Effective field theory for one-dimensional valence-bond-solid phases
and their symmetry protection
We investigate valence-bond-solid (VBS) phases in one-dimensional spin systems by an effective field theory developed by Schulz [Phys. Rev. B 34, 6372 (1986)]. While the distinction among the VBS phases are often understood in terms of different entanglement structures protected by certain symmetries, we adopt a different but more fundamental point of view, that is, different VBS phases are separated by a gap closing under certain symmetries. In this way, the effective field theory reproduces the known three symmetries: time reversal, bond-centered inversion, and dihedral group of spin rotations. It also predicts that there exists another symmetry: site-centered inversion combined with a spin rotation by . We demonstrate that the last symmetry gives distinct trivial phases, which cannot be characterized by their entanglement structure, in terms of a simple perturbative analysis in a spin chain. We also discuss several applications of the effective field theory to the phase transitions among VBS phases in microscopic models and an extension of the Lieb-Schultz-Mattis theorem to non-translational-invariant systems.
pacs:75.10.Kt, 75.10.Pq, 64.70.Tg
- I Introduction
- II Model
- III Bosonization and effective Hamiltonian
- IV Application to microscopic models
- V Symmetry protection of VBS phases
- VI Conclusion
- A Perturbative derivation of effective Hamiltonian
- B Compactifications of bosonic fields
Disordered ground states, which do not break any symmetry of the corresponding Hamiltonian, are classified into a single phase in the standard framework of the Landau-Ginzburg-Wilson (LGW) symmetry-breaking theory. However, it is now widely recognized that phase transitions can occur among those disordered states in quantum many-body systems. This suggests that there exists a variety of disordered quantum phases. Systematic understanding of these phases, usually referred to as topological phases, requires new concepts beyond the LGW theory.
While the standard characterization in terms of local order parameter fails for the topological phases, a more general way to classify the gapped quantum phases, based on the local unitary transformation (LUT), has been proposed in Ref. Chen et al. (2010). In this scheme, two gapped ground states of local Hamiltonians belong to the same phase if and only if the one is connected to the other by some LUT. If the two ground states belong to different phases, a gap closing (i.e. quantum phase transition) is necessary under any path that continuously connects the two Hamiltonians in the parameter space. In one dimension (1D), all the gapped ground states fall into a single phase unless some symmetry is imposed to the Hamiltonian and the LUT Chen et al. (2011a). Once some symmetry is imposed, there exist two classes of the quantum phases in 1D: conventional symmetry-breaking phase and symmetry-protected topological (SPT) phase Gu and Wen (2009); Chen et al. (2010). The latter phase does not break the imposed symmetry and thus cannot be characterized by local order parameters.
A notable example of SPT phases is the Haldane phase in a spin-1 chain Haldane (1983a, b). Its properties are well understood in terms of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state Affleck et al. (1987, 1988). Although there is no local order parameter due to the absence of symmetry breaking, the spin-1 Haldane phase typically has a nonlocal (string) order parameter den Nijs and Rommelse (1989); Kennedy and Tasaki (1992) and spin-1/2 gapless excitations at the ends of an open chain Kennedy (1990). While these features do not always exist once the AKLT Hamiltonian is perturbed, they can be traced back to a locally entangled nature of the state. In fact, this local entanglement can be directly observed as the two-fold degeneracy in the entanglement spectrum Pollmann et al. (2010). Furthermore, this entanglement cannot be removed as long as we keep one of three symmetries: time reversal, bond-centered inversion, and rotations around two orthogonal spin axes Pollmann et al. (2010, 2012). Thus, in the presence of these symmetries, the Haldane phase cannot be smoothly connected to unentangled states, that is direct-product states. Therefore the spin-1 Haldane phase is understood as an SPT phase.
The entangled nature of the Haldane phase is most conveniently described in terms of matrix-product state (MPS) Fannes et al. (1992); Klümper et al. (1992); Perez-Garcia et al. (2007). In the spin-1 Haldane phase, the aforementioned three symmetries act on the matrices with a nontrivial projective representation Pollmann et al. (2010), which enforces the degeneracy of the entanglement spectrum. In general, different SPT phases in 1D are labeled by different projective representations of a symmetry group Chen et al. (2011a, b); Schuch et al. (2011). Since the projective representation of corresponds to an element of a discrete group, the second-cohomology group , different SPT phases cannot be smoothly connected to each other; the projective representation is served as a topological invariant. In the language of MPS, an MPS must violate the injectivity to change this topological invariant Chen et al. (2011a, b); Schuch et al. (2011); Pollmann et al. (2012); this implies the existence of gap closing between different SPT phases. Since a topologically trivial state that can be smoothly connected to a direct-product state is characterized by the trivial projective representation (i.e. linear representation), it must be separated from the spin-1 Haldane phase by a gap closing.
A similar argument can be extended to valence-bond-solid (VBS) phases, which are constructed in a manner proposed by AKLT Affleck et al. (1988): On a lattice with spin-, we first decompose a spin- into spin-1/2’s on each site and then form spin singlets between nearest-neighboring sites. After distributing the singlet bonds over the whole lattice with keeping the lattice symmetry, we project the spin-1/2’s onto a spin- on each site. A state constructed in this way is called the VBS state and a phase that is adiabatically connected to a VBS state is called the VBS phase. VBS phases are realized as disordered ground states of various quasi-one-dimensional spin Hamiltonians: for examples, spin chains with single-site anisotropy Schulz (1986); den Nijs and Rommelse (1989); Tasaki (1991); Oshikawa (1992); Chen et al. (2003); Tonegawa et al. (2011); Kjäll et al. (2013), dimerized chains Affleck and Haldane (1987); Affleck et al. (1988); Arovas et al. (1988); Yamamoto (1997), and spin ladders White et al. (1994); Schulz (1996); Sierra (1996); Dell’Aringa et al. (1997); Reigrotzki et al. (1994); Hatano and Nishiyama (1995); Greven et al. (1996); Cabra et al. (1997, 1998a); Ramos and Xavier (2014). A phase transition between two VBS phases is observed when the parity of the number of singlets in one phase differs from that of the other under a certain spatial cut Kim et al. (2000). As in the case of the spin-1 Haldane phase, it is expected that this phase transition is due to the change of the topological invariant protected by one of the above three symmetries Pollmann et al. (2012).
Although the symmetry-protected nature of the VBS phases is well understood as the nontrivial projective representation on the MPS, going back to the original concept of the LUT, their nature in principle can be characterized by the presence or absence of gap closing between the two phases under a certain symmetry. The latter approach is particularly suitable for 1D systems, for which conformal field theory (CFT) provides a faithful description of gapless ground states, in contrast to that the MPS provides an efficient description of gapped ground states. We can also study the gapped ground states (with or without symmetry breaking) by perturbing the CFT in a quite controllable way by using the renormalization group. Thus we can directly examine the presence of gap closing from one phase to another in the field-theoretical approach. In this respect, the well-known classification of 1D interacting Majorana fermions with time reversal symmetry has been shown in both the MPS and field theory Fidkowski and Kitaev (2010, 2011). It is then natural to expect a similar field-theoretical approach for the VBS phases. In fact, Berg et al. Berg et al. (2008) have suggested the importance of an inversion symmetry for the stability of the spin-1 Haldane phase by a bosonization approach. However, their study did not yield the full identification of the symmetries later known by the MPS approach.
In this paper, we accomplish the field-theoretical approach to identify the full symmetries protecting the VBS phases in 1D spin systems. To this end, we adopt an effective low-energy theory proposed by Schulz Schulz (1986), which is a simple sine-Gordon theory obtained by Abelian bosonization. Surprisingly, this rather old theory actually captures essential properties of the VBS phases such as their symmetry protection. We first apply his theory to several spin systems, such as the spin chain, spin ladder, and those with a dimerization. We then show that, once different VBS phases are identified as gapped phases with different signs of the sine-Gordon coupling, his theory faithfully explains what symmetry preserves the gap closing between them. Those symmetries include the known three symmetries Pollmann et al. (2010): time reversal, bond-centered inversion, and dihedral group of spin rotations. Furthermore, there is another symmetry: site-centered inversion combined with a spin rotation by . Since this symmetry is not associated with the nontrivial projective representation Fuji et al. (2015), phases protected only by this symmetry are not distinguished by their entanglement structure. However if we adopt the first-principle definition of gapped phases, which states that they are distinguished by a gap closing, different VBS phases are still distinct even under the last symmetry. The effective theory also suggests that site-centered inversion symmetry forbids the unique gapped ground state for half-integer spin chains. This is a nontrivial extension of the Lieb-Schultz-Mattis theorem Lieb et al. (1961) to non-translational-invariant systems.
This paper is organized as follows. In Sec. II, we introduce an -leg spin-1/2 ladder model as a prototype of various quasi-1D spin Hamiltonians. In Sec. III, using Abelian bosonization technique, we revisit the low-energy effective theory for the ladder model, which is originally discussed by Schulz Schulz (1986). Although the original derivation was based on perturbation theory, we further argue its consistency with symmetry. The extension of the Lieb-Schultz-Mattis theorem is also discussed. In Sec. IV, we illustrate how the effective theory describes VBS phases for several well-known spin models. For those who just wish to appreciate how the effective theory describes different VBS phases, it will be enough to see Sec. IV.4 where the edge states are argued. Section V shows a main result of this paper: the low-energy theory faithfully identify the four symmetries protecting the VBS phases. We also draw a microscopic implication of the newly discovered symmetry. Section VI concludes this paper. Two appendices complement technical details of the analyses in the main text.
We consider an -leg spin-1/2 spin-ladder given by the Hamiltonian,
The first term represents decoupled spin-1/2 chains,
where is a spin-1/2 operator with a rung (leg) index (), is an antiferromagnetic intrachain exchange, and controls its uniaxial anisotropy. The second term represents interchain exchange couplings, which are generally written in the form,
Their coupling constants can be either ferromagnetic or antiferromagnetic. is taken to be a small integer such that the couplings will be short-ranged. A spin ladder model defined by is invariant under several symmetry operations: one-site translation , bond-centered inversion , time reversal , spin rotation around the axis, and rotation around the or axis. The last term in Eq. (1), , contains some perturbations that (partially or fully) break the above symmetries. In the following, we first identify the low-energy description of VBS phases in the ladder Hamiltonian and then examine the stability of gap closing between those VBS phases by adding the symmetry breaking perturbation .
Before proceeding, we here show examples of the simple limiting cases of our model (1) and briefly summarize their properties. If we take and , we have a ladder model only with perpendicular interchain couplings,
Depending on the boundary condition along the rung, we refer to this model as the spin tube if , while the open spin ladder if . Those ladder models are depicted in Fig. 1 (a) and (b). The case has been extensively studied Tsvelik (1991); Strong and Millis (1992); Barnes et al. (1993); Gopalan et al. (1994); Nishiyama et al. (1995); Strong and Millis (1994); Sénéchal (1995); Shelton et al. (1996); Orignac and Giamarchi (1998); Lecheminant and Orignac (2002); Hijii et al. (2005); Liu et al. (2012). There are also several systematic studies on -leg spin ladders and tubes White et al. (1994); Schulz (1996); Sierra (1996); Dell’Aringa et al. (1997); White et al. (1994); Reigrotzki et al. (1994); Hatano and Nishiyama (1995); Greven et al. (1996); Cabra et al. (1997, 1998a); Ramos and Xavier (2014). If the interchain couplings are ferromagnetic, the ground-state property is essentially the same as that of the spin- XXZ model. Thus, at the SU(2)-symmetric point and , we have the spin- Haldane phase for even while a gapless critical phase for odd . If the interchain couplings are antiferromagnetic, for even , the ground state is in a rung-singlet phase which is disordered and has a finite excitation gap. For odd , an open spin ladder has a gapless ground state, while a spin tube can have a gapped ground state with spontaneous breaking of translational invariance due to geometrical frustration.
Another example is a ladder mapping Timonen and Luther (1985); Schulz (1986); Timonen et al. (1991); White (1996); Legeza and Sólyom (1997); Kim and Sólyom (1999); Lecheminant et al. (2001) of the antiferromagnetic spin- XXZ chain,
where is a spin- operator on the site . If we decompose a spin- into spin-’s as
we can express as a sort of spin ladders with diagonal couplings (see Fig. 1 (c)),
Here the number of legs is . If the composite spins on each rung (6) are projected onto the fully symmetric sector with total spin , we will recover the physics of the single XXZ chain (5). However, even without th projection and even though we consider the model with a perturbatively small interchain coupling, we can still recover qualitatively the same low-energy physics as that of the XXZ chain Timonen and Luther (1985); Schulz (1986); Timonen et al. (1991); White (1996); Legeza and Sólyom (1997); Kim and Sólyom (1999); Lecheminant et al. (2001). The second term of Eq. (II) is nothing but Eq. (II) with , , and .
We can also introduce an on-site uniaxial anisotropy which is expressed as additional perpendicular couplings,
For integer and , this term drives the system into the so-called large- phase. This phase is adiabatically connected to a direct-product state of the states and has a finite gap.
Iii Bosonization and effective Hamiltonian
In this section, we apply the Abelian bosonization approach Giamarchi (2003); Gogolin et al. (1998); Cabra and Pujol (2004) to the ladder Hamiltonian (1) with . Following the discussion by Schulz Schulz (1986), we revisit an effective low-energy theory only with a single bosonic field. We further discuss its consistency with symmetry. We also propose an extension of the Lieb-Schultz-Mattis theorem for non-translational-invariant systems.
For , the decoupled chain part is a collection of critical spin-1/2 chains. In the continuum limit, is described by massless free bosons as
where we have introduced dual fields with respect to each chain, satisfying
and are the spin velocity and the Luttinger parameter,
and with the lattice spacing . In Eq. (9), we have neglected a marginally irrelevant term at since it is unimportant in the following discussion. Spin operators are expressed in terms of the bosonic fields as
where , , and are nonuniversal constants. Substituting these expressions into Eq. (II), we obtain
For brevity, we will collectively denote as . For instance, when we say that is relevant under renormalization group, all ’s are relevant. If we denote the scaling dimensions of as , they are given by , , , and . In general, the expression (III.1) is only valid for perturbatively small ’s. However, as long as there is a continuity between the weak-coupling and strong-coupling limits, we can use the above expression to investigate qualitative properties of the system for arbitrary strengths of ’s.
iii.2 Effective Hamiltonian
Since our model involves bosons and they are not decoupled in general, the analysis of the Hamiltonian is a formidable task. However, on the purpose to describe gapped disordered phases such as VBS phases, it is enough to see an effective Hamiltonian only with a single boson. To this end, let us introduce a center-of-mass field and relative fields with ,
If we add one extra dimension to and set , with forms an -dimensional orthogonal matrix satisfying
Their duals and are similarly defined as
The original chain fields are now represented as
In terms of these new fields, is rewritten as
We note that such a set of linear combinations of the fields is usually taken to diagonalize the marginal interactions, , and therefore not restrictive in the form (15). However, in order to derive the effective Hamiltonian only with a single bosonic field, it is essential to consider this particular form of (). For general Hamiltonians, this choice of linear combinations leaves some marginal interactions. Those interactions actually renormalize the original velocity and Luttinger parameter, but we assume that they do not affect the relevance of the coupling constants (III.1). In the following, we thus neglect the effect of the marginal coupling .
As seen from Eqs. (III.1) and (18), both the terms with and never involve the center-of-mass field () but may contain all the relative fields (). Our central assumption is to consider that is the most relevant coupling constant and reaches the strong-coupling limit faster than the other ’s. This is natural, in the sense that we must suppress any Ising antiferromagnetic order dominated by and , to favor gapped disordered phases. If the model is -symmetric or easy-plane anisotropic, this assumption is readily justified since has the smallest scaling dimension and the largest initial value. Once goes the strong-coupling limit, the relative fields are pinned and acquire masses. Correspondingly, their duals are strongly fluctuating. Thus we can integrate out (, ) and obtain an effective Hamiltonian only with the center-of-mass field (, ). As first shown by Schulz Schulz (1986), we obtain the effective Hamiltonian,
for even , while
for odd , where and are the renormalized velocity and Luttinger parameter. is the compactification radius of , which is explained in Sec. III.3. The vertex operator of is generated by -th (-th) order perturbation theory in for even (odd) . Even if accidentally vanishes, such a vertex is also generated by the same mechanism for , by replacing with its expectation value Kim and Sólyom (1999); Lecheminant et al. (2001). For completeness, we demonstrate the perturbative derivation of the effective Hamiltonians in Appendix A. Those effective Hamiltonians were also obtained directly from the spin- Heisenberg chain using non-Abelian bosonization through the CFT Cabra et al. (1998b); Nonne et al. (2011).
When the coupling or is relevant, the effective Hamiltonian describes a unique gapped ground state for even , which turns out to be in a VBS phase in Sec. IV, while a gapped degenerate ground state for odd . As we will see below, this becomes clear by considering the periodicity condition (compactification) of . We also give a non-perturbative argument for the validity of the effective Hamiltonian from the point of view of the symmetry.
iii.3 Compactifications of bosons and symmetry
In the effective Hamiltonians (III.2) and (III.2), in fact, the vertex operators of have the lowest scaling dimensions for each parity of , among those compatible with the compactification of the center-of-mass field and the symmetry of the Hamiltonian . To see this, we first return to the compactifications of the bosonic fields with respect to each chain,
Substitution of these relations into Eqs. (15) and (17) yields the compactifications of the new bosonic fields and . However, in deriving the effective Hamiltonian, we have assumed that were pinned and the relative fields could be integrated out. This modifies the compactification of the center-of-mass field from that for the fields remaining free. Indeed, we obtain the compactification for ,
This is demonstrated in Appendix B. As a consequence, this compactification forces vertex operators that can be added to the effective Hamiltonian to be of the form, , with some integers and .
Clearly, the vertex of Eq. (III.2) only has a single minimum of in the period , while that of Eq. (III.2) has two minima. The number of independent minima corresponds to the number of ground-state degeneracy once the vertex operator becomes relevant.
Further restrictions on the vertex operators come from the symmetry of the original Hamiltonian (1). All the symmetry operations considered in this paper are defined through individual operations on each spin- operator . From the bosonized forms of the spin operators (12), we can identify the corresponding symmetry transformation on the bosonic fields. In Table 1, we list the symmetries of the Hamiltonian (1) with and their transformations on the spin operators and the center-of-mass field. Symmetry operations on each chain field are also recovered by setting .
|Symmetry operation||Symbol||Transformation on spins||Transformation on field|
|Odd-site translation||trs||111 is an arbitrary odd integer.||,|
|Time reversal||222With complex conjugation.||,|
|rotation around axis||,||,|
|rotation around axis||,||,|
|rotation around axis||,||,|
We note that some symmetry transformations in Table 1 form larger symmetry groups. One such group consists of rotations around spin axes. For example, . This indeed means that the rotations around spin axes are elements of a dihedral group . Another is a 1D space group formed by an odd-site translation trs, bond-centered inversions , and site-centered inversion . This can be understood as follows: if we impose both the site-centered inversion symmetry with respect to the site and the bond-centered inversion symmetry with respect to the bond between and , those inversions automatically enforce the system to be invariant under the -site translation, i.e. (the dependence does not enter in the transformations on the bosonic field). Those transformation properties among the sets of symmetries are also reflected on the bosonic field, up to ambiguity from its compactification (23).
Although only rotations around spin axes are shown in Table 1, we have assumed the spin-rotational symmetry around the -axis in the Hamiltonian . In the bosonic language, this makes the effective Hamiltonian invariant under the shift with a real number in . Thus any vertex operator of is forbidden by the symmetry.
A symmetry constraint on the vertex operators of comes from the symmetries under . This restricts the vertex operators to even functions in , namely , . Another constraint arises from trs or . These symmetries leave the effective Hamiltonian invariant under the constant shift of by . For even , this shift can be absorbed in Eq. (23) so that for any positive integer is allowed. On the other hand, these symmetries only allow with even for odd . The above symmetry analysis is consistent with the effective Hamiltonians (III.2) and (III.2) derived by lowest-order perturbation theory, which only keep the vertex operators with the lowest scaling dimensions, i.e. the lowest values of . We note that the compatibility of the effective Hamiltonians with one-site translational invariance has already been pointed out in Ref. Lecheminant et al. (2001).
iii.4 An extension of the Lieb-Schultz-Mattis theorem
For half-odd-integer spin chains with one-site translational invariance and the spin-rotational symmetry, the Lieb-Schultz-Mattis theorem Lieb et al. (1961); Affleck and Lieb (1986) states that the ground state either has a gapless excitation or spontaneously breaks translational invariance in the thermodynamic limit. This theorem can also be understood by means of the bosonization approach Oshikawa et al. (1997). We first review the idea of Ref. Oshikawa et al. (1997) and then extend it to non-translational-invariant systems.
Within the effective Hamiltonian, if only one of vertex operators with and some real number is relevant, selects one of independent potential minima, resulting in a gapped degenerate ground state. On the other hand, if all the vertex operators are irrelevant, the ground state behaves as a free boson and thus is gapless. Therefore, if the effective Hamiltonian only allows the vertex operators with , the fate of the ground state is either gapless or degenerate with a finite excitation gap. This is indeed the case for half-odd-integer spin chains, or equivalently (in our approach), odd- spin-1/2 ladders with odd-site translational invariance. The ground-state degeneracy is accompanied by spontaneous breaking of translational invariance.
From our bosonization approach, the ground state is either gapless or degenerate, when the effective Hamiltonian only allows vertex operators with multiple potential minima corresponding to degenerate ground states with spontaneous symmetry breaking. For this condition to be satisfied, the symmetry is not necessarily required. As a consequence, we find that for half-odd-integer spin chains or odd- spin-1/2 ladders, the ground state is either gapless or degenerate, when the system has (i) trs and either or symmetries, or (ii) and either or symmetries. Both of those conditions leave the effective Hamiltonian invariant under
so that vertex operators lead to some degenerate ground state. We remark that for to transform in the above form, we must ensure that there is no magnetization. In fact, a finite magnetization in the axis modifies the transformation of and can lead to a unique gapped ground state Oshikawa et al. (1997). The absence of the magnetization is ensured by the symmetry under or . Although the present discussion based on the effective Hamiltonian is far from mathematical rigor, the same restriction on the ground state under the condition (i) has been proven by the MPS formalism Chen et al. (2011a). Actually, a slight modification of this MPS proof also leads to the condition (ii) 111This can be seen in Sec. VB4 of Ref. Chen et al. (2011a) with the following modification: Supposing that the inversion center is at the site , requires and thus at . This contradicts with the assumption that the symmetry acts on the physical Hilbert space with a nontrivial projective representation . Thus an MPS invariant under cannot be short-range correlated when a projective symmetry is imposed.. Although we here only consider or , the theorem can be generalized to any symmetry transforming in the nontrivial projective representation. A related discussion to the condition (ii) is also given in the point of view of a Berry phase associated with a local gauge twist Hirano et al. (2008).
Iv Application to microscopic models
In this section, we apply the effective Hamiltonian to several spin models. We especially focus on the integer-spin chain, even-leg spin ladder, and dimerized spin systems. These systems are known to exhibit VBS phases. We show that the distinction between different VBS phases is given by the sign of the coupling constant in the effective Hamiltonian (III.2). We also briefly discuss how to obtain the edge states.
iv.1 XXZ chain with integer spin
We first consider an XXZ chain with integer spin and an on-site uniaxial anisotropy,
Possible VBS phases in this model are schematically represented in Fig. 2 for and . For , its phase diagram has been extensively studied Schulz (1986); den Nijs and Rommelse (1989); Tasaki (1991); Chen et al. (2003) and there are two gapped disordered phases. One is the Haldane phase locating around the Heisenberg point and , and the other is the large- phase stabilized for sufficiently large . Both phases do not break any symmetry of the Hamiltonian, but they are separated by a Gaussian phase transition with central charge . For , the phase diagram of this model was also studied Schulz (1986); Schollwöck and Jolicoeur (1995); Schollwöck et al. (1996); Nomura and Kitazawa (1998); Aschauer and Schollwöck (1998); Tonegawa et al. (2011); Kjäll et al. (2013) and again the Haldane and large- phases were found. However, careful numerical simulations have indicated that there is no phase transition between these phases Tonegawa et al. (2011); Kjäll et al. (2013); Tzeng (2012). Thus, for , the Haldane and large- phases essentially belong to the same phase.
Such a difference between the and Haldane phases is extended to the difference between the odd- and even- Haldane phases Pollmann et al. (2012). This fact can be seen at the level of the effective Hamiltonian (III.2). Through the ladder mapping in Eq. (II), the initial coupling is given by
Assuming that , is most relevant and the relative fields can be integrated out. Applying -th order perturbation theory, we obtain the effective Hamiltonian (III.2) with the coupling constant Schulz (1986),
where is a nonuniversal constant. As demonstrated in Appendix A, it is reasonable to take the prefactor as some positive value.
If the effective coupling is relevant, the effective Hamiltonian leads to a unique gapped ground state without any symmetry breaking. We expect that this gapped ground state corresponds to the Haldane phase around while the large- phase for . For odd , increasing from zero, the coupling constant must change its sign. Thus, the Haldane and large- phases are naturally identified as the and regimes, respectively, and there manifestly exists a Gaussian transition at between those phases. For , this identification of the VBS phases by the sign of is justified by nonlocal order parameters Berg et al. (2008); Orignac and Giamarchi (1998); Nakamura (2003) and spin-1/2 edge states Lecheminant and Orignac (2002). On the other hand, for even , the coupling constant never changes its sign by increasing . Thus the Haldane and large- phases can adiabatically connect to each other by avoiding . From Eq. (28), it appears that there is a gapless point at , as predicted by Schulz Schulz (1986). But a finite is also generated from the coupling and then can open up a gap Kim and Sólyom (1999); Lecheminant et al. (2001). We note that a similar argument focusing on the power of the coupling has been done by Nonne et. al. Nonne et al. (2011) in the context of a 1D multi-component Hubbard model. Equation (28) may also indicate that the gap exponentially decays to zero in the semiclassical limit .
We also refer to the existence of the so-called intermediate- phase Oshikawa (1992), which is a realization of the spin-1 Haldane phase on spin-2 chains with uniaxial anisotropies. This phase is separated from the spin-2 Haldane and large- phases. Recent numerical simulations on the Hamiltonian (26) showed that it is absent Kjäll et al. (2013) or restricted in a quite narrow region on the parameter space Tonegawa et al. (2011); Tzeng (2012). At the level of the effective Hamiltonian, this subtlety on its existence can be seen from the fact that it requires higher-order perturbations to change the sign of . However, once we introduce a quartic anisotropy , the intermediate- phase is stabilized for a wide range of Kjäll et al. (2013); Okamoto et al. (2014). Since contributes to at the first order for , it is relatively easy to make the sign of positive, leading to the intermediate- phase. This also provides a strong evidence that the sign of is responsible for the distinction of VBS phases.
iv.2 Spin tube with even
As the second example, we consider an -leg spin tube with spin-1/2,
We here consider the even case. The ground state is in the rung-singlet phase for , which is smoothly connected to the direct-product state of singlets formed on each rung, while in the spin- Haldane phase for . Qualitative properties of the both phases can be understood in the strong-coupling limit .
Again, as in the previous example of the XXZ chain, we can see the difference between the odd- and even- Haldane phases in terms of the sign of the effective coupling constant,
where is a positive nonuniversal constant. For , the rung-singlet phase takes the negative sign of whereas the Haldane phase takes the opposite sign. This indicates that the rung-singlet and odd- Haldane phases belong to different phases separated by a gap closing. On the other hand, for , the rung-singlet and even- Haldane phases always share the same sign of and thus belong to the same phase.
In this model, to go from the region to the region, we have to pass through an obvious critical point at , corresponding to the decoupled critical chains. However, no matter what we take as a continuous path of parameters, we should observe a gap closing between the odd- Haldane and rung-singlet phases, according to the change of the sign of in the effective Hamiltonian (III.2). In fact, such a nontrivial phase transition has been observed for by introducing a diagonal exchange coupling Weihong et al. (1998); Kim et al. (2000); Fáth et al. (2001); Nersesyan and Tsvelik (2003); Starykh and Balents (2004); Kim et al. (2008) or a uniaxial anisotropy Liu et al. (2012). In contrast, we can find some path that connects the even- Haldane and rung-singlet phases without gap closing. In the spin tube (29) with , a direct observation of this adiabatic continuity has not been reported. Instead, in the two-coupled spin-1 chains,
(here is the spin-1 operator), the absence of the phase transition between the spin-2 Haldane and rung-singlet phases has been observed Todo et al. (2001); Berg et al. (2008); Pollmann et al. (2012). This is again explained in terms of the effective Hamiltonian (III.2) through the ladder mapping: the effective coupling takes the form with and hence does not change the sign.
The third example of interest is the open spin ladder with an explicit dimerization. As an example, we consider the -leg spin ladder (29) with a “columnar” dimerization,
The dimerization explicitly breaks both odd-site translational invariance and site-centered inversion symmetry. This extrinsic symmetry breaking does not affect the effective Hamiltonian for even but does for odd . It allows the vertex operator to be added to the effective Hamiltonian (III.2). Therefore, we can here deal with both the odd- and even- cases in the same effective Hamiltonian (III.2).
Eq. (32) is bosonized as
where is a nonuniversal coefficient. For , the dimerization contributes to the effective coupling as Cabra and Grynberg (1999)
where , according to a similar analysis to Appendix A. This implies that, for , the Haldane phase with is driven into another phase with by a strong dimerization Totsuka and Suzuki (1995); Martín-Delgado et al. (1996); Almeida et al. (2007). If we denote a VBS phase as the -VBS phase, which is adiabatically connected to the state with singlets on each odd bond and singlets on each even bond, the latter phase with is identified as the -VBS phase depicted in Fig. 3. On the other hand, if one starts from the rung-singlet [-VBS] phase for , no phase transition is expected by introducing the dimerization, since the sign of is unchanged.
For , the effective Hamiltonian takes the form of Eq. (III.2) and some VBS phases are expected to appear. The vertex operator in Eq. (III.2) is generated from perturbations such as and , and therefore we obtain the effective coupling,
with . For , we have a phase transition at some finite value of . This is consistent with a phase transition between the -VBS and -VBS phases, as found in Refs. Almeida et al. (2008a, b); Gibson et al. (2011). Schematic pictures of those phases are drawn in Fig. 3.
We note that the property of a VBS phase under some spatial transformation is also reflected in the effective Hamiltonian (III.2). As expected, phase transitions between two VBS phases only occur when the parity of the number of singlets under a spatial cut is changed. Every VBS phase realized on the odd- ladder changes this parity by odd-site translation or site-centered inversion. Thus, in the effective Hamiltonian (III.2), the sign of must be change under these operations. By construction, the corresponding symmetry operations in Table 1 indeed change the sign of , since is odd under those operations for odd . This is not the case for even ; since the parity of the number of singlets does not change under those operations, is also not affected.
In general, the columnar dimerization (32) gives the following leading contributions to the effective coupling :
for even , and
for odd , where and are positive nonuniversal constants. For , we can find at most distinct solutions for , although we have to determine the nonuniversal constants for the precise evaluation. This would coincide with the phase transitions and the VBS phases found in the dimerized spin- chain Affleck and Haldane (1987); Oshikawa (1992); Kato and Tanaka (1994); Yajima and Takahashi (1996); Yamanaka et al. (1996); Yamamoto (1997). The above discussion can also be applied to other shapes of the spin ladder and other configurations of the dimerization (e.g. staggered dimerization Martín-Delgado et al. (1996)).
iv.4 Edge states
We can also justify the above identifications of the VBS phases with the gapped phases of the effective Hamiltonian (III.2) by the existence of edge states. Although there exist related discussions from the non-linear sigma model Ng (1994) or the Majorana-fermion description Lecheminant and Orignac (2002), we could not find any discussion using Abelian bosonization. Therefore we here briefly explain how it works.
If we define the system on a finite segment , the open boundary condition corresponds to the Dirichlet boundary conditions on Lecheminant and Orignac (2002),
where are integers. Then the boundary condition for is given by
where is an integer. Let us first consider the even- case. For , is pinned at the potential minimum in the bulk, while it is locked into at the boundaries. (Recall that is defined modulo .) Thus there must be kinks near the boundaries, at which jumps by . Since these kinks break the symmetry corresponding to , , and , each edge gives two-fold ground-state degeneracy. On the other hand, the kinks do not necessarily break the symmetry involving or , since the coordinate is also transformed simultaneously. Therefore, the boundary degeneracy is not required for VBS phases protected only by those inversions Pollmann et al. (2010); Fuji et al. (2015). For , is pinned at in the bulk as well as the boundaries, so that it does not necessarily have kinks. For the odd- case, depending on the parity of , the locking position of differs by . This indicates that the position of the kink, i.e. the edge state, depends on where we introduce cuts in the system. This is also consistent with the VBS picture in Sec. IV.3.
V Symmetry protection of VBS phases
In the effective Hamiltonian (III.2), two distinct VBS phases are characterized by the different signs of and separated by a phase transition at . However, when some of the symmetries listed in Table 1 are broken by introducing the perturbation , we can add new vertex operators to the effective Hamiltonian (III.2). In order to see the role of symmetry on the distinction of the VBS phases, we examine the stability of the phase transition at in the presence of such extra vertex operators. For odd , we will explicitly break both the odd-site translational invariance trs and site-centered inversion symmetry as done in Sec. IV.3, so that the resulting effective Hamiltonian takes the form of Eq. (III.2) as for even . It turns out that one of , , , and is sufficient to maintain the distinction between the two gapped phases with and : the first three are known to protect the VBS phases from the entanglement point of view Pollmann et al. (2010), whereas the last one is not understood in the same way and separately discussed in Sec. V.3.
In the following, we assume that is small enough so that the effective Hamiltonian description with a single bosonic field is still valid (i.e. is most relevant and the relative fields can be integrated out). Furthermore, we assume certain uniformness of the perturbations; coupling constants of the vertex operators do not depend on the coordinate . This is practically done by keeping only the and components from the Fourier transform of the interaction, since the other components with incommensurate momenta vanish in the continuum limit. We do not consider random or point-like interactions since they cannot produce a uniform gap that remains finite for a sufficiently large system.
v.1 With symmetry
We first consider the case where the spin-rotational symmetry around the axis is preserved. This symmetry forbids any vertex operator of and makes the analysis much simpler. Although this case has been discussed by Berg et al. Berg et al. (2008) for , we here clarify the full set of symmetries protecting the VBS phase. From Table 1, we find that the three symmetry operations , , and (or ) take the same form of the transformation on ,
For even , also takes the same form up to the shift , which can be absorbed into the compactification of , given in Eq. (23). Once all these symmetries are explicitly broken, we can add a vertex operator odd in . Keeping only the most relevant operators, we obtain
If is relevant, is also relevant since they share the same scaling dimension. If we vary from to with fixed , we can continuously connect the two minima of , since we can unify the two vertex operators into a single one with and