Ground-state phase diagram of a spin-\frac{1}{2} frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases

Ground-state phase diagram of a spin- frustrated ferromagnetic XXZ chain:
Haldane dimer phase and gapped/gapless chiral phases

Shunsuke Furukawa Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan    Masahiro Sato Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258, Japan    Shigeki Onoda Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan    Akira Furusaki Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan
July 5, 2019

The ground-state phase diagram of a spin- XXZ chain with competing ferromagnetic nearest-neighbor () and antiferromagnetic second-neighbor () exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin- degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin- antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and Néel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and Néel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.

75.10.Jm, 75.10.Pq, 75.80.+q

I Introduction

The search for novel quantum states in frustrated magnets has been a subject of intensive theoretical and experimental research. One-dimensional (1D) systems offer unique laboratories for this search, as strong fluctuations enhance the tendency toward unconventional quantum states.Lecheminant05 () Among them, the 1D XXZ model with competing nearest-neighbor and second-neighbor interactions, defined by the Hamiltonian


provides a paradigmatic example expected to host rich variety of physics. Here represents the spin- operator at the site and parametrizes the XXZ exchange anisotropy. The model has frustration as far as is antiferromagnetic, irrespective of the sign of .

Early theoretical studies on the model (1) mostly considered the case when both and are antiferromagnetic.Majumdar69 (); Haldane82 (); Nomura94 (); White96 (); Nersesyan98 (); Hikihara01 () However, interest is now growing in the case of ferromagnetic and antiferromagnetic because of its relevance to quasi-1D edge-sharing cuprates. Among such cuprates, LiCuO (Refs. Masuda05, ; Park07, ), LiCuVO (Refs. Enderle05, ; Naito07, ), and PbCuSO(OH) (Ref. Yasui11, ), for example, exhibit multiferroic behaviors,Tokura10 (); Cheong07 () i.e., spiral magnetic orders and concomitant ferroelectric polarization at low temperatures. The negative sign of indeed plays a key role in stabilizing the vector chiral order responsible for these phenomena.FSO10 () By contrast, RbCuMoO (Ref. Hase04, ) shows no sign of magnetic order down to very low temperatures and may be considered as a candidate system for a spin liquid or a valence bond solid.

Figure 1: (Color online) The ground-state phase diagram of the model (1) with and . The vector chiral phase, which has a non-vanishing vector chirality (2), extends between the two boundaries with the “” symbols. These boundaries are determined as in Figs. 2 and 3 (see the vertical solid lines in these figures). Around the highly degenerate point , these two boundaries could not be determined accurately, but we expect both of them to continue to this point. It was also difficult to draw the boundary around the right top corner of the phase diagram. The onsets of (Haldane and even-parity) dimer and Néel orders occur inside the vector chiral phase, as indicated by the “” symbols (determined as in Figs. 14 and 15). Thus there are narrow intermediate phases (between the “” and “” symbols) where two kinds of orders coexist. The phase boundaries among the TLL, even-parity dimer, and Néel phases are determined in a previous work.FSF10 () On the right of the “” symbols, the even-parity dimer and Néel phases alternately appear when approaching the point : the first transition occurs at (“” symbols) and the second at (not shown).FSF10 () On the line where and , the ground state is ferromagnetic (FM). The “+” symbols inside the dimer and Néel phases indicate the Lifshitz line, on which the short-range spin correlation changes its character from incommensurate (IC) to commensurate (C); see Fig. 16.

In this paper, we study the ground-state properties of the spin- frustrated ferromagnetic XXZ chain (1) with and , by means of the infinite time evolving block decimation algorithm (iTEBD)Vidal07 () and effective field theories based on the bosonization methods. Previous works on the case with easy-plane anisotropy have discussed the competition among the vector chiral phase with gapless excitations and the dimer and Néel phases with gapped excitations. Tonegawa90 (); Somma01 (); Chubukov91 (); Nersesyan98 (); FSSO08 (); Sirker10 (); Itoi01 (); FSO10 (); FSF10 (); SFOF11 () The main goal of this paper is to present a conclusive phase diagram of the model (1), which is shown in Fig. 1, through detailed analyses that extends our previous works.FSO10 (); FSF10 (); SFOF11 () Firstly, we uncover the nature of the nonmagnetic phase around the SU(2)-symmetric case , which has long been controversial. We show that this phase has a dimer order associated with an emergent spin- degree of freedom on every other bond. We term this new phase the Haldane dimer phase. Secondly, we show the existence of narrow gapped phases that intervene between the gapless chiral phase and any one of gapped dimer and Néel phases. As weak inter-chain couplings are turned on, while the gapless chiral phase evolves into a spiral magnetic order, the Haldane dimer phase can be stabilized by a coupling with phonons due to the spin-Peierls mechanism. Our phase diagram may thus provide a useful starting point for understanding the competing phases in quasi-1D cuprates.

Let us briefly review previous results on the model (1) and summarize our new findings. While we are mainly concerned with the case of and in this paper, for comparison, we also review established results on the case of antiferromagnetic alongside.

In the classical limit , the ground state phase diagram of Eq. (1) does not depend on in the range . The ground state has ferromagnetic order for and antiferromagnetic (Néel) order for . For , the ground state is in a spiral magnetic ordered phase, in which the spins rotate by an incommensurate pitch angle along the spin chain. Except for the isotropic case , the spiral plane is fixed in the plane, and the vector chirality


has a non-vanishing uniform value independent of . Here stands for average in the ground state (with long-range order, if any).

In the ground state of the quantum spin- model, a long-range magnetic order with broken U(1) spin rotational symmetry is generally prohibited, unless the uniform magnetic susceptibility is divergent as in the case of ferromagnetism.Momoi96 () However, a long-range order (LRO) of the vector chirality that breaks only the parity symmetry can survive quantum fluctuations in the case of . Using the bosonization theory for and , Nersesyan et al.Nersesyan98 () predicted the appearance of the vector chiral phase with gapless excitations (as reviewed in Sec. IV.1.2). This gapless chiral phase shows the spatially uniform vector chirality and power-law decaying (incommensurate) spiral spin correlations; this phase may therefore be viewed as a quantum counterpart of the classical spiral phase. The gapless chiral phase competes with other quantum phases, in particular, valence bond solids driven by quantum fluctuations. In fact, for antiferromagnetic , a dimerized phase, in which the singlet state (written in the basis) is formed on dimerized bonds, appears in a large part of the classical spiral regime ,Majumdar69 (); Haldane82 (); Nomura94 (); White96 () and the gapless chiral phase appears only in a small region in the space spanned by and .Hikihara01 ()

The phase diagram for the case of ferromagnetic and easy-plane anisotropy is presented in Fig. 1. Early worksTonegawa90 (); Somma01 () mainly discussed the transition from the Tomonaga-Luttinger liquid (TLL) phase to a dimer phase with an even-parity unitChubukov91 () appearing for . Our recent worksFSO10 (); FSF10 () have uncovered a rich phase structure in an extended parameter space of and . In Ref. FSO10, , it was shown that the gapless chiral phase appears in a wide region, and survives up to the close vicinity of the isotropic case for (we also refer to Refs. FSSO08, ; Sirker10, for related earlier works). This remarkable stability of the gapless chiral phase for indicates that the sign of plays a crucial role in stabilizing the vector chirality and the associated ferroelectric polarization in multiferroic cuprates.Masuda05 (); Park07 (); Enderle05 (); Naito07 (); Yasui11 () In Ref. FSF10, , the instability of the TLL phase toward gapped phases was analyzed using the effective sine-Gordon theory combined with numerical diagonalization. It was found that the even-parity dimer phaseComment_dimer () discussed in Refs. Tonegawa90, ; Somma01, ; Chubukov91, and a Néel ordered phase appear alternately as is increased on the right side of the TLL phase in the phase diagram.

An important result of this paper is concerned with the nature of the nonmagnetic phase for around the SU(2)-symmetric case . Previous field-theoretical analysesNersesyan98 (); Itoi01 () have suggested that a dimer phase with a very small energy gap should appear in this region (as reviewed in Sec. III.1). However, neither a dimer order nor an energy gap has been detected in previous numerical studies. Using the iTEBD, which allow us to treat infinite-size systems directly, we present the first numerical evidence of a finite dimer order parameter


Remarkably, this dimer order is associated with ferromagnetic correlations of alternating strengths, in contrast to antiferromagnetic correlations in singlet dimers for . In this case, it is natural to interpret that effective spin- degrees of freedom emerge on the bonds with stronger ferromagnetic correlation, forming a valence bond solid stateAffleck87 () as in the Haldane chain.Haldane83 () We thus call this new phase the Haldane dimer phase.

We also present detailed analyses of the anisotropic case , extending our previous works.FSO10 (); FSF10 () In particular, we analyze the transition from the gapless chiral phase to each of the Haldane dimer, even-parity dimer, and Néel phases, and identify narrow intermediate gapped phases where two kinds of orders coexist (the regions between “” and “” symbols in Fig. 1). Furthermore, we describe how the properties of various phases can be captured in the language of the Abelian bosonizationGogolin98 (); Giamarchi04 () for , as summarized in Table 1.

The rest of the paper is organized as follows. In Sec. II, we present the numerical results on the order parameters and half-chain entanglement entropy, which provide the most basic information for identifying symmetry-broken phases. In Sec. III, we discuss in detail the dimer phases in the SU(2)-symmetric case from both field-theoretical Affleck88_review (); CFT96 (); Gogolin98 (); Giamarchi04 () and numerical analyses. In Sec. IV, we discuss the case with the easy-plane anisotropy . In particular, we review the effective field theory for the gapless chiral phase,Nersesyan98 () and, following Ref. Lecheminant01, , discuss its instability towards gapped chiral phases. The ranges of the gapped chiral phases are then determined numerically by analyzing the spin correlations. In Sec. V, we briefly describe how the quantum phases in the easy-axis caseTonegawa90 (); Igarashi89_FM (); Igarashi89_AFM () can be understood in the Abelian bosonization framework. In Sec. VI, we conclude the paper and discuss implications of our results for quasi-1D cuprates.

Ii Numerical analysis of order parameters

In this section, we present numerical results on several order parameters and half-chain entanglement entropy calculated by iTEBD. The vector chiral order parameter and the entanglement entropy are used to determine the boundaries of the region where the long-range vector chiral order exists (the “” symbols in Fig. 1). The numerical results in this section also suggest the existence of the narrow intermediate phases (between “” and “” symbols) in which the vector chiral order coexists with the dimer or Néel order. The precise ranges of these intermediate phases, however, will be determined in Sec. IV.2.2.

Before presenting the numerical results, let us briefly note characteristic features of our numerical method; for more detailed account of the method, see Supplementary Material of Ref. FSO10, . The iTEBD algorithmVidal07 () we employed is based on the periodic matrix product representation of many-body wave functions of an infinite system. It can directly address physical quantities in the thermodynamic limit, and is free from finite-size or boundary effects. The (variational) wave function is optimized to minimize the energy. The precision of the algorithm is controlled by the Schmidt rank , which gives the linear dimension of the matrices. We exploited the conservation of the total magnetization to achieve higher efficiency and precision of the calculations. When this algorithm is used in ordered phases, a variational state finally converges to a symmetry-broken state with an associated finite order parameter (if it is allowed by the periodicity of the matrix product state).note:iTEBD () In our implementation, we used a period-4 structure for the variational matrix product state. In this setting, the vector chiral, dimer, and Néel order parameters analyzed in this section can all be calculated through local quantities. In order to allow a finite vector chiral order parameter, the initial state must contain complex elements as a “seed” for the symmetry breaking.Okunishi08 ()

Figure 2: (Color online) (a) Chiral and (b,c) dimer order parameters and (d) half-chain entanglement entropy as functions of for fixed . These are calculated by the iTEBD with Schmidt ranks and . Panel (c) is a zoom of panel (b). In panel (d), and are defined for the bipartitions of the system at the bonds and , respectively. Solid vertical lines indicate the boundaries of the vector chiral phase, and are determined from the onsets of the vector chiral order parameter in panel (a) or more accurately from the peaks in the entanglement entropy in panel (d). Broken vertical lines indicate the transition points on which dimer orders set in. These points are difficult to locate within the analysis of dimer order parameters in panels (b,c), and are instead determined by the analysis of spin correlations in Fig. 14. Narrow intermediate phases exist between solid and broken vertical lines, where the vector chiral and dimer orders coexist. In the intermediate phase in , a dip in the entanglement entropy is seen in panel (d), as zoomed in the inset.
Figure 3: (Color online) (a) Chiral and (b) Néel order parameters and (c) half-chain entanglement entropy as a function of for fixed , calculated by the iTEBD. The solid and broken vertical lines indicate the onsets of the vector chiral and Néel orders, respectively. The former is determined by the peak position in the entanglement entropy in panel (c), while the latter is determined in Fig. 15. In the narrow intermediate phase in , the vector chiral and Néel orders coexist.

ii.1 Vector chiral order

Figures 2 and 3 present our numerical results along the vertical line and the horizontal line , respectively, in the phase diagram (Fig. 1). Let us first look at the vector chiral order parameter displayed in Figs. 2(a) and 3(a). This order parameter is always found to be spatially uniform along the spin chain in the present model, so we have fixed the site labels. By observing the rapid increase of , we find the onset of the vector chiral phase. It is natural to think that this rapid increase comes from the Ising nature of the transition with exponent for the spontaneous order parameter, as previously demonstrated in the XY case .Okunishi08 () To determine the transition points more precisely, however, we use the half-chain entanglement entropy explained next.

The half-chain von Neumann (vN) entanglement entropy is defined asVidal07 ()


where is a set of Schmidt coefficients associated with the decomposition of the infinite system into the left and right halves and is the Schmidt rank. As the system approaches a critical point characterized by a conformal field theory with a central charge , this quantity is known to diverge asVidal03 (); Calabrese04 ()


where is the correlation length and is a non-universal constant. In an iTEBD calculation with a finite Schmidt rank , the divergence of at the critical point is replaced by the increasing function of ,Pollmann09 ()


where is another non-universal constant. The calculated entanglement entropy is shown in Figs. 2(d) and 3(c). In Fig. 2(d), we plot two entropies and associated with the bipartitions of the system at the bonds and , since these bonds are inequivalent in the neighboring dimer phases. By finding peaks of , we can determine the boundaries of the vector chiral phase, more accurately than by using ; see the solid vertical lines in Figs. 2 and 3. In this way, we have determined the square symbols in Fig. 1. Although we could not extract from the current data of using Eq. (6) (which is expected to be satisfied for larger ), it is natural to expect that these critical points are characterized by the two-dimensional Ising universality class with (we again note that the critical exponent for this class was confirmed in the XY caseOkunishi08 ()).

In most part of the vector chiral phase, the entanglement entropy increases as a function of , indicating a critical nature. Indeed, in the effective field theory of Nersesyan et al.,Nersesyan98 () the gapless chiral phase has , and the increase of from the cases of to is roughly consistent with expected from Eq. (6) for . Near the boundaries (solid vertical lines), the entanglement entropy shows dips, whose implications will be discussed later.

ii.2 Dimer orders

Next we look at the and components of dimer order parameters,


The alternation of the sign of or along the spin chain would indicate some sort of dimer ordering. We assign the site labels in such a way that . The two order parameters are plotted in Figs. 2(b,c). We find that and are both finite and have mutually opposite signs for . By contrast, the two order parameters have small finite values of the same sign for ; in spite of the smallness, they are rather stable when the Schmidt rank is increased as seen in the zoomed plot in Fig. 2(c). These results indicate that the dimer phases in the two regions are of distinct types.

The nature of the dimer phase for can be easily understood as follows.Chubukov91 (); FSF10 () In the XY limit , the sign of in Eq. (1) can be reversed by performing the rotations of spins around the axis on every second sites. From the fact that the doubly degenerate ground states at are given by the products of singlet dimers, one finds, through the above -rotation transformation, that the exact ground states at are given by the dimer states whose unit is now replaced by (written in the basis). We note that this unit has the even parity with respect to the inversion about a bond center, in contrast to the odd parity of the singlet dimer at . The direct product states of even-parity dimers show . The mutually opposite signs of and and the approximate relation found for in Fig. 2(b) indicate that the even-parity nature of the dimer unit persists in this region. We thus call this phase the even-parity dimer phase.Comment_dimer () It is distinct from the singlet dimer phase appearing for , in which and show the same sign.

In the region in Fig. 2, and are both negative as in the singlet dimer phase. However, forming nearest-neighbor singlet dimers is unlikely for ferromagnetic . In Sec. III, we point out that the dimer order in this region is associated with ferromagnetic nearest-neighbor correlations of alternating strengths along the chain, in marked contrast to an antiferromagnetic correlation in a singlet dimer. A more detailed comparison of the dimer phases for and in the isotropic case () will be presented in Sec. III.

In the region of a finite vector chiral order () in Fig. 2, we find that the two dimer order parameters remain finite in the narrow regions between the solid and broken vertical lines. This indicates the existence of the chiral dimer phases (originally predicted in Ref. Lecheminant01, ), in which the vector chiral and dimer orders coexist and there are four-fold degenerate ground states below an excitation gap. In the entanglement entropy, a dip is seen in the interval , which also supports the existence of an intermediate gapped phase. The peaks in the entanglement entropy indicated by the solid lines in Fig. 2(d) correspond to the Ising critical point between two gapped phases. Between the two broken lines in Fig. 2, the dimer order parameters diminish and the entanglement entropy increases as we increase the Schmidt rank ; these features are consistent with the gapless chiral phase. The precise determination of the phase boundaries between gapped and gapless chiral phases is difficult within the analysis of the order parameters and entanglement entropy in Fig. 2; it will be done instead by analyzing spin correlation functions in Fig. 14 in Sec. IV.2.2.

ii.3 Néel order

The appearance of a Néel phase with spontaneous staggered magnetizations is discussed in detail in Ref. FSF10, . In Fig. 3(b), this Néel order is detected in the region by measuring . As in the case of the dimer phases, even in the region where the vector chiral order is finite (), the Néel order parameter remains finite. This indicates the existence of a narrow chiral Néel phase, in which the vector chiral and Néel orders coexist. The ground states in this phase should be four-fold degenerate with a finite excitation gap. In Fig. 3(c), a dip in the entanglement entropy can be found in this region, consistent with the expected gapped excitation spectrum. The precise determination of the transition point will be done in Fig. 15 in Sec. IV.2.2.

Iii Isotropic case

In this section, we present detailed analyses of the model (1) in the isotropic case . While it is known that the singlet dimer phase appears for ,Majumdar69 (); Haldane82 (); Nomura94 (); White96 (); Eggert96 () the nature of the nonmagnetic ground state in has not been well understood. In Sec. III.1, we summarize previous field-theoretical analysesNersesyan98 (); Itoi01 () for the weak-coupling limit , which predicted the appearance of dimer orders for both signs of . At first glance, this result may seem bizarre since the singlet dimerization on the bonds, as formed in the case of antiferromagnetic , is unlikely to occur in the case of ferromagnetic . In Sec. III.2, we present our numerical results and point out a remarkable difference between the and cases in the way how the system hosts the dimer order. This leads us to propose the picture of the “Haldane dimer phase” for the dimer phase with . Although the ground-state wave functions are largely different between the Haldane and singlet dimer phases, we argue that the two phases in fact share a common hidden order.

iii.1 Field-theoretical analyses

Here we summarize previous field-theoretical analysesNersesyan98 (); Itoi01 (); Allen97 (); Cabra00 (); Kim08 () for . In this regime, the model (1) can be viewed as two antiferromagnetic Heisenberg spin chains which are weakly coupled by the zigzag interchain coupling as in Fig. 4. We apply the Abelian and non-Abelian bosonization techniques to describe the two chains separately, and then treat the interchain coupling as a weak perturbation.

iii.1.1 Non-Abelian bosonization

We start from the non-Abelian bosonizationAffleck88_review (); CFT96 (); Gogolin98 () description of the isotropic model (1) with , and present the renormalization group (RG) analysis to identify (marginally) relevant perturbations.

In the limit , each isolated antiferromagnetic Heisenberg chain is described by the SU(2) Wess-Zumino-Witten (WZW) theory, with the spin velocity , perturbed by a marginally irrelevant backscattering term.Affleck88_review (); Gogolin98 (); Eggert96 () The spin operators in the -th chain () can be decomposed as


with and , where is the lattice spacing of each chain; see Fig. 4. The uniform and staggered components, and , have the scaling dimensions and , respectively. The former can be decomposed into chiral (right and left) components: . Another important operator is the (in-chain) staggered dimerization operator define by


which has the scaling dimension .

Figure 4: Zigzag chain picture for the - chain model (1). The axis indicates the coordinate for the continuum description.
Figure 5: Numerical solutions to the one-loop RG equations (16) for (a) and (b) . We set [see Eq. (14)]. It is found that the three coupling constants , , and are most relevant and grow under the RG, with asymptotically a simple ratio or for and , respectively. In the plots, factors are muliplied to and so that these ratios can be visually confirmed.

The inter-chain zigzag coupling produces at most marginal perturbations, in the RG sense, around the WZW fixed point; relevant perturbations such as are prohibited by the symmetry of the zigzag chain model. The symmetry-allowed marginal perturbations are summarized as


where runs over the following five operators:Itoi01 ()


Here is the backscattering term present in isolated chains. The zigzag coupling produces the current-current interactions, and , and the twist operator . The dimer twist operator is generated in the RG process as we see later. The bare coupling constants are given by


where was estimated in Ref. Eggert96, . All the operators in Eq. (11) have the scaling dimensions , and their competition in the RG flow must be analyzed carefully by deriving the RG equations. We define the dimensionless coupling constants


where is a dimensionless constant of order unity. Using the operator product expansions in the WZW theory,CFT96 (); Shelton96 (); Starykh04 (); Starykh05 (); Hikihara10 () the one-loop RG equationsCardy96 () are derived asNersesyan98 (); Itoi01 (); Cabra00 (); Kim08 ()


where the dot indicates the derivative () with respect to the change of the cutoff: . See Appendix A for the derivation of Eq. (16). We have ignored since it does not affect the flow of the other coupling constants at the one-loop level.

Numerical solutions to the RG equations (16) are presented in Fig. 5. For both signs of , the three coupling constants , , and finally grow to large values under the RG;Nersesyan98 (); Itoi01 () they asymptotically have the simple ratio or for and , respectively. Remarkably, finally grows with a positive sign for both signs of . For , in particular, it is initially negative but changes sign before starting to grow in the RG process. By contrast, retains the same sign as its initial value. The properties of the fixed points governed by large , , and are non-trivial. In fact, while the non-Abelian formalism allows us to derive the RG equations in a manifestly SU(2)-invariant form, it is often not very useful for discussing the physical roles of (marginally) relevant perturbations. In the next section, we proceed to the Abelian bosonization analysis to show that the positive development of induces a gapped state with a finite dimer order parameter .

As seen in Fig. 5(a) and (b), the coupling constants grow much more slowly for than for . This implies that for , the energy gap associated with the dimer order should be much smaller and the spin correlation length should be much larger. In fact, as argued by Itoi and Qin,Itoi01 () the correlation length becomes of astronomical scale [e.g., for the case of Fig. 5(b)]. Such a tiny gap or a large correlation length is very difficult to detect by any numerical investigation; the system effectively behaves like a gapless system even when the system size is macroscopically large. We stress, however, that this insight is based on the perturbative RG analysis for small , and it is possible that the energy gap grows to an observable magnitude as we increase . Our numerical result presented in Sec. III.2 indeed identifies a large but detectable correlation lengths around .

iii.1.2 Abelian bosonization

In this section, we use the Abelian bosonization formalism Giamarchi04 () to discuss the physical roles of the marginally relevant perturbations , , and identified in the non-Abelian analysis. Although the Abelian formalism obscures the SU(2) symmetry of the model, it has the advantage of simplifying identification of various orders with the pattern of locking of bosonic fields, as illustrated in Table 1.

Let us start from the two decoupled antiferromagnetic chains in the limit . We summarize the Abelian bosonization descriptionGogolin98 (); Giamarchi04 () of a single XXZ chain (), so that the same formulation can be used later in Sec. IV.1. Each decoupled XXZ chain labeled by is described by a Gaussian Hamiltonian


where the velocity and the TLL parameter are given by


The bosonic fields and satisfy the commutation relation


where is the step function


The spin and (in-chain) dimer operators are expressed in terms of the bosonic fields as


where , , (Refs. Hikihara98, ; Lukyanov97, ), and (Ref. Takayoshi10, ) are non-universal constants which depend on .

We now focus on the case , at which . To treat the coupled chains, it is useful to introduce the bosonic fields for symmetric and antisymmetric sectors:


The three perturbations found to grow in the non-Abelian analysis have the following expressions:Nersesyan98 (); Cabra00 (); Zarea04 (); Comment_O1_bos ()


Furthermore, the term, which is decoupled from the other terms in the RG equation (16), has the expression


The second lines of Eq. (25) and Eq. (26) can be combined with the Gaussian Hamiltonians (17) of the decoupled chains, leading to




Using the new Gaussian Hamiltonian , we can calculate the scaling dimension of the operators in Eq. (25). Specifically, the scaling dimension of and is given by and , respectively. In the non-Abelian analysis, we have seen that grows to a positive value in the RG flow irrespective of the sign of . Assuming , we find that the product of the two cosine operators in the first line of Eq. (25) (with scaling dimension ) is the most relevant term among those in Eq. (25). This term locks the bosonic fields at


These correspond respectively to finite positive or negative value of the dimer order parameter , since the (inter-chain) dimer operator is expressed as


In the last expression, the first term and the rest come from the and components of the spins, respectively. For the locking in Eq. (29), these components acquire both positive or both negative expectation values, in agreement with Fig. 2(c) and with the SU(2) symmetry of the model. It is worth noting that the locking positions of the two degenerate ground states in Eq. (29) are independent of the sign of in the isotropic case . The second most relevant terms in Eq. (25) are and with scaling dimension . As explained in Sec. IV.1, the former has the effect of inducing the incommensurability in spin correlations.Nersesyan98 () Since a finite energy gap opens due to in the dimer phases, the incommensurate spin correlations are expected to remain short-ranged.

iii.2 Numerical results and physical properties of dimer phases

In this section, we present numerical results on the model (1) in the isotropic case , and discuss physical properties of the dimer phases for different signs of . In agreement with the field-theoretical results reviewed in the previous section, we find that the dimer order parameter becomes finite for both signs of , and that there are doubly degenerate ground states with positive and negative . While we propose different physical pictures for the dimer orders in the and cases (Sec. III.2.1), we also discuss a hidden order common to the two cases (Sec. III.2.3). In the following, our numerical results (based on iTEBD with ) are presented for the ground state with .

iii.2.1 Local spin correlations

In Fig. 6(a), we plot nearest-neighbor spin correlations (with ) and the dimer order parameter for . While can be confirmed for both and , a notable difference between the two cases can be found in the signs of local spin correlations.

Figure 6: (Color online) (a) Nearest-neighbor spin correlations and the dimer order parameter , (b) the spin correlation length , and (c) the string correlation (34) with and , as a function of in the isotropic case . In panel (a), multiplied by is also plotted for (filled circular symbols). In panel (b), is too small to determine around .
Figure 7: (Color online) Sketches of (a) the singlet dimer state and (b) the Haldane dimer state. The thick lines indicate valence bonds. In (b), the encircled bonds indicate emergent spin-’s. From each of them, a valence bond emanate to each left and right; the wave function is given by a superposition of such valence bond covering states. Vertical cuts (dashed lines) are introduced to probe a hidden order; the number of valence bonds crossing with each cut is shown in the square. The alternation of odd and even numbers is found in both the states.

For , one of the following inequalities is always satisfied:


Namely, the system has a strong antiferromagnetic correlation on the bond and a weaker correlation on . In this case, it is natural to assume that singlet dimers are formed on the bonds , and are weakly correlated with each other, as schematically shown in Fig. 7(a). Hence we call this phase the singlet dimer phase. In particular, the ground state is exactly given by a direct product of singlet dimers at the Majumdar-Ghosh pointMajumdar69 () . In Fig. 6(a) we find that the weaker correlation changes the sign at this point.

By contrast, the following inequality is found to be satisfied when :


Namely, strong and weak ferromagnetic nearest-neighbor correlations alternate along the chain. This observation led us to propose that there should be emergent spin- degrees of freedom on the bonds that have stronger ferromagnetic correlation, as depicted by ellipses in Fig. 7(b). Since the total wave function is a spin singlet, such spin-’s are expected to form a valence bond solid stateAffleck87 () as in the spin- Haldane chain.Haldane83 () Namely, from each encircled bond in Fig. 7(b), two valence bonds emanate, one to the left and one to the right; the total wave function is obtained by superposing such valence bond covering states. We thus call the dimer phase with the Haldane dimer phase. The emergence of the Haldane chain physics in this phase is also supported by the presence of a hidden non-local order analyzed in Sec. III.2.3.

In Sec. III.1, it was argued that the marginal perturbation , which induces the dimer order, grows very slowly under the RG for and that the energy gap associated with the dimer order can be extremely small.Itoi01 () The result of Fig. 6(a) indicates that the dimer order parameter grows to a numerically detectable magnitude for intermediate values of , although the obtained values are much much smaller compared to the case (by a factor of around ). The weakness of the effect of in inducing the dimer order and the associated energy gap for is also seen in the spin correlation length discussed next.

iii.2.2 Spin correlation length

Figure 8: Determination of the spin correlation length , illustrated for . Assuming the asymptotic behavior (33), we plot the function , and tune such that the oscillation width of this function becomes as constant as possible as a function of .White96 () While the oscillations arise from the cosine factor in (33), it is difficult to extract the pitch angle from this figure; instead, calculations in Sec. IV.2.3 give .

We determine the spin correlation length in the dimer phases by using the method of Ref. White96, . Except at the Lifshitz point , the spin correlation function is expected to behave at long distances asWhite96 (); Nomura05 ()


In the incommensurate regions and , the pitch angle changes continuously from to , as will be discussed in Sec. IV.2.3 (see Fig. 16). For , is fixed at . To determine , we plot as a function of , and tune such that the amplitude of oscillations becomes as constant as possible, as illustrated in Fig. 8. While the coefficient in Eq. (33) is given by the oscillation amplitude in Fig. 8, it is not simple to determine which can fit these very rapid oscillations; instead it will be determined by calculating the spin structure factor in Fig. 13.

The calculated is plotted in Fig. 6(b). The data for are broadly in agreement with Ref. White96, .Comment_White96 () We find that the values of are much larger for than for , as anticipated from the magnitudes of the dimer order parameter in Fig. 6(a).

We use the above numerical data of the spin correlation length to infer the magnitude of the spin gap for . In general the spin gap should be inversely proportional to , with the proportionality constant being the spin velocity. From the data of Ref. White96, for , we extract an approximate relation . Applying the same relation to the case, we estimate the spin gap around to be roughly equal to . We note that this should be considered as a crude order of magnitude estimate.

iii.2.3 Hidden order

The singlet and Haldane dimer phases have different (local) features of short-range correlations as expressed in Eqs. (31) and (32). In spite of this local difference, the two phases in fact share a common non-local order, as we now explain. Let us count the number of valence bonds crossing the vertical cuts (dashed lines) depicted in Fig. 7. We find that even and odd numbers alternate in the same way in the two phases, when we take the ground state with . The existence of such a hidden non-local order can be probed numerically by calculating the string correlation functiondenNijs89 (); Tasaki91 (); Watanabe93 (); Nishiyama95 (); White96_ladder (); Kim00 ()


The intuition behind this expression is as follows. Consider a pair of spins on the bond , which the string correlation function (34) consists of. If an odd number of valence bonds cross any cut placed between the neighboring pairs, then the pattern of shows a hidden antiferromagnetic order, namely, alternation of and after removing all ’s (see figures in Refs. Nishiyama95, and Kim00, ). The correlation function (34) detects this hidden order and takes a non-vanishing value in the long-distance limit .

Figure 9: (Color online) String correlation function (34) for and in the isotropic case . For both values of , remains finite in the long-distance limit while decays to zero.

Figure 9 presents the numerical data of the string correlation functions (34) calculated with different starting points for the ground state with . We find that for both signs of , remains finite in the long-distance limit while decays to zero, in agreement with the even-odd structure in Fig. 7. We note that this behavior is also consistent with the bosonized expressions of the string correlationsNakamura03 ()


(with and being the two endpoints of the string) and the field locking position for the ground state with [see Eq. (29)]. The -dependence of for a long distance is shown in Fig. 6(c). Although the dimer order parameter shows a large difference in magnitude between the and cases, the values of the string correlation are rather comparable between the two cases.

Figure 10: (Color online) Entanglement spectra for (a) and (b) . The left and right panels are for the bipartition of the system at the bonds and , respectively. refers to the magnetization in the right half of the system. The lower entanglement level corresponds to the more important weight in the total state. We note that as an example, the exact singlet dimer ground state of Ref. Majumdar69, shows and for the two types of bipartition (with all the other levels at infinity).

Another way of probing the hidden order is to find the degeneracy in the entanglement spectrum.Pollmann10 () Using the Schmidt coefficients calculated in iTEBD, we plot the spectra in Fig. 10. Here the spectra are classified by the -component magnetization in the right half of the system (this classification is done in the process of our calculations to exploit the U(1) spin rotational symmetry for better efficiency). For the bipartition of the system at the bond (left panels), we find that the entanglement levels appear only for half-integer , and are all doubly degenerate due to the left-right symmetry around . By contrast, for the bipartition at (right panels), the entanglement levels appear only for integer , and non-degenerate levels are found for .Comment_ES () These features are found commonly for both signs of , and are consistent with the even-odd structure in Fig. 7.

In Fig. 7, we depicted short-range valence bonds only. However, the even-odd structure we discussed can be also defined in the presence of longer-range valence bonds. As the correlation length becomes longer, the weights of such longer-range valence bonds in the wave function would gradually grow while retaining the even-odd structure.Bonesteel89 () We expect that through this process, the Haldane dimer state of Fig. 7(b) smoothly changes into the exact resonating valence bond ground state at , in which valence bonds are uniformly distributed over all distances.Hamada88 ()

iii.2.4 Adiabatic connectivity to a ladder model

Figure 11: Expected phase diagram of the zigzag ladder model with alternating nearest-neighbor couplings and . and are assumed to be small. The solid diagonal line corresponds to the original model (1) (with ), and represents the first-order phase transition line in the current model. The vertical and horizontal dashed lines correspond to a usual ladder model (no phase transition on these lines). In four insets of zigzag ladders, thick lines indicate valence bonds, and ovals indicate the formation of effective spin-’s.

In order to gain further intuition about the two dimer phases, it is useful to introduce explicit bond alternation of the couplings in the Hamiltonian (1) (with ). Namely, we place inequivalent couplings and on the bonds and , respectively. Figure 11 displays an expected phase diagram for small and . This phase diagram can be obtainedKim00 (); Kim08 () by noticing that in the non-Abelian bosonization framework, the bond alternation induces the relevant term with scaling dimension in the Hamiltonian, which leads to the ground state where acquires a finite average with the same sign as that of . The limit