Quantum phase transitions in a strongly entangled spin-orbital chain: A field-theoretical approach

# Quantum phase transitions in a strongly entangled spin-orbital chain: A field-theoretical approach

Alexander Nersesyan The Abdus Salam International Centre for Theoretical Physics, 34100, Trieste, Italy
Andronikashvili Institute of Physics, Tamarashvili 6, 0177, Tbilisi, Georgia
Center of Condensed Mater Physics, ITP, Ilia State University, 0162, Tbilisi, Georgia
Gia-Wei Chern    Natalia B. Perkins Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA
###### Abstract

Motivated by recent experiments on quasi-1D vanadium oxides, we study quantum phase transitions in a one-dimensional spin-orbital model describing a Haldane chain and a classical Ising chain locally coupled by the relativistic spin-orbit interaction. By employing a field-theoretical approach, we analyze the topology of the ground-state phase diagram and identify the nature of the phase transitions. In the strong coupling limit, a long-range Néel order of entangled spin and orbital angular momentum appears in the ground state. We find that, depending on the relative scales of the spin and orbital gaps, the linear chain follows two distinct routes to reach the Néel state. First, when the orbital exchange is the dominating energy scale, a two-stage ordering takes place in which the magnetic transition is followed by melting of the orbital Ising order; both transitions belong to the two-dimensional Ising universality class. In the opposite limit, the low-energy orbital modes undergo a continuous reordering transition which represents a line of Gaussian critical points. On this line the orbital degrees of freedom form a Tomonaga-Luttinger liquid. We argue that the emergence of the Gaussian criticality results from merging of the two Ising transitions in the strong hybridization region where the characteristic spin and orbital energy scales become comparable. Finally, we show that, due to the spin-orbit coupling, an external magnetic field acting on the spins can induce an orbital Ising transition.

## I Introduction

Over the past decades, one-dimensional spin-orbital models have been a subject of intensive theoretical studies. The interest is to a large extent motivated by experimental discovery of unusual magnetic properties in various quasi-one-dimensional Mott insulators. axtell ; isobe The inter-dependence of spin and orbital degrees of freedom is usually described by the so-called Kugel-Khomskii Hamiltonian in which the effective spin exchange constant depends on the orbital configuration and vice versa. kk ; tokura Another mechanism of coupling spin and orbital degrees of freedom is the on-site relativistic spin-orbit (SO) interaction , where is the orbital angular momentum and is the coupling constant. In compounds with quenched orbital degrees of freedom, the presence of the SO term usually leads to the single-ion spin anisotropy where and denotes the energy scale of the crystal field which lifts the degenerate orbital states.

For systems with residual orbital degeneracy, on the other hand, the effect of the SO term is much less explored compared with the Kugel-Khomskii-type coupling. Due to the directional dependence of the orbital wave functions, the SU(2) symmetry of the Heisenberg spin exchange is expected to be broken in the presence of the SO interaction. The resultant spin anisotropy is likely to induce a long-range magnetic order in the spin sector. A more intriguing question is what happens to the orbital sector. To answer this question, one needs to consider the details of the interplay between the orbital exchange and the SO coupling. Here we consider the simplest case of a two-fold orbital degeneracy per site. Specifically, the two degenerate states could be the and orbitals in a tetragonal crystal field observed in several transition-metal compounds. We introduce pseudospin-1/2 operators () to describe the doublet orbital degrees of freedom assuming that correspond to the states and , respectively. Alternatively, one can also realize the double orbital degeneracy in the Mott-insulating phase of a 1D fermionic optical lattice where the eigenvectors of refers to and orbitals in an anisotropic potential.zhao ; wu Restricted to this doublet space, the orbital angular momentum operator . This can be easily seen by noting that the eigenstates of carry an angular momentum .

The exchange interaction between localized orbital degrees of freedom is characterized by its highly directional dependence: the interaction energy only depends on whether the relevant orbital is occupied for bonds of a given orientation. This is particularly true for interactions dominated by direct exchange mechanism. Denoting the relevant orbital projectors on a given bond as , where is an appropriate pseudospin-1/2 operator ( being a Pauli matrix), the orbital interaction is thus described by an Ising-type term . The well studied orbital compass model and Kitaev model both belong to this category.nussinov ; kitaev The quantum nature of these models comes from the fact that different operators , which do not commute with each other, are used for bonds of different types. To avoid unnecessary complications coming from the details of orbital interactions, we assume that there is only one type of bond in our 1D system and the orbital interaction is thus governed by a classical Ising Hamiltonian.

We incorporate these features into the following toy model of spin-orbital chain ():

 H = HS+Hτ+HSτ = JS∑nSn⋅Sn+1+Jτ∑nτznτzn+1+λ∑nτxnSzn.

Motivated by the recent experimental characterizations of quasi-1D vanadium oxides,mamiya ; reehuis ; onoda ; lee ; pieper ; niazi here we focus on the case of quantum spin with length . The above model thus describes a Haldane chain locally coupled to a classical Ising chain by the SO interaction . The role of the -term is two-fold: firstly it introduces anisotropy to the spin-1 subsystem, and secondly it endows quantum dynamics to the otherwise classical Ising chain.

Before turning to a detailed study of the phase diagram of model (I), we first discuss its connections to real compounds. As mentioned above, the interest in the toy model is partly motivated by the recent experimental progress on vanadium oxides which include spinel ZnVO mamiya ; reehuis ; onoda ; lee and quasi-1D CaVO. pieper ; niazi In both types of vanadates, the two electrons of V ions have a spin in accordance with Hund’s rule. In the low-temperature phase of both vanadates, the vanadium site embedded in a flattened VO octahedron has a tetragonal symmetry. This tetragonal crystal field splits the degenerate triplet into a singlet and a doublet. As one of the two electrons occupies the lower-energy state, a double orbital degeneracy arises as the second electron could occupy either or orbitals. The fact that the orbital is occupied everywhere also contributes to the formation of weakly coupled quasi-1D spin-1 chains in these compounds. chern1 On the other hand, the details of the orbital exchange depends on the geometry of the lattice and in the case of vanadium spinel the orbital interaction is of three-dimensional nature. The Ising orbital Hamiltonian in Eq. (I) thus should be regarded as an effective interaction in the mean-field sense. Nonetheless, the toy model provides a first step towards understanding the essential physics introduced by the SO coupling. Moreover, many conclusions of this paper can be applied to the case of quasi-1D compound CaVO where the vanadium ions form a zigzag chain.

It is instructive to first establish regions of stable massive phases. In the decoupling limit, , our model describes two gapped systems: a quantum spin-1 Heisenberg chain and a classical orbital Ising chain. The ground state of the spin sector is a disordered quantum spin liquid with a finite spectral gaphaldane , whereas the orbital ground state is characterized by a classical Néel order along the chain: . Quantum effects in the orbital sector induced by the SO coupling play a minor role. Obviously, just because of being gapped, both the spin-liquid phase and the orbital ordered state are stable as long as remains small. Consider now the opposite limit, . In the zeroth order approximation, the model is dominated by the single-ion term whose doubly degenerate eigenstates represent locally entangled spin and orbital degrees of freedom. Switching on small and leads to a staggered ordering of the and states along the chain. Physically, the large- ground state can be viewed as a simultaneous Néel ordering of spin and orbital angular momentum characterized by order parameters and such that and . The Ising order parameter vanishes identically in this phase.

These observations naturally lead to the following questions. How is the magnetically ordered Néel state at large connected to the disordered Haldane phase as ? What is the scenario for the orbital reorientation transition , which is of essentially quantum nature ? In this paper we employ the field-theoretical approach to address these questions. We first note that the one-dimensional model (I) is not exactly integrable. As a consequence, the regime of strong hybridization of the spin and orbital excitations, which is the case when , and are all of the same order, stays beyond the reach of approximate analytical methods. We thus will be mainly dealing with limiting cases and , in which one can integrate out the “fast” variables to obtain an effective action for the “slow” modes. Following this approach, we establish the topology and main features of the ground-state phase diagram in the accessible parts of the parameter space of the model. We were able to unambiguously identify the universality classes of quantum criticalities separating different massive phases. Using plausible arguments we comment on some features of the model in the regime of strong spin-orbital hybridization.

We demonstrate that the aforementioned reorientation transition can be realized in one of two possible ways. In the limit of large , we find a sequence of two quantum Ising transitions and an intermediate massive phase, sandwiched between these critical lines, in which both and are nonzero. This is consistent with the recent findings chern2 based on DMRG calculations and some analytical estimations. In the opposite limit, when the Haldane gap is the largest energy scale, integrating out the spin excitations yields an effective lowest-energy action for the orbital degrees of freedom, which shows that the crossover takes place as a single Gaussian quantum criticality. At this critical point, the orbital degrees of freedom display an extremely quantum behaviour: they are gapless and form a Tomonaga-Luttinger liquid. This is the main result of this paper. We bring about arguments suggesting that the emergence of the Gaussian critical line is the result of merging of the two Ising criticalities in the region of strong spin-orbital hybridization.

Any field-theoretical treatment of the model (I) must be based on a properly chosen contiuum description of the spin-1 antiferromagnetic Heisenberg chain. Its properties have been thoroughly studied, both analytically and numerically (see for a recent review Ref. mk, ). In what follows, the spin sector of the model (I) will be treated within the O(3)-symmetric Majorana field theory, proposed by Tsvelik: t

 HM=∑a=1,2,3[iv2(ξaL∂xξaL−ξaR∂xξaR)−imξaRξaL]+Hint.

Here is a degenerate triplet of real (Majorana) Fermi fields with a mass , the indices and label the chirality of the particles, and

 Hint=12g∑a(ξaRξaL)2

is a weak four-fermion interaction which can be treated perturbatively. The continuum theory (LABEL:Maj-Tsv) adequately describes the low-energy properties of the generalized spin-1 bilinear-biquadratic chain

 HS→¯HS=JS∑n[Sn⋅Sn+1−β(Sn⋅Sn+1)2]. (3)

in the vicinity of the critical point . babu This quantum criticality belongs to the universality class of the SU(2) Wess-Zumino-Novikov-Witten (WZNW) model with central charge .

At small deviations from criticality the Majorana mass determines the magnitude of the triplet gap, . The theory of a massive triplet of Majorana fermions is equivalent to a system of three degenerate noncritical 2D Ising models, with . This is one of the most appealing features of the theory because the most strongly fluctuating physical fields of the chain, namely the staggered magnetization and dimerization operators, have a simple local representation in terms of the Ising order and disorder parameters.t ; snt ; gnt It is this fact that greatly simplifies the analysis of the spin-orbital model (I). While the correspondence between the models (LABEL:Maj-Tsv) and (3) is well justified at , it is believed that the Majorana model (LABEL:Maj-Tsv) captures generic properties of the Haldane spin-liquid phase of the spin-1 chain, even though at large deviations from criticality () all parameters of the model should be treated as phenomenological.

The remainder of the paper is organized as follows. We start our discussion with Sec. II which contains a brief summary of known facts about the Majorana modelt that will be used in the rest of the paper. In Sec. III we consider the limit and by integrating out the ‘fast’ orbital modes, show that on increasing the SO coupling the system undergoes a sequence of two consecutive quantum Ising transitions in the spin and orbital sectors, respectively. In section IV we analyze the opposite limiting case, , and, by integrating over the ‘fast’ spin modes, show that there exists a single Gaussian transition in the orbital sector accompanied by a Neel ordering of the spins. We then conjecture on the topology of the ground-state phase diagram of the model. In Sec. V we show that spin-orbital hybridization effects near the orbital Gaussian transition lead to the appearance of a non-zero spectral weight well below the Haldane gap which can be detected by inelastic neutron scattering experiments and NMR measurements. In Sec. VI we comment on the role of an external magnetic field. We show that, through the SO interaction, a sufficiently strong magnetic field affects the orbital degrees of freedom and can lead to a quantum Ising transition in the orbital sector. Sec. VII contains a summary of the obtained results and conclusions. The paper has two appendices containing certain technical details.

## Ii Some facts about Majorana theory of spin-1 chain

In this Section, we provide some details about the O(3)-symmetric Majorana field theory,t Eq. (LABEL:Maj-Tsv), which represents the continuum limit of the biquadratic spin-1 model (3) at .

In the continuum description, the local spin density of the spin model (3) has contributions from the low-energy modes centered in momentum space at and :

 S(x)=IR(x)+IL(x)+(−1)x/a0N(x) (4)

The smooth part of the local magnetization, , is a sum of the level-2 chiral vector currents. The SU(2) Kac-Moody algebra of these currents is faithfully reproduced in terms of a triplet of massless Majorana fields zamo_fat :

 Iν=−i2(\boldmathξν×% \boldmathξν),   (ν=R,L) (5)

This fact is not surprising because, as already mentioned, the central charge of the SU(2) WZNW theory is , whereas that of the theory of a massless Majorana fermion (equivalently, critical 2D Ising model) is . At small deviations from criticality () the fermions acquire a mass. Strongly fluctuating fields of the spin-1 chain, the staggered magnetization and dimerization operator , are nonlocal in terms of the Majorana fields but admit a simple representation in terms of the order, , and disorder, , operators of the related noncritical Ising models:

 N ∼ (1/α)(σ1μ2μ3, μ1σ2μ3, μ1μ2σ3), ϵ ∼ (1/α)σ1σ2σ3, (6)

where is a short-distance cutoff of the continuum theory. These expressions together with their duals (i.e. their counterparts obtained by the duality transformation in all Ising copies, ) determine the vector and scalar parts of the WZNW 22 matrix field which is a primary scalar field with scaling dimension 3/8. It has been demonstrated in Ref. zamo_fat, that using the representation (6) and the short-distance operator product expansions for the Ising fields, one correctly reproduces all fusion rules of the SU(2) WZNW model. An equivalent way to make sure that this is indeed the case is to consider the four-Majorana representation of the weakly coupled spin-1/2 Heisenberg ladder snt ; gnt and take the limit of a infinite singlet Majorana mass to map the low-energy sector of the model on the O(3) theory (LABEL:Maj-Tsv).

In the spin-liquid phase of the spin chain (3), which is the case , the Majorana mass is positive, implying that the degenerate triplet of 2D Ising models is in a disordered phase: , . In particular, this implies that the O(3) symmetry remains unbroken, , and the ground state of the system is not spontaneously dimerized, .

The representation (6) proves to be very useful for calculating the dynamical spin correlation functions because the asymptotics of the Ising correlators and are well known both at criticality and in a noncritical regime. In the disordered phase (), the leading asymptotics of the Ising correlators are:

 ⟨μ(r)μ(0)⟩ ∼ (a/ξS)1/4[1+O(e−2r/ξS)], ⟨σ(r)σ(0)⟩ ∼ (a/ξS)1/4√ξS/r e−r/ξS (7)

where is the correlation length, and . (By duality, in the ordered phase () the asymptotics of the correlators in (7) must be interchanged.) Correspondingly, the dynamical correlation function

 ⟨N(r)N(0)⟩∼(a/ξS)3/4√ξS/r e−r/ξS. (8)

Its Fourier transform at and small describes a coherent excitation – a triplet magnon with the mass gap :

 Im χ(q,ω)∼m|ω|δ(ω−√(q−π)2v2+m2). (9)

Since the single-ion anisotropy lowers the original O(3) symmetry down to O(2) Z, one expects t that in the continuum theory it will induce anisotropy in the Majorana masses

 m1=m2≠m3,

as well as in the coupling constants parametrizing the four-fermion interaction:

 Hint→12∑a≠bgab(ξaRξaL)(ξbRξbL),   g13=g23≠g12 .

This can be checked by using the correspondence (4) and short-distance operator product expansions (OPE) for the physical fields. There will also appear anisotropy in the velocities, , but we will systematically neglect this effect. Thus, we have , with

 Hanis=Dα∫dx[I3(x)I3(x+α)+N3(x)N3(x+α)], (10)

where is a short-distance cutoff of the continuum theory. Using (5) and keeping only the Lorentz invariant terms (i.e. neglecting renormalization of the velocities) we can replace by . To treat the second term in the r.h.s. of (10), we need OPEs for the products of Ising operators: cft

 σ(z,¯z)σ(w,¯w) =1√2(α|z−w|)1/4[1−π|z−w|ε(w,¯w)], (11)
 μ(z,¯z)μ(w,¯w) =1√2(α|z−w|)1/4[1+π|z−w|ε(w,¯w)]. (12)

Here is the energy density (mass bilinear) of the Ising model, and are two-dimensional complex coordinates, and are their conjugates. From the above OPEs it follows that

 N3(x)N3(x+α)=i(π/α)(ξ1Rξ1L+ξ2Rξ2L−ξ3Rξ3L) −(π2C)[(ξ1Rξ1L)(ξ2Rξ2L)−(ξ1Rξ1L)(ξ3Rξ3L)−(ξ2Rξ2L)(ξ3Rξ3L)],

where is a nonuniversal constant. As a result,

 Hanis=−i∑a=1,2,3δma ξaRξaL+12∑a≠bδgij(ξaRξaL)(ξbRξbL), (13)

where

 δm1=δm2=−δm3=−(πC)D (14)

are corrections to the single-fermion masses, and are coupling constants of the induced interaction between the fermions. Smallness of the Majorana masses () implies that the additional mass renormalizations caused by the interaction in (13) are relatively small, , so that the main effect of the single-ion anisotropy is the additive renormalization of the fermionic masses, , with given by Eq.(14).

The cases and correspond to an easy-plane and easy-axis anisotropy, respectively. The spin anisotropy (18) induced by the spin-orbit coupling is of the easy-axis type. At the singlet Majorana fermion, , is the lightest, . Increasing anisotropy drives the system towards an Ising criticality at , where . At the system occurs in a new phase where the Ising doublet remains disordered while the singlet Ising system becomes ordered. It then immediately follows from the representation (6) that the new phase is characterized by a Néel long-range order with . Transverse spin fluctuations, as well as fluctuations of dimerization, are incoherent in this phase.

## Iii Two Ising transitions in the \boldmathΔS\boldmath≪J\boldmathτ limit

Now we turn to our model (I). Let us consider the case when, in the absence of spin-orbit coupling, the orbital gap is the largest: . The orbital pseudospins then represent the ‘fast’ subsystem and can be integrated out. Assuming that , we treat the spin-orbit coupling perturbatively. In this case, the zero order Hamiltonian describes decoupled spin and orbital systems, while the spin-orbit interaction denotes perturbation. Defining the interaction representation for all operators according to (here denotes imaginary time), the interaction term in the Euclidian action is given by

 SSτ=λ∑n∫dτ τxn(τ)Szn(τ). (15)

The first nonvanishing correction to the effective action in the spin sector is of the second order in :

Averaging in the right-hand side of (III) goes over configurations of the classical Ising chain . The correlation function is calculated in Appendix A. It is spatially ultralocal (because there are no propagating excitations in the classical Ising model) and rapidly decaying at the characteristic time , which is much shorter than the spin correlation time :

 ⟨τxn(τ1)τxm(τ2)⟩τ=δnmexp(−4Jτ|τ1−τ2|). (17)

Passing to new variables, and , and integrating over yields a correction to the effective spin action which has the form of a single-ion spin anisotropy. Thus in the second order in , the spin Hamiltonian acquires an additional term

 Hani=−λ24Jτ∑n(Szn)2. (18)

The anisotropy splits the Majorana triplet into a doublet and singlet , with masses

 m1=m2=m+πCλ24Jτ,   m3=m−πCλ24Jτ, (19)

where is a nonuniversal positive constant. The anisotropy is of the easy-axis type, so that the singlet mode has a smaller mass gap.

As long as all the masses remain positive, the system maintains the properties of an anisotropic Haldane’s spin-liquid. The dynamical spin susceptibilities calculated at small and (see Sec. II),

 Im χxx(q,ω)=Im χyy(q,ω) (20) ∼m1|ω|δ(ω−√(q−π)2v2+m21), Im χzz(q,ω)∼m3|ω|δ(ω−√(q−π)2v2+m23),

indicate the existence of the and optical magnons with mass gaps and , respectively. Increasing the spin-orbital coupling leads eventually to an Ising criticality at , where . At the system occurs in a long-range ordered Néel phase with staggered magnetization , in which the -symmetry of model (18) is spontaneously broken. Using the Ising-model representation (6) of the staggered magnetization of the spin-1 chain, we find that at the order parameter follows a power-law increase:

 ζ(λ)∼(λ−λc1λc1)1/8. (21)

The transverse spin fluctuations become incoherent in this phase. The situation here is entirely similar to that in the spontaneously dimerized massive phase of a two-chain spin-1/2 laddergnt ; nt , where the dimerization kinks make spin fluctuations incoherent. In the present case, the spontaneously broken symmetry of the Neel phase leads to the existence of pairs of massive topological kinks contributing to a broad continuum with a threshold at (the details of calculation can be found in Ref.gnt, ):

 Im χxx(q,ω) (22) ∼1√m1|m3|θ(ω2−(q−π)2v2−(m1+|m3|)2)√ω2−(q−π)2v2−(m1+|m3|)2.

In the Néel phase, the orbital sector acquires quantum dynamics because antiferromagnetic ordering of the spins generates an effective transverse magnetic field which transforms the classical Ising model to a quantum Ising chain. At the spin-orbit term takes the form

 HSτ=−h∑n(−1)nτxn+H′Sτ, (23)

where and accounts for fluctuations. Since both the orbital and spin sectors are gapped, the main effect of this term is a renormalization of the mass gaps and group velocities. The transverse field gives rise to quantum fluctuations which decrease the classical value of and, at the same time, lead to a staggered ordering of the orbital pseudospins in the transverse direction. Since the orbital sector has a finite susceptibility with respect to a transverse staggered field, in the right vicinity of the critical point follows the same power-law increase as but with a smaller amplitude:

 ηx∼(hJτ)∼√ΔSJτ(λ−λc1λc1)1/8. (24)

This result is in a good agreement with previously obtained numerical results for order parameters (See Fig. 4(a) in Ref. chern2, ).

Performing an inhomogeneous -rotation of the pseudospins around the -axis, , , we find that at the effective model in the orbital sector reduces to a ferromagnetic Ising chain in a uniform transverse (pseudo)magnetic field:

 Hτ;eff=−Jτ∑nτznτzn+1−h∑nτxn. (25)

Notice that the restriction , which was imposed in the derivation of the effective Hamiltonian in the spin sector, now can be released because the spin sector is assumed to be in the Néel phase.

At , i.e. at where satisfies the equation

 λc2 ζ(λc2)=Jτ, (26)

the model (25) undergoes a 2D Ising transition kogut ; gnt to a massive disordered phase with . This quantum critical point can be reached when is further increased in the region . It is clear from (26) that is of the order of or greater than . It is reasonable to assume that for such values of the Néel magnetization is close to its nominal value, , implying that . We see that the two Ising transitions are well separated:

 λc2/λc1∼(Jτ/ΔS)1/2≫1. (27)

Thus, in the limit , the ground-state phase diagram of the model (I) consists of three gapped phases separated by two Ising criticalities, one in the spin sector () and the other in the orbital sector (). At the spin sector represents an anisotropic spin-liquid while in the orbital sector there is a Néel-like ordering of the pseudospins: . At the orbital degrees of freedom reveal their quantum nature: the onset of the spin Néel order () is accompanied by the emergence of the transverse component of the staggered pseudospin density: . Upon increasing , the staggered orbital order parameter undergoes a continuous rotation from the -direction to -direction. At a quantum Ising transition takes place in the orbital sector where vanishes. At both sectors are long-range ordered, with order parameters . The dependence of order parameters on is schematically shown in Fig. 2(a); this picture is in full qualitative agreement with the results of the recent numerical studies. chern2

The crossover between the small and large limits studied in this section corresponds to path 1 on the phase diagram shown in Fig. 1. The path is located in the region . Starting from the massive phase I and moving along this path we first observe the spin-Ising transition (I II) to the Néel phase. Long-range ordering of the spins induces quantum reconstruction of the initialy classical orbital sector (i.e. generation of a nonzero ). The orbital-Ising transition (II III) takes place inside the spin Néel phase. Of course, feedback effects (that is, orbit affecting spin) become inreasingly important upon deviating from the critical curve into phases II and III, especially in the vicinity of the orbital transition where the spin-orbit coupling is very strong, . In this region the behavior of the spin degrees of freedom is not expected to follow that of an isolated anisotropic spin-1 chain in the Néel phase since the effect of an “explicit” staggered magnetic field becomes important. We will see a pattern of such behavior in the opposite limit of “heavy” spins, which is discussed in the next section.

## Iv Gaussian criticality at J\boldmathτ\boldmath≪\boldmathΔS

In this section we turn to the opposite limiting case: . Now the spin degrees of freedom constitute the “fast” subsystem and can be integrated out to generate an effective action in the orbital sector. We will show that, in this regime, the intermediate massive phase where the orbital order parameter undergoes a continuous rotation from to no longer exists. Going along path 2, Fig. 1, which is located in the region , we find that the two massive phases, I and III, are separated by a single Gaussian critical line characterized by central charge . On this line the vector vanishes, the orbital degrees of freedom become gapless and represent a spinless Tomonaga-Luttinger liquid characterized by power-law orbital correlations.

At the spin-1 subsystem represents a disordered, isotropic spin liquid. Therefore the first nonzero correction to the low-energy effective action in the orbital sector appears in the second order in :

 ΔS(2)τ=−16⟨S2Sτ⟩S (28) =−12λ2∑nm∫dτ1∫dτ2 ⟨Sn(τ1)Sm(τ2)⟩S τxn(τ1)τxm(τ2),

where means averaging over the massive spin degrees of freedom. According to the decomposition of the spin density, Eq. (4), the correlation function in (IV) has the structure:

 ⟨Sl(τ)S0(0)⟩=(−1)lf1(r/ξS)+f2(r/ξS). (29)

Here is the spin correlation length and is the Euclidian two-dimensional radius-vector. and are smooth functions with the following asymptotic behaviour gnt

 f1(x)=C1x−1/2e−x,  f2(x)=C2x−1e−2x  (x≫1), (30)

where and are nonuniversal constants. DMRG calculations show sorensen that ; for this reason the contribution of the smooth part of the spin correlation function can be neglected in (IV).

Integrating over the relative time we find that the spin-orbit coupling generates a pseudospin -exchange with the following structure:

 H′τ=∑n∑l≥1(−1)l+1J′τ(l)τxnτxn+l (31)

Here the exchange couplings exponentially decay with the separation , , so the summation in (31) actually extends up to . In the Heisenberg model is of the order of a few lattice spacings, so for a qualitative understanding it would be sufficient to consider the term as the leading one and treat the term as a correction. Making a rotation in the pseudospin space, , , we pass to the conventional notations and write down the effective Hamiltonian for the orbital degrees of freedom as a perturbed XY spin-1/2 chain:

 Heffτ=∑n(Jxτxnτxn+1+Jyτynτyn+1)+H′τ. (32)

where

 H′τ=−J′x∑nτxnτxn+2+⋯. (33)

Here , and . By order of magnitude .

In the absence of the perturbation , the model (32) represents a spin-1/2 XY chain which for any nonzero anisotropy in the basal plane () has a Néel long-range order in the ground state and a massive excitation spectrum. This follows from the Jordan-Wigner transformation

 τzn=2a†nan−1,   τ+n=τxn+% iτyn=2a†neiπ∑j

which maps the XY chain onto a model of complex spinless fermions with a Cooper pairing:lsm

 Heffτ = (Jx+Jy)∑n(a†nan+1+h.c.) (35) + (Jx−Jy)∑n(a†na†n+1+h.c.).

By increasing (equivalently, decreasing ) the model (35) can be driven to a XX quantum critical point, , i.e. , where the the system acquires a continuous U(1) symmetry. At this point the Jordan-Wigner fermions become massless and the system undergoes a continuous quantum transition.

The transition is associated with reorientation of the pseudospins. Away from the Gaussian criticality the effective orbital Hamiltoian is invariant under transformations: , . In massive phases this symmetry is spontaneously broken. Making a back rotation from to we conclude that at () , , while at () , . Both and vanish at the critical point, so contrary to the case , here there is no region of their coexistence.

The passage to the continuum limit for the model (32) based on Abelian bosonization is discussed in Appendix B. There we show that the perturbation adds a marginal four-fermion interaction to the free-fermion model (LABEL:XY-cont). In the spin-chain language, this is equivalent to adding a weak ferromagnetic -coupling. In the limit of weak XY anisotropy, , the low-energy properties of the orbital sector are described by a quantum sine-Gordon model (all notations are explained in Appendix B)

 H=u2[KΠ2+1K(∂xΦ)2]+2γπαcos√4πΘ, (36)

where

 γ∼Jτ(λ−λcλc),   K=1+2g+O(g2). (37)

The U(1) criticality is reached at where, due to a finite value of , the orbital degrees of freedom represent a Tomonaga-Luttinger liquid. Close to the criticality, the spectral gap in the orbital sector scales as the renormalized mass of the sine-Gordon model (36):

 Morb∼∣∣λ−λcλc∣∣K2K−1. (38)

Strongly fluctuating physical fields acquire coupling dependent scaling dimensions. In particular, according to the bosonization rules,gnt the staggered pseudospin densities are expressed in terms of the vertex operators,

 (−1)nτxn ≡ nx(x)∼sin√πΘ(x), (−1)nτzn ≡ nz(x)∼cos√πΘ(x), (39)

both with scaling dimension . This anomalous dimension determines the power-law behaviour of the average staggered densities close to the criticality:

 ηz(λ)∼(λc−λ)1/4K, λ<λc ηx(λ)∼(λ−λc)1/4K, λ>λc. (40)

A finite staggered pseudospin magnetization at generates an effective external staggered magnetic field in the spin sector:

 HS→¯H=HS+H′S,   H′S=−hS∑n(−1)nSzn, (41)

where . The spectrum of the Hamiltonian is always massive. This can be easily understood within the Majorana model (LABEL:Maj-Tsv). According to (6), in the continuum limit, the sign-alternating component of the spin magnetization, , can be expressed in terms of the order and disorder fields of the degenerate triplet of 2D disordered Ising models: . In the leading order, the magnetic interaction gives rise to an effective magnetic field applied to the third Ising system: . The latter always stays off-critical.

Since in the Haldane phase the spin correlations are short-ranged, close to the transition point the induced staggered magnetization can be estimated using linear response theory. Therefore, at , follows the same power-law increase as that of but with a smaller amplitude:

 ζ∼hSΔS∼(JτΔS)1/2(λ−λcλc)1/4K (42)

So, in the part of the phase C, Fig. 1, where , the -orbital order, being the result of a spontaneous breakdown of a symmetry , acts as an effective staggered magnetic field applied to the spins and induces their Néel alignment. This fact is reflected in a coupling dependent, nonuniversal exponent characterizing the increase of the staggered magnetization at . The order parameters as functions of in the limit is schematically shown in Fig. 2(b).

As already mentioned, the absence of a small parameter in the regime of strong hybridization, , makes the analysis of the phase diagram in this region not easily accessible by analytical tools. Nevertheless some plausible arguments can be put forward to comment on the topology of the phase diagram. It is tempting to treat the curve as a single critical line going throughout the whole phase plane (). If so, we then can expect that there exists a special singular point located in the region . This expectation is based on the fact that at limit the transition is of the Ising type and the spontaneous spin magnetization below the critical curve follows the law with a universal critical exponent, whereas at the spin magnetization has a different, nonuniversal exponent, . Continuity considerations make it very appealing to suggest that at the special point the Tomonaga-Luttinger liquid parameter takes the value , and the two power laws match. Since the central charges of two Ising and one Gaussian criticalities satisfy the relation , the singular point must be a point where the two Ising critical curves merge into a single Gaussian one.

## V Dynamical sin susceptibility and NMR relaxation rate in the vicinity of Gaussian criticality

It may seem at the first sight that, in the regime , the spin degrees of freedom which have been integrated out remain massive across the orbital Gaussian transition, and the spectral weight of the staggered spin fluctuations is only nonzero in the high-energy region . However, this conclusion is only correct for the zeroth-order definition of the spin field , given by Eq. (6), with respect to the spin-orbit interaction. In fact, the staggered magnetization hybridizes with low-energy orbital modes via SO coupling already in the first order in and thus acquires a low-energy projection which contributes to a nonzero spectral weight displayed by the dynamical spin susceptibility at energies well below the Haldane gap.

To find the low-energy projection of the field , we must fuse the local operator with the perturbative part of the total action. Keeping in mind that close to and at the Gaussian criticality most strongly fluctuating fields are the staggered components of the orbital polarization, we approximate the SO part of the Euclidian action by the expression

 SSτ≃λa0vS∫d2r Nz(r)nx(r), (43)

where is the two-dimensional radius vector (here is the imaginary time). We thus construct

 NzP(r)=⟨e−SSτNz(r)⟩ =Nz0(r)−λa0vS∫d2r1⟨Nz0(r)Nz0(r1)⟩S nx(r1) + O(λ2), (44)

where averaging is done over the unperturbed, high-energy spin modes. For simplicity, here we neglect the anisotropy of the spin-liquid phase of the S=1 chain and use formula (8). The spin correlation function is short-ranged. Treating the spin correlation length as a new lattice constant (new ultraviolet cutoff) and being interested in the infrared asymptotics , we can replace in (44) by . The integral

 ∫d2\boldmathρ ⟨Nz0(% \boldmathρ)Nz0(0)⟩S (45) ∼1a20(a/ξS)3/4∫∞0dρ ρ√ξs/ρ e−ρ/ξS∼(ξS/a0)5/4.

So the first-order low-energy projection of the staggered magnetization is proportional to

 NzP(r)∼λΔS(ξSa0)1/4nx(r). (46)

This result clarifies the essence of the hybridization effect: close to the Gaussian criticality the spin fluctuations acquire a finite spectral weight in the low-energy region, , , which is contributed by orbital fluctuations and can be probed in magnetic inelastic neutron scattering experiments and NMR measurements.

Away from but close to the Gaussian criticality the behavior of the dynamical spin susceptibility is determined by the excitation spectrum of the sine-Gordon model for the dual field, Eq.(36). Since , it consists of kinks, antikinks carrying the mass , and their bound states (breathers) with masses (see e.g. Ref. gnt, )

 Mj=2Morbsin(πj/2ν), j=1,2,…ν−1,  ν=2K−1 (47)

Since is small, there will be only the first breather in the spectrum, with mass . The sine-Gordon model is integrable, and the asymptotics of its correlation functions in the massive regime have been calculated using the form-factor approach (see for a recent review essler, ). Here we utilize some of the known results. At the operator has a nonzero matrix element between the vacuum and the first breather state. This form-factor contributes to a coherent peak in the dynamical spin susceptibility at frequencies much smaller than than the Haldane gap:

 Imχ(q,ω,T=0) = A(λ/ΔS)2δ[ω