Exotic criticality in the dimerized spin-1 $XXZ$ chain with single-ion anisotropy

We consider the dimerized spin-1 $XXZ$ chain with single-ion anisotropy $D$. In absence of an explicit dimerization there are three phases: a large-$D$, an antiferromagnetically ordered and a Haldane phase. This phase structure persists up to a critical dimerization, above which the Haldane phase disappears. We show that for weak dimerization the phases are separated by Gaussian and Ising quantum phase transitions. One of the Ising transitions terminates in a critical point in the universality class of the dilute Ising model. We comment on the relevance of our results to experiments on quasi-one-dimensional anisotropic spin-1 quantum magnets.


Introduction
It is well established that quantum effects in one-dimensional antiferromagnetic (AFM) spin systems lead to interesting physical phenomena. While a uniform Heisenberg chain is gapless for half-integer spins, an exotic ground state with a finite gap appears for integer spins [1]. For spins S = 1, this Haldane phase can be understood in the framework of the Affleck-Kennedy-Lieb-Tasaki model [2,3], whose exact ground state can be constructed in terms of valence bonds, i.e., singlet pairs of S = 1/2 spins. Meanwhile, the Haldane phase is recognized as a symmetry-protected topological (SPT) state [4,5] and attracts continued attention from both theoretical and experimental points of view. For instance, the Haldane gap was confirmed experimentally in a compound with Ni 2+ ions Ni(C 2 H 8 N 2 ) 2 NO 2 (ClO 4 ) [6,7], in which a small value of the single-ion anisotropy D was reported [8]. A minimal model for the description of such anisotropic spin-1 chains iŝ where (Ŝ j ·Ŝ j+1 ) ∆ =Ŝ x jŜ x j+1 +Ŝ y jŜ y j+1 + ∆Ŝ z jŜ z j+1 . Assuming a positive exchange parameter J > 0 and ∆ > 0, the ground-state phase diagram exhibits three gapped phases [9]. At the isotropic point (D = 0 and ∆ = 1) the model is in a Haldane phase. A sufficiently strong single-ion anisotropy D/J induces a Gaussian quantum phase transition (QPT) with central charge c = 1 to a topologically trivial large-D (LD) phase. On the other hand, increasing ∆ for fixed D = 0 from the isotropic point leads to a Ising QPT with c = 1/2 to a long-range ordered AFM phase. At larger values of ∆ and D there is a first order transition between the LD and AFM phases.
A natural extension of the spin-1 XXZ chain (1) is the introduction of an explicit bond alternationĤ Interestingly this model realizes dimerized versions of the same three phases as the one described by Eq. (1), namely, dimerized Haldane (D-H), AFM (D-AFM) and LD (D-LD) phases.
The case D = 0 has been studied previously [10,11] and it was found that the D-H to D-LD transition is again of the Gaussian type, but the entire D-AFM-phase boundary, including the transition to the D-LD phase, belongs to the Ising universality class. A key question is how the criticality at the phase boundary changes, if both D and δ are finite. Earlier studies of half-filled Hubbard-type models realizing SPT insulating and long-range ordered (chargedensity-wave) phases [12][13][14] indicated a transition line that is separated into continuous Ising and first-order QPTs. The meeting point of these lines belongs to the tricritical Ising universality class with c = 7/10, which can be described by the second minimal model of conformal field theory [15,16].
In this paper, we determine and analyze the ground-state phase diagram of the extended model (2) by means of field theory and matrix-product-state based density-matrix renormalization group (DMRG) [17,18] techniques, focusing on the quantum criticality at the phase boundaries. By calculating the central charge c, we provide compelling evidence for the existence of a critical point in the tricritical Ising universality class. Field-theory predictions for the phases and the nature of the phase boundaries of the model (2) with both single-ion anisotropy D and bond alternation δ are shown to be in excellent agreement with numerical simulations. Finally, we discuss the relevance of our results to experiments on dimerized spin-1 materials [19].

Ground-state phase diagram
Let us first describe the numerical method we have used to determine the phase boundaries of the model (2). By means of the infinite DMRG (iDMRG) [20] a characteristic correlation length ξ χ can be calculated. While this ξ χ is always finite for fixed bond dimension χ, it strongly peaks at a critical point and therefore allows for an accurate determination of QPT points, see Appendix B. This approach was already applied to half-filled Hubbard-type models [12][13][14].
In order to identify the different continuous phase transitions occurring in the model (2), we calculate the corresponding central charges c via the entanglement entropy. For a critical system with L sites and periodic boundary conditions, the von Neumann entanglement entropy of a contiguous block of sites with the rest of the system is S L ( ) = c 3 ln L π sin π L + s 1 , where s 1 is a non-universal constant [21]. An accurate determination of the central charge is possible by using the relation [13,22] where in view of the explicit dimerization the doubled unit cell has been taken into account. Calculating the central charge numerically via Eq. (3), the universality classes of the QPT points are confirmed; this is demonstrated in Appendix B. For iDMRG simulations typical truncation errors are 10 −12 , using bond dimensions χ up to 1600. In the case of finite-system DMRG calculations with periodic boundary conditions, e.g., by estimating the central charge via Eq. (3), the maximal truncation errors are about 10 −9 , with χ up to 6000. In the following, combining field theory and DMRG, we discuss various QPTs, including the direct Ising transition from the D-LD to the D-AFM phase.

Field-theory approach
In order to obtain a field-theory description of the model in the vicinity of the various phase transition lines we consider the Hamiltonian which differs from Eq. (2) by an additional biquadratic exchange term. A field-theory description of the model (4) can be constructed in the vicinity of the Takhtajan-Babujian point [23,24] (α = 0, δ = 0, D = 0, ∆ = 1 and ∆ = 1) following Ref. [25]. This leads to a Hamiltonian density of the form whereL a andR a are left and right moving Majorana fermions,σ a are three Ising order parameter fields andĴ The parameter λ inĤ is proportional to the dimerization δ and by virtue of the U(1) symmetry of the microscopic Hamiltonian (4) we have v 1 = v 2 , m 1 = m 2 ≡ m, and g 1 = g 2 ≡ g. The masses m and m 3 are functions of D and α. The functional form of this dependence is only known in the vicinity of the Takhtajan-Babujian point and in what follows we therefore take m 3 and m as free parameters, which we adjust in order to recover the structure of the phase diagram obtained by DMRG. Our main working assumption is that the field theory (4) remains a good description of the low-energy degrees of freedom in the vicinity of the various phase transition lines in the microscopic model even far away in parameter space from the Takhtajan-Babujian point. We note that an alternative way of deriving a field theory proposed by Schulz [26] leads to equivalent results. A third approach would be to develop a field-theory description around the SU(3) symmetric point of the spin-1 chain [27][28][29][30], but we do not pursue this here. The relation between lattice spin operators and continuum fields iŝ where x = ja 0 (a 0 is the lattice spacing). The smooth components of the spin operators are proportional to the currentsM a (x) ∝Ĵ a (x), whilen a (x) are expressed in terms of Ising order and disorder operators asn In order to facilitate comparisons between field-theory and iDMRG results for the lattice model it is useful to define lattice operatorŝ At long distances we havem It is convenient to use the U(1) symmetry to bosonizê In terms of the corresponding canonical Bose fieldΦ =φ L +φ R and the dual fieldΘ =φ R −φ L the field theory (5) reads: where K is the Luttinger liquid (LL) parameter.

Renormalization group analysis
The most relevant perturbation is always the dimerization, and concomitantly at weak coupling the λ term reaches strong coupling first under the renormalization group (RG) flow. This results in a non-zero dimerization For later convenience we define a lattice version of the normal-ordered dimerization operator To see what happens after the dimerization perturbation has reached strong coupling we consider the next most relevant operators, which are the Majorana mass term and the costerm in the bosonic sector. Assuming that we have m > 0, what happens then depends on the sign of the Majorana mass term m 3 . If it is positive the third Ising model is in its disordered phase σ 3 (x) = 0, while m 3 < 0 implies that σ 3 (x) = 0. In the latter case the strong coupling RG fixed point is amenable to a mean-field analysis. The termĤ int coupling the bosonic and fermionic sectors can be decoupled, e.g.
This leads to a mean-field description in terms of an Ising model in a longitudinal field and a double sine-Gordon model [31,32] where The classical ground state of the double sine-Gordon model is obtained by solving Importantly, this tells us that for m > 0 we have which in turn implies that Hence the strong coupling RG fixed point describes a phase where antiferromagnetic order coexists with dimerization. This is the D-AFM phase identified above by the DMRG.
In the other phases the RG fixed points again occur at strong coupling but cannot be analyzed in terms of a simple mean-field argument. However, the field theory nevertheless allows for a description of the various transition lines as shown in what follows.

D-LD D-AFM phase transition line
This corresponds to the situation where the bosonic sector remains gapped, while the third Ising model undergoes a transition between a disordered phase σ 3 = 0 on the D-LD side and an ordered phase σ 3 = 0 on the D-AFM side of the phase diagram. As a result the D-LD D-AFM phase transition is in the universality class of the two-dimensional Ising model. In the vicinity of the transition we may project onto the low-energy Ising degrees of freedom following e.g. Ref. [33]. Details are given in Appendix A. This yieldŝ Along the phase transition line we thus have n z jn z j+ and The predictions (29)-(32) are compared to iDMRG simulations below.

D-H D-AFM phase transition line
The D-AFM to D-H transition is described by the same scenario as discussed above, since it also belongs to the Ising universality class with c = 1/2. Accordingly, Eqs. (29)-(32) are valid on this transition line as well.

D-H D-LD phase transition line
As we cross from the D-AFM into the D-H phase at fixed ∆ by increasing D the (effective) Majorana mass m 3 increases. Assuming that this relation continues to hold, the characteristic energy scale in the Majorana sector can eventually become large compared to that of the bosonic sector and it is then justified to integrate out the Majorana sector. This leads to an effective low-energy description in terms of a sine-Gordon model The main effect of integrating out the Majorana sector is the renormalization of the sine-Gordon coupling. Importantly, m * can vanish for particular values of D, which corresponds to a phase transition line described by a LL characterized by the LL parameter K. The low-energy projections of the lattice spin operators along this line arê This gives the following field-theory predictions for power-law decays of two-point functions

DMRG analysis
In this section, we examine various two-point correlation functions of the lattice Hamiltonian (2) using iDMRG in order to prove the field-theory predictions described in the last section. Then, the topological properties of each phase are discussed by simulating topological order parameters. These values can also be extracted from the long-distance behavior of the smooth part of the spin-spin correlation function (42), that is, the LL parameter determines the amplitude of the correlation function but not the exponent. We calculate the longitudinal spin correlation function and isolate the smooth component from a Fourier transformed structure factor for q ≈ 0, where q = 2π/L. The LL parameter is determined as K = lim q→0 πS(q)/q [34].

Topological order parameters
Let us now explore the topological properties of the phases of the model (2). Following Vidal [35], we use the infinite matrix-product-state representation formed by χ × χ matrices Γ σ and a positive real, diagonal matrix Λ: where the index σ labels the basis states of the local Hilbert spaces. The Γ σ and Λ are assumed to be in the canonical form: If |ψ is invariant under an internal symmetry represented by a unitary matrix Σ σσ , then the transformed Γ σ matrices satisfy [5,36] σ Here U is a unitary matrix that commutes with Λ, and e iθ is a phase factor. In the case of time reversal symmetry (inversion symmetry), Γ σ on the left-hand side is replaced by its complex conjugate Γ † σ (its transpose Γ T σ ). Exploiting the properties of the matrices U each SPT phase can be classified [5]: In the case of time reversal (inverse) symmetry the matrices satisfy U T U * T = ±1 (U I U * I = ±1), and the sign can be used to distinguish different SPT phases. In presence of a Z 2 × Z 2 symmetry the order parameter is given by where we use the symmetry operationsR x = exp(iπ jŜ x j ) andR z = exp(iπ jŜ z j ) to calculate U x and U z .
In the presence of dimerization the unit cell consists of two sites, which we have to block together in order to apply the above description. For the model (2), blocking sites across weak bonds gives the same values of the order parameters as blocking across strong bonds. Figure 4 shows the iDMRG results for the order parameters in case of inverse and Z 2 × Z 2 symmetries. If U x and U z commute (O Z 2 ×Z 2 = 1), the system is in a trivial phase, i.e., a site-factorizable LD state, whereas if they anticommute (O Z 2 ×Z 2 = −1), the system realizes a non-trivial Haldane state. If the symmetry is broken, we set O Z 2 ×Z 2 = 0. Obviously, the order parameter O Z 2 ×Z 2 changes its sign only if a phase transition occurs between D-LD and D-H phases. O I behaves similarly to O Z 2 ×Z 2 , i.e., O I = ±1 for the two symmetric phases, and O I = 0 in the D-AFM phase.
To summarize this subsection, dimerization does not affect the topological properties of the system (2), so that the D-H (D-LD) phase remains a non-trivial (trivial) SPT phase as in the system without dimerization (1).

Relevance to experiments
Let us finally relate our findings with experimental results. There are several realizations of spin-1 bond-alternating chains, such as Ni(C 9 H 2 4N 4 ) (NO 2 )ClO 4 [37,38] and [Ni(333-tet)(µ-N 3 ) n ](ClO 4 ) n [39][40][41]. Most remarkably, in the latter material a logarithmic decrease of the susceptibility was observed at low temperature, indicating a vanishing excitation gap [19]. Comparing quantum Monte-Carlo simulations with experimental data suggested that the material is described by a Hamiltonian of the form (2) with δ = 0.25, ∆ = 1 and D/J = 0. Totsuka et al. [42] determined the critical point for D = 0 numerically and obtained δ c = 0.25 ± 0.01 and c = 1, while results by Kitazawa and Nomura [11] suggested that δ c = 0.2598. Importantly these parameter sets are close to the location of the point where the Gaussian and Ising phase transitions merge [10,11].
In the following, we therefore determine the ground-state phase diagram of the model (2) for δ = 0.25 and reexamine the magnetic susceptibility of the above mentioned nickel compound using the infinite time-evolving block decimation (iTEBD) [35]. Figure 5(a) displays the corresponding phase diagram of the model (2). Although the extent of the Haldane phase is significantly reduced, the Gaussian and Ising transition lines can still be detected numerically. As shown in Fig. 5(b) the experimental data of the magnetic susceptibility for [Ni(333-tet)(µ-N 3 ) n ](ClO 4 ) n can be fitted most successfully for ∆ = 1 and D/J = 0.02, taking the reported small single-ion anisotropy D/J < 0.1 [19] into account. On the other hand, the numerical data at the Gaussian transition point for fixed ∆ = 1 deviates from experimental ones in the lower-temperature regime. Thus, this nickel compound may be even closer to the Ising transition line than to the c = 1 transition line considered so far. It would be interesting to investigate signatures of the Ising QPT experimentally, e.g., by inelastic neutron scattering, where the corresponding dynamical structure factor can be calculated numerically, see Ref. [43].

Summary and Conclusions
In this work we investigated the ground-state phase diagram and quantum criticality of the dimerized spin-1 XXZ chain with single-ion anisotropy D, employing a combination of analytical and numerical techniques. For weak dimerization (δ 0.26) and single-ion anisotropy, the symmetry-protected topological Haldane phase survives and the transition between the D-LD and D-AFM phases, which is always of first order in the absence of dimerization, be-

A Low-energy projections of operators
Let us denote the Euclidean action corresponding to the Hamiltonian density (14) by where S 3 and S B involve only Ising and bosonic degrees of freedom respectively and S int describes the interaction between the two sectors. In the regimes where the mass scale associated with S 3 is much smaller (larger) than the one associated with S B and where S int can be treated as a perturbation, we may integrate out the bosonic (fermionic) degrees of freedom, see e.g. Ref. [33].

A.1 Integrating out the bosonic degrees of freedom
This case pertains to the transition lines between the D-AFM phase and the D-LD and D-H phases. In these cases the low-energy projection of a general local operator is given bŷ where Φ denotes the average with respect to the bosonic action S B . As we have assumed that the parameter m is positive, we have This implies that the low-energy projection of the dimerization operator iŝ In the last line we have used that the expectation value in the bosonic sector decays exponentially in the Euclidean distance r = (x − y) 2 + v 2 τ 2 , which in turn allows us to employ the operator product expansion in the Ising sector Finally we have fixed the constant part in the low-energy projection by using that it must give the correct expectation value of the dimerization operator. Similarly we obtain The leading contribution to the low-energy projection ofn z j occurs at orderÔ(λ ) 0 of our procedure and givesn

A.2 Integrating out the fermionic degrees of freedom
This case pertains to the transition line between the D-LD and D-H phases. Here we havê where 3 denotes the average with respect to the Majorana action S 3 . On the transition line we have m 3 > 0 which implies An immediate consequence of (58) is that The low-energy projections of other operators can be worked out as beforê Here we have used that which permits us to employ operator product expansions in the bosonic sector. The projection of the dimerization operator iŝ

B Determination of phase boundaries
In this section, we explain how the QPT points and their universality classes are determined within the (i)DMRG method. Since the QPTs are the only points in the considered parameter region where the system becomes critical, they are easily obtained by simulating the correlation length ξ χ , as demonstrated in Figs. 6(a) and (b) for δ = 0.1 with fixed D/J = 1 and 3, respectively. The divergence of the physical correlation length at a QPT is reflected by a pronounced peak of ξ χ whose height increases with the bond dimension χ. From the peak positions for large enough χ, we pinpoint the phase transition with an accuracy of at least three digits. For D/J = 1 the transitions occur at ∆ c1 1.135 and ∆ c2 1.789 [see Fig. 6(a)], while there is only one Ising transition at ∆ c 3.303 [see Fig. 6(b)].
The central charge c * (L) calculated by DMRG also exhibits a peak structure around the critical points [see Figs. 6(c) and (d)]. These peaks become more distinct with increasing system size L. From the heights of the peaks at large L, we obtain the central charges c = 1 and c = 1/2, which are consistent with Gaussian-and Ising-type transitions, respectively. Moreover, the positions of the peaks agree with the QPT points estimated from the correlation length.

C Ground-state phase diagram for strong dimerization
With increasing dimerization the D-H phase is reduced, and it disappears for δ 0.26 [11] if we limit ourselves to the parameter region J > 0 and δ > 0.