Instantons in Chiral Magnets

TL;DR

This paper systematically constructs and analyzes instanton solutions in one-dimensional anti-ferromagnetic chiral magnets, revealing diverse soliton behaviors across different phases and mapping models with Dzyaloshinskii-Moriya interaction to simpler forms.

Contribution

It provides a comprehensive construction of instantons in chiral magnets and explores their properties across various phases, including phase diagrams and mappings to related models.

Findings

Instantons manifest as domain walls, bions, dislocations, and isolated solitons.

Phase diagram includes easy-axis, helical, and tricritical points with distinct instanton behaviors.

Models with Dzyaloshinskii-Moriya interaction can be mapped to simpler models without it.

Abstract

We exhaustively construct instanton solutions and elucidate their properties in one-dimensional anti-ferromagnetic chiral magnets based on the $O(3)$ nonlinear sigma model description of spin chains with the Dzyaloshinskii-Moriya (DM) interaction. By introducing an easy-axis potential and a staggered magnetic field, we obtain a phase diagram consisting of ground-state phases with two points (or one point) in the easy-axis dominant cases, a helical modulation at a fixed latitude of the sphere, and a tricritical point allowing helical modulations at an arbitrary latitude. We find that instantons (or skyrmions in two-dimensional Euclidean space) appear as composite solitons in different fashions in these phases: temporal domain walls or wall-antiwall pairs (bions) in the easy-axis dominant cases, dislocations (or phase slips) with fractional instanton numbers in the helical state, and isolated instantons and calorons living on the top of the helical modulation at the tricritical point. We also show that the models with DM interaction and an easy-plane potential can be mapped into those without them, providing a useful tool to investigate the model with the DM interaction.

Tables

Table 1. Table 1: The OPMs and associated homotopy groups.

	OPM $G/H$	${\pi }_0$	${\pi }_1$	${\pi }_2$	${\pi }_3$
(I)	$S^1$	0	$\mathbb{Z}$	0	0
(a)	${\mathbb{Z}}_2$	${\mathbb{Z}}_2$	0	0	0
Tri-critical pt.	$O\left(3\right)/O\left(2\right)\simeq S^2$	0	0	$\mathbb{Z}$	$\mathbb{Z}$
(II) (III) (b)(c)	1pt	0	0	0	0

Equations

$$\displaystyle E[n]=\int\mathrm{d}x\bigg{[}$$

$$\displaystyle+\mu\frac{1-(n^{3})^{2}}{2}+Bn^{3}\bigg{]}.$$

$$E[n]=\int \mathrm{d}x\left[\frac{(D_x{n}^a{)}^2}{2}+(\mu -{\kappa }^2)\frac{1-(n^3{)}^2}{2}+B{n}^3\right].$$

$$D_x{n}^a=0(a=1,2,3),$$

$$n^1+\mathrm{i}{n}^2=A{e}^{-\mathrm{i}\kappa x},n^3=\pm \sqrt{1-|A{|}^2}$$

$$\begin{array}{rl}\mathcal{L}=& \frac{1}{2}({\partial }_i{n}^a{)}^2+\kappa (n^1{\partial }_x{n}^2-n^2{\partial }_x{n}^1)\\ & +\frac{\mu }{2}[1-(n^3{)}^2]+B{n}^3.\end{array}$$

$$v=\frac{n^1+\mathrm{i}{n}^2}{1+n^3},n^3=\frac{1-|v{|}^2}{1+|v{|}^2}.$$

$$\mathcal{L}=$$

$${\rho }_Q=\frac{1}{4\pi }{ϵ}_{abc}{n}^a{\partial }_x{n}^b{\partial }_{\tau }{n}^c=\frac{1}{\pi }\frac{{\partial }_{z}v{\partial }_{\bar{z}}\bar{v}-{\partial }_{\bar{z}}v{\partial }_z\bar{v}}{(1+|v{|}^2{)}^2},$$

$$\displaystyle\!\!\!\int\mathrm{d}x\,{\cal L}=2\int\mathrm{d}x\bigg{[}\frac{|(\partial_{x}+\eta)v|^{2}}{(1+|v|^{2})^{2}}+\partial_{x}\frac{\sqrt{\mu-\kappa^{2}}}{1+|v|^{2}}\bigg{]}$$

$$\int \mathrm{d}x\mathcal{L}\ge 2\sqrt{\mu -{\kappa }^2}{\left[\frac{1}{1+|v{|}^2}\right]}_{x=-\infty }^{x=\infty }.$$

$$({\partial }_x+\mathrm{i}\kappa )v+\sqrt{\mu -{\kappa }^2}v=0$$

$$v=C{e}^{-\mathrm{i}\kappa x-\sqrt{\mu -{\kappa }^2}x},C\in \mathbb{C}.$$

$$({\partial }_x+\mathrm{i}\kappa )v=0,{\partial }_{\tau }v+\sqrt{\mu -{\kappa }^2}v=0$$

$$v=C{e}^{-\mathrm{i}\kappa x-\sqrt{\mu -{\kappa }^2}\tau },C\in \mathbb{C}$$

$$v=e^{-\mathrm{i}\kappa (x-x_0)}\frac{\sqrt{B}}{\omega }\sinh \omega (\tau -{\tau }_0),$$

$$v=e^{-\mathrm{i}\kappa x}\frac{z}{|z|}f(|z|),$$

$$\int {\mathrm{d}}^{2}x{\mathcal{L}}_{\mathrm{cr}}\ge \pm (4\pi Q+\int {\mathrm{d}}^{2}x\kappa {\partial }_{\tau }{n}^3).$$

$${\partial }_{\tau }{n}_a\mp {ϵ}_{abc}{n}^b{D}_x{n}^c=0\text{mbox}or{\partial }_{\bar{z}}v=-\mathrm{i}\kappa v/2$$

$$v(z,\bar{z})=e^{-\mathrm{i}\kappa x}w(z),$$

$$v=e^{-\mathrm{i}\kappa x}\frac{p(z)}{q(z)},$$

$$v=e^{-\mathrm{i}\kappa x}\frac{p(e^{\mathrm{i}\frac{2\pi z}{L}})}{q(e^{\mathrm{i}\frac{2\pi z}{L}})}$$

$$n^1+\mathrm{i}{n}^2=({\hat{n}}^1+\mathrm{i}{\hat{n}}^2)e^{-\mathrm{i}\kappa x},n^3={\hat{n}}^3.$$

$${\mathcal{L}}_{\mathrm{woDM}}(\hat{n})$$

Full text

Introduction.— Topological excitations (topological solitons and instantons) play key roles in various systems from particle physics Rajaraman (1982); Coleman (1988); Manton and Sutcliffe (2004) and cosmology Vilenkin and Shellard (2000) to condensed matter systems Volovik (2003); Nelson (2002). Amongst various examples, magnets allow magnetic skyrmions and domain walls in spin systems Bogdanov and Yablonskii (1989); Bogdanov and Hubert (1994); Rößler et al. (2006); Binz et al. (2006), and in particular, chiral magnets with the Dzyaloshinskii-Moriya (DM) interaction Dzyaloshinsky (1958); Moriya (1960) is a representative where topological excitations play a pivotal role for applications to nano-devices such as magnetic memories. Recent theoretical and experimental developments have confirmed that there exists the so-called chiral soliton lattice phase — aligning helical domain walls —in one-dimensional spin chains Togawa et al. (2012); Kishine and Ovchinnikov (2015); Togawa et al. (2016). More recently, a skyrmion lattice in two-dimensional (2D) chiral magnets and associated peculiar transport phenomena have been experimentally observed Mühlbauer et al. (2009); Yu et al. (2010); Heinze et al. (2011); Nagaosa and Tokura (2013). Furthermore, magnetic monopoles have been recently paid much attentions. They appear for unwinding a skyrmion line in a skyrmion lattice Milde et al. (2013) and form a stable crystal in certain parameter region Kanazawa et al. (2016). One of recent interesting theoretical developments might be the finding of a critical coupling at which the strength of DM interaction and Zeeman magnetic field or magnetic anisotropy are balanced Barton-Singer et al. (2018); Adam et al. (2019a, b); Schroers (2019), analogous to superconductors at the critical coupling between types I and II. In this case, it allows so-called Bogomol’yi-Prasad-Sommerfield (BPS) topological solitons Bogomolny (1976); Prasad and Sommerfield (1975), i. e. the most stable configurations with a fixed boundary condition (or a topological sector), which were originally found for magnetic monopoles and other topological solitons in high energy physics and now are realized in superconductors at the critical coupling.

Despite of such experimental and theoretical developments, instantons Belavin et al. (1975); Polyakov and Belavin (1975) (see Refs. Rajaraman (1982); Coleman (1988); Manton and Sutcliffe (2004)) – classical solutions of Euclidian field theory and one of the most crucial theoretical concepts to understand physical properties of quantum systems –have never been studied thus far in chiral magnets. They represent nonperturbative quantum effects coming from nontrivial saddle point solutions of the Euclidian path integral. With the help of instantons, we can understand several key results of physical systems such as a ground-state property of non-abelian gauge theory ’t Hooft (1976a, b); Jackiw and Rebbi (1976); Callan et al. (1976); Schäfer and Shuryak (1998) and nonlinear sigma models.

In this Letter, we work out, for the first time, instantons in chiral magnets. After presenting a phase diagram, we exhaustively provide instanton solutions in one-dimensional anti-ferromagnetic spin chains with the DM interaction: temporal domain walls, domain wall-antidomain wall pair (called bions), vortices or dislocations as fractional instantons (called merons), and BPS instantons and calorons at the critical coupling. Although most experiments focus on ferromagnetic chiral magnets, anti-ferromagnetic chiral magnets also exhibit rich behaviors as we show below.

Model and ground state.— Magnetic spin texture in quantum spin-chain is described in the continuum limit by a unit spin vector $n^{a}\,(a=1,2,3)$ with $\sum_{a}n^{a}n^{a}=1$. The energy functional for one-dimensional spin-chain involving the DM interaction with the strength $\kappa$ in addition to the kinetic term, easy-axis potential and staggered magnetic field is given at low-energy as a form of the $O(3)$ or ${\mathbb{C}}P^{1}$ sigma model:

[TABLE]

For $\mu>0$, the potential favors for $n^{a}$ to point to the north ($n^{3}=1$) or the south ($n^{3}=-1$) pole (easy-axis). For $\mu<0$, it favors the equator ($n^{3}=0$) (easy-plane). The term $Bn^{3}$ is a staggered magnetic field, but, for simplicity, we call $B$ as a magnetic field.

Since the DM interaction can be regarded as a background gauge field Schroers (2019), the energy can be rewritten by defining a covariant derivative $D_{x}n^{a}\equiv\partial_{x}n^{a}-\kappa\epsilon^{3ab}n^{b}$ as

[TABLE]

Since the first nonnegative term must vanish for a ground state, we obtain

[TABLE]

which has the helical-state solution given by

[TABLE]

with $A\in{\mathbb{C}}$ satisfying $|A|\leq 1$. This is the spatially modulated state, where $n^{a}$ rotates at a constant latitude of the $S^{2}$ target space with the wave number $\kappa$ along the spatial direction $x$. The actual minimum of $E[n]$ depends on the value of the easy-axis potential $\mu$ and the magnetic field $B$. By examining the minimum of the potential as a function of $n^{3}$, we find three critical lines emanating from the tri-critical point at $\mu=\kappa^{2},\,B=0$ as illustrated in Fig. 1: the line (a) along $B=0,\,\mu>\kappa^{2}$, the line (b) along $\kappa^{2}-\mu=B>0$, and the line (c) along $\kappa^{2}-\mu=-B>0$. The ground state of the chiral magnet is helical states in the region (I) the south pole in the region (II) and the north pole in the region (III), respectively (See Fig. 1). At the tri-critical point $B=0,\,\mu=\kappa^{2}$, all the helical states with $|A|\leq 1$ become the ground states with the same energy. The topology of the order parameter manifold (OPM) is different in various regions as summrized in Table 1: $S^{2}$ at the tri-critical point, two discrete points (the north and south poles) along the line (a), $S^{1}$ in the region (I), a point (north pole) in the region (II) including the line (b), and a point (south pole) in the region (III) including the line (c). It is interesting to observe that the DM interaction tends to favor easy-plane configuration, so that helical ground states can be ground states even in the presence of the easy-axis potential with $\mu>0$.

The order of phase transition for the critical lines are as follows: The first-order phase transition occurs on the line (a), since the global minimum of the energy jumps from one local minimum to the other across (a). The second-order phase transition occurs on the lines (b) and (c), where the second derivative of the ground state energy density with respect to $B$ is discontinuous. It is also notable that the tri-critical point, which is a switching point of the first- and second-order transitions, has a larger symmetry as we will discuss in details later.

In order to explore instanton solutions for the anti-ferromagnetic material111 Effective theory of ferromagnetic material involves the first order term in time derivative, instead of the second order. , we introduce the imaginary time $\tau$ and consider the following Euclidean 2D Lagrangian $(i=0,1)$

[TABLE]

Note that this Euclidean 2D model should also be useful to describe the energy density of a 2+1D magnetic material (both ferromagnetic and anti-ferromagnetic) which has an anisotropic uniaxial DM interaction only in one spatial direction $x$ rather than two spatial directions. It is also convenient to use the stereographic projection of the target space $n^{a}\in S^{2}$ to a complex plane $v\in{\mathbb{C}}$ through

[TABLE]

The $O(3)$ sigma model with the DM interaction becomes

[TABLE]

with $v\overset{\leftrightarrow}{\partial}\bar{v}=v\partial_{x}\bar{v}-\bar{v}\partial_{x}v$. The instanton number density $\rho_{Q}$ is defined by

[TABLE]

where we used the complex coordinates $z=x+\mathrm{i}\tau,\,\bar{z}=x-\mathrm{i}\tau$. The intergration of this quantity in the Euclidean 2D space yields the instanton number charactrizing the second homotopy group $\pi_{2}$. Below, we show that instanton solutions exist in all the phases, sometimes as composite objects even when the OPS does not have a nontrivial $\pi_{2}$.

Domain walls on the line (a).— In this case the OPM consists of two points, suggesting that there exists a domain wall solution connecting these two discrete ground states. For instance, we can impose a boundary condition $n_{3}(\tau,x=\pm\infty)=\pm 1$ at the left and right spatial infinities. For a $\tau$-independent configuration, we can make a Bogomol’nyi completion Bogomolny (1976); Prasad and Sommerfield (1975) of the energy to find

[TABLE]

with $\eta=\sqrt{\mu-\kappa^{2}}+\mathrm{i}\kappa$. Since the first term is positive semi-definite, the surface term provides the lower bound of the energy:

[TABLE]

The equality holds when the BPS equation

[TABLE]

is satisfied. This equation gives the following domain wall solution with a complex integration constant $C$

[TABLE]

This is the lowest energy configuration satisfying the boundary condition. This does not carry an instanton number in contrast to the temporal case discussed below.

Similarly we can construct a domain wall solution in the temporal direction, regarded as an instanton, for the boundary condition $n_{3}(\tau=\pm\infty,x)=\pm 1$. The Bogomol’nyi bound for the action per unit $x$ is obtained by replacing $x\to\tau$ in Eq. (10). This bound is saturated if the BPS equations

[TABLE]

are satisfied. The domain wall solution is

[TABLE]

with unit instanton number and action $2\sqrt{\mu-\kappa^{2}}$ per length $\Delta x=2\pi/\kappa$. It is interesting to note that as shown in Fig. 2 (a), helical states show up only in the vicinity of the wall even though it is hidden in the ground states222 Since in 2+1D, a domain wall carrying a skyrmion number (equivalent to the instanton number for Euclidean 2D) density induced by a modulating $U(1)$ phase on the wall is called a domain wall skyrmion Nitta (2012); Kobayashi and Nitta (2013a), we may call this solution a domain wall instanton. ($n^{3}=\pm 1$).

Bions in the phases (II) and (III).— If the magnetic field is turned on ($B\not=0$) in the region (II) or (III), we find that the linear term in $n^{3}$ allows non-BPS wall-antiwall solutions, so-called bion solutions Fujimori et al. (2016, 2017a). For example, the simplest solution with a single wall-antiwall pair is given by

[TABLE]

where $\omega=\sqrt{\mu-\kappa^{2}+B}$ and $\tau_{0}$ and $x_{0}$ are the position moduli parameters. These solutions play a vital role in nonperturbative effects and resurgence theory Dunne and Unsal (2013, 2016); Misumi et al. (2014, 2015a, 2015b); Fujimori et al. (2016, 2017a, 2017b) recently being extensively considered in field theory.

Fractional instantons in helical state (I).— In this case, the OPM is $S^{1}$ at latitude $\sqrt{(\kappa^{2}-\mu+B)/(\kappa^{2}-\mu-B)}$. Therefore, we expect to find a vortex characterized by a non-trivial winding number around $S^{1}$. It carries a fractional instanton number Nitta and Vinci (2012) as is known for skyrmions with easy-plane potential Jaykka and Speight (2010); Kobayashi and Nitta (2013b), and is sometimes called a meron. A single-winding configuration takes the form

[TABLE]

where $z=x+\mathrm{i}\tau,\bar{z}=x-\mathrm{i}\tau$ are the complex coordinates and $f(|z|)$ is a profile function satisfying $f(0)=0$ and $f(\infty)=\sqrt{(\kappa^{2}-\mu+B)/(\kappa^{2}-\mu-B)}$. Numerically solving field equations we find a fractional instanton. Fig. 3 (a) shows the profile of a fractional instanton, which also exhibits a dislocation of the phase (or a phase slip) as shown in Fig. 3 (b).

Instantons in the tri-critical point.— The richest array of instantons is obtained at the tri-critical point $\mu=\kappa^{2},\,B=0$, where the Euclidean 2D action is bounded by the instanton number $Q=\int d^{2}x\,\rho_{Q}$ as

[TABLE]

The bound is saturated if and only if the BPS equation

[TABLE]

is satisfied. Let us consider solutions with a nonnegative instanton number $Q$ (by taking the upper sign of the bound). Hence all the BPS solutions are obtained as

[TABLE]

with an arbitrary holomorphic function $w(z)$.

The BPS equation has the general $l$-instanton BPS solution given by a rational function of degree $l$

[TABLE]

where $p(z),\,q(z)$ are mutually coprime polynomials of degree $m,\,n$ with $l=\max(m,n)$ [Fig. 4 (a)]. We can also construct general $l$-caloron (periodic instantons)-solutions in the periodicity interval $x\sim x+L$ as

[TABLE]

with a rational map of degree $l$ [Fig. 4 (b)]. Let us note that all these caloron solutions satisfy a twisted boundary condition $v(x+L,\tau)=e^{-\mathrm{i}\kappa L}v(x,\tau)$. By changing a moduli parameter, this caloron can be split into a pair of a temporal domain wall and antidomain wall with fractional instanton numbers, separated at arbitrary distance with the same energy Eto et al. (2005, 2006a, 2006b).

Equivalence theorem.— Let us now show that there exists a one-to-one mapping between the above model (5) and the usual $O(3)$ sigma model. For that purpose, inspired by the helical ground states (4), we define new variables333 When our system is put in the finite interval $0\leq x\leq L$, we also need to take account of the change of the boundary condition.

$\hat{n}^{a}$ as

[TABLE]

In terms of the new variables, the $O(3)$ sigma model with the DM interaction can be rewritten into that without the DM interaction ${\cal L}_{\mathrm{woDM}}(\hat{n})={\cal L}(n)$ :

[TABLE]

Note that the strength of the easy-axis potential is reduced from the original one. From this equivalence theorem, we find that all instanton solutions in the $O(3)$ sigma model with the DM interaction has one-to-one correspondence with those of the $O(3)$ sigma model without the DM interaction. This indicates that the original model (5) possesses a kind of hidden symmetry (known as modified symmetry Ohashi et al. (2017); Takahashi et al. (2017)), which enables us to compactly summarize our finding on the ground states and instanton solutions (See Table 1). Nevertheless, it should be emphasized that the physical variable $n^{a}$ must be used to find the real magnetic texture of chiral magnets.

Summary and Discussion.— We have clarified possible instanton solutions for the one-dimensional anti-ferromagnetic spin chain in the presence of the DM interaction. Depending on the phases, we have exhausted all possible instanton solutions including temporal domain walls with the instanton number density distributed in the temporal direction, vortices (dislocations or phase slips) as fractional instantons and BPS instantons and caloron at the critical coupling. We have also shown that the model with the DM interaction is equivalent to the model without the DM interaction.

Our results have ienergy. mplications both for theoretical and experimental researches. The instanton solutions in chiral magnets, which were not discussed before, give a novel theoretical insight into the anti-ferromagnetic spin chains and our methodology to obtain them based on the equivalence theorem can be applied broadly in the related studies. The phase diagram in Fig. 1 with the variety of instantons can help us to understand physics which would be observed in future experiments on chiral magnets with controllable DM interaction Siegfried et al. (2015); Koretsune et al. (2015); Belabbes et al. (2016); Ma et al. (2016) or easy-axis potential.

There are several interesting avenues related to this work. While we have considered one-dimensional chiral magnets in this letter, we can generalize our approach to higher-dimensional systems Barton-Singer et al. (2018); Adam et al. (2019a, b); Schroers (2019). In particular, the 2D model allows a topologically conserved skyrmion current and Hopf terms, thus involves rich theoretical structures. Although we have only focused on the ground state at the mean field level and instanton solutions interpolating them, it is also interesting to consider generic quantum aspects of the systems; e. g. a generalization of the Haldane conjecture Haldane (1983a, b) in chiral anti-ferromagnetic chains, and deconfined quantum criticality in $2+1$-dimensional systems Haldane (1988); Read and Sachdev (1989); Senthil et al. (2004). The possible ’t Hooft anomaly — field theoretical manifestation of the Lieb-Schultz-Mattis theorem Lieb et al. (1961); Affleck and Lieb (1986) — together with semi-classics analyses including resurgence theory in the sigma models Dunne and Unsal (2013, 2016); Misumi et al. (2014, 2015a, 2015b); Fujimori et al. (2016, 2017a, 2017b) will shed light on the detailed quantum aspects of chiral magnets.