L y , at the thermo-dynamic limit . The number of low-lying levels in theTOS columns agree with the full SU (2)-symmetry break-ing in the thermodynamic limit. That is N TOS S = (2 S +1)for all S = 0 , , , ,
4, as previously observed by Kolley et al. . For low-lying Nambu-Goldstone modes betweenthe TOS columns, we suggest the triangular-shape dis-persion patterns of Fig. 16(c) are signs for the formation9 -1 -0.5 0 0.5 102.557.51012.5 (a) -1 -0.5 0 0.5 102.557.51012.5 (b) -1 -0.5 0 0.5 1k/ π - l og ( λ / λ ) S=0S=1S=2S=3S=4 (c)
FIG. 17. (Color online) iDMRG momentum-resolved ES ofthe columnar order, J = 0 .
5, for (a) YC8, (b) YC10 and(c) YC12 structures of the THM versus Y -direction momenta(the reference momentum is fixed as k TOS [ λ ] = ± π ). Boxesemphasize TOS columns at the unique momenta of k TOS ν =0 , π . Dashed-lines are guides to the eyes and connect theNambu-Goldstone modes of the ES for the first few levels onthe top of the TOS levels. of sine-like structures, however, due to relatively smallsize of L y , the k n -resolution does not suffice to discernmore details.In Fig. 17, we present the momentum-resolved ES ofthe columnar order for different widths of the YC struc-ture. The presence of two characterizing TOS columns(note that k TOS [ λ ] = ± π -columns are the same) is clearfor all system widths, as predicted by the theory for a N s = 2-state. As before, the low-lying levels inside theTOS columns have a clear gap to the higher levels andobserved to converge to a finite value, linearly with L y ,at the thermodynamic limit . The partial breaking of SU (2) to U (1) symmetry can be confirmed by the levelcounting of N TOS S = 1 for low-lying S = 0 , , , , L y = 12 system, Fig. 17(c).In Fig. 18, we present the momentum-resolved ES ofan ASL state on a L y = 6 cylinder. Clearly, there is nosignature for the presence of TOS columns, which sug-gests the nonmagnetic nature of the phase. In addition,we observe no non-trivial degeneracy of low-lying ES lev-els. So, there exist no fractionalization of symmetries toidentify SPT and/or some intrinsic topological ordering -1 -0.5 0 0.5 1k/ π - l og ( λ / λ ) S=0S=1S=2S=3S=4No TOS columns
FIG. 18. (Color online) iDMRG momentum-resolved ES ofan ASL state, J = 0 . Y -direction momenta (the reference momentum is fixedas k [ λ ] = ± π ). with anyonic excitations (see also 85). IX. TIME-REVERSALSYMMETRY-BREAKING AND THEROBUSTNESS OF THE TOPOLOGICAL PHASEAGAINST THE CHIRALITY
The existence of the time-reversal symmetry is a keyfeature of H J , Eq. (1). A chiral groundstate sponta-neously breaks time-reversal, τ , and parity reflection, P ,symmetry, but respects the combined P τ -symmetry. Af-ter consistent numerical observations of a nonmagneticphase in the J - J THM phase diagram (cf. Sec. I andIII), the natural question is, whether the new state sta-bilizes due to SSB of τ , which would result in a CSL. Fora scenario in which the true groundstate in the SL phaseregion is Z topological ordered (advocated by DMRGresults ), we already investigated the chirality ofanyonic sectors in detail, using direct measurement of the τ -operator expectation values and calculating a scalarchiral order parameter, O χ = 1 L u (cid:88) (cid:104) i,j,k (cid:105) ( S i × S j ) · S k , (34)where (cid:104) i, j, k (cid:105) represent a NN triangular plaquette andthe sum goes over the wavefunction unit cell. We dis-covered that the topological sectors are all τ -symmetricas the O χ -values observed to be small and decreasingrapidly to numerically vanishing magnitudes at the ther-modynamic limit of m → ∞ (furthermore, ˆb and ˆf-sector are fractionalizing time-reversal symmetry). How-ever, Hu et al. determined the ˆi-sector groundstateas strongly prone to the chirality by adding directional( a ± ◦ -axis) anisotropy to the Hamiltonian. This is, in0 J χ S ca l a r C h i r a l O r d e r P a r a m e t e r Extrapolation results versus J χ Individual extrapolations versus ε m J χ critical =0.0014(1) FIG. 19. (Color online) iDMRG results for the scalar chiralorder parameter, O χ , Eq. (34), versus J χ for the groundstatesof H χ , Eq. (35), constructed from a YC8-ˆi sector. Each data-point represents a O χ [ m → ∞ , J χ ], which is the result of aseparate extrapolation on individual O χ versus ε m toward thethermodynamic limit of ε m → m → ∞ ). The red line isour attempted fit of ˜ b + ˜ b J ˜ b χ to the black circles, excludingthe first two J χ points (where the chirality is zero within theerror-bars), which is used to estimate the phase transitionwhen O χ [ m → ∞ , J critical χ ] = 0. A zoom-in plot is presentedin the inset, as a guide to the eyes. part, leading another question of our interest: is the SLphase robust against perturbing H J with a term thatexplicitly breaks the τ -symmetry and forms a chiral long-range order? To answer this question, one can study the J - J - J χ model, H χ = H J + J χ (cid:88) (cid:104) i,j,k (cid:105) ( S i × S j ) · S k , (35)where (cid:104) i, j, k (cid:105) indicates the sum over all NN triangularplaquettes in a Hamiltonian unit cell. The phase diagramof H χ is previously studied using variational QMC andED techniques, however, no clear result has emerged onthe nature of the J χ → to the chiral field by adiabaticallyadding a J χ -term to H J , as in Eq. (35), and findingnew groundstates using the SU (2)-symmetric iMPS andiDMRG methods.We present our results for the extrapolated O χ in thethermodynamic limit of m → ∞ in Fig. 19. We no-tice that, within our resolution, upon varying J χ , thereis at least one (significant) point exposed to nonzerochiral perturbations, but has negligible O χ ( m → ∞ )within the error-bars. This means that the topologicalSL phase is robust against chirality and one needs toprovide τ -symmetry-breaking terms larger than a finite-value, namely J critical χ , to impose a chiral groundstate.To further predict this small J critical χ , we applied a fit of ˜ b + ˜ b J ˜ b χ to the data and find J critical χ = ( − b b ) b =0 . second order phase transition toward the CSL phase.This is consistent with the suggestion from Wietek andL¨auchli , and may clarify the results of Hu et al. ,where it is unclear if O χ would be zero or not in the J χ → X. CONCLUSION
We have presented comprehensive results for the phasediagram of the J - J Heisenberg model on triangularlattices, using infinite-length YC structures. Using theBinder ratio of the magnetization order parameter, U r ,Eq. (31), and TOS columns of the momentum-resolvedES, we have obtained phase boundaries and character-ized the nature of the symmetry breaking magnetic or-der. We found that the Binder ratio reliably detectsphase boundaries between magnetically ordered states,even when using SU (2) symmetry, where the order pa-rameter itself is zero by construction. We identifiedthe 120 ◦ -ordered groundstate as a three-sublattice LROwith full SU (2)-symmetry-breaking in the thermody-namic limit; the columnar-ordered groundstate as a two-sublattice LRO with partial SU (2)-symmetry-breakingat the thermodynamic limit, and confirm the nonmag-netic nature of the SL states on infinite cylinders ofwidths up to 12 sites. In addition, we have discoveredthe stabilization of a new ASL phase, with power-lawcorrelation lengths, for width-6 infinite cylinders. Wehave pinpointed the phase transitions between the infi-nite cylinder’s groundstates of the THM, precisely, us-ing the Binder ratios. The transitions are relativelyclose to the phase boundaries found from the direct mea-surements of the local order parameters using fDMRGon L y = 3 , , , S EE = a ( L y ) + ( α + α L y ) log( ξ ), a mixture ofthe area-law and the quantum critical behavior, as ex-pected for the magnetic phases built by the inherentlyone-dimensional SU (2)-symmetric iMPS ansatz. To thebest of our knowledge, a set of numerical tools to ef-ficiently distinguish and classify LROs were previouslyabsent in the SU (2)-symmetric iDMRG literature. Con-sidering the advantages of SU (2)-symmetric calculations,we suggest that the proposed methods can be appliedwidely to detect symmetry broken states using the iMPS.Finally, to unravel the true nature of time-reversalsymmetry-breaking in the topological SL, we have in-vestigated the robustness of YC8-ˆi sector under perturb-ing H J with a chiral term, Eq. (35) (it was previouslysuggested that YC8-ˆi states are prone to become chi-ral under applying bond anisotropies to the Hamilto-nian). The results of the scalar chiral order parameter, O χ ( m → ∞ ), versus J χ can be fitted using ˜ b +˜ b J ˜ b χ with1high accuracy and shows the existence of a continuousphase transition to the CSL phase at small, but non-zero, J critical χ = 0 . ACKNOWLEDGMENTS
The authors would like to thank Jason Pillay for use-ful discussions. This work has been supported by theAustralian Research Council (ARC) Centre of Excellence for Engineered Quantum Systems, grant CE110001013.I.P.M. also acknowledges support from the ARC FutureFellowships scheme, FT140100625.
Notes added. – After completing this work, a relatedpaper appeared in which the authors study the phasediagram of the J - J - J χ model, Eq. (35), on finite- L x cylinders using the SU (2)-symmetric fDMRG algorithm.In agreement to Sec. IX results, Gong et al. find a smoothphase transition from J - J SL to a CSL at a small butfinite chiral coupling strength ( J critical χ ≈ .
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