Kaleidoscope of quantum phases in a long-range interacting spin-1 chain
Zhe-Xuan Gong, Mohammad F. Maghrebi, Anzi Hu, Michael Foss-Feig, Phillip Richerme, Christopher Monroe, Alexey V. Gorshkov
KKaleidoscope of quantum phases in a long-range interacting spin-1 chain
Z. -X. Gong,
1, 2, ∗ M. F. Maghrebi,
1, 2
A. Hu,
1, 3
M. Foss-Feig,
1, 2
P. Richerme,
1, 4
C. Monroe,
1, 2 and A. V. Gorshkov
1, 2 Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA Joint Center for Quantum Information and Computer Science,NIST/University of Maryland, College Park, Maryland 20742, USA Department of Physics, American University, Washington, DC 20016, USA Department of Physics, Indiana University, Bloomington, Indiana, 47405, USA
Motivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study ofthe phase diagram of a spin-1 chain with XXZ-type interactions that decay as /r α , using a combination offinite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-rangeinteractions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase,a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-rangeinteractions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phaseboundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact theentire phase diagram. Importantly, for α (cid:46) , long-range interactions destroy the Haldane phase, break theconformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a U (1) continuoussymmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains.We show that the main signatures of all five phases found could be observed experimentally in the near future. PACS numbers: 75.10.Jm, 75.10.Pq, 03.65.Vf, 05.30.Rt
The study of quantum phase transitions in low-dimensionalspin systems has been a major theme in condensed matterphysics for many years [1]. A well-known implication ofMermin and Wagner’s famous results [2] on finite temperaturequantum systems is that, for a large class of one-dimensionalquantum spin systems, long-range order is forbidden even atzero temperature. This absence of classical order promotesquantum fluctuations to a central role, and they often deter-mine the qualitative properties of the quantum ground state.An important example, first conjectured by Haldane [3, 4], isthat a spin-1 antiferromagnetic Heisenberg chain possesses adisordered phase with an energy gap to bulk excitations, lateridentified as a symmetry protected topological phase [5, 6].Its spin-1/2 counterpart, despite possessing the same classicallimit, has a disordered ground state with gapless excitations,and is described by a conformal field theory (CFT) [7].Experimentally, such quantum phase transitions have beenexplored in quasi-1D materials, and more recently using artifi-cial materials designed through the careful control of atomic,molecular, and optical (AMO) systems [8–11]. These AMOsystems are usually well-isolated from the environment, ofsystem parameters, and make possible both measurement andcontrol at the individual lattice-site level. A distinctive fea-ture of AMO systems is that interactions between particlesare often long-ranged, decaying as a power-law with dis-tance ( /r α ). The exponent α varies widely amongst differentAMO systems, ranging from α = 6 for van de Waals interac-tions in Rydberg atoms, to α = 3 for polar molecules andmagnetic atoms, to α = 0 for atoms coupled to cavities [11–19]. The effect of long-range interactions can be tuned byeither changing the dimensionality of the system, e.g. for neu-tral atoms or molecules in optical lattices, or by directly (andoften continuously) altering the value of α , e.g. in trappedions or cold atoms coupled to photonic crystals [14]. Theavailability of tunable long-range interactions creates an en- tirely new degree of freedom—absent in typical condensed-matter systems—for inducing quantum phase transitions, andcan potentially lead to novel quantum phases [20–22].While long-range interacting classical models have beenstudied in considerable detail for some time [23–27], thereis a relative lack of in-depth studies of quantum phase tran-sitions in long-range interacting systems, despite the emerg-ing experimental prospects for studying both their equilibriumand non-equilibrium properties [15, 16, 18, 28–34]. One rea-son is that many analytically solvable lattice models becomeintractable when interactions are no longer short-ranged, awell-known example being the spin-1/2 XXZ model. In ad-dition, to properly incorporate long-range interactions in low-energy effective theories, existing field theoretic treatmentsneed to be modified and usually become much more com-plicated [35, 36]. Though numerically exact techniques forquantum systems have been adapted to treat long-range inter-actions, significant challenges remain in the numerical calcu-lation of phase diagrams. In particular, power-law decayinginteractions generally lead to a divergent correlation length[31, 37], and a much larger system size or a much higher pre-cision is typically required to faithfully describe the propertiesof the system in the long-wavelength limit. Several authorshave performed analytical studies of non-interacting bosonicand fermionic systems with long-range hopping and pairing[32, 34, 38, 39], but there have been relatively few numericalstudies of non-integrable systems, and those that exist haveprimarily focused on spin-1/2 chains [20, 28, 40–42].In this manuscript, we carry out a detailed study of a spin-1 chain with tunable XXZ interactions that decay monotoni-cally as /r α , for all α > . Our study is largely motivatedby imminent trapped-ion based experiments that can simulatethis model with widely tunable index α [43–45]. In the ab-sence of long-range interactions, the choice of spin-1 overspin-1/2 allows us to have four distinct quantum phases by a r X i v : . [ c ond - m a t . s t r- e l ] O c t varying the anisotropy of the interactions: a ferromagnetic(FM) phase and an antiferromagnetic (AFM) Ising phase thatare both gapped and long-range ordered, a disordered gaplessphase (the XY phase), and a gapped and topologically orderedphase (the Haldane phase). By using a combination of den-sity matrix renormalization group (DMRG) calculations, spinwave analysis, and field theory, we obtain the phase diagramfor arbitrary anisotropy and all α > , with both ferromag-netic and antiferromagnetic interactions. Our key observationis that, when interactions in all spatial directions are antifer-romagnetic, long-range interactions are frustrated, leading toprimarily quantitative changes to the phase boundaries com-pared to the short-range interacting chain. Interestingly, wefind that the topological Haldane phase is robust under long-range interactions with any α > [46]. However, when theinteractions in the x − y plane become ferromagnetic, we finda number of significant modifications to the phase diagram:(1) The Haldane phase is destroyed at a finite α due to a clos-ing of the bulk excitation gap; (2) The gapless XY phase, de-scribed by a CFT with central charge c = 1 , disappears when α (cid:46) due to a breakdown of conformal symmetry [32, 34];(3) The disappearance of the XY phase heralds the emergenceof a new phase at α (cid:46) (continuous-symmetry breaking, orCSB) in which the spins order in the xy plane, spontaneouslybreaking a U (1) symmetry and possessing gapless excitations(Nambu-Goldstone modes); (4) Novel tricritical points, withno direct analogue in short-range interacting 1D models, ap-pear at the intersection of the Haldane, CSB, and XY/AFMphases.The manuscript is organized as follows. In Sec. I, we intro-duce the model Hamiltonian and present complete phase dia-grams for the ferromagnetic and antiferromagnetic cases. InSec. II, we study the boundary of the FM phase, where a spin-wave approximation is found to be asymptotically exact in thelarge-system limit. In Sec. III, we determine both the XY-to-Haldane and Haldane-to-AFM transition lines accuratelyusing DMRG calculations, and use field theory arguments toexplain the effect of long-range interactions on the boundaryof the Haldane phase. In Sec. IV, we introduce the new CSBphase and explain its emergence using spin-wave theory. Theboundary between the CSB and XY phases is determined bya numerical calculation of central charge. In Sec. V, we showthat all five phases possess distinct signatures that could beobserved in near-future trapped ion quantum simulations withchains of 16 spins. Finally, we conclude the work in Sec. VIand comment on a number of open questions. I. MODEL HAMILTONIAN AND PHASE DIAGRAMS
We consider the following spin-1 Hamiltonian with long-range XXZ interactions in a 1D open-boundary chain: H = (cid:88) i>j i − j ) α [ J xy ( S xi S xj + S yi S yj ) + J z S zi S zj ] . (1) Here J z ∈ ( −∞ , ∞ ) and α ∈ (0 , ∞ ) are allowed to varycontinuously, and we consider both the J xy = 1 (anti-ferromagnetic) and J xy = − (ferromagnetic) cases. We notethat, for < α < , Eq. (1) does not have a well-definedthermodynamic limit when J xy and/or J z is ferromagnetic,since the ground-state energy-density diverges. To make theground-state energy extensive, we may impose an energyrenormalization factor N α − , first introduced by Kac [47],when taking the thermodynamic or continuum limit (here N is the chain length). For finite-size numerical calculations,we do not need to implement the Kac renormalization for < α < since ground-state properties are unaffected byenergy renormalization [48].Figure 1 shows our full phase diagram for both J xy = 1 and J xy = − , with actual phase boundaries plotted us-ing the results of calculations discussed in the following sec-tions. The nearest-neighbor interaction limit is achieved at α → ∞ ( /α = 0 ). In this limit, the Hamiltonian in Eq. (1)with J xy = − is equivalent to the one with J xy = 1 , ascan be seen by performing a local unitary transformation thatflips every other spin in the x − y plane while preserving thespin commutation relations: S x,yi → ( − i S x,yi . The differ-ent ground-state phases of this short-range Hamiltonian havebeen well-studied [49–51]. Notably, Haldane first conjectured[3, 4] that for λ < J z < λ , a disordered gapped phase (theHaldane phase) will emerge. At J z = λ , the ground stateundergoes a second-order phase transition from the Haldanephase to an AFM phase, which belongs to the same univer-sality class as the 2D Ising model. The value λ ≈ . hasbeen found by various numerical techniques including Monte-Carlo [52], exact diagonalization [53], and DMRG [54–56].At J z = λ , a Berezinskii-Kosterlitz-Thouless (BKT) transi-tion intervenes between the Haldane phase and a gapless dis-ordered XY phase at J z < λ . The value of λ is theoreti-cally predicted to be exactly zero after mapping Eq. (1) (for α = ∞ ) to a field theory model using bosonization [57]. Thisprediction is supported by conformal field theory arguments[58] and a level spectroscopy method based on a renormaliza-tion group analysis and the SU (2) /Z symmetry of the BKTtransition [50, 59–61]. Numerically, λ ≈ has been verifiedvia finite-size scaling [53, 62, 63] and DMRG [54]. Finally,at J z = λ = − , a first-order phase transition from the XYphase to a ferromagnetic Ising phase takes place [50, 55, 64].We now introduce our results for the long-range interactingcase ( /α > ). For J xy = 1 and J z > , long-range inter-actions are frustrated and the Haldane-to-AFM phase transi-tion point λ ( α ) increases moderately as α decreases, with-out changing the universality class of the transition. For suf-ficiently small J z < , the ferromagnetic long-range inter-actions along the z direction eventually favor a ferromagneticground state, inducing a first-order transition at λ ( α ) . Themagnitude of the critical coupling, | λ ( α ) | , decreases mono-tonically from (at α = ∞ ) to (for all α ≤ ) in thethermodynamic limit. The XY-to-Haldane phase boundary λ ( α ) becomes negative for finite α , similar to the XXZ spin-1 chain with next-nearest-neighbor interactions [65], eventu-ally terminating in a tricritical point at the intersection of FM,Haldane, and XY phases. The entire XY phase (includingthe XY-to-Haldane phase boundary) has conformal symmetrywith c = 1 , and the XY-to-Haldane phase boundary remains aBKT transition until it terminates at the tricritical point.For J xy = − , where long-range interactions in the x − y plane are not frustrated, the phase diagram [Fig. 1(b)] shows anumber of important qualitative differences from the nearest-neighbor phase diagram as α is decreased. First, the XY-to-Haldane phase boundary bends significantly toward positive J z , and we find the Haldane phase to terminate at α ≈ for J z = 1 . Second, we expect the XY phase to disappear for α (cid:46) due to the breakdown of conformal symmetry [32, 34].Third, for α (cid:46) a new CSB phase emerges—this is not inviolation of the Mermin-Wagner theorem, as it no longer ap-plies for this range of interactions [2, 40, 66–69]. The CSB-to-AFM phase transition is expected to be first-order, sinceat large J z and small α , quantum fluctuations play negligi-ble roles for both the N´eel-ordered state and the ordered CSBstate. This behavior is similar to the transition between theAFM phase and the large- D phase (where a large positiveanisotropy term D (cid:80) i ( S zi ) causes all spins to stay in the | S zi = 0 (cid:105) state) reported in Refs. [54, 55, 64]. The Haldanephase has a c = 1 critical phase boundary with the XY phase,a c = 0 . phase boundary with the AFM phase [56], and apossibly exotic phase boundary with the CSB phase, a bound-ary that is not described by a 1+1D CFT. II. FM PHASE AND ITS BOUNDARY
Because the ferromagnetic state with all spins polarizedalong ± z (or an arbitrary superposition of these two states)is an exact eigenstate of the Hamiltonian for any value of α and J z , we expect a first-order quantum phase transition atthe boundary of the FM phase. The FM state has an energy E FM = J z (cid:80) i>j ( i − j ) − α , and the phase transition out of thisstate, defining the critical line J z = λ ( α ) , occurs when someother eigenstate with no ferromagnetic order appears with alower energy. The dependence of λ on α can be estimatedusing the following intuitive argument. For a given J z < ,the energy density of the ferromagnetic state in the thermody-namic limit is given by (cid:15) FM = J z ζ ( α ) [ ζ ( α ) ≡ (cid:80) ∞ r =1 r − α is the Riemann zeta function], which diverges as α → .For J xy = 1 , the magnitude of the energy density arisingfrom the term (cid:80) i>j ( S xi S xj + S yi S yj ) / ( i − j ) α can be at most η ( α ) ≡ (cid:80) ∞ r =1 ( − r − /r α (the Dirichlet eta function), withthis value obtained for any state that is N´eel-ordered alongsome direction in the x − y plane. The competition betweenthe energy of these two classical states gives a critical point J z ≈ − η ( α ) /ζ ( α ) , which smoothly varies from J z = − at α = ∞ to J z = 0 at α = 1 . For J xy = − , the situation isquite different, because the polarized state along any directionin the x − y plane has an energy density equal to − ζ ( α ) , andthus we naively expect the phase boundary to be at J z = − for all α > . -1 -0.5 0 0.5 1 1.5 2 J z / α (a) FMXY Haldane AFM λ λ λ -1 0 1 2 3 J z / α (b) FM XY Haldane
AFMCSB
Figure 1: Proposed phase diagram for (a) J xy = 1 and (b) J xy = − . Five different phases are identified: a ferromagnetic (FM) Isingphase, an antiferromagnetic (AFM) Ising phase, a disordered XYphase, a topological Haldane phase, and a continuous symmetrybreaking (CSB) phase. At α = ∞ , the transition points are de-noted by J z = λ , , in (a). The FM-to-XY, FM-to-CSB, and CSB-to-AFM transitions are first order (green line); the XY-to-Haldanetransition is BKT type with central charge c = 1 (purple line); theHaldane-to-AFM transition is second order with c = 0 . (yellowline); the CSB-to-XY transition (white dashed line) has c = 1 , but isa BKT-like transition corresponding to a universality class differentfrom the XY-to-Haldane transition [69]; the CSB-to-Haldane transi-tion (black dashed lines) appears to be an exotic continuous phasetransition not described by a 1+1D CFT. The location of solid tran-sition lines are expected to be accurate in the thermodynamic limit,while the location of dashed transition lines may be inaccurate dueto finite-size effects in our numerics. More formally, the boundary can be calculated via a spin-wave analysis. We treat the spin state that is polarized alongthe + z direction as the vacuum state with no excitations, andapply the Holstein-Primakoff transformation (for spin ) tomap spin excitations (spin-waves) into bosons: S zi = 1 − a † i a i , S + i ≡ S xi + iS yi = √ a † i (1 − a † i a i / / . In the weak excita-tion limit, (cid:104) a † i a i (cid:105) (cid:28) , we can approximate S + i ≈ √ a † i , andour Hamiltonian becomes H sw ≈ (cid:88) i>j − J z ( a † i a i + a † j a j ) + J xy ( a † i a j + a † j a i )( i − j ) α , (2) -0.8 -0.6 -0.4 -0.2 0 J z α N = 10 Exact N = 10 Spin wave N = 20 Exact N = 20 Spin wave Figure 2: Comparison of the (first-order) transition point out of theFM phase calculated using finite-size DMRG and spin-wave theoryfor J xy = 1 . The DMRG result is regarded as exact since its error isfar below the resolution of the plot. where we have ignored the interaction terms a † i a i a † j a j since (cid:104) a † i a i (cid:105) , (cid:104) a † j a j (cid:105) (cid:28) is assumed. Assuming for the moment pe-riodic boundary conditions, this quadratic Hamiltonian can bediagonalized by Fourier transformation, H sw = 2 (cid:80) k ω k c † k c k ,with the following dispersion relation ( q ≡ πk/N ) for an in-finite system ω ( q ) = − J z ∞ (cid:88) r =1 r − α + J xy ∞ (cid:88) r =1 cos( qr ) /r α . (3)If ω min ≡ min ω ( q ) > , then the ground state of H sw isthe vacuum state of all modes k , and (cid:104) a † i a i (cid:105) = 0 for all i ,consistent with the approximation (cid:104) a † i a i (cid:105) (cid:28) . If ω min < ,then the ground state has an extensive number of spin exci-tations and the spin-wave approximation should break down,and we do not expect the polarized state in the z directionto be the quantum ground state. The ω min = 0 condi-tion thus sets the phase boundary for H sw . For J xy = 1 , ω min = ω ( q = π ) = − J z ζ ( α ) − η ( α ) , leading to a criti-cal line of J z = − η ( α ) /ζ ( α ) . For J xy = − , ω min = ω ( q =0) = (1 − J z ) ζ ( α ) , leading to a critical line at J z = − , inde-pendent of α . These results exactly match with the previousintuitive arguments.Note that we can estimate the transition point of a finitechain without translational invariance by numerically diago-nalizing H sw . However, we note that the spin-wave analy-sis is not necessarily exact in this case. The interactions be-tween bosons that we have ignored can make multi-particleeigenstates have a lower energy than the vacuum state, de-spite the fact that the single-particle excitation spectrum has afinite-size gap. In other words, the condition ω min > onlyguarantees the ferromagnetic state to be the ground state ofthe non-interacting Hamiltonian H sw [Eq. (2)], but not of theoriginal Hamiltonian [Eq. (1)]. This effect of interactions canindeed be observed for finite-size systems. In Fig. 2, we showthat for J xy = 1 , the critical J z obtained by exact numerical calculations of Eq. (1) with N = 10 spins is slightly smallerthan spin-wave prediction given by the condition ω min = 0 for < α < ∞ . To the contrary, for J xy = − we find thatthe spin-wave prediction is exact for any number of spins andfor all α > .It is also interesting to note that the deviation of the transi-tion point due to the spin-wave approximation decreases withincreasing α , and vanishes in the α → ∞ limit, showing thatlong-range interactions are playing an important role. How-ever, using a finite-size DMRG algorithm [70–73], we findthat as N increases, the deviation caused by the spin-wave ap-proximation decreases quickly (Fig. 2). In addition, by usingan infinite-size DMRG algorithm [73, 74], we find that thespin-wave prediction of the transition line J z = − η ( α ) /ζ ( α ) [the green line in Fig. 3(a)] is correct within our numericalprecision, strongly suggesting that the spin-wave predictionbecomes exact in the thermodynamic limit. III. HALDANE PHASE AND ITS BOUNDARY
The existence of the Haldane phase in a spin-1 XXZ chainmakes the phase diagram much richer than that of a spin-1/2 XXZ chain. We focus first on the XY-to-Haldane phaseboundary λ ( α ) . The transition out of the Haldane phaseis signaled by a vanishing of the string-order correlationfunction S ξij ≡ (cid:104) S ξi S ξj (cid:81) i AFMCSB J z -1 0 1 2 / α Figure 3: Infinite-size DMRG calculation of S zij ≡(cid:104) S zi S zj (cid:81) i The celebrated Mermin-Wagner theorem rigorously rulesout continuous symmetry breaking in 1D and 2D quantum andclassical spin systems at finite temperature, as long as the in-teractions satisfy the convergence condition (cid:80) i>j J ij r ij < ∞ in the thermodynamic limit ( r ij and J ij are respectivelythe distance and coupling strength between sites i and j ) [2].The long-distance properties of 1D spin systems at zero tem-perature can often be related to those of a 2D classical modelat finite temperature; however, in the process of this mapping,the long-range interactions are only inherited by one of thetwo spatial directions in the classical model, and the Mermin-Wagner convergence condition will be satisfied for interac-tions decaying faster than /r . Thus we expect no continu-ous symmetry breaking in the ground state of our HamiltonianEq. (1) for α > . Indeed, we have found exclusively disor-dered or discrete ( Z ) symmetry breaking phases for α > inour phase diagrams (Fig. 1). Continuous symmetry breakingcan (and does) appear when α < . To gain a better un-derstanding of the robustness of symmetry breaking states toquantum fluctuations, below we carry out a spin-wave analy-sis [88].We start by considering the J xy = − case, and take thestate with all spins polarized along the + x direction as the vac-uum state. With this choice of vacuum, and assuming that thedensity of spin waves is small ( (cid:104) a † i a i (cid:105) (cid:28) in the followingexpressions), the Holstein-Primakoff mapping is now S xi =1 − a † i a i , S yi ≈ ( a † i + a i ) / √ , S zi ≈ ( a † i − a i ) /i √ . Underthis mapping, and dropping terms that are quartic in bosonicoperators (again based on the assumption that (cid:104) a † i a i (cid:105) (cid:28) ), H becomes H swx = N/ (cid:88) k = − N/ (cid:0) a † k a − k (cid:1) (cid:18) ω k µ k µ k ω k (cid:19) (cid:18) a k a †− k (cid:19) ; (10) ω k = N/ (cid:88) r =1 J r + J z − N/ (cid:88) r =1 J r cos( 2 πkN r ) , (11) µ k = − J z + 12 N/ (cid:88) r =1 J r cos( 2 πkN r ) , (12)where a k = √ N (cid:80) j e i πjk/N a j . H swx can be diagonalizedwith a Bogoliubov transformation, yielding non-interactingBogoliubov quasi-particles with a spectrum ν k . Importantly,when | ω k | > | µ k | , ν k > and the vacuum is dynamicallystable. When | ω k | < | µ k | , ν k is imaginary and the sys-tem is dynamically unstable indicating that we have madethe wrong choice of a classical ground state. Using the ex-pressions for ω k and µ k in Eqs. (11) and (12), we find that | ω k | > | µ k | is satisfied for all k (cid:54) = 0 modes if and only if − ≤ J z < ζ ( α ) /η ( α ) . This is because when J z < − ,the classical ground state is ferromagnetic in z direction, andwhen J z > ζ ( α ) /η ( α ) the classical ground state is N´eel or-dered along the z direction.Because the Bogoliubov quasiparticles consist of both par-ticles and holes, the ground state of H swx can have a finite oreven divergent density of spin excitations, measured by (cid:104) a † i a i (cid:105) = 1 N (cid:88) k (cid:54) =0 12 ([1 − µ k /ω k ] − / − (13) N →∞ −−−−→ π ˆ π − π dq (cid:16) [1 − µ ( q ) /ω ( q )] − / − (cid:17) . The integrand [1 − µ ( q ) /ω ( q )] − / above diverges at q = 0 , and whether or not the integral is infrared diver-gent depends on the value of α . We find that for α > , [1 − µ ( q ) /ω ( q )] − / ∝ | q | − to leading order in q , andtherefore (cid:104) a † i a i (cid:105) ∼ ln( N ) diverges as N → ∞ . This means (a) XY Haldane AFMCSB J z / α (b) No CSB AFMCSB J z -1 0 1 2 / α Figure 4: Continuous symmetry breaking for J xy = − . (a)iDMRG calculation of (cid:104) S + i S − j (cid:105) for | i − j | = 500 , with bond di-mension χ = 200 (at this separation and bond dimension, the resultsare well converged). Long-range order in the x − y plane is increas-ingly favored as α decreases, but we can not extract a sharp phaseboundary between the CSB and XY phase because an impracticallylarge bond dimension is needed to accurately extract the simultane-ous χ, | i − j | → ∞ limit of (cid:104) S + i S − j (cid:105) . The white line denotingthe boundary of the AFM phase is from Fig. 3(a). (b) Spin-waveexcitation density (cid:104) a † i a i (cid:105) calculated using Eq. (13) for N = 1001 spins. For J z > ζ ( α ) /η ( α ) imaginary frequencies appear in the Bo-goliubov spectrum, indicating a classical instability toward the AFMphase. We set (cid:104) a † i a i (cid:105) = 1 in this region, as well as in regions where (cid:104) a † i a i (cid:105) > . For α > , (cid:104) a † i a i (cid:105) → ∞ as N → ∞ , thus no CSBphase is expected (boundary shown by the white dashed line). the long-range ferromagnetic order along the x direction is de-stroyed by quantum fluctuations in the thermodynamic limit;we expect that lim | i − j |→∞ (cid:104) S + i S − j (cid:105) = 0 , and the system willbe disordered (either Haldane or XY). For α < , we find that [1 − µ ( q ) /ω ( q )] − / ∝ | q | − ( α − / to leading order in q ,and the integral is infrared convergent. The excitation den-sity (cid:104) a † i a i (cid:105) converges to a finite constant, so we expect a CSBphase with lim | i − j |→∞ (cid:104) S + i S − j (cid:105) (cid:54) = 0 . However, when (cid:104) a † i a i (cid:105) converges to a constant on the order of , the spin-wave ap-proximation is not expected to be accurate, and it is possiblethat the actual ground state of H remains disordered for α slightly less than .For J xy = 1 , classically the spins prefer to anti-align in the x − y plane. Expanding around this classical state with thesame spin-wave approximation, both µ ( q ) and ω ( q ) becomefully analytic due to an additional alternating sign ( − r inEqs. (11) and (12). As a result, [1 − µ ( q ) /ω ( q )] − / alwaysexhibits a | q | − divergence at small q , and continuous sym-metry breaking is forbidden for all α > .From our infinite-size DMRG calculations, we see that (cid:104) S + i S − j (cid:105) ∼ / | i − j | η decays with a rather slow power law inthe XY phase (e.g. η = 0 . at J z = 0 and α = ∞ ; η is non-universal and depends on J z and α ). At the maximum sepa-ration that we can calculate accurately, (cid:104) S + i S − j (cid:105) only showsa crossover from the XY phase to the CSB phase [Fig. 4(a)].This crossover can in fact be qualitatively reproduced usingthe above spin-wave theory by calculating the spin-wave ex-citation density (cid:104) a † i a i (cid:105) for a finite system size [Fig. 4(b)].Further numerical evidence of the CSB phase is obtainedby calculating the effective central charge c eff as a functionof α and J z , which can be obtained by calculating the half-chain entanglement entropy S for two chains with differenttotal lengths N and N using a finite-size DMRG algorithm.Explicitly, for large N and N , we have c eff ≈ S ( N ) − S ( N )ln( N ) − ln( N ) . (14)In the XY phase (including its boundaries) and at theboundary between the Haldane and AFM phases, we ex-pect 1+1D conformal symmetry in the underlying field theorymodel [56, 58], with c eff being the actual central charge rep-resenting the conformal anomaly [78]. In the Haldane, FM,and AFM phases, no 1+1D conformal symmetry exists due tothe presence of a gap. Although the CSB phase is gapless, weexpect a breakdown of 1+1D conformal symmetry due to the /r α long-range interactions that become relevant in the RGsense for α (cid:46) [32, 34, 69]. We emphasize that in phaseswith no conformal symmetry, c eff does not have the meaningof the central charge and is used only as a diagnostic here tonumerically find phase boundaries.We identify the XY-to-CSB phase boundary in Fig. 5 asthe place where c eff starts to become appreciably (5-10%)larger than . Due to finite-size effects, c eff changes contin-uously for continuous phase transitions, and we are not ableto obtain the precise location of the XY-to-CSB phase bound-ary. Nevertheless, we find good agreement with the XY-to-CSB phase boundary predicted by spin-wave theory, espe-cially near J z = − , where spin-wave theory should be al-most exact. Together with perturbative field theory calcula-tions presented in Ref. [69], we expect the phase boundaryin Fig. 5 to be accurate within a few percent. A CSB-XY-Haldane tricritical point is found at α ≈ . and J z ≈ . .From Ref. [69], it follows that the XY-to-CSB transition isa BKT-like transition that belongs to a universality class dif-ferent from the XY-to-Haldane BKT transition. The Haldane-to-CSB transition is somewhat exotic, because the Haldane phase maps to a high-temperature disordered phase in a 2Dclassical model [82], and in the absence of long-range inter-actions , the CSB phase exists in 2D only at zero temperature[2] and is unlikely to undergo a phase transition directly toa high-temperature disordered phase. We also argue that theCSB-to-Haldane transition is not described by a 1+1D CFT,as supported by our numerical calculations shown in Fig. 5(b),where c eff changes smoothly (at least for finite chains) from avalue larger than to during the CSB-to-Haldane transition. J z / α XY Haldane AFMCSB (a) J z c e ff (b) α = 3 . α = 2 . α = 2 . Figure 5: Calculation of the effective central charge c eff as a functionof J z and α for J xy = − , extracted from finite-size DMRG calcu-lations with N = 100 , N = 110 , and a maximum bond dimensionof . (a) The black squares (fitted by the black line) show where c eff starts to deviate from when going from the XY to the CSBphase. The purple line and white line are from Fig. 3, and show theboundaries of the Haldane phase. (The calculation of c eff is inaccu-rate in predicting the location of the XY-to-Haldane transition due tostrong finite-size effects [50, 75–77].) For better contrast, locationswith c > are shown with the color corresponding to c = 2 . (b)For our finite-size chains, the XY-to-Haldane BKT phase transitionis signaled by a continuous drop of c eff from to ( α = 3 . ). TheHaldane-to-AFM phase transition is identified by a peak with valuearound . in c eff ( α = 3 . and α = 2 . ). The CSB-to-Haldanetransition is expected to be continuous and not associated with a cen-tral charge ( α = 2 . ). The CSB-to-AFM transition has a sharp peakin c eff ( α = 2 . ), an indication of a first-order transition [56]. The CSB-to-AFM phase transition is very likely to be first-order, similar to the transition between the large- D and AFMphases studied in Refs. [55, 64], despite the existence of quan-tum fluctuations in both phases. As shown in Fig. 5, we ob-serve a sharp peak in c eff at small α s when J z is varied, indi-cating a first order transition [56], with further evidence thatincludes jumps in sub-lattice magnetization and spin-spin cor-relation across the CSB-to-AFM transition (not shown). V. EXPERIMENTAL DETECTION It was theoretically proposed in Refs. [43, 44] that theHamiltonian we consider can be simulated (for widely tun-able J z and < α < ) by using microwave field gradientsor optical dipole forces to induce spin-spin interactions in achain of trapped ions. The simulation of Eq. (1) with J xy = 1 and J z = 0 was experimentally demonstrated for a few ionswith α tuned around [45], where the ground state was adi-abatically prepared by slowly ramping down an extra single-ion anisotropy term D ( t ) (cid:80) i ( S zi ) , with D ( t ) > . As thesystem size increases, the energy gap separating the groundstate from the rest of the spectrum will become progressivelysmaller near the point where a phase transition between the“large- D ” phase and the XY/Haldane/FM/AFM phase occursin the thermodynamic limit [64]. To avoid a slow ground statepreparation process, we can adiabatically ramp down a stag-gered magnetic field in the z direction, h ( t ) (cid:80) Ni =1 ( − i S zi ,with h ( t ) > [43, 44]. By preparing an initial state thatis the highest excited state of the staggered field Hamilto-nian, the same adiabatically ramping process will lead us tothe ground state of the Hamiltonian Eq. (1) with the oppo-site sign of both J xy and J z . As discussed in Ref. [44], thespin correlation functions (cid:104) S zi S zj (cid:105) and the string-order cor-relation S zij ≡ (cid:104) S zi S zj (cid:81) i By tuning the anisotropy J z / | J xy | and the power-lawexponent α , we have explored a rich variety of quantumphases—and the transitions between them—in a long-rangeinteracting spin-1 XXZ chain. For J xy = − , long-range in-teractions give rise to a rather unusual phase diagram due tothe emergence of a continuous symmetry breaking phase inone spatial dimension. Because the CSB phase cannot hap-pen in short-range interacting 1D spin-system, the nature ofthe phase transitions into and out of it is rather interesting;an in-depth study of the universality class of the CSB-to-XYtransition was carried out in a separate work [69], where asimilar transition in the long-range interacting spin-1/2 XXZchain is analyzed. On the other hand, the CSB-to-Haldanetransition, absent in spin-1/2 chains, requires further study tobe understood thoroughly. The CSB-Haldane-AFM tricriticalpoint is reminiscent of the tricritical point at the intersectionof the large- D , Haldane and AFM phases, which has been re-lated to the integrable Takhtajan-Babujian model described byan SU (2) Wess-Zumino-Witten (WZW) model with centralcharge c = 3 / [56, 89–92]. Additional numerical calcula-tions are needed to accurately determine the central chargeat the CSB-Haldane-AFM tricritical point. Generalizationsof our model to include single-ion anisotropy and a magneticfield are readily achievable in current trapped-ion experiments[44, 45]. Understanding these exotic quantum phase transi-tions—induced by long-range interactions that are highly tun-able in current experiments—requires the confrontation of nu-merous theoretical and numerical challenges, and motivatesexperimental quantum simulation of the model using AMOsystems. Acknowledgements We thank G. Pupillo, D. Vodola, L. Lepori, A. Turner,J. Pixley, M. Wall, P. Hess, A. Lee, J. Smith, A. 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