Magnetic field-induced deformation of the spin-density wave microphases in Ca₃Co₂O₆

Frustrated magnets can have a manifold of nearly degenerate low-energy states from which interesting phenomena may emerge, such as exotic magnetic and nonmagnetic orders, topological order, liquid-like or even glassy behavior, and so on, varying from one material to another.Lacroix2011 Even a classical system can host unconventional quasiparticles, such as skyrmions,Bogdanov2020 solitons, and monopoles in spin ice,Bramwell01; Paulsen2014; Bramwell20 and they may crystallize into novel spin textures like skyrmion crystals Okubo2012; Leonov2015 or soliton crystals.Bak82; Selke88 Such emergent crystalline states can often be sensitive to external perturbations, which makes them attractive as potential devices in some cases.Fert2017 They can also provide a platform to study far-from-equilibrium dynamics due to metastable states.Kudasov06; Paulsen2014

Since the late 90’s,Fjellvag96; Aasland97; Kageyama97; Kageyama97b the frustrated triangular Ising magnet Ca₃Co₂O₆ (CCO) has long been known for an intriguing combination of extremely slow spin dynamics and peculiar magnetic phases, such as metamagnetic magnetization steps and an incommensurate spin-density wave (SDW) state,Agrestini08a; Mazzoli09; Agrestini08b; Moyoshi11; Fleck10; Paddison14; Motoya18 the latter of which can be seen as a soliton crystal.Kamiya12 In CCO, trigonal prismatic Co³⁺ $S=2$ sites form ferromagnetic Ising chains running along the $c$ axis, while arranged in a triangular lattice in the $ab$ plane coupled by weak antiferromagnetic interactions (Fig. 1).Leedahl19 Below spin-freezing temperature $T_{\mathrm{SF}}\simeq 5\,\mathrm{K}$, CCO exhibits striking evenly-spaced metamagnetic magnetization steps,Kageyama97 whose origin has been a subject of long-time debates.Kageyama97; Kageyama97b; Maignan00; Hardy04b; Maignan04; Moyoshi11; Kim18 Interestingly, while the step heights are sensitive to protocol details such as sweep rate of the external magnetic field, the transition magnetic fields ($\simeq$1.2 T, 2.4 T, and 3.6 T, with additional steps at higher fields) are rather robust.Hardy04b Although some theory invoked an analogy with quantum tunneling in molecular magnets,Maignan04 an alternative scenario is that peculiar frustration in CCO causes a non-equilibrium phenomenon.Kudasov06 In the so-called “rigid chain” model,Yao06a; Yao06b; Kudasov08; Qin09; Soto09; Zukovic12 each ferromagnetic chain is replaced by an effective Ising spin on a two-dimensional antiferromagnetic triangular lattice. Based on this mapping, it was argued that the metamagnetic transition steps in CCO may arise from the same kind of degenerate manifold as in the two-dimensional triangular lattice Ising model.Kudasov06

However, the origin of the slow dynamics can be more intricate than suggested by the rigid chain picture. More recently, resonant x-ray Agrestini08a; Mazzoli09 and neutron spectroscopies Agrestini08b; Moyoshi11; Fleck10; Paddison14; Motoya18 revealed the SDW order below $T_{\mathrm{SDW}}\simeq$ 25 K, which has a three-sublattice structure and a very long modulation wavelength $\lambda_{\mathrm{SDW}}\!\simeq\!10^{3}$ $\AA$ ($\simeq\!\!10^{2}$ magnetic sites) along the $c$ axis. It was found that $\lambda_{\mathrm{SDW}}$ increases as temperature $T$ is lowered and the corresponding relaxation time grows substantially. Eventually, the system starts to deviate from equilibrium below $T\lesssim$ 13 K,Moyoshi11 which is much higher than $T_{\mathrm{SF}}$ for the appearance of the metamagnetic magnetization steps. Since the spin chains are not ferromagnetically ordered in the SDW state, the interpretation of the rigid chain picture is, if not questionable, more subtle than originally proposed.

Indeed, the SDW phase may hold the key to understanding the peculiar slow dynamics at low temperatures. It was recently demonstrated that the slow spin dynamics can be bypassed by an elaborate field-cooling protocol, where every in-field measurement is performed after a separate cooling in the target magnetic field.Nekrashevic21 Remarkably, the protocol allowed for reaching the 1/3 magnetization plateau down to $T=2\,\mathrm{K}<T_{\mathrm{SF}}$ without being suffered from metastable states, which was in good agreement with MC simulations in equilibrium. Since the SDW order is believed to disappear and replaced by a ferrimagnetic state at high magnetic fields, the new experiment may suggest that the spin relaxation at low temperatures may be influenced by the extent to which the system has been through the low-field SDW phase during the cooling. In fact, it is known that the SDW order is accompanied by short-range order indicative of spin disordering,Agrestini11 which could be due to the combination of the $T$-dependent ordering wavevector and the slowness in the relaxation to follow the variation. Moyoshi11

As mentioned above, the observed SDW state is essentially a soliton crystal as in the axial next-nearest-neighbor Ising (ANNNI) model,Kamiya12 a prototypical model for spontaneous superstructures due to competition between nearest and next nearest neighbor Ising interactions in one direction of a square or cubic lattice.Bak82; Selke88 The $T$-dependent change of $\lambda_{\mathrm{SDW}}$ Moyoshi11 corresponds to different magnetic microphases,Kamiya12 similar to other self-organizing modulated phases in physical and chemical systems.Seul1995 Thus, CCO may provide a rare intersection where the ANNNI model phenomenology Bak82; Selke88 meets out-of-equilibrium physics in a solid state system with only short-range interactions. To elucidate this conjecture in CCO and similar materials such as Ca₃Co_2-xMn_xO₆,Zapf18; Kim18 Sr₂Ca₂CoMn₂O₉,Hardy18 and Ca₃CoRhO₆,Niitaka01 it is a matter of paramount importance to investigate the magnetic field-induced deformation of the SDW state and incommensurate-commensurate (IC-C) transitions under the condition much closer to equilibrium than ever reached before. The goal of this work to present an equilibrium in-field phase diagram for a realistic three-dimensional (3D) lattice spin model for CCO, thereby providing a theoretical guide for experiments. We address both model-specific and universal physics by combining mean-field theory (Sec. III), MC simulations (Sec. IV), and Ginzburg-Landau (GL) theory (Sec. V).

In CCO, $S=2$ spins have large easy-axis anisotropy,Kageyama97 which permits a description by an effective classical Ising model,

where $\sigma^{z}_{i}=\pm 1$, $h=g\mu_{B}SH$ with $g$, $\mu_{B}$, and $H$ being the $g$-factor, the Bohr magneton, and a magnetic field, respectively, and $\langle{ij}\rangle_{\nu}$ denotes neighboring sites connected by $J_{\nu}$, $\nu\in\{1,2,3\}$. $J_{1}<0$ is the intrachain ferromagnetic interaction and $J_{2}$ ($J_{3}$) is the antiferromagnetic interchain interaction shifted by 1/3 (2/3) lattice parameters along the $c$ axis (Fig. 1). An ab initio study suggested $\lvert{J_{1}}\rvert\gg J_{2}\simeq J_{3}$ Fresard04 and $J_{1}=-23.9(2)$ K and $J_{2}+J_{3}=2.3(2)$ K was reported by an NMR experiment, which further suggested $J_{2}=1.1$ K and $J_{3}=1.2$ K to explain the SDW ordering wavevector.Allodi14 Below, for simplicity, we assume $J_{2}/\lvert{J_{1}}\rvert=J_{3}/\lvert{J_{1}}\rvert$ and denote the ratio by $\kappa$; in relation with CCO, $\kappa\simeq 0.048$.

In CCO, $J_{2}$ and $J_{3}$ compete with $J_{1}$ after a few steps along a spiral path due to the vertical shifts of the interchain interactions.Chapon09 This spiral structure is the key to realize the same kind of geometrical frustration as in the ANNNI model Bak82; Selke88 despite the apparent structural differences. We first briefly discuss a heuristic mean-field theory in zero field by assuming a ferromagnetic order in each $ab$ plane, which are separated by 1/3 lattice parameters from each other along the $c$ axis (Fig. 1).Kamiya12 The mean field equation for the magnetization $m_{l}$ of layer $l$ is

where $h_{l}=-J_{1}(\left\langle{m_{l+3}}\right\rangle+\left\langle{m_{l-3}}\right\rangle)-3J_{2}(\left\langle{m_{l+1}}\right\rangle+\left\langle{m_{l-1}}\right\rangle)-3J_{3}(\left\langle{m_{l+2}}\right\rangle+\left\langle{m_{l-2}}\right\rangle)$. In this quasi-one-dimensional description, $J_{1}$, $J_{2}$, and $J_{3}$ serve as the effective third, first, and second neighbor interactions, respectively, realizing a very similar situation as in the prototypical ANNNI model [Fig. 1(c)]. The reason for assuming an in-plane ferromagnetic order, even though the interchain interactions $J_{2}$ and $J_{3}$ are much smaller than the intrachain interaction $J_{1}$, is that the energy scale associated with the competition between SDW states with different wavelengths along the $c$ axis can be even smaller, as will be discussed by using a sine-Gordon model. In Fig. 2(a), we show the mean field $(\kappa,T)$ phase diagram, by extending the previous work.Kamiya12 Below the SDW transition temperature $T_{\mathrm{SDW}}(\kappa)$, the ordering wavevector $\mathbf{Q}=(0,0,Q_{3})$ varies quasi-continuously. Eventually, there is a lock-in IC-C transition at $T=T_{\mathrm{IC-C}}$, below which the magnetic unit cell of the mean field solution is $\uparrow\uparrow\downarrow$ or $\downarrow\downarrow\uparrow$ for $\kappa<1$ and $\uparrow\uparrow\downarrow\downarrow$ for $\kappa>1$ in the effective one-dimensional description in Fig. 1(c). We find that quasi-continuous changes of $Q_{3}(\kappa,T)$ dominate the overall phase diagram, corresponding to numerous microphases of soliton lattice states, especially for relatively small $\kappa$.Kamiya12 Meanwhile, distinct discontinuous changes of $Q_{3}(\kappa,T)$ are also seen in the region with relatively large $\kappa$, where a few relatively extended commensurate states are found. However, the latter case has little significance in relation with CCO, where $\kappa$ has been suggested to be very small.

Finally, for a more universal description of the commensurate and incommensurate phases in CCO and similar materials, we consider a GL theory. As will be shown below, the theory is essentially in the same form as that for the ANNNI model.Aharony1981; Bak80 Interestingly, however, the GL theory for ANNNI-like models in magnetic fields Yokoi81 has yet been thoroughly discussed in the literature, despite its experimental relevance.

We use a complex order parameter $\psi(\mathbf{r})\sim e^{-i\mathbf{q}_{\text{com}}\cdot\mathbf{r}}\sigma^{z}(\mathbf{r})$, $\mathbf{q}_{\text{com}}=(0,0,2\pi)$, which can be formally introduced by using the Hubbard-Stratonovich transformation. $\psi(\mathbf{r})$ describes the local three-sublattice order in a coarse-grained way, which can be either the FIM order or the PDA order depending on the phase factor. For $h=0$, we find the following GL Hamiltonian,

where $t$, $u_{4}$, $u_{6}$, and $v_{6}$ are GL coefficients and $\bm{\epsilon}=(0,0,\epsilon)$. A crucial point of the present theory is that the wavevector $\mathbf{q}_{\text{com}}$ of the local order described by $\psi(\mathbf{r})$ is slightly shifted from the minima $\mathbf{q}=\pm\mathbf{q}_{\text{min}}=\pm(\mathbf{q}_{\text{com}}+\bm{\epsilon})$ of the Fourier transform $J(\mathbf{q})$ of the exchange interactions, as briefly mentioned before. For this reason, $\mathscr{H}_{\,h=0}$ includes the vector potential-like (but constant) contribution $-i\bm{\epsilon}$ in the gradient term. $\mathscr{H}_{\,h=0}$ also includes the six-order term ($v_{6}$) as the leading Umklapp term, the order of which is determined by the size of the magnetic sublattice of the commensurate order and time reversal symmetry; a factor of three comes from $3\mathbf{q}_{\text{com}}\equiv 0$ and time reversal symmetry requires another factor of two.

As is clear from the origin, the gradient term acts as adding a momentum $\bm{\epsilon}$ to $\psi$, in favor of the three-sublattice long-wavelength SDW state $\langle{\psi(\mathbf{r})}\rangle\sim\text{(const.)}\times e^{i\bm{\epsilon}\cdot\mathbf{r}}$. Although the SDW state thus appears to benefit from the exchange energy, the sinusoidal modulation made of localized Ising-like moments also requires entropy contribution to the free energy. In contrast, because of the shift of $\mathbf{q}_{\text{com}}$ from the minima of $J(\mathbf{q})$, the commensurate three-sublattice ordered state, $\langle{\psi(\mathbf{r})}\rangle\sim\text{const.}$, may appear not to acquire the full energy gain of the exchange interaction. However, the state is quite compatible with the Ising anisotropy. In the GL theory (10), the Umklapp term plays the role of the entropy contribution related with the Ising anisotropy. While the commensurate state may be favored by the Umklapp term by adjusting its constant phase factor, the incommensurate SDW states generally gain no corresponding contribution because of the phase cancellation in the integral over the space. In fact, we could rederive the GL theory in terms of $\phi_{c}(\mathbf{r})\sim\sigma^{z}(\mathbf{r})\,e^{-i\mathbf{q}_{\text{min}}\cdot\mathbf{r}}$ instead of $\psi(\mathbf{r})\sim\sigma^{z}(\mathbf{r})\,e^{-i\mathbf{q}_{\text{com}}\cdot\mathbf{r}}$, and the result is an ordinary $\phi_{c}^{4}$-theory for the one-component complex order parameter without Umklapp terms if $\mathbf{q}_{\text{min}}$ is incommensurate. Hence, the key role in the GL theory (10) is played by the competition between the gradient and the Umklapp terms favoring incommensuration and commensuration, respectively. Thus, although somewhat different in appearance, the Hamiltonian of this system (1) realizes essentially the same kind of situation as the classic ANNNI model.Bak80

At $T=T_{\mathrm{SDW}}$, critical fluctuations renders $\psi(\mathbf{r})$ nonzero with the additional momentum $\bm{\epsilon}$, resulting in the SDW state with the ordering wavevector $\mathbf{Q}=\mathbf{q}_{\text{com}}+\bm{\epsilon}=\mathbf{q}_{\text{min}}$. In other words, we expect a condensation of the softest mode $\phi_{c}(\mathbf{r})\sim\sigma^{z}(\mathbf{r})\,e^{-i\mathbf{q}_{\text{min}}\cdot\mathbf{r}}$ rather than $\psi(\mathbf{r})\sim\sigma^{z}(\mathbf{r})\,e^{-i\mathbf{q}_{\text{com}}\cdot\mathbf{r}}$. Since the Umklapp $v_{6}$ term has no effect for the incommensurate SDW states, the transition, which breaks translation symmetry along the $c$ axis, will be in the 3D XY universality class (emergent U(1) symmetry). The observed main peak of the specific heat, which exhibits a sign of smearing [Fig. 3(a)], is consistent with the negative exponent $\alpha=-0.0146(8)<0$ for the XY universality class.Campostrini2001

For $T<T_{\mathrm{SDW}}$, the competition between the gradient term and the Umklapp term sets in, which affects the phase factor of $\langle{\psi(\mathbf{r})}\rangle$, thereby $\mathbf{Q}(T<T_{\mathrm{SDW}})$. To see this, we may write $\psi(\mathbf{r})=A(\mathbf{r})e^{i\theta(\mathbf{r})}$ and apply a mean-field decoupling for the massive amplitude fluctuation (“Higgs”) mode $\delta A(\mathbf{r})=A(\mathbf{r})-\langle{A}\rangle$ and the phase mode $\theta(\mathbf{r})$.Bak80 The result is the following sine-Gordon model for $\theta(\mathbf{r})$,

The gradient term tends to drift the phase, which can lead to a plethora of soliton lattices through the competition against the cosine term,Bak80; Chaikin-Lubensky1995 in good agreement with our mean field and MC studies. In the meantime, because the order parameter amplitude $\langle{A}\rangle$ increases as $T$ is lowered below $T_{\mathrm{SDW}}$, the strength of the cosine term is enhanced. Consequently, the model at low $T$ is expected to undergo a lock-in transition eventually. For $v_{6}>0$ ($v_{6}<0$), the phase is locked-in at $\theta=2n\pi/6$ [$\theta=(2n+1)\pi/6$] with an integer $0\leq n<6$, corresponding to the FIM ($\uparrow\uparrow\downarrow$ or $\downarrow\downarrow\uparrow$) state and the PDA state,Blankschtein84; Coppersmith85; Isakov03; Heinonen89; Bunker93; Lin14 respectively. Although our mean field calculation implies $v_{6}>0$ while our unbiased MC simulation suggests $v_{6}<0$ in zero field, the subtle discrepancy does not require a serious attention as $v_{6}$ is generated through fluctuations.

For $h\neq 0$, the uniform $\mathbf{q}=0$ component $m(\mathbf{r})$ is allowed by symmetry and may be induced by the magnetic field. Consequently, in addition to $v_{6}$, a lower-order Umklapp term appears in the GL Hamiltonian. We find

as the leading-order contribution with the new coupling constant $w_{4}$. The total effective GL Hamiltonian for $h\neq 0$ is

where $\mathscr{H}_{\,\mathbf{q}=0,\,m}=\int\!\mathrm{d}^{3}\mathbf{r}\,\bigl{[}\frac{1}{2}\bigl{(}c\nabla m(\mathbf{r})\bigr{)}^{2}+\mu^{2}m(\mathbf{r})^{2}-hm(\mathbf{r})\bigr{]}$ is the noninteracting part for $m(\mathbf{r})$ with $c>0$ and $\mu^{2}>0$ being the gapped spin wave parameters near $\mathbf{q}=0$. By a similar mean field decoupling as in the $h=0$ case, we find a new term in the sine-Gordon model,

The field-induced Umklapp term has the following consequences. Firstly, because of the reduced symmetry in the $\theta$ space, only a subset of the FIM states is favored by $\Delta\mathscr{H}^{\prime}_{\,0\mathrm{-}\mathbf{q}_{\text{com}},\,\theta}$ among the three-sublattice ordered states. Depending on the sign of $w_{4}$, the favored states are $\uparrow\uparrow\downarrow$ or $\downarrow\downarrow\uparrow$, each of which is three-fold degenerate, though $\uparrow\uparrow\downarrow$ is naturally anticipated for $h>0$. Secondly, as the prefactor is $\propto\langle{A}\rangle^{3}$, as opposed to $\propto\langle{A}\rangle^{6}$ in zero field, the strength of the field-induced Umklapp term is expected to grow faster for $T<T_{\mathrm{SDW}}$. Moreover, since the prefactor is $\propto\langle{m}\rangle$, we expect that this trend is further enhanced for larger $h$. Therefore, the region of the incommensurate SDW microphases is expected to become narrower for larger $h$, in excellent agreement with our MC results (Fig. 2). Thirdly, considering that our MC simulation shows the PDA state at $h=0$ below $T_{\mathrm{IC-C}}$, the zero- and the field-induced Umklapp terms ($\sim\cos 6\theta,\,\cos 3\theta$, respectively) must compete against each other in the present system. The competition opens a possibility of a kind of mixed phase below $T_{\mathrm{IC-C}}$, though our MC results shows no evidence down to an extremely low field, $h/\lvert{J_{1}}\rvert=0.025$. Finally, near $T=T_{\mathrm{SDW}}$, the present GL theory suggests the condensation of the softest mode $\phi_{c}(\mathbf{r})\sim\sigma^{z}(\mathbf{r})\,e^{-i\mathbf{q}_{\text{min}}\cdot\mathbf{r}}$ rather than $\psi(\mathbf{r})\sim\sigma^{z}(\mathbf{r})\,e^{-i\mathbf{q}_{\text{com}}\cdot\mathbf{r}}$, as in the case of zero magnetic field. Hence, we expect the emergent U(1) symmetry at $T=T_{\mathrm{SDW}}$ also for $h\neq 0$, because Umklapp terms disappear from the GL theory in terms of the incommensurate critical mode $\phi_{c}$. The theory thus predicts $\mathbf{Q}(T=T_{\mathrm{SDW}})=\mathbf{q}_{\text{min}}$ for any magnetic field, which is indeed consistent with our MC results at low magnetic fields. However, the simulation closer to the magnetic field-induced multicritical point $h_{\mathrm{LP}}\simeq 0.2\lvert{J_{1}}\rvert$ might point to a deviation from this behavior, suggesting that $\mathbf{Q}(T=T_{\mathrm{SDW}})$ approaches towards $\mathbf{q}_{\text{com}}$ for the larger systems we investigated [Fig. 3(c)]. This observation may be an indication that the multicritical point is a Lifshitz point induced by a magnetic field.

To summarize, we presented the magnetic phase diagram in equilibrium of the 3D spin model for the frustrated quasi-one-dimensional triangular Ising antiferromagnet Ca₃Co₂O₆. We identified the region of incommensurate SDW microphases in a magnetic field (Fig. 2). We found the deformation of SDW microphases as a function of $T$, characterized by the temperature dependence of the ordering wavevector $\mathbf{Q}(T)$, occurring much more rapidly in a magnetic field than in zero field (Fig. 3). The deformation eventually leads to the IC-C transition into the PDA (FIM) state for $h=0$ ($h\neq 0$). Between the PDA and the FIM phases, there may be a mixed phase in an extremely low-field regime, though not confirmed in this work. The GL theory we derived includes different symmetry-allowed Umklapp terms for $h=0$ and $h\neq 0$. The GL theory allowed for further deriving an effective sine-Gordon model that provides a qualitative explanation of the observed magnetic field-induced deformation of the SDW microphases. Moreover, these effective theories demonstrate that the present system can be seen as an incarnation of the classic ANNNI model,Bak80 despite different appearance of the lattice structure and the complicated network of the exchange interactions.

Finally, we discuss the relation between the theoretical phase diagram in this work and the previous experiments. As mentioned in Introduction, the material is known for the intriguing combination of the slow relaxation phenomena and the long-wavelength SDW order. As the recent field-cooling study suggested,Nekrashevic21 the slow relaxation at low temperature may be greatly influenced by the cooling process passing through the low-field SDW phase at intermediate temperature. To verify the conjectured relation experimentally, the challenge is that experiments under equilibrium conditions are known to be notoriously difficult for Ca₃Co₂O₆. For example, a resonant X-ray experiments reported a field-induced IC-C transition at 5 K,Mazzoli09 which is unfortunately most likely non-equilibrium because the temperature is too low. However, at intermediate temperatures above $T_{\mathrm{SF}}$, there are some experiments that seem to capture the desired physics of the field-induced deformation of the SDW phase and the IC-C transition. For example, a $\mu$SR measurements at 20 K reported a magnetic field-induced phase transition at around 0.4 T.Takeshita2007 Although the original interpretation of the result suggested a PDA-FIM transition, the obtained phase boundary is very similar to the SDW-FIM transition line shown in the present work. Since the SDW phase was not confirmed back then, it is quite possible that the anomaly in the $\mu$SR experiment is the sign of the IC-C transition induced by the magnetic field. We also note that a similar phase diagram was obtained also by weak anomaly in the magnetic entropy change.Lampen14 We thus believe that further experiments studying the field-induced IC-C transition in Ca₃Co₂O₆, such as neutron scattering and other spectroscopies focusing on the low-field regime, will be very promising, especially when combined with the recently proposed field-cooling protocol.Nekrashevic21 Such experiments may provide further insights in the peculiar slow dynamics and out-of-equilibrium behaviors in Ca₃Co₂O₆ and related materials.