Non-Hermitian Casimir effect of magnons

There has been a growing interest in non-Hermitian quantum mechanics. The key concepts of quantum mechanics are quantum fluctuations. Quantum fluctuations of quantum fields confined in a finite-size system induce the zero-point energy shift. This quantum phenomenon, the Casimir effect, is one of the most striking phenomena of quantum mechanics in the sense that there are no classical analogs and has been attracting much attention beyond the hierarchy of energy scales, ranging from elementary particle physics to condensed matter physics, together with photonics. However, the non-Hermitian extension of the Casimir effect and the application to spintronics have not yet been investigated enough, although exploring energy sources and developing energy-efficient nanodevices are its central issues. Here we fill this gap. By developing a magnonic analog of the Casimir effect into non-Hermitian systems, we show that this non-Hermitian Casimir effect of magnons is enhanced as the Gilbert damping constant (i.e., the energy dissipation rate) increases. When the damping constant exceeds a critical value, the non-Hermitian Casimir effect of magnons exhibits an oscillating behavior, including a beating one, as a function of the film thickness and is characterized by the exceptional point. Our result suggests that energy dissipation serves as a key ingredient of Casimir engineering.


I. INTRODUCTION
Recently, non-Hermitian quantum mechanics has been drawing considerable attention [1].The important concepts of quantum mechanics are quantum fluctuations.Quantum fluctuations of quantum fields under spatial boundary conditions realize a zero-point energy shift.This quantum effect which arises from the zero-point energy, the Casimir effect [2][3][4][5], is one of the most striking phenomena of quantum mechanics in the sense that there are no classical analogs.Although the original platform for the Casimir effect [2][3][4][5] is the photon field [6], the concept can be extended to various fields such as scalar, tensor, and spinor fields [7][8][9][10][11][12][13][14][15].Thanks to this universal property, the Casimir effects have been investigated in various research areas [16] beyond the hierarchy of energy scales [7][8][9][10][11][12][13][14][15], ranging from elementary particle physics to condensed matter physics, together with photonics.However, the non-Hermitian extension of the Casimir effect and the application to spintronics remain missing ingredients, although exploring energy sources and developing the potential for energy-efficient nanodevices are the central issues of spintronics [17][18][19][20][21].
Here we fill this gap.The Casimir effects are characterized by the energy dispersion relation.We therefore incorporate the effect of energy dissipation on spins into the energy dispersion relation of magnons through the Gilbert damping constant [22] and thus develop a magnonic analog of the Casimir effect [23], called the magnonic Casimir effect (see Fig. 1) [24], into non-Hermitian systems.
We then show that this non-Hermitian extension of the magnonic Casimir effect, which we call the magnonic non-Hermitian Casimir effect, is enhanced as the Gilbert damping constant increases.When the damping constant exceeds a critical value, the magnonic non-Hermitian Casimir effect exhibits an oscillating behavior as a function of the film thickness and is characterized by the exceptional point [25] (EP).We refer to this behavior as the magnonic EP-induced Casimir oscillation.We emphasize that this magnonic EP-induced Casimir oscillation is absent in the dissipationless system of magnons.The magnonic EPinduced Casimir oscillation exhibits a beating behavior in the antiferromagnets (AFMs) where the degeneracy between two kinds of magnons is lifted.Our result suggests that energy dissipation serves as a new handle on Casimir engineering [26] to control and manipulate the Casimir effect of magnons.Thus, we pave a way for magnonic Casimir engineering through the utilization of energy dissipation.

A. System
We consider the insulating AFMs of two-sublattice systems in three dimensions described by the Hamiltonian, where S i = (S x i , S y i , S z i ) is the spin operator at the site i, J > 0 parametrizes the antiferromagnetic exchange interaction between the nearest-neighbor spins ⟨i, j⟩, K h > 0 is the hard-axis anisotropy, and K e > 0 is the easyaxis anisotropy.These are generally K h /J ≪ 1 and K e /J ≪ 1.The AFMs have the Néel magnetic order and there exists the zero-point energy [27,28].Through- out this study, we work under the assumption that the Néel phase remains stable in the presence of energy dissipation.Elementary magnetic excitations are two kinds of magnons σ = ±, acoustic mode for σ = + and optical mode for σ = −.
By incorporating the effect of energy dissipation on spins into the energy dispersion relation of magnons through the two-coupled Landau-Lifshitz-Gilbert equation where the value of the Gilbert damping constant α > 0 for each sublattice is identical to each other, we study the low-energy magnon dynamics [29] described by the energy dispersion relation ϵ σ,k,α ∈ C of Re(ϵ σ,k,α ) ≥ 0 and the wavenumber k = (k x , k y , k z ) ∈ R [30] in the long wavelength limit as [31] and where k := |k|, the length of a magnetic unit cell is a, the spin moment in a magnetic unit cell is S, and the others are material-dependent parameters which are independent of the wavenumber, In the absence of the hard-axis anisotropy K h = 0, two kinds of magnons σ = ± are in degenerate states, whereas the degeneracy is lifted by K h > 0. Note that, in general, the effect of dipolar interactions is negligibly small in AFMs, and we neglect it throughout this study.The Gilbert damping constant α is a dimensionless constant, and the energy dissipation rate increases as the Gilbert damping constant grows.In the dissipationless system [23], the Gilbert damping constant is zero α = 0.The dissipative system of α > 0 described by Eq. ( 2) can be regarded as a non-Hermitian system for magnons in the sense that the energy dispersion takes a complex value.Note that the constant term in Eq. ( 2), −iαC, is independent of the wavenumber and just shifts the purely imaginary part of the magnon energy dispersion ϵ σ,k,α .For this reason [Eq.(10a)], the constant term, −iαC, is not relevant to the magnonic Casimir effect.We then define the magnon energy gap of Eq. ( 2) as ∆ σ,α := Re(ϵ σ,k=0,α ), i.e.,

B. Magnonic exceptional point
When the damping constant α is small and (E σ,k=0,α ) 2 > 0, E σ,k=0,α takes a real value and decreases as α increases.This results in Thus, the magnon energy gap decreases as the damping constant increases [32] [compare the solid line with the dashed one in the left panel of Fig. 2 (i)].When the damping constant is large enough, the magnon energy gap vanishes ∆ σ,α = 0 at α = α cri σ , where there exists the gapless magnon mode which behaves like a relativistic particle with the linear energy dispersion.From the property of Eq. ( 6), we call (i) α ≤ α cri σ the gap-melting regime.When the damping constant exceeds the critical value α cri σ , i.e., α > α cri σ , E σ,k=0,α takes a purely imaginary value as (E σ,k=0,α ) 2 < 0. In this regime, the real part of the magnon energy dispersion remains zero Re(ϵ σ,k,α ) = 0 for the region 0 whereas Re(ϵ σ,k,α ) > 0 for k > k cri σ,α [see the highlighted in yellow in the left panel of Figs. 2 (ii) and (iii)].The critical point k cri σ,α can be regarded as the EP [32] for the wavenumber k, and we refer to it as the magnonic EP.As the value of the damping constant becomes larger, that of the EP increases At the EP k = k cri σ,α , the group velocity v σ,k,α := Re[∂ϵ σ,k,α /(∂ℏk)] becomes discontinuous [see the solid lines in the left panel of Figs. 2 (ii) and (iii)].In the vicinity of the EP, the group velocity becomes much larger than the usual such as in the gap-melting regime (i) [compare the solid lines in the left panel of Figs. 2 (ii) and (iii) with the one of Fig. 2 (i)].

C. Magnonic Casimir energy
The magnonic analog of the Casimir energy, called the magnonic Casimir energy [23], is characterized by the energy dispersion relation of magnons.Therefore, by incorporating the effect of energy dissipation on spins into the energy dispersion relation of magnons through the Gilbert damping constant [Eq.( 2)], a non-Hermitian extension of the magnonic Casimir effect can be developed.We remark that the Casimir energy induced by quantum fields on the lattice, such as the magnonic Casimir energy [23], can be defined by using the lattice regularization [33][34][35][36][37][38][39].In this study, we focus on thin films confined in the z direction (Fig. 1).In the twosublattice systems, the wavenumber on the lattice is replaced as (ak j ) 2 → 2[1 − cos(ak j )] along the j axis for j = x, y, z.Here by taking into account the Brillouin zone (BZ), we set the boundary condition for the z direction in wavenumber space so that it is discretized as k z → πn/L z , i.e., ak z → πn/N z , where L z := aN z is the film thickness, N j ∈ N is the number of magnetic unit cells along the j axis, and n = 1, 2, ..., 2N z .Thus, the magnonic Casimir energy E Cas [23] per the number of magnetic unit cells on the surface for N z is defined as the difference between the zero-point energy E sum 0 for the discrete energy ϵ σ,k,α,n due to discrete k z [Eq.( 10b)] and the one E int 0 for the continuous energy ϵ σ,k,α [Eqs.(10c) and ( 2)] as follows [33][34][35][36][37][38][39]: where the integral is over the first BZ, and the factor 1/2 in Eqs.(10b) and (10c) arises from the zero-point energy of the scalar field.We remark that [29] assuming thin films of N z ≪ N x , N y (Fig. 1), the zero-point energy in the thin film of the thickness N z is E sum 0 (N z )N x N y and consists of two parts as Then, to see the film thickness dependence of E Cas (N z ), we introduce the rescaled Casimir energy and call Cas the magnonic Casimir coefficient in the sense that Cas N z −b .Note that the zero-point energy arises from quantum fluctuations and does exist even at zero temperature.The zero-point energy defined at zero temperature does not depend on the Bose-distribution function [Eqs.(10b) and (10c)].Throughout this work, we focus on zero temperature [29].
When α < α cri σ=+ , the magnon energy gap for both σ = ± is nonzero ∆ σ=±,α > 0 and both magnons σ = ± are the gapped modes.For each gapped mode, the absolute value of the magnonic Casimir coefficient C [3] Cas decreases and approaches asymptotically to zero as the film thickness increases.We emphasize that the magnon energy gap decreases as the damping constant α increases [Eq.( 6)].Then, the magnitude of the magnonic Casimir energy and its coefficient increase as the value of the damping constant becomes larger and approaches to the critical value α → α cri σ=+ [see the middle panel of Fig. 2 (i)].
When α = α cri σ=+ , the magnon σ = − remains the gapped mode, whereas the magnon energy gap for σ = + vanishes ∆ σ=+,α = 0 and the magnon σ = + becomes the gapless mode which behaves like a relativistic particle with the linear energy dispersion.In the gapless mode, the magnonic Casimir coefficient C [3] Cas approaches asymptotically to a nonzero constant as the film thickness increases.The behavior of the gapless magnon mode is analogous to the conventional Casimir effect of a massless scalar field in continuous space [42] except for adependent lattice effects, whereas that of the gapped magnon modes is similar to the Casimir effect known for massive degrees of freedom [42,43].

Oscillating regime
(ii) Oscillating regime α cri σ=+ < α < α cri σ=− .The magnonic Casimir energy takes a complex value as shown in the middle and right panels of Fig. (ii), see also Eq. ( 11).There is one EP, e.g., ak cri σ=+,α=0.04∼ 0.039 1 for α = 0.04 [see the left panel of Fig. 2 (ii)].Then, the magnonic non-Hermitian Casimir effect exhibits an oscillating behavior as a function of N z for the film thickness L z := aN z .An intuitive explanation for the oscillation of the magnonic non-Hermitian Casimir effect and its relation to the EP is given as follows: Through the lattice regularization, the magnonic Casimir energy is defined as the difference [Eq.(10a)] between the zero-point energy with the discrete wavenumber k z [Eq.( 10b)] and the one with the continuous wavenumber [Eq.(10c)].On the lattice, the wavenumber k z under the boundary condition is discretized in units of π/aN z as k z → (π/aN z )n.As the film thickness N z increases, the unit becomes smaller, and finally, it matches the EP as π/aN z = k cri σ,α , i.e., N z = π/ak cri σ,α , where the magnonic non-Hermitian Casimir effect is enhanced due to the EP.Then, the magnonic non-Hermitian Casimir effect is periodically enhanced where the film thickness N z is multiples of π/ak cri σ,α .Thus, the oscillating behavior of the magnonic non-Hermitian Casimir effect stems from the EP, k cri σ,α , and the oscillation is characterized in units of π/ak cri σ,α .We refer to this oscillating behavior as the magnonic EPinduced Casimir oscillation.The period of this Casimir oscillation is As an example, the period is Λ Cas σ=+,α=0.04 ∼ 80.4 for α = 0.04.This agrees with the numerical result in the middle and right panels of Fig. 2 (ii), see the highlighted in red.We call (ii) α cri σ=+ < α < α cri σ=− the oscillating regime.The middle and right panels of Fig. 2 (ii) show that the magnonic EP-induced Casimir oscillation is characterized by its Casimir coefficient C

[b]
Cas of b = 1.5.We remark that the beating behavior is absent in the uniaxial AFMs of K h = 0 and K e > 0 where two kinds of magnons σ = ± are in degenerate states [29].

Imaginary part of the Casimir energy
Here, we discuss the meaning of the imaginary part of the Casimir energy.The (complex) Casimir energy is defined by the zero-point energy which is the sum of all the possible (complex) eigenvalues.The real part of the zero-point energy originates from the sum of the real parts of the eigenvalues, whereas the imaginary part of the zero-point energy is defined as the sum of imaginary parts of eigenvalues.Since the imaginary parts of eigenvalues are formally regarded as the decay width (or the inverse of a lifetime) of an unstable particle, the imaginary part of the zero-point energy is the sum of all the possible decay widths.Hence, if the decay width of an unstable particle depends on the wavenumber, and the width in the thin film and that in the bulk are different from each other, then the imaginary part of the Casimir energy can be nonzero.In this work, since we focus on magnons in the geometry of Fig. 1, the imaginary part of magnonic Casimir energy represents the L z -dependence of the sum of magnon decay widths.

A. Magnonic Casimir engineering
The Gilbert damping can be enhanced and controlled by the established experimental techniques of spintronics such as spin pumping [29].In addition, microfabrication technology can control the film thickness and manipulate the magnonic non-Hermitian Casimir effect.The Casimir pressure of magnons, which stems from the real part of its Casimir energy, contributes to the internal pressure of thin films.We find from the middle panel of Figs. 2 (ii) and (iii) that depending on the film thickness, the sign of the real part of the magnonic Casimir coefficient changes.This means that by tuning the film thickness, we can control and manipulate the direction of the magnonic Casimir pressure as well as the magnitude thanks to the EP-induced Casimir oscillation.Thus, our study utilizing energy dissipation, the magnonic non-Hermitian Casimir effect, provides the new principles of nanoscale devices, such as highly sensitive pressure sensors and magnon transistors [44], and paves a way for magnonic Casimir engineering.

B. Conclusion
We have shown that as the Gilbert damping constant increases, the non-Hermitian Casimir effect of magnons in antiferromagnets is enhanced and exhibits the oscillating behavior which stems from the exceptional point.This exceptional point-induced Casimir oscillation also exhibits the beating behavior when the degeneracy between two kinds of magnons is lifted.These magnonic Casimir oscillations are absent in the dissipationless system of magnons.Thus, we have shown that energy dissipation serves as a new handle on Casimir engineering.

C. Outlook
In this paper following Ref.[31], the effect of dissipation is incorporated into the energy dispersion relation of magnons through the Landau-Lifshitz-Gilbert equation.It will be intriguing to find the quantum effect of dissipation on magnonic non-Hermitian Casimir effect, beyond the Landau-Lifshitz-Gilbert equation, by using quantum master equation [21,[45][46][47].We also remark that dipolar interactions contribute to the form of the dispersion relation [48] and play a crucial role in magnonic Casimir effect in ferrimagnets [23].Hence, taking dipolar interactions into account, it will be interesting to develop this study, magnonic non-Hermitian Casimir effect in antiferromagnets, into ferrimagnets.We leave these advanced studies for future works.

Methods
Numerical calculation was performed by using the software Wolfram Mathematica.
In the main text, following the Casimir energy for photon fields (i.e., quantum fields in continuous space) [2], the magnonic Casimir energy is defined as in Eqs.(10a), (10b), and (10c) through the lattice regularization.In contrast to the Casimir effect for photon fields (i.e., quantum fields in continuous space), the magnonic Casimir energy is induced by its quantum field on the lattice, and there is no ultraviolet divergence in each component [see Eqs.Appendix S-II: The hard-axis anisotropy 1.In the absence of the hard-axis anisotropy In the main text, we have considered NiO.NiO is a biaxial AFM of K h > 0 and K e > 0: There exist not only the easy-axis anisotropy K e = 0.001 71829 meV but also the hard-axis anisotropy K h = 0.039 5212 meV, see the main text for other parameter values.Here, by changing only the value of K h to K h = 0 with leaving other parameter values unchanged, we estimate the magnonic Casimir effect and provide some details about its behavior in the absence of the hard-axis anisotropy.
Figure S1 shows the magnon energy dispersion ϵ σ,k,α for the gap-melting regime (i) in the absence of the hard-axis anisotropy K h = 0. Figure S2 shows the real part of the magnonic Casimir energy Re(E Cas ) for the gap-melting regime (i) in the absence of the hard-axis anisotropy K h = 0 and that in the presence of hard-axis anisotropy K h = 0.039 5212 meV.The latter is the same as the middle panel of Fig. 2 (i).
In the absence of the hard-axis anisotropy K h = 0, two kinds of magnons σ = ± are in degenerate states [see Eq. ( 2)].This results in α cri σ=+ = α cri σ=− = 0.008 54 [see Eq. ( 7)].When the damping constant reaches the critical value α = α cri σ=+ = α cri σ=− = 0.008 54, the magnon energy gaps for both σ = ± vanish, ∆ σ=±,α = 0, and both magnons σ = ± become the gapless modes which behave like relativistic particles with the linear energy dispersion (see the solid line in Fig. S1).Then, the magnonic Casimir coefficient C [3] Cas asymptotically approaches to a nonzero constant as the film thickness increases (see Fig. S2), which means that its Casimir energy exhibits the behavior of E Cas ∝ 1/N z 3 .Figure S2 also shows that the magnitude of the magnonic Casimir energy and its coefficient for K h = 0 become larger than that for K h = 0.039 5212 meV.As an example, Cr 2 O 3 can be regarded as a uniaxial AFM of K h = 0 and K e > 0, where two kinds of magnons σ = ± are in degenerate states.Hence, the magnonic EPinduced Casimir oscillation is one type, and its beating behavior is absent in Cr 2 O 3 .Magnonic non-Hermitian Casimir effects in the AFMs of K e > 0 are summarized in Table T1.Note that Ref. [60] reported the experimental realization of sub-terahertz spin pumping in Cr 2 O 3 , and Ref. [61] reported that in NiO.

Appendix S-III: Remarks on observation
In the main text, we have explained that the Gilbert damping can be enhanced and controlled by the use of the established experimental techniques of spintronics such as spin pumping.Here we add remarks on observation of our theoretical prediction.We expect that the magnonic Casimir effect in the AFMs can be experimentally observed in principle through measurement of magnetization.The reason is as follows.
External magnetic fields induce magnetostriction, which can be regarded as a kind of lattice deformation, and its correction for the length of a magnetic unit cell a is characterized by the magnetostriction constant [62][63][64][65][66][67][68][69].The magnonic Casimir energy of the AFMs does not depend on external magnetic fields usually, whereas the magnonic Casimir effect is influenced by magnetostriction, and its correction for the magnonic Casimir energy depends on magnetic fields and contributes to magnetization.Thus, although the correction is small, the magnonic Casimir effect in the AFMs can be experimentally observed in principle through measurement of magnetization and its film thickness dependence by using external magnetic fields (i.e., magnetostriction).
We remark that the magnetic-field derivative of the real part of the Helmholtz free energy is magnetization.At zero temperature, assuming thin films of N z ≪ N x , N y (see Fig. 1), the Helmholtz free energy of quantum fields for magnons in the thin film of the thickness N z is E sum .Since E Cas (N z ) exhibits an oscillating and a beating behavior as a function of the film thickness in the regimes (ii) and (iii), respectively [see Eq. ( 11) and the middle panels of Figs. 2 (ii) and (iii)], the Helmholtz free energy of the thin film shows a different N z -dependence from the linear-in-N z behavior.In other words, magnetization of the thin film exhibits an oscillating or a beating behavior as a function of the film thickness due to the magnonic non-Hermitian Casimir effect.Hence, our prediction, the non-Hermitian Casimir effect of magnons, can be observed in principle through measurement of magnetization, its oscillating or beating behavior as a function of the film thickness.
FIG. 1. Schematic of magnonic Casimir effect.Magnonic Casimir effect arises from quantum vacuum fluctuations of magnon fields.
and 0 < C ∈ R: The parameters are given as[31]

FIG. 2 .
FIG. 2. Plots of the magnon energy dispersion ϵ σ,k,α , the real part of the magnonic Casimir energy Re(ECas), and the imaginary part Im(ECas) for NiO in (i) the gap-melting regime, (ii) the oscillating regime, and (iii) the beating regime.Inset: Each magnonic Casimir coefficient C [b] (10b) and (10c)].Here we remark that the Casimir energy induced by quantum fields on the lattice, such as the magnonic Casimir energy E Cas (N z ) [see Eq. (10a)], plays a key role in finding the film thickness dependence of the zero-point energy in the thin film (see Fig. 1).The zero-point energy in the thin film of the thickness N z is E sum 0 (N z )N x N y [see Eq. (10b)] and consists of two parts as E sum 0 (N z ) = E Cas (N z ) + E int 0 (N z ) [see Eq. (10a)], where E int 0 (N z ) exhibits the behavior of E int 0 (N z ) ∝ N z [see Eq. (10c)].

0(
N z )N x N y [see Eq. (10b)] and consists of two parts as E sum 0 (N z ) = E Cas (N z )+E int 0 (N z ) [see Eq. (10a)], where E int 0 (N z ) exhibits the linear-in-N z behavior as E int 0 (N z ) ∝ N z [see Eq. (10c)]