# Easy-plane antiferromagnet in tilted field: gap in magnon spectrum and
susceptibility

Artemiy S. Sherbakov<EMAIL_ADDRESS>National Research Center
“Kurchatov Institute” B.P. Konstantinov Petersburg Nuclear Physics Institute,
Gatchina 188300, Russia  Oleg I. Utesov<EMAIL_ADDRESS>Saint Petersburg
State University, Ulyanovskaya 1, St. Petersburg 198504, Russia  National
Research Center “Kurchatov Institute” B.P. Konstantinov Petersburg Nuclear
Physics Institute, Gatchina 188300, Russia  St. Petersburg School of Physics,
Mathematics, and Computer Science, HSE University, St. Petersburg 190008,
Russia

###### Abstract

Motivated by recent experimental data on dichloro-tetrakis thiourea-nickel
(DTN) [Soldatov _et al_ , Phys. Rev. B 101, 104410 (2020)], a model of
antiferromagnet on a tetragonal lattice with single-ion easy-plane anisotropy
in the tilted external magnetic field is considered. Using the smallness of
the in-plane field component, we analytically address field dependence of the
energy gap in “acoustic” magnon mode and transverse uniform magnetic
susceptibility in the ordered phase. It is shown that the former is non-
monotonic due to quantum fluctuations, which was indeed observed
experimentally. The latter is essentially dependent on the “optical” magnon
rate of decay on two magnons. At magnetic fields close to the one which
corresponds to the center of the ordered phase, it leads to experimentally
observed dynamical diamagnetism phenomenon.

##  I Introduction

Quantum phase transitions and quantum phases of matter are among the most
extensively studied subjects of contemporary condensed matter physics [1]. In
this context, quantum magnets play an important role in revealing versatile
physics [2, 3, 4]. Importantly, in such compounds one can add impurities in
more or less controllable way [5, 6, 7], which allows studying effects of
quenched disorder and so-called “dirty-boson” systems [8, 9]. Moreover, such
elusive phases as Bose and Mott glasses can be observed in disordered quantum
magnets[6].

Particular compounds DTN [dichloro-tetrakis thiourea-nickel, NiCl2-4SC(NH2)2]
and its bromine-doped counterpart DTNX have been studied for last several
decades (see, e.g., Refs. [10, 11, 12, 13, 6, 14, 15] ). However, they are
still able to reveal peculiar features. Their properties can be described
using the model of weakly coupled antiferromagnetic (AF) chains with spin
$S=1$ and strong single-ion easy-plane anisotropy [16], see Fig. 1. The latter
is responsible for DTN interesting magnetic properties. In the external
magnetic field $h$ along the tetragonal axis, at $h<h_{c1}$ the system is in
singlet paramagnetic phase, at $h_{c1}<h<h_{c2}$ ordered canted AF (CAF) phase
emerges, whereas at $h>h_{c2}$ the fully polarized gapped state can be
observed [10, 16]. Critical fields $h_{c1},\,h_{c2}$ and their vicinity can be
discussed in terms of elementary excitations (triplons and magnons,
respectively) Bose-Einstein condensation [17, 18, 16, 19, 3, 12]. The
intermediate CAF phase not very close to the critical fields can be described
as a usual magnetically ordered phase and studied, e.g., by means of electron
spin resonance technique [11, 20, 21]. Interestingly, such experiments reveal,
for instance, nontrivial field variation of the “optical” (high-energy) magnon
frequency. In our previous work [22] it was explained in the framework of
conventional diagrammatic $1/S$ expansion, where the role of strong quantum
fluctuations was highlighted.

Figure 1: In the present paper, a model of an antiferromagnet relevant to DTN
and DTNX is considered. It consists of relatively weakly coupled
antiferromagnetic chains. There is also large easy-plane single-ion
anisotropy. In moderate external fields along the $z$ axis, the spins are
canted towards the field direction. Moreover, the slightly tilted towards the
$x$ axis field leads to rotational symmetry breaking and to an average $S^{x}$
component, which results in the emergence of the gap in the magnon spectrum.

In the present paper, we continue our theoretical discussion [22]. Here we
address the case of the tilted magnetic field realized experimentally [21].
Using the Kubo formalism for linear response and $1/S$ expansion, we show that
induced by the tilted field in-plane spin component inherits peculiar field-
behavior of the optical magnon frequency. This leads to experimentally
observed field dependence of the gap in the “acoustic” (low-energy) branch of
the spectrum. Moreover, we show that the optical branch near the center of the
ordered phase acquires significant damping due to two-magnon processes. So,
our theory quantitatively supports qualitative discussion of the dynamic
diamagnetism phenomenon observed in Ref. [21].

The rest of the paper is organized as follows. In Sec. II we formulate the
model, briefly recall our previous results, and employ our approach for the
gap in acoustic branch derivation. Section III is devoted to homogeneous
transverse susceptibility in the tilted field and the dynamical diamagnetism
effect. We present our conclusions in Sec. IV. Three appendices contain
cumbersome formulas and some other details of calculations.

##  II Gap in magnon spectrum

###  II.1 Problem formulation

We consider a model of an antiferromagnet with single-ion easy-plane
anisotropy in the canted external magnetic field on a simple tetragonal
lattice[23], see Fig. 1. For definiteness, we take the magnetic field in the
$xz$ plane ($xy$ is the easy plane). The corresponding Hamiltonian reads

$\mathcal{H}=\dfrac{1}{2}\sum_{<i,j>}J_{ij}\textbf{S}_{i}\cdot\textbf{S}_{j}+D\sum_{i}({S_{i}^{z}})^{2}-h_{z}\sum_{i}S_{i}^{z}-h_{x}\sum_{i}S_{i}^{x}.$
(1)

Here the external magnetic field is taken in the energy units
($h=g\mu_{B}|\mathbf{H}|$) and $D>0$ indicates the easy-plane anisotropy.
Since our aim is to address experimentally relevant case [20, 21] of small
tilt angles $\alpha$, we will treat the part of the Hamiltonian with
$h_{x}\approx\alpha h$ as a small perturbation. Hence, the starting point of
our analysis will be a theory from our previous paper [22]. Let us briefly
recall its main findings.

We considered the case of the external field along the tetragonal axis. We
mainly discussed the “optical” magnon branch and its energy in the center of
the Brillouin zone $\Delta$ (which can be probed by ESR [20, 21]) employing
Holstein-Primakoff representation of spin operators via bosonic ones [24] in
the alternating coordinate frame suitable for the canted antiferromagnetic
ordering [25]. Using quasiclassical $1/S$-series diagrammatic approach, we
found a strong renormalization of the magnon spectrum due to the first order
in $1/S$ quantum corrections, which lead to experimentally observed [20, 21]
nonmonotonous dependence $\Delta(h)$.

Despite being small, the transverse magnetic field $h_{x}$ has an important
effect on the system: it breaks the rotational symmetry of the easy $xy$
plane. This results in the disappearance of the Goldstone mode, the
corresponding magnon branch becomes gapped and will be called “quasi-
Goldstone” below.

Our main goal is to analytically describe the properties of the emergent gap
(as a function of $h$ and $\alpha$) and the magnetic susceptibility of the
system. For this sake, we start from the ground state at $\alpha=0$ and
immediately observe that $h_{x}$ terms in the Hamiltonian (1) provide linear
in bosonic operators terms (equivalent to the ground state instability), which
in the reciprocal space read

$\displaystyle\frac{h_{x}\sqrt{SN}}{\sqrt{2}}\left(a^{\dagger}_{\mathbf{k}=0}+a_{\mathbf{k}=0}\right),$
(2)

see Appendix A for some details. We eliminate these terms using the following
shift in operators:

$a_{\mathbf{k}=0}=b_{\mathbf{k}=0}+\rho,\quad
a^{\dagger}_{\mathbf{k}=0}=b^{\dagger}_{\mathbf{k}=0}+\rho.$ (3)

Parameter $\rho$ is related to the linear response of the system to $h_{x}$
and can be found directly from the Hamiltonian or using the Kubo
susceptibility. In any case, we found that $\rho\propto\alpha$. Importantly,
$\rho\neq 0$ leads to new terms in the Hamiltonian, which will be used for
discussing the gap in the quasi-Goldstone mode and the magnetic susceptibility
in the tilted field. Note that the smallness of $\alpha$ allows us to omit
terms $\propto\rho^{n},n>2$ when recalculating the Hamiltonian for nonzero
$h_{x}$.

###  II.2 Calculation of $\rho$

Here we derive an analytical expression for the crucial parameter $\rho$.
Importantly, it is simply related to the average $S^{x}$ value, which is the
same for all spins. Explicitly:

$\langle S^{x}\rangle=\frac{1}{N}\sum_{i}\langle
S^{x}_{i}\rangle=\frac{\sqrt{S}}{\sqrt{2}N}\sum_{i}\langle
a^{\dagger}_{i}+a_{i}\rangle\\\ =\frac{\sqrt{S}}{\sqrt{2N}}\langle
a^{\dagger}_{\mathbf{k=0}}+a_{\mathbf{k=0}}\rangle=\sqrt{2}S\frac{\rho}{\sqrt{SN}}.$
(4)

Parameter $\rho$ can be found by making linear terms in the Hamiltonian to be
zero [see Eq. (2)] using the shift (3). However, in order to take into account
quantum corrections, we will use a more general method of the Kubo
susceptibility:

$\displaystyle\chi=\langle
S^{x}(\mathbf{q},\omega)S^{x}(-\mathbf{q},\omega)\rangle\Bigr{|}_{\mathbf{q}=0,\omega=0,h_{x}=0}=$
$\displaystyle=\frac{S}{2}\langle(a^{\dagger}_{\mathbf{q}}+a_{-\mathbf{q}})(a^{\dagger}_{-\mathbf{q}}+a_{\mathbf{q}})\rangle_{\omega}\Bigr{|}_{\mathbf{q}=0,\omega=0,h_{x}=0}=$
$\displaystyle=-\frac{S}{2}(F_{k=0}+G_{k=0}+F^{\dagger}_{k=0}+G_{k=0}).$ (5)

So, $\chi$ is the sum of retarded magnon Green’s functions taken at
$k=(\omega=0,\mathbf{k}=0)$, see Appendix A for details. The latter can be
obtained at $h_{x}=0$ using the conventional diagrammatic approach.

In the linear spin-wave theory we have

$\displaystyle\chi_{0}=\frac{1}{2SJ_{\mathbf{0}}},$ (6)

which yields

$\displaystyle\langle S^{x}\rangle=\dfrac{h_{x}}{2SJ_{\mathbf{0}}},$ (7)
$\displaystyle\frac{\rho}{\sqrt{NS}}=\dfrac{h_{x}}{2\sqrt{2}SJ_{\mathbf{0}}}.$
(8)

One sees that the linear spin-wave theory predicts constant susceptibility.

When $1/S$ corrections are taken into account, the susceptibility behavior
becomes much more interesting. Using Eq. (5) with renormalized by magnon
interaction Green’s functions, we obtain

$\displaystyle\chi_{1/S}(h)=\frac{P(h)}{2\Delta^{2}(h)},$ (9)

where

$\displaystyle P(h)$ $\displaystyle=$ $\displaystyle
2E_{\mathbf{k}}-2B_{\mathbf{k}}+\Sigma_{k}+\Sigma_{-k}-2\Pi_{k}\Bigr{|}_{k=0},$
(10) $\displaystyle\Delta(h)$ $\displaystyle=$
$\displaystyle\varepsilon_{\mathbf{k}}+(\Sigma_{k}-\Sigma_{-k})/2$ (11)
$\displaystyle+\left[E_{\mathbf{k}}(\Sigma_{k}+\Sigma_{-k})-2B_{\mathbf{k}}\Pi_{k}\right]/2\varepsilon_{\mathbf{k}}\Bigr{|}_{k=0}.$

Here $\varepsilon_{\mathbf{k}}$ is the magnon spectrum of the linear theory
and, as one can see, we use the on-shell approximation for self-energy parts.
Importantly, $\Delta(h)$ is renormalized energy of the optical magnon with
$\mathbf{k}=0$, which was studied in detail in Ref. [22] and has nonmonotonic
behavior as a function of $h$ due to the $1/S$ corrections. This effect is
also pronounced in the susceptibility and $\langle S^{x}\rangle$, as it is
shown in Figs. 2 and 3 where the linear spin-wave theory results are
contrasted to the ones, obtained using the first $1/S$ corrections. We report
drastic differences in the corresponding results, which are inherited from
$\Delta(h)$.

Figure 2: Comparison between magnetic susceptibility calculated in the linear
theory $\chi_{0}$ (black line) and the one, including $1/S$ corrections to
Green’s functions, $\chi_{1/S}$, (blue line) as functions of magnetic field
$h$. Parameters $\alpha=\pi/60$, $D=8$ K, $J_{c}=2$ K, $J_{a}=0.2$ K were
used.  Figure 3: The dependence of the average spin component $S^{x}$ on $h$
in the tilted field case, calculated using Kubo formalism in linear theory
(black line) and including quantum corrections (blue line). Parameters
$\alpha=\pi/60$, $D=8$ K, $J_{c}=2$ K, $J_{a}=0.2$ K were used.

###  II.3 Gap at $h_{x}\neq 0$.

After the shift in Bose-operators (3), multiple terms $\propto\rho^{n},\,n\geq
1$ emerge in the Hamiltonian in addition to those discussed in Appendix A.
Some details are presented in Appendix B.

Linear in Bose-operators terms are canceled by proper choice of $\rho$ (see
previous Subsec.). We also neglect “decay” terms $\delta\mathcal{H}_{3}$ (and
the higher order in Bose-operators ones) because they provide small in $1/S$
corrections. Moreover, the subject of the calculations below is magnon with
$\mathbf{k}=\mathbf{k}_{0}$ and of minimal energy, which cannot acquire a
finite lifetime due to such processes.

So, we focus on corrections to the bilinear part of the Hamiltonian.
Importantly, they include normal and umklapp parts, which read

$\displaystyle\delta\mathcal{H}^{N}_{2}$ $\displaystyle=$ $\displaystyle
S\sum_{\mathbf{k}}\overline{E}_{\mathbf{k}}b_{\mathbf{k}}^{\dagger}b_{\mathbf{k}}+S\sum_{\mathbf{k}}\overline{B}_{\mathbf{k}}\frac{b_{\mathbf{k}}b_{-\mathbf{k}}+b_{\mathbf{k}}^{\dagger}b_{-\mathbf{k}}^{\dagger}}{2},$
(12) $\displaystyle\delta\mathcal{H}_{2}^{U}$ $\displaystyle=$ $\displaystyle
iS\sum_{\mathbf{k}}X_{\mathbf{k}}\left(b_{\mathbf{k}}^{\dagger}b_{\mathbf{k}-\mathbf{k_{0}}}-b_{\mathbf{k}-\mathbf{k_{0}}}^{\dagger}b_{\mathbf{k}}\right)$
$\displaystyle+iS\sum_{\mathbf{k}}Y_{\mathbf{k}}\frac{b_{\mathbf{k}}b_{\mathbf{k_{0}}-\mathbf{k}}-b_{\mathbf{k}}^{\dagger}b_{\mathbf{k_{0}}-\mathbf{k}}^{\dagger}}{2},$

respectively. Explicit formulas for the coefficients are presented in Appendix
B.

Next, one can see that the normal corrections can be directly plugged into the
equation for the magnon spectrum of the linear theory (29), whereas the
umklapps require some additional calculations. They contribute to both normal
and anomalous self-energy parts; corresponding diagrams are shown in Fig. 4.

Figure 4: Umklapp terms-induced corrections to the self-energy parts
$\Sigma^{U}$ (left) and $\Pi^{U}$ (right). $X_{\mathbf{k}}$ and
$Y_{\mathbf{k}}$ denote the corresponding umklapp amplitudes, see Eqs. (40)
and (41).

However, at $\mathbf{k=k_{0}}$ only the normal terms contribute to the
spectrum since the umklapp corrections cancel each other (see Appendix B). So,
the equation for the gap in the spectrum reads:

$\displaystyle\omega_{\mathbf{k=k_{0}}}=\Delta_{h_{x}}=\sqrt{2E_{\mathbf{k}_{0}}(\overline{E}_{\mathbf{k}_{0}}+\overline{B}_{\mathbf{k}_{0}})}$
(14)

Using the expressions for $\overline{E}_{\mathbf{k}_{0}}$ (38) and
$\overline{B}_{\mathbf{k}_{0}}$ (39) and taking the value of parameter $\rho$
calculated within the Kubo formalism, we get

$\displaystyle\Delta_{h_{x}}=\sqrt{2E_{\mathbf{k_{0}}}A_{h}}\dfrac{S\rho}{\sqrt{NS}},$
(15)

where

$\displaystyle
A_{h}=\dfrac{h_{x}\sqrt{NS}}{S\rho}+8U_{\mathbf{0k_{0}0-k_{0}}}+12V_{\mathbf{0k_{0}0}}+8U_{\mathbf{k_{0}k_{0}00}}+12V_{\mathbf{0k_{0}k_{0}}}.$

The first term in Eq. (II.3) is much smaller than the others and it is
significant only near the critical fields $h_{c1}$ and $h_{c2}$. In the
intermediate field region (on which we mainly focus) it can be safely
neglected. Finally, we obtain the following result for the gap value:

$\displaystyle\Delta_{h_{x}}=\sqrt{E_{\mathbf{k_{0}}}A_{h}}\langle
S^{x}\rangle.$ (17)

Results of our calculations are presented in Fig. 5 for a particular parameter
set. Note that the gap is proportional to the average $S^{x}$ value (cf. Fig.
3). Moreover, its behavior is similar to the one, observed experimentally [20,
21].

Figure 5: Field dependencies of frequencies of “quasi-Goldstone”
($\mathbf{k}=\mathbf{k}_{0}$; black curve) and “optical”
($\mathbf{k}=\mathbf{0}$; blue curve) magnons. Parameters $\alpha=\pi/60$,
$D=8.0$K, $J_{c}=2$K, and $J_{a}=0.2$K were used.

##  III Magnetic susceptibility

Here we discuss the uniform transverse magnetic susceptibility of the system
in the tilted external field. A crucial point here is the significant damping
of the optical magnon with $\mathbf{k}=0$, which is addressed below.

###  III.1 Magnon damping

In our previous work [22], we were particularly interested in the energy of
the optical magnon with $\mathbf{k}=0$. For our current purposes, it’s damping
also becomes important. Theoretically, the finite magnon lifetime is connected
with the possibility for it to decay; in our case on two other magnons.

Explicitly, the damping comes from the imaginary part of magnon Green’s
function self-energies [see Eq. (35)]. Moreover, here we can neglect small
$h_{x}$-induced terms in the Hamiltonian. Separating the imaginary part of the
corresponding denominator from the real one, we get it in the first order in
$1/S$:

$\displaystyle\Gamma(k)=\mathrm{Im}\left[\dfrac{\Sigma_{k}-\Sigma_{-k}}{2}+\dfrac{E_{\mathbf{k}}(\Sigma_{k}+\Sigma_{-k})-2B_{\mathbf{k}}\Pi_{k}}{2\varepsilon_{k}}\right].$

Corresponding imaginary parts of normal and anomalous self-energy parts are
due to $\mathcal{H}_{3}$ terms [see Eq. (A)] where a magnon with momentum
$\mathbf{k}$ can decay on magnons with momenta $\mathbf{k-p+k_{0}}$ and
$\mathbf{p}$. Below, we only discuss the damping of magnon with
$\mathbf{k}=\mathbf{0}$ and use the on-shell approximation. The corresponding
damping is $h$-dependent, we denote it as $\Gamma(h)$.

Figure 6: “Optical” magnon with $\mathbf{k}=\mathbf{0}$ damping $\Gamma(h)$ as
a function of external magnetic field. Parameters $D=8$ K, $J_{c}=2$ K, and
$J_{a}=0.2$ K were used.

Our calculations show that $\Gamma(h)$ becomes zero at the critical fields
$h_{c1}$, $h_{c2}$ and has a maximum with significant value in between (near
the ordered phase center). This result is illustrated in Fig. 6 for particular
parameters set .

###  III.2 Susceptibility

The generalization of Eq. (II.2) for the static transverse spin susceptibility
is the sum of retarded Green’s functions:

$\displaystyle\chi^{\perp}(\omega,\mathbf{k})=-\dfrac{1}{2}\left(G^{R}_{\omega,\mathbf{k}}+G^{R}_{-\omega,-\mathbf{k}}+F^{R}_{\omega,\mathbf{k}}+F^{R}_{-\omega,-\mathbf{k}}\right),$

where one should take into account corrections from magnetic field $h_{x}$.

In our calculations, in Eq. (III.2) we use the first order in $1/S$ self-
energies, which account for the umklapps. The result is the following (see
Appendix C for the details):

$\displaystyle\mathrm{Re}\left[\chi^{\perp}(\omega,\mathbf{k=0})\right]=-\mathrm{Re}\Biggl{\\{}\frac{P(h)+2\overline{E}_{\mathbf{k}=0}-2\overline{B}_{\mathbf{k}=0}}{\omega^{2}-(\Delta-i\Gamma)^{2}}$
$\displaystyle+\frac{K}{[(\omega+i\delta)^{2}-\Delta^{2}_{h_{x}}][\omega^{2}-(\Delta-i\Gamma)^{2}]}\Biggr{\\}}.$
(20)

Importantly, this formula has two types of poles, corresponding to “stable”
acoustic magnon ($\delta\rightarrow+0$) and to damped optical magnon. So, an
interplay between them determines uniform susceptibility as a function of
frequency $\chi^{\perp}(\omega)$. Moreover, the magnitude of optical magnon
damping $\Gamma$ plays a crucial role here, see Fig. 7.

In the absence of the damping ($\Gamma=0$), the real part of the
susceptibility at $\omega>0$ has two hyperbolic features associated with
acoustic and optical magnon energies. However, when the damping grows up (what
happens near the ordered phase center, see Fig. 6) the qualitative behavior of
$\mathrm{Re}\,\chi^{\perp}(\omega)$ changes. Region of
$\mathrm{Re}\,\chi^{\perp}(\omega)>0$, pronounced near $\omega=\Delta$ at
small $\Gamma$, fades away. Thus, in our analytical consideration, we observe
the change of the susceptibility sign in that frequency domain, which was
called “dynamic diamagnetism” in Ref. [21]. Moreover, our interpretation
agrees with the one from this paper on a qualitative level, since two-magnon
processes are the reason for large $\Gamma$ in our calculations.

Figure 7: The real part of spin susceptibility $\chi^{\perp}$ at $H_{X}=7.3$T
for different damping constants $\Gamma$ (III.1). Parameters $\alpha=\pi/60$,
$D=8$ K, $J_{c}=2$ K, and $J_{a}=0.2$ K were used.

##  IV Discussion and Conclusions

We study antiferromagnet with large single-ion easy-plane anisotropy in the
tilted external field at small tilt angles in the ordered canted phase. In
this case, the in-plane field component is small and we use linear response
theory (Kubo formalism) to calculate the corresponding corrections to the
ground state. This allows us to obtain corrections to the Hamiltonian, which
appear due to the in-plane field component. Also, the latter breaks the
rotational symmetry with respect to the tetragonal axis. Importantly, this
leads to the gap in magnon spectrum emergence, which is non-monotonic under
the external field variation. Theoretically, this feature is inherited from
also nonmonotonic behavior of the energy of the optical magnon in the center
of the Brillouin zone. It was studied in our previous paper [22] and connected
with strong quantum fluctuations-induced renormalization of this energy.
Moreover, we show that our result for the gap field-dependence is in good
qualitative agreement with the experimental findings of Ref. [20], where DTN
was studied.

After establishing the ground state in the tilted field, our formalism also
allows for studying transverse spin susceptibility. It is shown that there are
two crucial points in this derivation. The first one is the significant
damping of the optical magnons, which is connected with decay on two magnons.
The second one is accurate accounting for the umklapp terms in the
Hamiltonian.

As a result, using realistic DTN parameters, we theoretically explain the
effect of dynamic diamagnetism, which was experimentally observed in Ref.
[21]. Moreover, two-magnon processes plays important role in our derivation,
which lies in agreement with the interpretation proposed in that paper.

It is pertinent to mention here, that the best fit of the experimental data
was not our goal, and we use some realistic parameters of DTN just for
illustration purposes. Furthermore, due to the quasi-1D nature of DTN, the
dependence of the results on the parameter values is strong and the question
of higher order corrections in the $1/S$ series can be also raised.
Nevertheless, we find that our qualitative conclusions are true in a wide
range of parameters. Moreover, we can report that the used parameters result
in a semi-quantitative agreement with the experimental data.

Our theory and its results should be compared with the approach developed in
Ref. [26]. The advantages of the latter are its elegance and simple analytical
formulas as the yield. However, it is valid only for magnetic fields close to
the center of the ordered phase and relies on $J/D$ smallness. In our
approach, the calculations are much more cumbersome, but the methods are more
general. The latter allows us to address the transverse susceptibility in the
same framework as the gap in the magnon spectrum.

###### Acknowledgements.

We are grateful to A.I. Smirnov for stimulating discussions. The reported
study was supported by the Foundation for the Advancement of Theoretical
Physics and Mathematics “BASIS”.

##  Appendix A Hamiltonian at $h_{x}=0$

Here we briefly remind the basic technique, which was used in our previous
study [22] (see also Refs. [25, 12]). Hamiltonian (1) can be transformed into
bosonic one using conventional Holstein-Primakoff spin-operators
representation [24] in the approximate form (which allows taking into account
first $1/S$ corrections):

$\displaystyle
S_{i}^{x^{\prime}}+iS_{i}^{y^{\prime}}=\sqrt{2S}a^{\dagger}_{i}\sqrt{1-\dfrac{a^{\dagger}_{i}a_{i}}{2S}}\approx\sqrt{2S}a^{\dagger}_{i}\left(1-\dfrac{a^{\dagger}_{i}a_{i}}{4S}\right),$
$\displaystyle
S_{i}^{x^{\prime}}-iS_{i}^{y^{\prime}}=\sqrt{2S}\sqrt{1-\dfrac{a^{\dagger}_{i}a_{i}}{2S}}a_{i}\approx\sqrt{2S}\left(1-\dfrac{a^{\dagger}_{i}a_{i}}{4S}\right)a_{i},$
$\displaystyle S_{i}^{z^{\prime}}=-S+a^{\dagger}_{i}a_{i}$ (21)

in a local coordinate frame, suitable for canted antiferromagnetic order
description [$\mathbf{k}_{0}=(\pi,\pi,\pi)$ is the AF vector]:

$\displaystyle S_{i}^{x}=S_{i}^{x^{\prime}},$ $\displaystyle
S_{i}^{y}=S_{i}^{y^{\prime}}\cos{\theta}+S_{i}^{z^{\prime}}\exp(i\textbf{k}_{0}\textbf{R}_{i})\sin{\theta},$
$\displaystyle
S_{i}^{z}=S_{i}^{z^{\prime}}\cos{\theta}-S_{i}^{y^{\prime}}\exp(i\textbf{k}_{0}\textbf{R}_{i})\sin{\theta}.$

As a result we have $\mathcal{H}=\sum^{4}_{n=0}\mathcal{H}_{n}$; each term
contains $n$ Bose-operators. Explicitly:

$\displaystyle\mathcal{H}_{1}=iS^{3/2}\sqrt{N}\left(La_{\mathbf{k_{0}}}-La_{\mathbf{k_{0}}}^{\dagger}\right),$
(23) $\displaystyle
L=\dfrac{1}{\sqrt{2}}\left[(2J_{0}+2\widetilde{D})\cos\theta-\dfrac{h_{z}}{S}\right]\sin\theta.$
(24)

This term can be eliminated by taking proper canting angle $\theta$, which
depends on the external field. Next, the bilinear part is the following:

$\displaystyle\mathcal{H}_{2}=S\sum_{\mathbf{k}}a_{\mathbf{k}}^{\dagger}a_{\mathbf{k}}E_{\mathbf{k}}+\frac{S}{2}\sum_{\mathbf{k}}(a_{\mathbf{k}}a_{-\mathbf{k}}+a_{\mathbf{k}}^{\dagger}a_{-\mathbf{k}}^{\dagger})B_{\mathbf{k}},$
(25) $\displaystyle
E_{\mathbf{k}}=S(J_{0}+J_{\mathbf{k}})\cos^{2}\theta+S(J_{0}+D)\sin^{2}\theta,$
(26) $\displaystyle B_{\mathbf{k}}=S(J_{\mathbf{k}}-D)\sin^{2}\theta.$ (27)

Here the Fourier transform of the exchange interaction reads

$\displaystyle
J_{\mathbf{k}}=2\left[J_{c}\cos{k_{z}}+J_{a}(\cos{k_{x}}+\cos{k_{y}})\right].$
(28)

So, the magnon spectrum in the linear theory is given by:

$\displaystyle\varepsilon_{\mathbf{k}}=\sqrt{{E^{2}_{\mathbf{k}}}-{B^{2}_{\mathbf{k}}}}.$
(29)

Since the exchange interaction Fourier transform $J_{\mathbf{k}}$ obeys the
property $J_{\mathbf{0}}=-J_{\mathbf{k}_{0}}$, this spectrum is gapless with
$\varepsilon_{\mathbf{k}_{0}}=0$ in agreement with the Goldstone’s theorem.

We also have the interaction part of the Hamiltonian. It consists of two
terms, the first one is

$\displaystyle\mathcal{H}_{3}$ $\displaystyle=$ $\displaystyle
i\dfrac{\sqrt{S}\sin\theta\cos\theta}{4\sqrt{2N}}\sum_{123=k_{0}}(a_{1}^{\dagger}a_{2}^{\dagger}a_{-3}-a_{-3}^{\dagger}a_{2}a_{1})\dfrac{(V_{\mathbf{1}}+V_{\mathbf{2}})}{2},$
$\displaystyle V_{\mathbf{k}}$ $\displaystyle=$
$\displaystyle\left(2J_{\mathbf{k}_{0}}-8J_{\mathbf{k}}+10D-\dfrac{h}{S\cos\theta}\right),$
(30)

where we denote $\textbf{k}_{1},\textbf{k}_{2},\textbf{k}_{3}$ as $1,2,3$, and
the momentum conservation law reads

$\displaystyle\textbf{k}_{1}+\textbf{k}_{2}+\textbf{k}_{3}=\textbf{k}_{0}.$
(31)

The second one is the following:

$\displaystyle\mathcal{H}_{4}=\dfrac{1}{N}\sum_{1234=0}U_{\mathbf{1234}}a_{1}^{\dagger}a_{2}^{\dagger}a_{-3}a_{-4}+$
$\displaystyle+\dfrac{1}{N}\sum_{1234=0}V_{\mathbf{123}}(a_{1}^{\dagger}a_{2}^{\dagger}a{3}^{\dagger}a_{-4}+a_{-4}^{\dagger}a_{3}a_{2}a_{1})$
(32)

with the conventional momentum conservation law. Here

$\displaystyle U_{\mathbf{1234}}$ $\displaystyle=$
$\displaystyle\Biggl{[}\dfrac{1-2\sin^{2}\theta}{2}J_{\mathbf{4}-\mathbf{1}}-\dfrac{\cos^{2}\theta}{4}(J_{\mathbf{1}}+J_{\mathbf{4}})$
$\displaystyle+D\left(1-\dfrac{3}{2}\sin^{2}\theta\right)\Bigg{]},$
$\displaystyle V_{\mathbf{123}}$ $\displaystyle=$
$\displaystyle\left[\dfrac{D\sin^{2}\theta}{4}-\dfrac{(J_{\mathbf{1}}+J_{\mathbf{2}}+J_{\mathbf{3}})\sin^{2}\theta}{12}\right].$
(34)

For non-linear corrections to magnon spectrum analysis, it is convenient to
use normal ($G_{k}=\langle
a_{\textbf{k}},a_{\textbf{k}}^{\dagger}\rangle_{\omega}$) and anomalous
($F^{\dagger}_{k}=\langle
a_{-\textbf{k}}^{\dagger},a_{\textbf{k}}^{\dagger}\rangle_{\omega}$) Green’s
functions. The solution of the system of Dyson’s equation for these functions
reads

$\displaystyle G_{k}=\dfrac{\omega+E_{\textbf{k}}+\Sigma_{-k}}{D_{\omega,k}},$
(35) $\displaystyle F_{k}=-\dfrac{B_{\textbf{k}}+\Pi_{k}}{D_{\omega,k}},$ (36)
$\displaystyle
D_{k}=\omega^{2}-\varepsilon^{2}_{\textbf{k}}-E_{\textbf{k}}(\Sigma_{k}+\Sigma_{-k})+2B_{\textbf{k}}\Pi_{k}+$
$\displaystyle+\omega(\Sigma_{k}-\Sigma_{-k}),$ (37)

where $\Sigma_{k},\Pi_{k}$ are normal and anomalous self-energy parts,
respectively. They can be calculated perturbatively.

##  Appendix B Corrections to the Hamiltonian at small $h_{x}$.

When the external magnetic field is tilted, nonzero $h_{x}$ appears. It leads
to uniform $\langle S^{x}\rangle$ component. In our theory, it is related to
the parameter $\rho$. Nonzero $\rho$ provides various additional terms in the
Hamiltonian, which can be altogether denoted as $\delta\mathcal{H}$.

Linear terms $\delta\mathcal{H}_{1}$ vanish when $\rho$ is properly chosen.

The bilinear part of $\delta\mathcal{H}_{2}$ can be divided onto normal and
umklapp contributions [see Eqs. (12) and (II.3)]. Explicit expressions for the
corresponding coefficients in $\delta\mathcal{H}^{N}_{2}$ are the following:

$\displaystyle\overline{E}_{\mathbf{k}}=\dfrac{h_{x}\rho}{2S\sqrt{2NS}}+8U_{\mathbf{0k0-k}}\dfrac{\rho^{2}}{NS}+12V_{\mathbf{0k0}}\dfrac{\rho^{2}}{NS},$
(38)
$\displaystyle\overline{B}_{\mathbf{k}}=\dfrac{h_{x}\rho}{2S\sqrt{2NS}}+8U_{\mathbf{kk00}}\dfrac{\rho^{2}}{NS}+12V_{\mathbf{0k-k}}\dfrac{\rho^{2}}{NS}.$
(39)

These terms are analogous to the contributions in Eq. (25). For
$\delta\mathcal{H}^{U}_{2}$ the corresponding equations read

$\displaystyle
X_{\mathbf{k}}=\sin\theta\cos\theta(V_{\mathbf{0}}+V_{\mathbf{k}})\dfrac{\rho}{4\sqrt{2NS}},$
(40) $\displaystyle
Y_{\mathbf{k}}=\sin\theta\cos\theta(V_{\mathbf{k}-\mathbf{k}_{0}}+V_{\mathbf{k}})\dfrac{\rho}{4\sqrt{2NS}}.$
(41)

Importantly, the umklapps provide corrections to the normal and anomalous
self-energy parts $\sim\rho^{2}$ (they are combined in pairs, see Fig. 4):

$\displaystyle\Sigma^{U}_{\mathbf{k}}=X^{2}_{\mathbf{k}}G_{\omega,\mathbf{k-k_{0}}}+Y^{2}_{\mathbf{k}}G_{-\omega,\mathbf{k_{0}-k}}-$
$\displaystyle-
X_{\mathbf{k}}Y_{\mathbf{k}}F_{\omega,\mathbf{k-k_{0}}}-X_{\mathbf{k}}Y_{\mathbf{k}}F_{-\omega,\mathbf{k_{0}-k}}$
(42)
$\displaystyle\Pi^{U}_{\mathbf{k}}=X^{2}_{\mathbf{k}}F_{\omega,\mathbf{k-k_{0}}}+Y^{2}_{\mathbf{k}}F_{-\omega,\mathbf{k_{0}-k}}-$
$\displaystyle-
X_{\mathbf{k}}Y_{\mathbf{k}}G_{\omega,\mathbf{k-k_{0}}}-X_{\mathbf{k}}Y_{\mathbf{k}}G_{-\omega,\mathbf{k_{0}-k}}$
(43)

However, one can see that at the momentum $\mathbf{k}=\mathbf{k}_{0}$
condition $X_{\mathbf{k}_{0}}=Y_{\mathbf{k}_{0}}$ is satisfied, so
$\Sigma^{U}_{\mathbf{k_{0}}}=-\Pi^{U}_{\mathbf{k_{0}}}$. This means that there
is no correction to the energy of magnon with $\mathbf{k}=\mathbf{k}_{0}$ from
the umklapp processes and only the normal terms are important in this context.

##  Appendix C Derivation of transverse susceptibility in tilted field

Here, for calculations, it is also convenient to use normal and anomalous
Green’s functions:

$G_{k}=\langle
b_{\textbf{k}},b_{\textbf{k}}^{\dagger}\rangle_{\omega};\,F^{\dagger}_{k}=\langle
b_{-\textbf{k}}^{\dagger},b_{\textbf{k}}^{\dagger}\rangle_{\omega},$ (44)

where $k=(\omega,\textbf{k})$. These functions obey the following system of
Dyson’s equations

$\displaystyle
G_{k}=G^{0}_{k}+G^{0}_{k}\,\Sigma_{k}\,G_{k}+G^{0}_{k}(B_{\textbf{k}}+\overline{B}_{\mathbf{k}}+\Pi_{k})F_{k},$
(45) $\displaystyle
F_{k}=G^{0}_{-k}(B_{\textbf{k}}+\overline{B}_{\mathbf{k}}+\Pi_{k})G_{k}+G^{0}_{-k}\,\Sigma_{-k}\,F_{k},$

where the bare Green’s function reads

$\displaystyle G^{0}=\dfrac{1}{\omega-
E_{\textbf{k}}-\overline{E}_{\mathbf{k}}+i\delta}.$ (46)

After solving the system of equations (45) we obtain the following expression
for normal and anomalous Green’s functions

$\displaystyle
G_{k}=\dfrac{\omega+E_{\textbf{k}}+\overline{E}_{\mathbf{k}}+\Sigma_{k}}{D_{\omega,k}};$
(47) $\displaystyle
F_{k}=-\dfrac{B_{\textbf{k}}+\overline{B}_{\mathbf{k}}+\Pi_{k}}{D_{\omega,k}}.$
(48)

Here the denominator reads

$\displaystyle
D_{\omega,k}=\omega^{2}-\varepsilon^{2}_{\textbf{k}}-E_{\textbf{k}}(\Sigma_{k}+\Sigma_{-k}+2\overline{E}_{\mathbf{k}})$
$\displaystyle+2B_{\textbf{k}}(\Pi_{k}+\overline{B}_{\mathbf{k}})+\omega(\Sigma_{k}-\Sigma_{-k}).$
(49)

To calculate the susceptibility (III.2), it is necessary to use the retarded
Green’s functions $G^{R}$ and $F^{R}$, which are related to the causal ones
(47) by the following relation:

$\displaystyle
G^{R}_{\omega,\mathbf{k}}=\theta(\omega)G_{\omega,\mathbf{k}}+\theta(-\omega)G^{*}_{\omega,\mathbf{k}},$
(50)

and the same for $F$. The real parts of retarded functions are equal to the
corresponding causal ones. So, we get

$\displaystyle\mathrm{Re}[\chi^{\perp}_{\omega,\mathbf{k}}]=-\mathrm{Re}\Biggl{\\{}$
$\displaystyle\frac{2E_{\mathbf{k}}+\Sigma_{k}+\Sigma_{-k}-2B_{\mathbf{k}}-2\Pi_{k}}{\omega^{2}-(\Delta-i\Gamma)^{2}}+$
(51)
$\displaystyle+\frac{2\overline{E}_{\mathbf{k}}-2\overline{B}_{\mathbf{k}}+2\Sigma^{U}_{\mathbf{k}}-2\Pi^{U}_{\mathbf{k}}}{\omega^{2}-(\Delta-i\Gamma)^{2}}\Biggr{\\}}.$

Using Eqs. (B) and (B) at $\mathbf{k}=0$ and Eq. (10), the real part of the
uniform transverse spin susceptibility can be finally expressed as

$\displaystyle\mathrm{Re}[\chi^{\perp}_{\omega,\mathbf{k=0}}]=-\mathrm{Re}\Biggl{\\{}\frac{P(h)+2\overline{E}_{\mathbf{k}=0}-2\overline{B}_{\mathbf{k}=0}}{\omega^{2}-(\Delta-i\Gamma)^{2}}+$
$\displaystyle\frac{K}{[(\omega+i\delta)^{2}-\Delta^{2}_{h_{x}}][\omega^{2}-(\Delta-i\Gamma)^{2}]}\Biggr{\\}},$

where

$\displaystyle
K=2(X_{\mathbf{k=0}}+Y_{\mathbf{k=0}})^{2}(\overline{E}_{\mathbf{k_{0}}}+\overline{B}_{\mathbf{k_{0}}}).$
(53)

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