Easy-Plane Antiferromagnet in Tilted Field: Gap in Magnon Spectrum and Susceptibility

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.

Appendices

APPENDIX A

HAMILTONIAN AT h _x = 0

Here we briefly remind the basic technique, which was used in our previous study [22] (see also [12, 25]). 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):

$$\begin{gathered} S_{i}^{{x'}} + iS_{i}^{{y'}} = \sqrt {2S} a_{i}^{\dag }\sqrt {1 - \frac{{a_{i}^{\dag }{{a}_{i}}}}{{2S}}} \approx \sqrt {2S} a_{i}^{\dag }\left( {1 - \frac{{a_{i}^{\dag }{{a}_{i}}}}{{4S}}} \right), \\ S_{i}^{{x'}} - iS_{i}^{{y'}} = \sqrt {2S} \sqrt {1 - \frac{{a_{i}^{\dag }{{a}_{i}}}}{{2S}}} {{a}_{i}} \approx \sqrt {2S} \left( {1 - \frac{{a_{i}^{\dag }{{a}_{i}}}}{{4S}}} \right){{a}_{i}}, \\ S_{i}^{{z'}} = - S + a_{i}^{\dag }{{a}_{i}} \\ \end{gathered} $$

(A.1)

in a local coordinate frame, suitable for canted antifer-romagnetic order description [k₀ = (π, π, π) is the AF vector]:

$$\begin{gathered} S_{i}^{x} = S_{i}^{{x'}}, \\ S_{i}^{y} = S_{i}^{{y'}}\cos \theta + S_{i}^{{z'}}\exp (i{{{\mathbf{k}}}_{0}} \cdot {{{\mathbf{R}}}_{0}})\sin \theta , \\ S_{i}^{z} = S_{i}^{{z'}}\cos \theta - S_{i}^{{y'}}\exp (i{{{\mathbf{k}}}_{0}} \cdot {{{\mathbf{R}}}_{0}})\sin \theta . \\ \end{gathered} $$

(A.2)

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

$${{\mathcal{H}}_{1}} = i{{S}^{{3/2}}}\sqrt N (L{{a}_{{{{{\mathbf{k}}}_{0}}}}} - La_{{{{{\mathbf{k}}}_{0}}}}^{\dag }),$$

(A.3)

$$L = \frac{1}{{\sqrt 2 }}\left[ {(2{{J}_{0}} + 3\tilde {D})\cos \theta - \frac{{{{h}_{z}}}}{S}} \right]\sin \theta .$$

(A.4)

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

$${{\mathcal{H}}_{2}} = S\sum\limits_{\mathbf{k}}^{} {a_{{\mathbf{k}}}^{\dag }{{a}_{{\mathbf{k}}}}{{E}_{{\mathbf{k}}}}} + \frac{S}{2}\sum\limits_{\mathbf{k}}^{} {({{a}_{{\mathbf{k}}}}{{a}_{{ - {\mathbf{k}}}}} + a_{{\mathbf{k}}}^{\dag }a_{{ - {\mathbf{k}}}}^{\dag }){{B}_{{\mathbf{k}}}},} $$

(A.5)

$${{E}_{{\mathbf{k}}}} = S({{J}_{0}} + {{J}_{{\mathbf{k}}}}){{\cos }^{2}}\theta + S({{J}_{0}} + D){{\sin }^{2}}\theta ,$$

(A.6)

$${{B}_{{\mathbf{k}}}} = S({{J}_{{\mathbf{k}}}} - D){{\sin }^{2}}\theta .$$

(A.7)

Here the Fourier transform of the exchange interaction reads

$${{J}_{{\mathbf{k}}}} = 2[{{J}_{c}}\cos {{k}_{z}} + {{J}_{a}}(\cos {{k}_{x}} + \cos {{k}_{y}})].$$

(A.8)

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

$${{\varepsilon }_{{\mathbf{k}}}} = \sqrt {E_{{\mathbf{k}}}^{2} - B_{{\mathbf{k}}}^{2}} .$$

(A.9)

Since the exchange interaction Fourier transform J_k obeys the property J₀ = –${{J}_{{{{{\mathbf{k}}}_{0}}}}}$, 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

$$\begin{gathered} {{\mathcal{H}}_{3}} = i\frac{{\sqrt S \sin \theta \cos \theta }}{{4\sqrt {2N} }} \\ \times \sum\limits_{123 = {{k}_{0}}}^{} {(a_{1}^{\dag }a_{2}^{\dag }{{a}_{{ - 3}}} - a_{{ - 3}}^{\dag }{{a}_{2}}{{a}_{1}})\frac{{({{V}_{1}} + {{V}_{2}})}}{2}} , \\ {{V}_{{\mathbf{k}}}} = \left( {2{{J}_{{{{{\mathbf{k}}}_{0}}}}} - 8{{J}_{{\mathbf{k}}}} + 10D - \frac{h}{{S\cos \theta }}} \right), \\ \end{gathered} $$

(A.10)

where we denote k₁, k₂, k₃ as 1, 2, 3, and the momentum conservation law reads

$${{{\mathbf{k}}}_{1}} + {{{\mathbf{k}}}_{2}} + {{{\mathbf{k}}}_{3}} = {{{\mathbf{k}}}_{0}}.$$

(A.11)

The second one is the following:

$$\begin{gathered} {{\mathcal{H}}_{4}} = \frac{1}{N}\sum\limits_{1234 = 0}^{} {{{U}_{{1234}}}a_{1}^{\dag }a_{2}^{\dag }{{a}_{{ - 3}}}{{a}_{{ - 4}}}} \\ + \frac{1}{N}\sum\limits_{1234 = 0}^{} {{{V}_{{123}}}(a_{1}^{\dag }a_{2}^{\dag }a_{3}^{\dag }{{a}_{{ - 4}}} + a_{{ - 4}}^{\dag }{{a}_{3}}{{a}_{2}}{{a}_{1}})} \\ \end{gathered} $$

(A.12)

with the conventional momentum conservation law. Here

$$\begin{gathered} {{U}_{{1234}}} = \left[ {\frac{{1 - 2{{{\sin }}^{2}}\theta }}{2}{{J}_{{4 - 1}}} - \frac{{{{{\cos }}^{2}}\theta }}{4}} \right.({{J}_{1}} + {{J}_{4}}) \\ \left. {\, + D\left( {1 - \frac{3}{2}{{{\sin }}^{2}}\theta } \right)} \right], \\ \end{gathered} $$

(A.13)

$${{V}_{{123}}} = \left[ {\frac{{D{{{\sin }}^{2}}\theta }}{4} - \frac{{({{J}_{1}} + {{J}_{2}} + {{J}_{3}}){{{\sin }}^{2}}\theta }}{{12}}} \right].$$

(A.14)

For nonlinear corrections to magnon spectrum analysis, it is convenient to use normal (G_k = 〈a_k, $a_{{\mathbf{k}}}^{\dag }$〉_ω) and anomalous ($F_{k}^{\dag }$ = 〈$a_{{ - {\mathbf{k}}}}^{\dag }$, $a_{{\mathbf{k}}}^{\dag }$〉_ω) Green’s functions. The solution of the system of Dyson’s equation for these functions reads

$${{G}_{k}} = \frac{{\omega + {{E}_{{\mathbf{k}}}} + {{\Sigma }_{{ - k}}}}}{{{{D}_{{\omega ,k}}}}},$$

(A.15)

$${{F}_{k}} = - \frac{{{{B}_{{\mathbf{k}}}} + {{\Pi }_{k}}}}{{{{D}_{{\omega ,k}}}}},$$

(A.16)

$$\begin{gathered} {{D}_{k}} = {{\omega }^{2}} - \varepsilon _{{\mathbf{k}}}^{2} - {{E}_{{\mathbf{k}}}}({{\Sigma }_{k}} + {{\Sigma }_{{ - k}}}) \\ \, + 2{{B}_{{\mathbf{k}}}}{{\Pi }_{k}} + \omega ({{\Sigma }_{k}} - {{\Sigma }_{{ - k}}}), \\ \end{gathered} $$

(A.17)

where Σ_k, Π_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 〈S^x〉 component. In our theory, it is related to the parameter ρ. Nonzero ρ provides various additional terms in the Hamiltonian, which can be altogether denoted as δ$\mathcal{H}$.

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

The bilinear part of δ${{\mathcal{H}}_{2}}$ can be divided onto normal and umklapp contributions (see Eqs. (12) and (13)). Explicit expressions for the corresponding coefficients in δ$\mathcal{H}_{2}^{N}$ are the following:

$${{\bar {E}}_{{\mathbf{k}}}} = \frac{{{{h}_{x}}\rho }}{{2S\sqrt {2NS} }} + 8{{U}_{{0{\mathbf{k}}0 - {\mathbf{k}}}}}\frac{{{{\rho }^{2}}}}{{NS}} + 12{{V}_{{0{\mathbf{k}}0}}}\frac{{{{\rho }^{2}}}}{{NS}},$$

(B.1)

$${{\bar {B}}_{{\mathbf{k}}}} = \frac{{{{h}_{x}}\rho }}{{2S\sqrt {2NS} }} + 8{{U}_{{{\mathbf{kk}}00}}}\frac{{{{\rho }^{2}}}}{{NS}} + 12{{V}_{{0{\mathbf{k}} - {\mathbf{k}}}}}\frac{{{{\rho }^{2}}}}{{NS}}.$$

(B.2)

These terms are analogous to the contributions in Eq. (A5). For δ$\mathcal{H}_{2}^{N}$ the corresponding equations read

$${{X}_{{\mathbf{k}}}} = \sin \theta \cos \theta ({{V}_{0}} + {{V}_{{\mathbf{k}}}})\frac{\rho }{{4\sqrt {2NS} }},$$

(B.3)

$${{Y}_{{\mathbf{k}}}} = \sin \theta \cos \theta ({{V}_{{{\mathbf{k}} - {{{\mathbf{k}}}_{0}}}}} + {{V}_{{\mathbf{k}}}})\frac{\rho }{{4\sqrt {2NS} }}.$$

(B.4)

Importantly, the umklapps provide corrections to the normal and anomalous self-energy parts ρ² (they are combined in pairs, see Fig. 4):

$$\begin{gathered} \Sigma _{{\mathbf{k}}}^{U} = X_{{\mathbf{k}}}^{2}{{G}_{{\omega ,{\mathbf{k}} - {{{\mathbf{k}}}_{0}}}}} + Y_{{\mathbf{k}}}^{2}{{G}_{{ - \omega ,{{{\mathbf{k}}}_{0}} - {\mathbf{k}}}}} \\ \, - {{X}_{{\mathbf{k}}}}{{Y}_{{\mathbf{k}}}}{{F}_{{\omega ,{\mathbf{k}} - {{{\mathbf{k}}}_{0}}}}} - {{X}_{{\mathbf{k}}}}{{Y}_{{\mathbf{k}}}}{{F}_{{ - \omega ,{{{\mathbf{k}}}_{0}} - {\mathbf{k}}}}}, \\ \end{gathered} $$

(B.5)

$$\begin{gathered} \Pi _{{\mathbf{k}}}^{U} = X_{{\mathbf{k}}}^{2}{{F}_{{\omega ,{\mathbf{k}} - {{{\mathbf{k}}}_{0}}}}} + Y_{{\mathbf{k}}}^{2}{{F}_{{ - \omega ,{{{\mathbf{k}}}_{0}} - {\mathbf{k}}}}} \\ \, - {{X}_{{\mathbf{k}}}}{{Y}_{{\mathbf{k}}}}{{G}_{{\omega ,{\mathbf{k}} - {{{\mathbf{k}}}_{0}}}}} - {{X}_{{\mathbf{k}}}}{{Y}_{{\mathbf{k}}}}{{G}_{{ - \omega ,{{{\mathbf{k}}}_{0}} - {\mathbf{k}}}}}. \\ \end{gathered} $$

(B.6)

However, one can see that at the momentum k = k₀ condition ${{X}_{{{{{\mathbf{k}}}_{0}}}}}$ = ${{Y}_{{{{{\mathbf{k}}}_{0}}}}}$ is satisfied, so $\Sigma _{{{{{\mathbf{k}}}_{0}}}}^{U}$ = –$\Pi _{{{{{\mathbf{k}}}_{0}}}}^{U}$. This means that there is no correction to the energy of magnon with k = k₀ 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:

$${{G}_{k}} = {{\langle {{b}_{{\mathbf{k}}}},b_{{\mathbf{k}}}^{\dag }\rangle }_{\omega }};\quad F_{k}^{\dag } = {{\langle b_{{ - {\mathbf{k}}}}^{\dag },b_{{\mathbf{k}}}^{\dag }\rangle }_{\omega }},$$

(C.1)

where k = (ω, k). These functions obey the following system of Dyson’s equations

$$\begin{gathered} {{G}_{k}} = G_{k}^{0} + G_{k}^{0}{{\Sigma }_{k}}{{G}_{k}} + G_{k}^{0}({{B}_{{\mathbf{k}}}} + {{{\bar {B}}}_{{\mathbf{k}}}} + {{\Pi }_{k}}){{F}_{k}}, \\ {{F}_{k}} = G_{{ - k}}^{0}({{B}_{{\mathbf{k}}}} + {{{\bar {B}}}_{{\mathbf{k}}}} + {{\Pi }_{k}}){{G}_{k}} + G_{{ - k}}^{0}{{\Sigma }_{{ - k}}}{{F}_{k}}, \\ \end{gathered} $$

(C.2)

where the bare Green’s function reads

$${{G}^{0}} = \frac{1}{{\omega - {{E}_{{\mathbf{k}}}} - {{{\bar {E}}}_{{\mathbf{k}}}} + i\delta }}.$$

(C.3)

After solving the system of equations (C.2) we obtain the following expression for normal and anomalous Green’s functions

$${{G}_{k}} = \frac{{\omega + {{E}_{{\mathbf{k}}}} + {{{\bar {E}}}_{{\mathbf{k}}}} + {{\Sigma }_{k}}}}{{{{D}_{{\omega ,k}}}}},$$

(C.4)

$${{F}_{k}} = - \frac{{{{B}_{{\mathbf{k}}}} + {{{\bar {B}}}_{{\mathbf{k}}}} + {{\Pi }_{k}}}}{{{{D}_{{\omega ,k}}}}}.$$

(C.5)

Here the denominator reads

$$\begin{gathered} {{D}_{{\omega ,k}}} = {{\omega }^{2}} - \varepsilon _{{\mathbf{k}}}^{2} - {{E}_{{\mathbf{k}}}}({{\Sigma }_{k}} + {{\Sigma }_{{ - k}}} + 2{{{\bar {E}}}_{{\mathbf{k}}}}) \\ \, + 2{{B}_{{\mathbf{k}}}}({{\Pi }_{k}} + {{{\bar {B}}}_{{\mathbf{k}}}}) + \omega ({{\Sigma }_{k}} - {{\Sigma }_{{ - k}}}). \\ \end{gathered} $$

(C.6)

To calculate the susceptibility (19), it is necessary to use the retarded Green’s functions G^R and F^R, which are related to the causal ones (C.4) by the following relation:

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

(C.7)

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

$$\begin{gathered} \operatorname{Re} \text{[}\chi _{{\omega ,{\mathbf{k}}}}^{ \bot }] = - \operatorname{Re} \left\{ {\frac{{2{{E}_{{\mathbf{k}}}} + {{\Sigma }_{k}} + {{\Sigma }_{{ - k}}} - 2{{B}_{{\mathbf{k}}}} - 2{{\Pi }_{k}}}}{{{{\omega }^{2}} - {{{(\Delta - i\Gamma )}}^{2}}}}} \right. \\ \left. { + \frac{{2{{{\bar {E}}}_{{\mathbf{k}}}} - 2{{{\bar {B}}}_{{\mathbf{k}}}} + 2\Sigma _{{\mathbf{k}}}^{U} - 2\Pi _{{\mathbf{k}}}^{U}}}{{{{\omega }^{2}} - {{{(\Delta - i\Gamma )}}^{2}}}}} \right\}. \\ \end{gathered} $$

(C.8)

Using Eqs. (B.5) and (B.6) at k = 0 and Eq. (10), the real part of the uniform transverse spin susceptibility can be finally expressed as

$$\begin{gathered} \operatorname{Re} \text{[}\chi _{{\omega ,{\mathbf{k}} = 0}}^{ \bot }] = - \operatorname{Re} \left\{ {\frac{{P(h) + 2{{{\bar {E}}}_{{{\mathbf{k}} = 0}}} - 2{{{\bar {B}}}_{{{\mathbf{k}} = 0}}}}}{{{{\omega }^{2}} - {{{(\Delta - i\Gamma )}}^{2}}}}} \right. \\ \left. { + \frac{K}{{[{{{(\omega + i\delta )}}^{2}} - \Delta _{{{{h}_{x}}}}^{2}][{{\omega }^{2}} - {{{(\Delta - i\Gamma )}}^{2}}]}}} \right\}, \\ \end{gathered} $$

(C.9)

where

$$K = 2{{({{X}_{{{\mathbf{k}} = 0}}} + {{Y}_{{{\mathbf{k}} = 0}}})}^{2}}({{\bar {E}}_{{{{{\mathbf{k}}}_{0}}}}} + {{\bar {B}}_{{{{{\mathbf{k}}}_{0}}}}}).$$

(C.10)