Field-induced dimer orders in quantum spin chains

In this paper I discuss a quantum spin-$1/2$ chain with the following Hamiltonian:

	$\displaystyle\mathcal{H}$	$\displaystyle=J\sum_{j}\bm{S}_{j}\cdot\bm{S}_{j+1}-h_{0}\sum_{j}S_{j}^{z}-h_{2}\sum_{j}(-1)^{j}S_{j}^{x}$
		$\displaystyle\qquad-h_{4}\sum_{j}\delta_{j}S_{j}^{z},$		(1)

where $\bm{S}_{j}$ is the $S=1/2$ spin operator, $J>0$ is the antiferromagnetic exchange coupling, and $\delta_{j}=1,1,-1,-1$ for respectively, $j=0,1,2,3\mod 4$. Parameters $h_{0}$, $h_{2}$, and $h_{4}$ denote the uniform magnetic field, the twofold staggered field, and the fourfold screw field, respectively. Note that $\delta_{j}$ has a simple expression,

$\displaystyle\delta_{j}$

$\displaystyle=\sqrt{2}\cos\biggl{(}\pi\frac{2j-1}{4}\biggr{)}.$

(2)

Throughout this paper, I employ the unit of $\hbar=a=1$ unless otherwise stated, where $a$ is the lattice spacing.

The spin-chain model (1) is related to a model proposed for $\mathrm{[Cu(pym)(H_{2}O)_{4}]SiF_{6}\cdot H_{2}O}$ Liu et al. (2019) but differs from the latter in three points: field directions, a weak exchange anisotropy, and a uniform Dzyaloshinskii-Moriya (DM) interaction. In Ref. Liu et al. (2019), $h_{0}$ and $h_{2}$ are applied in the $x$ and the $y$ directions, respectively. The model used in Ref. Liu et al. (2019) contains a weak XXZ interaction and the uniform DM interaction. Those differences hardly affect the ground state of $\mathrm{[Cu(pym)(H_{2}O)_{4}]SiF_{6}\cdot H_{2}O}$, which are to be clarified in Sec. VI.

In the compound $\mathrm{[Cu(pym)(H_{2}O)_{4}]SiF_{6}\cdot H_{2}O}$ Liu et al. (2019), the twofold staggered field $h_{2}$ and the fourfold screw field $h_{4}$ originate from the $g$ tensor of electrons and are thus proportional to the externally applied uniform magnetic field $h_{0}$. In this section, I first deal with an unrealistic but simplest situation with $h_{0}=h_{2}=0$ and $h_{4}\not=0$ in Sec. II.2. I will discuss a more realistic situation with $h_{2}\propto h_{0}$ and $h_{4}\propto h_{0}$ later in Sec. V.

The fourfold screw field $h_{4}$ can generate an excitation gap all by itself to the ground state of the spin chain. To show this, I set $h_{0}=h_{2}=0$ and discuss the $h_{4}$ dependence of the lowest-energy excitation gap from the ground state. The Hamiltonian is thus simplified as

$\displaystyle\mathcal{H}_{4}$

$\displaystyle=J\sum_{j}\bm{S}_{j}\cdot\bm{S}_{j+1}-h_{4}\sum_{j}\delta_{j}S_{j}^{z}.$

(3)

When $h_{4}=0$, the ground state of the model (3) is gapless Giamarchi (2004). Figure 2 shows numerical results on the excitation gap obtained by using the density-matrix renormalization group (DMRG) method with the ITensor C++ library ITe , where I used the bond dimension $\chi=400$ and the truncation error cutoff $1\times 10^{-10}$. Note that all the DMRG calculations in this paper were performed with the open boundary condition. The DMRG result implies that an infinitesimal $h_{4}/J$ immediately opens the excitation gap between the ground state and the lowest-energy excited state,

$\displaystyle\Delta\propto J\biggl{(}\frac{h_{4}}{J}\biggr{)}^{1.34}.$

(4)

Similarly to the twofold staggered field, the fourfold screw field induces an excitation gap with a power law. However, the power $1.34$ differs from that, $2/3$, of the twofold staggered field Oshikawa and Affleck (1997); Affleck and Oshikawa (1999).

Unlike the twofold staggered field, the fourfold screw field $h_{4}$ induces no Néel order. Instead, $h_{4}$ induces dimer orders (Figs. 3 and 4),

	$\displaystyle D_{\perp}$	$\displaystyle=\frac{1}{L}\sum_{j}\braket{(-1)^{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})},$		(5)
	$\displaystyle D_{\parallel}$	$\displaystyle=\frac{1}{L}\sum_{j}\braket{(-1)^{j}S_{j}^{z}S_{j+1}^{z}},$		(6)

where $L$ is the length of the spin chain. The dimer order parameters (5) and (6) show different power-law dependence on $h_{4}$. DMRG results (Figs. 3 and 4) imply

	$\displaystyle D_{\perp}$	$\displaystyle\propto\biggl{(}\frac{h_{4}}{J}\biggr{)}^{0.672},$		(7)
	$\displaystyle D_{\parallel}$	$\displaystyle\propto\biggl{(}\frac{h_{4}}{J}\biggr{)}^{1.07}.$		(8)

Induction of $D_{\parallel}$ by $h_{4}$ is easily understandable. Let us recall that $h_{4}$ is coupled to an operator,

$\displaystyle f_{j}^{z}$

$\displaystyle=\sqrt{2}\cos\biggl{(}\pi\frac{2j-1}{4}\biggr{)}S_{j}^{z}.$

(9)

The fourfold screw field induces the uniform $f^{z}$ order:

$\displaystyle\sum_{j}\braket{f_{j}^{z}}\not=0.$

(10)

The longitudinal dimer order parameter $D_{\parallel}$ is written in terms of $f_{j}^{z}$ as

$\displaystyle D_{\parallel}$

$\displaystyle=\frac{1}{L}\sum_{j}\braket{f_{j}^{z}f_{j+1}^{z}}.$

(11)

Nonzero $D_{\parallel}$ follows immediately from the uniform $f_{j}^{z}$ order (10). However, the induction of the transverse dimer order (7) is nontrivial.

Performing the Fourier transformation on Eq. (14), I obtain

	$\displaystyle\mathcal{H}_{\rm XY}$	$\displaystyle=\sum_{k}\epsilon(k)c_{k}^{\dagger}c_{k}$
		$\displaystyle\qquad-\frac{h_{4}}{\sqrt{2}}\sum_{k}\bigl{(}e^{-\pi i/4}c_{k}^{\dagger}c_{k+\frac{\pi}{2}}+e^{\pi i/4}c_{k+\frac{\pi}{2}}^{\dagger}c_{k}\bigr{)}$
		$\displaystyle\qquad+\mathrm{const.}$		(15)

where $\epsilon(k)=-(J/2)\cos k$ and $k\in(-\pi,\pi]$ is the wave number. When $J_{4}=0$, the spinless fermion is free and has the simple cosine dispersion. Since the total magnetization is zero in the XY chain, the cosine band is half occupied and the Fermi points are located at $\pm\pi/2$.

Let us make some observations on effects of the fourfold screw field on the free spinless fermion chain in the TL-liquid phase. Particle-hole excitations are the fundamental low-energy excitation in the TL-liquid phase. An operator $\rho_{q}=\sum_{k}c_{k+q}^{\dagger}c_{k}$, which creates a particle-hole excitation with a wave number $q$, can be written as a superposition of bosonic creation and annihilation operators of the TL liquid Giamarchi (2004). When the second line of Eq. (15) acts on the ground state of the XY model, particle-hole excitations with wave numbers $q=\pm\pi/2$ are generated [Fig. 5 (b)]. Apparently, such an $h_{4}$ term hardly affects the low-energy physics of the XY model because these particle-hole excitations have large excitation energies of $O(J)$. However, applying the $h_{4}$ term twice to the ground state, I can generate low-energy particle-hole excitations with the wave number $q=\pi$ [Figs. 5 (a,c)]. These observations show that though the fourfold screw field is highly irrelevant, it will generates a relevant interaction in a second-order perturbation process.

To confirm the perturbative generation of the relevant interaction, I derive a simple effective Hamiltonian that governs the low-energy physics of the fermion chain (15). Among several options to derive such a low-energy effective Hamiltonian is to use the Schrieffer-Wolff canonical transformation Schrieffer and Wolff (1966); Slagle and Kim (2017). The generic theory is explained in Appendix A.1. Here, I briefly summarize the derivation. First, I perform a canonical transformation,

$\displaystyle\mathcal{H}^{\prime}_{\rm XY}:=e^{\eta}\mathcal{H}_{\rm XY}e^{-\eta}$

(16)

with an antiunitary operator $\eta$. Two Hamiltonians $\mathcal{H}_{\rm XY}$ and $\mathcal{H}^{\prime}_{\rm XY}$ have one-to-one corresponding lists of eigenstates with exactly the same eigenenergies. Next, I perform a perturbative expanison by using a projection operator $P$ onto a low-energy subspace $\{\ket{\phi_{k}}\}_{k}$ with $k\in R$ and

$\displaystyle R=\bigl{\{}k\in(-\pi,\pi]|\,0\leq\bigl{|}|k|-|k_{F}|\bigr{|}<\Lambda\bigr{\}},$

(17)

where $\Lambda$ is a cutoff in the wave number and assumed as $\Lambda\ll 1$. Here, $\ket{\phi_{k}}$ is an eigenstate of the unperturbed Hamiltonian, Eq. (15) for $h_{4}=0$, with the total wave number $k$. $P$ can explicitly be written as $P=\sum_{k\in R}\ket{\phi_{k}}\bra{\phi_{k}}$. The projection onto the low-energy subspace leads to the effective Hamiltonian

$\displaystyle\tilde{\mathcal{H}}_{\rm XY}$

$\displaystyle=P\mathcal{H}^{\prime}_{\rm XY}P.$

(18)

Choosing $\eta$ properly, I can simplify the perturbative expansion of the right hand side of Eq. (18). The effective Hamiltonian up to the second order of $h_{4}/J$ is then given by (see Appendix A.2)

	$\displaystyle\tilde{\mathcal{H}}_{\rm XY}$
	$\displaystyle=\sum_{k\in R}\epsilon(k)c_{k}^{\dagger}c_{k}-i\frac{h_{4}^{2}}{4}\sum_{k\in R}(c_{k+\pi}^{\dagger}c_{k}-c_{k}^{\dagger}c_{k+\pi})$
	$\displaystyle\qquad\times\biggl{(}\frac{1}{\epsilon(k)-\epsilon(k+\frac{\pi}{2})}+\frac{1}{\epsilon(k+\pi)-\epsilon(k+\frac{\pi}{2})}\biggr{)}$
	$\displaystyle\quad+\frac{h_{4}^{2}}{2}\sum_{k\in R}c_{k}^{\dagger}c_{k}\biggl{(}\frac{1}{\epsilon(k)-\epsilon(k-\frac{\pi}{2})}+\frac{1}{\epsilon(k)-\epsilon(k+\frac{\pi}{2})}\biggr{)}.$		(19)

Since the cutoff $\Lambda$ is small enough, the kinetic term of Eq. (19) can be linearized around the Fermi surface Giamarchi (2004). Creation operators $c_{k}^{\dagger}$ at $k\approx\pi/2$ and $k\approx-\pi/2$ are replaced to those of different species, which I denote as $c_{k,R}^{\dagger}$ and $c_{k,L}^{\dagger}$, respectively. $R$ and $L$ refer to right movers and left movers of fermions. The low-energy Hamiltonian thus turns out to be

	$\displaystyle\tilde{\mathcal{H}}_{\rm XY}$	$\displaystyle\approx\sum_{k\in R}\{v_{F}(k-k_{F})c_{k,R}^{\dagger}c_{k,R}-v_{F}(k+k_{F})c_{k,L}^{\dagger}c_{k,L}\}$
		$\displaystyle\qquad+i\frac{h_{4}^{2}}{J}\sum_{k\in R}(c_{k,R}^{\dagger}c_{k,L}-c_{k,L}^{\dagger}c_{k,R}),$		(20)

where $v_{F}$ is the Fermi velocity. Note that the last term of Eq. (19) was discarded in the linearized Hamiltonian (20) because the coefficient of $c_{k}^{\dagger}c_{k}$ is $O(\Lambda h_{4}^{2}/J)$ and thus negligibly small for $k\in R$. The Hamiltonian (20) is diagonalized in terms of Majorana fermions,

$\displaystyle\xi_{k,\nu}$

$\displaystyle=\frac{c_{k,\nu}+c_{k,\nu}^{\dagger}}{\sqrt{2}},\quad\chi_{k,\nu}=\frac{c_{k,\nu}-c_{k,\nu}^{\dagger}}{\sqrt{2}i}$

(21)

The second line of Eq. (20) becomes mass terms,

$\displaystyle i\frac{h_{4}^{2}}{J}\sum_{k\in R}(c_{k,R}^{\dagger}c_{k,L}-c_{k,L}^{\dagger}c_{k,R})$

$\displaystyle=i\frac{h_{4}^{2}}{J}\sum_{k\in R}(\xi_{R}\xi_{L}+\chi_{R}\chi_{L}),$

(22)

which indicate that these Majorana fermions have the excitation gap $\Delta=h_{4}^{2}/J$ Shelton et al. (1996). Note that the gap does not reproduce the power law (4). This discrepancy of the power is attributed to interactions of fermions. I will come back to this point in Sec. IV.

The mass terms (22) are nothing but the bond alternation $J\delta_{\perp}\sum_{j}(-1)^{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})$ with $\delta_{\perp}=(h_{4}/J)^{2}$ Giamarchi (2004):

	$\displaystyle\sum_{j}$	$\displaystyle(-1)^{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})$
		$\displaystyle=-\frac{1}{2}\sum_{j}(-1)^{j}(c_{j}^{\dagger}c_{j+1}+c_{j+1}^{\dagger}c_{j})$
		$\displaystyle\approx i\sum_{k\in R}(c_{k,R}^{\dagger}c_{k,L}-c_{k,L}^{\dagger}c_{k,R}).$		(23)

For comparison, the following is the twofold staggered field term in terms of fermions.

$\displaystyle h_{2}\sum_{j}(-1)^{j}S_{j}^{z}\approx h_{2}\sum_{k\in R}(c_{k,R}^{\dagger}c_{k,L}+c_{k,L}^{\dagger}c_{k,R}).$

(24)

Here, I comment on effects of the canonical transformation (16) on observables. The canonical transformation also transforms an operator, say, $\mathcal{O}$, in the original model $\mathcal{H}_{\rm XY}$ to

$\displaystyle\tilde{\mathcal{O}}$

$\displaystyle=e^{\eta}\mathcal{O}e^{-\eta}.$

(25)

Note that one observes $\braket{\mathcal{O}}$ in experiments, not $\braket{\tilde{\mathcal{O}}}$. Equations (23) and (24) refer to the latter. According to the generic framework of Appendix A.1, the operator $e^{\eta}$ can be expanded with $h_{4}/J$,

$\displaystyle\tilde{\mathcal{O}}=\mathcal{O}+[\eta,\mathcal{O}]+[\eta,\mathcal{O}]+\frac{1}{2}[\eta,[\eta,\mathcal{O}]]+\cdots.$

(26)

A relation $\braket{\tilde{\mathcal{O}}}\approx\braket{\mathcal{O}}$ is valid for $h_{4}/J\ll 1$. I can thus basically identify $\mathcal{O}$ and $\tilde{\mathcal{O}}$ but their small discrepancy, $[\eta_{1},\mathcal{O}]$, would affect dynamics of spin chains (see Appendix. B).

I showed that the fourfold screw field yields the bond alternation instead of the twofold staggered field. Actually, the bond-centered inversion symmetry of the spin chain (3) forbids the twofold staggered field from emerging in the effective Hamiltonian (20).

The uniform spin chain is symmetric under two types of spatial inversions: the site-centered inversion $I_{s}$ and the bond-centered inversion $I_{b}$. These spatial inversions act on spins as $I_{s}:\bm{S}_{j}\mapsto\bm{S}_{-j}$ and $I_{b}:\bm{S}_{j}\mapsto\bm{S}_{1-j}$. The twofold staggered field is invariant under $I_{s}$ but not under $I_{b}$. On the other hand, the bond alternation and the fourfold screw field are invariant under $I_{b}$ but not under $I_{s}$. In general, a low-energy effective Hamiltonian keeps the symmetries that the original Hamiltonian possesses. In this sense, the low-energy effective Hamiltonian of the spin chain (3) cannot have the twofold staggered field term that breaks the bond-centered inversion symmetry of the original Hamiltonian (14).

On the basis of the fact that the fourfold screw field induces the transverse dimer order (34), here, I investigate the realistic case with $h_{2}$ and $h_{4}$ proportional to the uniform field $h_{0}$. I assume two proportional coefficients

$\displaystyle\alpha_{2}=h_{2}/h_{0},\qquad\alpha_{4}=h_{4}/h_{0},$

(39)

are both constant. DMRG results for the dimer order parameters and the Néel order parameter are shown in Figs. 6, 7, and 8 for $(\alpha_{2},\alpha_{4})=(0.8,0.4)$.

To understand DMRG results, I replace the Hamiltonian (1) to the low-energy effective Hamiltonian,

	$\displaystyle\tilde{\mathcal{H}}$	$\displaystyle=J\sum_{j}\bm{S}_{j}\cdot\bm{S}_{j+1}-h_{0}\sum_{j}S_{j}^{z}-h_{2}\sum_{j}(-1)^{j}S_{j}^{x}$
		$\displaystyle\qquad+J\delta_{\perp}\sum_{j}(-1)^{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})$		(40)

with $\delta_{\perp}\propto(h_{4}/J)^{2}$. I can immediately bosonize it.

	$\displaystyle\tilde{\mathcal{H}}$	$\displaystyle=\frac{v}{2}\int dx\bigl{\{}(\partial_{x}\theta)^{2}+(\partial_{x}\phi)^{2}\bigr{\}}-\frac{h_{0}}{\sqrt{2\pi}}\int dx\,\partial_{x}\phi$
		$\displaystyle\qquad-g_{2}\int dx\,\cos(\sqrt{2\pi}\theta)+g_{4}\int dx\,\cos(\sqrt{2\pi}\phi),$		(41)

where $g_{2}=b_{0}h_{2}$ and $g_{4}=d_{xy}J\delta_{\perp}$. This complex Hamiltonian consists of two parts. The first line of Eq. (41) favors the gapless TL-liquid ground state for small $h_{0}/J$. The second line represents potential terms of $\phi$ and $\theta$ that give rise to an excitation gap. In general, the scaling dimensions of $\cos(\sqrt{2\pi}\theta)$ and $\cos(\sqrt{2\pi}\phi)$ are $1/4K$ and $K$, respectively. Here, $K$ is a parameter called the Luttinger parameter that signifies strength of interactions Giamarchi (2004). The XY and the Heisenberg chains have $K=1$ and $K=1/2$, respectively. In the latter case, two cosine interactions are equally relevant. Therefore, Néel and dimer orders can coexist in the ground state from the viewpoint of the renormalization group (RG).

The coupling constant $g_{2}$ of $\cos(\sqrt{2\pi}\theta)$, whose bare value is $b_{0}h_{2}$, is increasing in the course of iterative RG transformations. $g_{2}$ follows the RG equation,

$\displaystyle\frac{dg_{2}(\ell)}{d\ell}$

$\displaystyle\approx\frac{3}{2}g_{2}(\ell).$

(42)

Here, $\ell$ characterizes the effective short-distance cutoff $a(\ell)=ae^{\ell}$. Note that $a$ is the lattice spacing which was set to be unity. The RG transformation of Eq. (42) is terminated when $a(\ell)$ reaches a correlation length of the lowest-energy excitation, $v/\Delta$.

Despite the same value of scaling dimensions, behaviors of the RG transformation of $g_{4}$ differ from that of $g_{2}$. This is due to the Zeeman energy which competes with the transverse bond alternation $\cos(\sqrt{2\pi}\phi)$. I can absorb the Zeeman energy in Eq. (41) into the kinetic term by shifting $\phi\to\phi+h_{0}/\sqrt{2\pi}v$. The $\phi$ shift introduces an incommensurate oscillation to the transverse bond alternation term, $\cos(\sqrt{2\pi}\phi)\to\cos(\sqrt{2\pi}\phi+h_{0}x/v)$. When the wave length $v/h_{0}$ is much longer than the short-distance cutoff $a(\ell)$, the incommensurate oscillation is negligible and then the RG equation of $g_{4}$ is simply

$\displaystyle\frac{dg_{4}(\ell)}{d\ell}$

$\displaystyle\approx\frac{3}{2}g_{4}(\ell).$

(43)

Otherwise, the incommensurate oscillation is rapid enough to eliminate $g_{4}$:

$\displaystyle g_{4}(\ell)=0.$

(44)

The same argument can be found in Ref. Affleck and Oshikawa (1999).

Now I can classify into two cases the strong-coupling limit that the RG flow eventually reaches. (i) When $v/\Delta\ll v/h_{0}$, the coupling constants $g_{2}(\ell)$ and $g_{4}(\ell)$ grow equally following Eqs. (42) and (43) and eventually reach $O(1)$. Then the Néel and the dimer orders coexist in the ground state. (ii) When $v/\Delta\gg v/h_{0}$, the coupling constant $g_{4}(\ell)$ vanishes because of the rapid incommensurate oscillation [Eq. (44)]. Then, the ground state has only the Néel order.

The correlation length $v/\Delta$ and the wave length $v/h_{0}$ are easily compared. If $\alpha_{2}/\alpha_{4}=0$, the gap becomes $\Delta\propto(h_{4}/J)^{2/3}\propto(h_{0}/J)^{4/3}$. When $h_{0}/J\ll 1$, the gap $\Delta\propto(h_{0}/J)^{4/3}$ never exceeds $h_{0}/J$, in other words, $v/\Delta\gg v/h_{0}$. Then the ground state does not have the dimer order. On the other hand, if $\alpha_{2}/\alpha_{4}$ is finite, the gap is a complex function of $h_{0}/J$. Still, in the limit $h_{0}/J\to 0$, the gap is reduced to the simple form of $\Delta\propto(h_{0}/J)^{2/3}$, which is much larger than $h_{0}/J$. In other words, $v/\Delta\ll v/h_{0}$ is valid and the first scenario comes true. Therefore, finite $|\alpha_{2}/\alpha_{4}|$ is necessary for the coexistence of the Néel and the dimer orders.

There is one remaining problem in the RG analysis on the coexistence of the Néel and the dimer orders. The transverse Néel order $(-1)^{j}S_{j}^{x}\approx\cos(\sqrt{2\pi}\theta)$ and the transverse dimer order $\cos(\sqrt{2\pi}\phi)$ seem to compete with each other since $\phi$ and $\theta$ are noncommutative. However, this competition is an artifact of the Abelian bosonization and these orders are cooperative Garate and Affleck (2010); Chan et al. (2017); Jin and Starykh (2017).

To avoid the artifact, I rewrite the Hamiltonian (40) as

	$\displaystyle\tilde{\mathcal{H}}$	$\displaystyle=J\sum_{j}\biggl{\{}1+\frac{2\delta_{\perp}}{3}(-1)^{j}\biggr{\}}\bm{S}_{j}\cdot\bm{S}_{j+1}-h_{2}\sum_{j}(-1)^{j}S_{j}^{x}$
		$\displaystyle\qquad-\frac{J\delta_{\perp}}{3}\sum_{j}(-1)^{j}(2S_{j}^{z}S_{j+1}^{z}-S_{j}^{x}S_{j+1}^{x}-S_{j}^{y}S_{j+1}^{y}),$		(45)

where I assume finite $\alpha_{2}/\alpha_{4}$. According to the RG analysis, the uniform Zeeman energy is negligible for finite $\alpha_{2}/\alpha_{4}$. Here, I simply put $h_{0}=0$ from the beginning. Note that the second line of Eq. (45) yields only irrelevant interactions for the relation (37) and is discarded hereafter.

Instead of the Abelian bosonization, I employ the non-Abelian bosonization approach Affleck and Haldane (1987); Gogolin et al. (2004). In the non-Abelian bosonization language, the effective Hamiltonian (41) for $h_{0}=0$ is written as

	$\displaystyle\tilde{\mathcal{H}}$	$\displaystyle=\frac{2\pi v}{3}\int dx\,(\bm{J}_{R}\cdot\bm{J}_{R}+\bm{J}_{L}\cdot\bm{J}_{L})$
		$\displaystyle\qquad+\frac{d_{xy}J\delta_{\perp}}{3}\int dx\,\operatorname{tr}(g)-i\frac{b_{0}h_{2}}{2}\int dx\,\operatorname{tr}(g\sigma^{x}).$		(46)

Here, the spin operator $\bm{S}_{j}$ is represented as

$\displaystyle\bm{S}_{j}=\bm{J}_{R}+\bm{J}_{L}-\frac{ib_{0}}{2}\operatorname{tr}(g\bm{\sigma}),$

(47)

with $\mathrm{SU(2)}$ currents $\bm{J}_{R}$ and $\bm{J}_{L}$, a fundamental field $g\in\mathrm{SU(2)}$, the Pauli matrices $\bm{\sigma}=(\sigma^{x}\,\sigma^{y}\,\sigma^{z})^{T}$ Affleck and Haldane (1987); Gogolin et al. (2004). The matrix $g\in\mathrm{SU(2)}$ is simply related to the U(1) bosons,

$\displaystyle g$

$\displaystyle=\begin{pmatrix}e^{i\sqrt{2\pi}\phi}&ie^{-i\sqrt{2\pi}\theta}\\ ie^{i\sqrt{2\pi}\theta}&e^{-i\sqrt{2\pi}\phi}\end{pmatrix}.$

(48)

Since global rotations keep the excitation spectrum unchanged, I perform a global $\pi/2$ rotation in the spin space as $(\sigma^{x},\,\sigma^{y},\,\sigma^{z})\to(\sigma^{z},\,\sigma^{y},\,-\sigma^{x})$. The rotation transforms the Hamiltonian (46) into

	$\displaystyle\tilde{\mathcal{H}}$	$\displaystyle=\frac{2\pi v}{3}\int dx\,(\bm{J}_{R}\cdot\bm{J}_{R}+\bm{J}_{L}\cdot\bm{J}_{L})$
		$\displaystyle\qquad+\frac{d_{xy}J\delta_{\perp}}{3}\int dx\,\operatorname{tr}(g)-i\frac{b_{0}h_{2}}{2}\int dx\,\operatorname{tr}(g\sigma^{z}).$		(49)

Translating it to the Abelian bosonizaion language, I can express this Hamiltonian as

	$\displaystyle\tilde{\mathcal{H}}$	$\displaystyle=\frac{v}{2}\int dx\bigl{\{}(\partial_{x}\theta)^{2}+(\partial_{x}\phi)^{2}\bigr{\}}$
		$\displaystyle\quad+\frac{2d_{xy}J\delta_{\perp}}{3}\int dx\cos(\sqrt{2\pi}\phi)-b_{0}h_{2}\int dx\,\sin(\sqrt{2\pi}\phi)$
		$\displaystyle=\frac{v}{2}\int dx\bigl{\{}(\partial_{x}\theta)^{2}+(\partial_{x}\phi)^{2}\bigr{\}}+g\int dx\cos(\sqrt{2\pi}\phi+\alpha),$		(50)

with the coupling constant $g=\sqrt{(2d_{xy}J\delta_{\perp}/3)^{2}+(b_{0}h_{2})^{2}}$ and the phase shift $\alpha=\tan^{-1}(3b_{0}h_{2}/2d_{xy}J\delta_{\perp})$. The complex model (40) is thus reduced to the simple sine-Gordon model (50).

The incommensurate phase shift $\alpha$ realizes the coexistence of the Néel order,

$\displaystyle N_{x}:=\sum_{j}(-1)^{j}\braket{S_{j}^{x}}/L,$

(51)

and the transverse dimer order (5). Their ground-state averages are given by

	$\displaystyle N_{x}$	$\displaystyle\propto\sqrt{\Delta/J}\sin\alpha\propto(g/J)^{1/3}\sin\alpha,$		(52)
	$\displaystyle D_{\perp}$	$\displaystyle\propto\sqrt{\Delta/J}\cos\alpha\propto(g/J)^{1/3}\cos\alpha.$		(53)

For $h_{0}/J\to 0$, the angle $\alpha$ approaches $\pi/2$. Therefore, at low fields $h_{0}/J\ll 1$, $N_{x}$ and $D_{\perp}$ are expected to follow power laws:

$\displaystyle N_{x}$

$\displaystyle\propto(h_{0}/J)^{1/3},\quad D_{\perp}\propto(h_{0}/J)^{4/3}.$

(54)

The power law (54) is qualitatively consistent with Figs. 6 and 8 at low fields. For weak magnetic fields, the longitudinal and the transverse dimer order parameters are almost equal. On the other hand, they are much smaller than the Néel order parameter $N_{x}$. If we assume $\delta_{\perp}=C_{\perp}{h_{4}}^{2}$, the bosonized theory (50) predicts a ratio $D_{\perp}/N_{x}$ given by

	$\displaystyle\frac{D_{\perp}}{N_{x}}$	$\displaystyle=\frac{d_{xy}}{b_{0}}\cot\alpha$
		$\displaystyle=\frac{2C_{\perp}}{3}\biggl{(}\frac{d_{xy}}{b_{0}}\biggr{)}^{2}\frac{{\alpha_{4}}^{2}}{\alpha_{2}}\frac{h_{0}}{J}.$		(55)

The right hand side is approximately estimated as $0.028C_{\perp}h_{0}/J$ for the parameters used in DMRG. If $C_{\perp}\approx 0.7$, the ratio (55) is roughly consistent with the DMRG data of Figs. 6 and 8 for $h_{0}/J<0.3$.

The model (1) that I have dealt with so far is similar to a model proposed for the spin-chain compound $\mathrm{[Cu(pym)(H_{2}O)_{4}]SiF_{6}\cdot H_{2}O}$ with the following Hamiltonian: Liu et al. (2019)

	$\displaystyle\mathcal{H}_{\rm exp}$	$\displaystyle=J\sum_{j}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}+\lambda S_{j}^{z}S_{j+1}^{z})$
		$\displaystyle\qquad+h_{0}\sum_{j}S_{j}^{x}+h_{2}\sum_{j}(-1)^{j}S_{j}^{y}+h_{4}\sum_{j}\delta_{j}S_{j}^{z}$
		$\displaystyle\qquad+D_{u}\sum_{j}(S_{j}^{x}S_{j+1}^{y}-S_{j}^{y}S_{j+1}^{x}),$		(56)

where $\lambda\approx 1$. There are three differences in two models (1) and (56): field directions, the weak exchange anisotropy, and the uniform DM interaction. In this section, I investigate effects of these differences one by one and discuss an experimental feasibility of the field-induced transverse dimer order.