A Correcting sine-Gordon

# Excitations and quasi-one-dimensionality in field-induced nematic and spin density wave states

## Abstract

We study the excitation spectrum and dynamical response functions for several quasi-one-dimensional spin systems in magnetic fields without dipolar spin order transverse to the field. This includes both nematic phases, which harbor “hidden” breaking of spin-rotation symmetry about the field and have been argued to occur in high fields in certain frustrated chain systems with competing ferromagnetic and antiferromagnetic interactions, and spin density wave states, in which spin-rotation symmetry is truly unbroken. Using bosonization, field theory, and exact results on the integrable sine-Gordon model, we establish the collective mode structure of these states, and show how they can be distinguished experimentally.

## I Introduction

Much of the research in frustrated quantum magnets has focused on the elusive quest for magnetically disordered phases with highly entangled ground states: quantum spin liquids Balents (2010). Somewhat intermediate between these rare beasts and commonplace antiferromagnets are moderately exotic phases of antiferromagnets in strong magnetic fields which exhibit no dipolar magnetic order transverse to the field, contrary to typical spin-flop antiferromagnetic states. One such state, the Spin Nematic (SN), has received a particularly high degree of theoretical attention Andreev and Grishchuk (1984); Tsunetsugu and Arikawa (2006); Penc and Lauchli (2011). Argued to occur in some quasi-one-dimensional strongly frustrated insulators with competing ferromagnetic and antiferromagnetic interactions Chubukov (1991), the SN phase has a “hidden” order which breaks spin-rotation symmetry about the magnetic field despite the lack of transverse spontaneous local moments. A less celebrated but competitive state in such systems is the collinear Spin Density Wave (SDW) Starykh et al. (2010); Chen et al. (2013), which develops magnetic order but with spontaneous moments, whose magnitude is spatially modulated, entirely along the magnetic field direction. Both types of phases are strongly quantum, i.e. cannot occur in classical models with moments of fixed length at zero temperature. The absence of transverse moments in both phases may lead the two to be confused experimentally, and one of the reasons for the present study is to clearly define the characteristics that distinguish them in laboratory measurements.

A spin nematic is usually defined as a state without any spontaneous dipolar order, i.e. so that in a magnetic field along , , but with quadrupolar order, , for nearby sites . Such a nematic breaks the spin rotation symmetry about the field axis, but in a more non-trivial way than a usual canted antiferromagnet. The spin nematics relevant to this paper are based on the frustrated Heisenberg chain with ferromagnetic nearest-neighbor coupling and antiferromagnetic second-neighbor coupling, in a strong magnetic field. For a region of parameters, the single magnon excitations with of the fully saturated high field state are bound into pairs with . Roughly, these latter excitations “condense” upon lowering the field, leading to a spin nematic state Hikihara et al. (2008); Sudan et al. (2009); Zhitomirsky and Tsunetsugu (2010). Some caution should be exercised, however, since in one dimension true condensation is not possible, and spontaneous breaking of rotational symmetry about the field cannot occur. A sharp characterization of the one-dimensional (1d) spin nematic is, rather than nematic order, the presence of a gap to excitations. The 1d SN state may be thought of more properly as a bose liquid of particles, and hence has not only power-law nematic order but also power-law density fluctuations of those bosons Hikihara et al. (2008) (see Sec. II.2.2). The latter is just power-law SDW correlations. Inter-chain couplings can stabilize either long-range nematic or SDW order. One of our results is that, in fact SDW order is typically more stable, and true nematic long-range order occurs only in a narrow range of applied fields very close to the fully saturated magnetization.

More generally, SDW order also occurs in frustrated 1d systems from other mechanisms, unrelated to magnon pairing and 1d spin nematicity. Thus we will spend considerable time in this paper discussing the properties of the SDW. At the level of order parameter, an SDW state is described by the expectation value

 ⟨Szi⟩=M+Re[Φeiksdw⋅ri]+⋯ (1)

where the ellipses represent higher order harmonics that may be present, or small effects from spin-orbit coupling etc. SDW states are are relatively common in itinerant systems with Fermi surface instabilities Grüner (1994), but much less so at low temperature in insulating spin systems, which tend to behave classically and hence possess magnetic moments of fixed length. From the point of view of symmetry, the SDW breaks no global symmetries (time reversal symmetry is broken and the axis is already selected by a magnetic field), but instead breaks translational symmetry. Consequently, its only low energy mode is expected to be the pseudo-Goldstone mode of these broken translations, known as a phason. The phason is a purely longitudinal mode, as it corresponds to the phase of above and hence a modultion only of . This is also unusual in the context of insulating magnets, as the low energy collective modes are usually spin waves, which are transverse excitations, associated with small rotations of the spins away from their ordered axes. In spin wave theory, indeed, longitudinal modes are typically expected to be highly damped, and hence either undefined or hard to observe Affleck and Wellman (1992); Schulz (1996). In SDW states, they can instead control the low energy spectral weight in a scattering experiment. The SDW state also has transverse excitations, as we discuss in Sec. III.2.2, but these exhibit a spectral gap which is generally non-zero. They can be distinguished from the phasons by their polarization and their location in momentum space.

In this paper, we focus primarily on the excitations of SDW and 2d spin nematic states. We show how to use the tools of one dimensional field theory, combined with the random phase approximation (RPA) and other methods to obtain both excitations and their contributions to different components of the dynamical and momentum dependent spin susceptibilities in a quantitative fashion. This analysis is greatly facilitated by the use of copious exact results on the excitations and correlation functions of the one dimensional sine-Gordon model Essler et al. (2003); Essler and Konik (2004); Gogolin et al. (2004). The results for the excitations of SDW states can also be easily extended to describe magnetization plateaux, which can be viewed as SDW states pinned by the commensurate lattice potential Starykh et al. (2010). Most of the results for SDW excitations carry over directly to such plateaux, with the main modification that the phason develops a small gap due to pinning.

In experiment, inelastic neutron scattering is a powerful way to study the SDW and 2d spin nematic states, and for convenience we summarize several distinguishing features identified from our analysis here. Both states have linearly dispersion gapless modes: phasons in the SDW case and the Goldstone modes (“quadrupolar waves”) in the nematic case Shindou et al. (2013); Bar’yakhtar et al. (2013); Smerald and Shannon (2013). In the structure factor, the phason appears with greatest weight at the SDW wavevector, which is in general incommensurate and away from the zone center and boundary. Here it gives a pole contribution whose weight diverges as as the energy of the pole approaches zero. The phason also contributes, although much more weakly, in the vicinity of the zone center, with a pole whose weight vanishes as the wave vector approaches zero. For the nematic, there is no divergent gapless contribution, and the gapless mode appears only at the zone center. The weights of the zone center contributions, though they both vanish on approaching , differ in the angular dependence of the weight of the low energy pole. Another distinction is in the gapped portion of the spectrum. In the SDW case, the lowest gapped excitation, which carries a relatively large spectral weight, occurs usually at , and occurs in the transverse () channel (a caveat here is that, in the SDW arising out of 1d spin nematic chains, this is not the case, and the transverse excitation at is pushed to high energy). In the nematic, the lowest energy gapped excitations occur instead at the incommensurate value , and excitations at appear only at much larger energies.

The rest of the paper is structured as follows. In Sec. II, we introduce bosonization and one-dimensional effective field theories in a general fashion which can be applied to both SDW and spin nematic states, in several different physical contexts. In Sec. III we derive the excitations and structure factor of the SDW phase, and in Sec. IV we do the same for the 2d nematic phase. We conclude in Sec. V with a Discussion of other ways to compare SDW and spin nematic phases, and of existing experiments. Several appendices contain technical details to support the results in the main text.

## Ii One dimensional effective theory

In this section, we introduce the standard bosonization description which applies to many critical one dimensional systems, and establish notations to be used in the rest of the paper. A unified formalism of this type applies to several distinct physical situations, which we delineate below.

To justify the bosonization treatment, we will consider a quasi-one-dimensional geometry, composed of spin chains or ladders, coupled together by somewhat weaker exchange interactions between these one dimensional units. Each unit is characterized by some exchange scale , presumed the largest in the problem, which sets a temperature scale , such that a low energy effective description of the one dimensional units applies for . Interactions amongst the one-dimensional units can them be described in terms of the low energy field theory, i.e. bosonization. These interactions, , induce ordering with a temperature , where the exponent is in general depedent upon more details of the interactions between and within the one-dimensional subsystems. Specific cases will be discussed below.

### ii.1 Bosonization for 1d Bose liquids

The low energy physics of a great variety of one dimensional spin systems can be described by bosonization in terms of free scalar bosonic field theory. We introduce one such field theory per one dimensional unit or chain, indexing these units by a discrete variable . We presume spin rotational symmetry about the axis, which allows but does not require a magnetic field along this axis.

Due to the symmetry, we may view a spin-1/2 system as a Bose liquid, mapping for example the state to the vacuum, the state to a (hard core) boson, and thereby to boson creation/annihilation operators. The Bose liquid language has an advantage in that it allows for a unified view of ordinary antiferromagnetic spin chains and the more exotic one dimensional nematic (see below). Therefore we present first the bosonized form for the theory of a Bose liquid, and then give specific applications of this to different spin systems.

For a 1d Bose liquid, the fundamental operators are the density field and creation/annihilation fields , which are bosonized (yes, we are bosonizing bosons!) according to

 ny(x) = ¯¯¯n+1β∂xφy−A1sin[2πβφy(x)−ksdwx], ψy(x) = A3e−iβθy(x)+... (2)

Here continuous runs along the chain, and we have introduced the slowly-varying “phase” fields which are continuous functions of and time . The parameter depends upon details of the Bose liquid; it is also often convenient to introduce the “compactification radius” . , or equivalently , determines the long-distance behavior of the 1d correlation functions. The modulation wavevector is that of an incipient Bose solid at the average Bose density , which is . It is sometimes convenient to define the “charge density wave” order parameter for these bosons,

 Φy(x)=e−i2πβφy, (3)

so that

 ny(x)=¯¯¯n+1β∂xφy−iA12(Φy(x)eiksdwx−h.c.). (4)

For spin systems, becomes the spin density wave order parameter. To keep the presentation symmetric, we also define the “superfluid” or XY order parameter , so that

 ψy(x)=A3Ψy(x). (5)

The conjugate fields obey the commutation relation

 [θy(x),φy′(x′)]=−iΘ(x−x′)δyy′. (6)

where is the Heavyside step-function. Their dynamics is described by the free field Hamiltonian

 H0=∑y∫dxv2{(∂xθy)2+(∂xφy)2}. (7)

This describes a single bosonic mode for each : a central charge conformal field theory, also known as a Luther-Emery liquid or Luttinger liquid. The Hamiltonian contains a single parameter , which gives the velocity of excitations which propagate relativistically, and which again depends upon microscopic details.

Such a Luttinger liquid is characterized by algebraic correlations, which are simply obtained from the above free field theory, the most prominent of which are

 ⟨ny(x)ny(0)⟩c = 12A21cos[ksdwx] |x|−2Δz, (8) ⟨ψy(x)ψ†y(0)⟩ = A23 |x|−2Δ⊥. (9)

Their power-law decay is controlled by the scaling dimensions and . Here we gave only the leading terms in (8) and (9), omitting corrections which decay faster with distance.

For the case of many spin chains, including the XXZ chain in a field along , we can simply apply the above bosonization rules taking

 Szy(x) = 12−ny(x), (10) S+y(x) = (−1)xψy(x). (11)

In that case, , where is the uniform magnetization, and hence . For the isotropic Heisenberg chain, monotonically decreases from at zero magnetization () to at the full saturation . This shows that in the presence of external magnetic field transverse spin fluctuations are more relevant (decay slower) than the longitudinal ones, for . At the same time the wave vector of longitudinal spin fluctuations shifts with magnetization continuously, as , toward the Brillouin zone center, while that of the transverse fluctuations, , remains fixed at the Brillouin zone boundary.

As discussed above, two-dimensional order appears as a result of residual inter-chain interactions which are described by a perturbing Hamiltonian . To understand under which conditions SDW can emerge from , it is instructive to start by considering the simplest case of non-frustrated inter-chain coupling

 H′non−fr=J′∑x,ySy(x)⋅Sy+1(x)→ (12) →∑y∫dx γsdwcos[2π(φy−φy+1)/β] +γxycos[β(θy−θy+1)].

Here we rewrote the first line in an appropriate low-energy form with the help of the representation (10), and we defined continuum inter-chain coupling constants and , which are of the same order. Since the fields on different chains are not correlated with each other at leading order (7), the scaling dimension of the SDW (cone) term in (12) is simply . Since in the case of isotropic Heisenberg chains for all , as argued above, the second term in the above equation becomes parametrically stronger than the first under the renormalization group (RG) flow. As a result, the interchain interaction (12) reduces to the xy term which implies two-dimensional order, via spontaneous symmetry breaking, in the plane perpendicular to the external magnetic field. This is a familiar canted antiferromagnet, or spin-flop two sublattice ordered state. Note that is completely uniform in this phase.

The absence of an SDW phase noted here clearly follows from the condition . We observe that this may break down in three ways. First, for spin chains other than the simple Heisenberg one, the inequality may be violated in favor of the opposite situation. Second, for yet more exotic spin chains (or ladders), the relation between spin operators and those of the effective Bose gas may differ from that in Eqs. (10). Finally, third, the interactions between chains may differ from those in Eq. (12). We will encounter all these situations below.

### ii.2 Physical realizations

We now consider three different microscopic lattice models that lead to dominant SDW interactions. These models represent physically different ways of achieving the inequality . In general, the models we consider have, in their bosonized continuum limits, a Hamiltonian of the form , with describing decoupled chains as in Eq. (7), and the inter-chain coupling of the form

 H′ = ∑y∫dx{12γsdw(Φ†yΦy+1+Φ†y+1Φy) (13) +12γxy(Ψ†yΨy+1+Ψ†y+1Ψy) +12γ′xy(Ψ†yi∂xΨy+1−Ψ†y+1i∂xΨy).

The different models are distinguished by the values of the couplings and by the value of the chain interaction parameter .

#### Ising anisotropy

The most straightforward route to is provided by arranging . This occurs by keeping the same unfrustrated rectangular arrangement of spin-1/2 chains discussed above, but replacing the Heisenberg chains with XXZ ones with Ising anisotropy,

 HIsing = J∑x,y(Sxx,ySxx+1,y+Syx,ySyx+1,y+δSzx,ySzx+1,y), + J′∑x,y(Sxx,ySxx,y+1+Syx,ySyx,y+1+δSzx,ySzx,y+1),

where parameterizes Ising anisotropy, and for simplicity we have taken the same anisotropy in the inter-chain coupling , though this is not very important. In zero magnetic field, even in the absence of inter-chain coupling, such a chain orders spontaneously (at zero temperature, ) into one of the two Néel states, with spins ordered along the easy Ising () axis. The the non-frustrated interchain exchange then immediately selects the staggered arrangement of Néel order of adjacent chains, further stabilizing the antiferromagnet for low but non-zero temperature.

However, a sufficiently strong magnetic field, applied along the axis, breaks the gap, driving the XXZ chains into gapless Luttinger liquid state again Okunishi and Suzuki (2007). For small , the problem can then be treated by bosonization and has the general form found above in Eq. (13), with , , and . More importantly, the Ising anisotropy increases relative to the Heisenberg chain. Indeed, it turns out that the critical indices of this state (parametrized in Okunishi and Suzuki (2007) by instead of our ) do have the desired property that for in the finite range . The critical magnetization , separating and regimes, increases with increasing anisotropy .

It is clear that interchain interaction then stabilizes the two-dimensional SDW state in (approximately) the same magnetization interval because here immediately implies . The exact value of the critical magnetization separating the two-dimensional SDW and cone states (with non-zero ) depends on many details and is not rigorously known. A reasonable estimate can be made by the chain mean field theory (CMFT), using the precise forms of the longitudinal and transverse spin susceptibilities as well as small (of the order ) corrections to magnetization caused by the interchain exchange . We disregard all these complications in order not to overload the discussion.

It appears that spin-1/2 antiferromagnet BaCoVO realizes exactly this situation Kimura et al. (2008a). Static SDW order has been observed in several neutron and sound-attenuation studies (refs).

#### Spin-nematic chains

A second route to the collinear SDW is to suppress the leading xy instability altogether, by driving the individual spin chain into a completely different phase. This occurs in the model derived from LiVCuO, in which the one-dimensional chains are not XXZ like but instead incorporate ferromagnetic nearest-neighbor exchange and antiferromagnetic next-nearest exchange Enderle et al. (2005); Nishimoto et al. (2012). Such chains (which can also be “folded” into zig-zag ladders) have distinct behavior which is not captured by Eqs. (10).

Extensive research into this interesting chain geometry, dating back to 1991 Chubukov (1991), has found that the spectrum of the fully magnetized chain contains, in addition to usual single magnon states, tightly bound magnon pairs (in fact, three- and four-magnon complexes exists in some parameter range as well Hikihara et al. (2008); Sudan et al. (2009)). Importanly, these two-magnon pairs lie below the two-magnon continuum. As the magnetic field is reduced to the critical one, the gap for the two-magnon states vanishes while the single magnon gap remains non-zero. For , therefore, one obtains not a Bose liquid of single magnons (which is the physical content of Eqs. (10)), but rather a Bose liquid of magnon pairsHikihara et al. (2008); Sato et al. (2013). In such a liquid, Eqs. (10) is replaced by

 Szy(x) ∼ 12−2ny(x), S+y(x)S+y(x+1) ∼ ψy(x). (15)

where now annihilates a magnon pair, and counts the magnon pairs. The appearance of the operator quadratic in above indicates the existence of critical “spin nematic” correlations. Since a gap for single magnons (single spin flips) remains, the low energy projection of the single spin-flip operator vanishes

 S±y(x)∼‘‘0". (16)

For a single chain, this is still a Luttinger liquid state, but simple XY correlations decay exponentially instead of as a power law. The density correlations in this Bose liquid remain critical, and hence from Eq. (II.2.2) so do those of .

With this understanding, we see that even simple unfrustrated exchange interactions coupling the chains are “projected” onto dominantly Ising interactions, which strongly favor an SDW ground state. Specifically, we have again the form in Eq. (13), but with and . The strong suppression of all single spin-flip operators suggests that, unlike in the previous case, the SDW state extends up to very close to the saturation value .

Unusual functional form of is due to the fact that it describes coupling of the nematic fields of different chains. Such a coupling, involving four spin operators, see (II.2.2), is simply absent in the lattice model. It is, however, generated by quantum fluctuations in second order in the inter-chain exchange, which explains its peculiar form (the proportionality constant is non-trivial Sato et al. (2013) and not determined here). We will see that this can stabilize a true 2d SN near the saturation field – see Sec. IV.2. But away from a narrow region near saturation, the SDW state indeed dominates as naïvely expected.

#### Spatially anisotropic triangular lattice antiferromagnet

In the above two examples, we modified the interactions on the individual chains from the Heisenberg type. A third way to stabilize the SDW phase is to retain the simple nearest-neighbor Heisenberg form for the chain Hamiltonian, but modify explicitly the interactions between chains in a manner that frustrates the competing XY order. This occurs naturally for the situation of a spatially anisotropic triangular lattice Starykh and Balents (2007); Starykh et al. (2010). In this case, each spin is coupled symmetrically to two neighbors on adjacent chains, which frustrates the inter-chain interactions. Specifically, the interchain coupling reads

 H′frust = (17) J′∑x,ySy(x)⋅(Sy+1(x−1/2)+Sy+1(x+1/2)).

Note that this Hamiltonian is written in a cartesian basis in which spins on, say, odd chains are located at the integer positions while those on the even chains are at the half-integer locations . Bosonization of (17) gives again the form of Eq. (13), but with due to frustration. The other two interactions are and .

The SDW term retains its form but its coupling constant reflects frustration as well, for . The SDW coupling resists the appearance of the derivative which occurs for the XY term, as a result of the shift of the longitudinal wave vector from its commensurate value for finite . It is this shift that makes SDW interaction more relevant than the XY one. While the SDW scaling dimension remains , that of the XY interaction increases to . The addition of reflects the derivative in the term of Eq. (13).

Since in a rather wide range of magnetization, approximately for , interchain frustration stabilizes collinear SDW order Starykh et al. (2010).

## Iii Excitations of collinear SDW state

In this section, we discuss the excitation spectrum of the collinear SDW state, and its manifestation in the magnetic structure factor (or wavevector dependent spin susceptibility). The magnetic excitations are collective modes, strongly influenced by symmetry. In an applied magnetic field, the only symmetries of the Hamiltonian are rotation symmetry about the field, and the space group symmetries of the lattice. Notably, the collinear SDW state preserves the former symmetry, and in the absence of broken continuous symmetry, lacks a Goldstone mode. Thus there are no acoustic transverse spin waves. Instead, we expect gapped transverse excitations. Given the highly quantum nature of the SDW phase in the quasi-1d, situation discussed here, there is in fact no a priori reason these excitations may be treated semiclassically in the traditional spin wave fashion. Instead, in the following, we will obtain the gapped excitations from a purely quantum treatment based on knowledge of the integrable 1d sine-Gordon model.

The collinear SDW does, however, break translation symmetry, and in particular exhibits incommensurate order (see Eq. (1)). Although translational symmetry is discrete, in cases of incommensurate order it is known to behave in some respects like a continuous symmetry and consequently the collinear SDW state supports a phason mode, which is the “pseudo-Goldstone” mode of broken translation symmetry. Physically this mode – which is acoustic – appears because of the vanishing energy cost for uniformly “sliding” the incommensurate density wave. In the bosonization framework, the elevation of the discrete lattice translation symmetry to an effectively continuous one appears in an emergent continuous symmetry of Eq. (12): invariance under . While it is well-known in SDW-ordered metals, the phason excitation is perhaps less familiar in magnetically ordered insulators. We now turn to the detailed exposition of the excitation spectrum, including both phason and gapped modes. For simplicity, we focus here on zero-temperature () properties and apply CMFT to the problem. An alternative derivation of the phason dispersion, based on the Ginzburg-Landau (GL) action, is sketched in Appendix B.

### iii.1 Single chain excitations

In this subsection, we present the chain mean field theory which approximates the problem of the 2d system by a self-consistent set of independent chains, specifically 1+1d sine-Gordon models. We describe the gapped excitations occuring within individual such chains. The effects of two-dimensionality on the spectrum, and especially the emergence of the low energy phason mode, is discussed in the following subsection.

#### Chain mean field theory

Focusing on the SDW state, we drop the and terms in Eq. (13), and make the mean field replacement (neglecting a constant), with

 H′MF → =∑y∫dx12γsdw(⟨Φ†y⟩Φy+1+Φ†y⟨Φy+1⟩ (18) +⟨Φ†y+1⟩Φy+Φ†y+1⟨Φy⟩)−const..

With the ansatz

 ⟨Φy⟩=¯¯¯¯Φ(−1)y, (19)

we then obtain

 H′MF = −γsdw¯¯¯¯Φ∑y∫dx(−1)y(Φy+Φ†y), (20)

where we took real.

The chain mean field theory (CMFT) has now reduced the system to a problem of decoupled chains. It can be brought into a simple standard form by expressing it in terms of the bosonized fields, and making the shift , which gives finally , where

 HsG=∫dx v2[(∂xφ)2+(∂xθ)2]−2μcos[2πβφ]. (21)

Here and the self-consistency requirement in Eq. (19) becomes

 ¯¯¯¯Φ=⟨ei2πβφ⟩sG=⟨cos2πβφ⟩sG. (22)

Our notation here closely follows Refs.[Essler and Konik, 2004; Gogolin et al., 2004], which describe many technical details important for the subsequent analysis.

#### Mass spectrum of the sine-Gordon model

The excitations of the sine-Gordon model in the massive phase () come in two varieties: solitons and antisolitons, which are domain walls connecting degenerate vacua (minima of the cosine), and breathers, which are bound states of solitons and antisolitons. The number of breathers is determined by the dimensionless parameter , such that ( denotes closest to integer such that ). The minimum energy of each breather – the mass in the relativistic sense – is given by the formula

 mn=2mssin[π2ξn] for n=1,2,...[1ξ], (23)

expressed here in terms of the fundamental soliton mass .

In the case of the spatially anisotropic triangular lattice, ranges from at to at the saturation, . The breather masses are plotted in Fig. 4 versus . For , there are two breather modes. When the magnetization is increased to this value, the upper breather reaches the energy of the two-soliton continuum and merges with it. Hence, when , there is only a single breather.

The soliton mass is determined by the coupling constant via the exact relation, Zamolodchikov (1995)

 μ = vΓ(18πR2)πΓ(1−18πR2)(msv√πΓ(1+ξ2)2Γ(ξ2))2−1/(4πR2) (24) ∼v(ms/v)2−1/(4πR2).

The scaling shown in the second line can be understood by simple renormalization group arguments. The relevant cosine operator in (21) grows under the RG according to , where is the initial value of the coupling constant and is the logarithmic RG variable, so that the running energy scale is . The coefficient reaches strong coupling at such that . Solving this for , one obtains the energy , which indeed matches the last line of (24). The value of the the exact solution in the first line of (24) is that it also provides with exact numerical prefactor.

#### Self-consistency

To determine the overall scale of the excitation spectrum, we require the soliton mass or . This is obtained from the self-consistency condition . The expectation value defining is readily obtained from the relation

 ¯¯¯¯Φ = =−12∂F(μ)∂μ, (25)

where is the ground state energy density of . Eq. (25) follows from first order perturbation theory in changes of .

At the scaling level, as it is an energy density, we expect , and using Eq. (24) one obtains

 ms∼v(γsdw/v)2πR2/(4πR2−1). (26)

This power can be understood from RG arguments, which indicate it is correct beyond CMFT. Under the RG, the SDW coupling grows according to , with , which defines a scale by the condition that reaches strong coupling, i.e. becomes of order . Then using , we obtain Eq. (26).

To go beyond scaling and obtain the prefactor and hence an absolute number for , we turn to the exact solution of the sine-Gordon model. The standard result in the literature is . It is, however, insufficient in the present case due to the obvious (and unphysical) divergence of in the () limit, i.e. in the limit of .

This divergence is analyzed and cured, with the help of nominally less relevant terms, in the Appendix A. We present the result here. To obtain the soliton mass, one first solves for from the equation

 (μv)1−ξ=1+ξ8tan[πξ2]A21A1+ξ2(γsdwv)× (27) ×(1−18tan[π1+ξ]A4(1+ξ)1A22 Q(1−ξ)(1+ξ)(γsdwv))−1.

The soliton mass is then obtained as

 ms=vA1(μvA2)(1+ξ)/2. (28)

This procedure allows us to explicitly determine the soliton mass as a function of for a given coupling constant of the original spin problem. For illustrative purposes, we plot the result for the case of the spatially anisotropic triangular lattice, for which , with (chosen arbitrarily) in Figure 5.

### iii.2 Spin susceptibilities

The aim of this subsection is to show how the excitations described in the prior section, which are excitations already on a single chain, and the collective modes, which appear only when the full 2d dynamics are considered, appear in the physical dynamical susceptibilities, i.e. the components of the dynamical structure factor measured in inelastic neutron scattering. Formally these are defined as the linear response quantities,

 Xμν(k,ω)=δSμ(k,ω)δhν(k,ω)∣∣∣h(k,ω)=0, (29)

where is an oscillating infinitesimal applied Zeeman field at wavevector and frequency . By the usual linear response theory, this is minus the retarded correlation function of spin operators

 Xμν(k,ω)∼i∫∞0dte(iω−ϵ)t⟨[Sμ(k,t),Sν(−k,0)]⟩, (30)

where .

We distinguish two types of susceptibilities. The longitudinal susceptibility describes the dynamical correlations of spin components along the applied field and the SDW polarization. Using the bosonization rule of Eqs. (4) and (10), we see that this is related to correlations of the SDW order parameter . Hence we define the bosonized equivalent, , of the longitudinal susceptibility

 Xzz(k=(ksdw+q,π+qy),ω)∼χzz(q,qy,ω), (31)

and hence

 χzz(q,qy,ω) = i∫∞0dt∫dx∑yeiqx+iqyy+(iω−ϵ)t (32) ×⟨[Φy(x,t),Φ†0(0,0)]⟩.

Note that in , gives the shift of the momentum along the chain from the SDW one, i.e. , while is measured from due to the shift of field by made in deriving (21). Moreover, the continuum formula in Eq. (32) describes only the contributions to the susceptibility at low energy near . Other contributions may apply elsewhere. For example, contribution from the vicinity of is described by the hermitian conjugate of the expression in Eq. (32), while in that near , the operator in Eq. (II.1) or Eq.(4) contributes. We neglect it here because, since this operator has larger scaling dimension than , it gives a subdominant contribution in the sense of smaller integrated weight in (i.e. the weight near is smaller than that near ).

The transverse susceptibility describes the spin components , normal to the field and the SDW axis. Using the bosonization rule in Eqs. (II.1,3), we find

 Xxx(k=(π+q,ky),ω) = Xyy(k=(π+q,ky),ω) (33) ∼ χxy(q,ky,ω),

with

 χxy(q,qy,ω) = i∫∞0dt∫dx∑yeiqx+ikyy+(iω−ϵ)t (34) ×⟨[Ψy(x,t),Ψ†0(0,0)]⟩.

As for the longitudinal one, we have defined the continuum transverse susceptibility in such a way that gives a shift in momentum relative to some offset, but with a different offset from the one used in the longitudinal susceptibility. Here , i.e. on passing from the longitudinal to transverse susceptibility. This difference originates from the distinct momenta of singular response of a one dimensional spin system in the two channels. It must be noted that while we can study this object, defined by Eq. (34), also for the case of the SDW formed from SN chains, in that case it is not the true transverse spin susceptibility. Due to the definition of for the SN case, it instead represents the nematic susceptibility.

In the following, we obtain these quantities using the RPA approximation, which expresses these 2d dynamical susceptibilities in terms of the 1d dynamic susceptibilities of the individual decoupled chains we obtained in the CMFT approximation.

#### Susceptibilities of the sine-Gordon model

We now obtain the 1d dynamical susceptibilities. These are by construction independent of . According to bosonization, the longitudinal and transverse susceptibilities are related to correlations of exponentials of and fields, respectively. The corresponding correlations of the sine-Gordon model may be calculated via the form-factor expansion which is described in great detail in Ref. Essler and Konik, 2004. Here we present key results from this reference as adapted for our needs.

Longitudinal susceptibility: The longitudinal susceptibility is obtained from the two-point correlation function of in Eq. (32). What excitations are created by this effective longitudinal spin operator? Since is local in (see Eq. (3)), it cannot generate topological excitations with non-zero soliton number. Instead, acting on the ground state, it generates gapped excitations corresponding to breathers, unbound soliton-antisoliton pairs, and also higher energy states such as multiple breather states. The largest contribution, however, comes simply from the first breather (in the notation of Ref. Essler and Konik, 2004). In the approximation in which only this excitation contributes, the longitudinal susceptibility has a single simple pole,

 χzz1d(q,ω)=CzZzm21+v2q2−ω2−iϵ. (35)

The mass is given by in (23). Note that in the whole magnetization range , when , the first breather’s mass exceeds that of the soliton, . The residue is determined by the soliton mass , while the factor collects all numerical coefficients and depends smoothly on the magnetization . The second breather does not contribute because it only connects states of the same parity while is odd under parity.

The continuum soliton-antisoliton states become available for . In the form factor expansion of Ref. Essler and Konik, 2004, this contribution was denoted . We consider energies close to the threshold, , where . With some analysis of formula in that reference, we find that the contribution to the dynamic structure factor of the single chain starts smoothly as . This is in accord with the general behavior expected for the two particle contribution to correlation functions of one dimensional systems in the situation where the particles experience attractive interactions (which must be the case here since bound states (breathers) form). In general, for , all other contributions will occur inside the two soliton continuum, and we expect that mixing with the continuum will remove any sharp features at higher energies (though this mixing may be controlled by deviations from integrability). The end result is that Eq. (35) should be supplemented by the continuum contribution for , which extends smoothly to higher energies.

Transverse susceptibility: The transverse susceptibility is obtained from correlations of as in Eq. (34). The field is not local in the variables, and indeed can be expressed as an integral of the canonical momentum conjugate to . Consequently, it creates soliton and antisoliton defects in , and some algebra shows that it changes the topological charge by . Hence the lowest energy contribution to the transverse susceptibility is simply that of single solitons, and again has a pole form. Thus

 χxy1d(q,ω)=CxyZxym2s+v2q2−ω2−iϵ. (36)

Here while includes all numerical coefficients and smooth dependence on and magnetization . Note that so that the first onset of spectral weight in the chain occurs here in the transverse correlation function rather than the longitudinal one.

Corrections to this form account for multi-particle contributions to . These can be of soliton-breather (B) and of soliton-soliton-antisoliton () types, as is schematically shown in eq.3.73 of Ref.Essler and Konik, 2004. They appear at energy and . Thus the continuum contribution for the transverse susceptibility occurs above the one for the longitudinal one. We do not pursue it further here. The spectral content of equations (35) and (36) is schematically depicted in Fig. 1.

It is instructive to compare the excitation structure found here with the “dual” sine-Gordon problem which has been frequently discussed in other problems of one-dimensional magnetism, in which the ordering is transverse, so in (21) is replaced by . In that case,Essler et al. (2003) the parameter ranges from at zero magnetization, , to at , resulting in many more breathers (up to 7) peeling off of the soliton-antisoliton continuum with an increasing number with increasing magnetization. In parallel with this, the spectral composition of different excitations branches changes accordingly: the breathers contribute near momentum , while solitons (antisolitons) contribute near momentum () – see for example Fig.1 of Ref.Essler et al., 2003.

#### Susceptibility of 2d SDW phase

The single chain approximation is not sufficient for describing two-dimensional (2d) spin correlations. At the single chain level, all spin excitations have a gap, there is no dispersion transverse to the chains (i.e. dependence upon ), and there are no Goldstone (spin wave) modes. These deficiencies are easily fixed, however, with the help of a simple random-phase approximation (RPA) in the interchain couplings, as suggested by Schulz and developed in great details by Essler and Tsvelik.

We apply the RPA approximation directly to the continuum problem of correlations of and . This gives expressions for the 2d susceptibilities directly from the single-chain susceptibilities, described above:

 χα2d(q,ky,ω)=χα1d(q,ω)1+2γα(q,ky)χα1d(q,ω). (37)

Here describes the two channels, is the Fourier transform of the interchain interaction in the -channel:

 γzz(q,ky) = γsdwcosky, (38) γxy(q,ky) = [γxy−qγ′xy]cosky. (39)

The parameters , , and are collected for convenience in Table I. Using them, and Eqs. (38,37,36,35), one can obtain the two dimensional susceptibility for any of the three models discussed here.

As an example, we discuss this now in some detail for the case of the spatially anisotropic triangular antiferromagnet. Applying Eq. (38) and Table I, we obtain and . We see that owing to the additional factor of in this term, which ultimately arose from inter-chain frustration.

Hence in the ordered two-dimensional SDW state

 χzz2d(q,ky,ω)=((χzz1d(q,ω))−1+2γzz(ky))−1 (40) =CzZz[(m21+2CzZzJ′A21sin(πM)cos[ky])+v2q2−ω2].

As written, this expression is characterized by a finite, albeit renormalized and -dependent, gap in the spin excitation spectrum, and does not seem to describe a gapless phason mode. This shortcoming is of course due to the approximate nature of the RPA expression (37). Since the phason is a Goldstone mode which is required by the very existence of the 2d SDW order, we follow Schulz and simply require that the gap must close at some appropriate . Clearly for this happens at . This reflects the preference of SDWs on adjacent chains to order out of phase due to repulsive (antiferromagnetic) interactions between them.

To check the consistency of this procedure we need to make sure that both terms in the expression for scale in the same way with – and this is exactly what we find. While in accordance with (26), it is also easy to see that the interchain term follows the same power law. Thus the two terms are of the same order and our requirement simply fixes the overall numerical coefficient of the longitudinal susceptibility.

Hence, in the vicinity of ordering momentum , we have, with and ,

 χzz2d(q,π+qy,ω)∼Zzz;2d(v2q2+v2⊥q2y)−ω2, (41)

with , when . The phason has linear dispersion

 ω=√v2q2+v2⊥q2y (42)

with strongly anisotropic velocity. Its transverse (inter-chain) velocity is much smaller than .

In the transverse channel we have instead

 χxy2d(q,ky,ω)= CxyZxy~m