Causality and stability analysis for the minimal causal spin hydrodynamics

We perform the linear analysis of causality and stability for a minimal extended spin hydrodynamics up to second order of the gradient expansion. The first order spin hydrodynamics, with a rank-3 spin tensor being antisymmetric for only the last two indices, are proved to be acausal and unstable. We then consider the minimal causal spin hydrodynamics up to second order of the gradient expansion. We derive the necessary causality and stability conditions for this minimal causal spin hydrodynamics. Interestingly, the satisfaction of the stability conditions relies on the equations of state for the spin density and chemical potentials. Moreover, different with the conventional relativistic dissipative hydrodynamics, the stability of the theory seems to be broken at the finite wave-vector when the stability conditions are fulfilled at small and large wave-vector limits. It implies that the behavior in small and large wave-vector limits may be insufficient to determine the stability conditions for spin hydrodynamics in linear mode analysis.

So far, the spin hydrodynamics in the first order of the gradient expansion, with a rank-3 spin tensor that exhibits antisymmetry solely in its last two indices, has been established [45,48].Before simulating the spin hydrodynamics, it is necessary to investigate the theory's causality and stability, as is done in conventional hydrodynamics.In fact, the first order conventional relativistic hydrodynamics at Landau frame in the gradient expansion are always acausal and unstable, e.g., see the discussions in Refs.[156][157][158][159]. Therefore, the question whether the first order spin hydrodynamics can be casual or stable arises.Several studies conclude that the spin hydrodynamics up to the first order in the gradient expansion may be acausal and unstable in the linear modes analysis [70,71].In the early study [45], the authors have modified the constitutive relations for the anti-symmetric part of energy momentum tensor through the equations of motion for the fluid and the stability conditions of this first order theory in the rest frame of fluid seem to be satisfied in the linear modes analysis.Later on, the Ref. [71] shows that this first order theory may be acausal, while Ref. [70] finds the stability conditions (which corresponds to Eq. (46) in this work) may not be satisfied.
In this work, we systematically investigate the linear causality and stability for the spin hydrodynamics proposed in Refs.[45,48].Our findings indicate that the spin hydrodynamics up to the first order in the gradient expansion is acausal and unstable even when using the replacement mentioned by Ref. [45].The acausal and unstable modes can usually be removed when extending the theory up to the second order in the gradient expansion.Therefore, we follow the method outlined in the conventional hydrodynamics [158][159][160][161][162] to consider the minimal causal spin hydrodynamics.It is sufficient to see whether the causality and stability can be recovered up to the second order in the gradient expansion [158][159][160][161][162].
We then analyze the causality and stability for this minimal extended theory.
The paper is organized as follows.We first review the first order spin hydrodynamics introduced in Refs.[45,48] in Sec.II and show it is acausal and unstable in Sec III.In Sec IV, we consider the minimal causal spin hydrodynamics following the method outlined in the conventional hydrodynamics.In Sec V, we analyze the causality and stability for the minimal causal spin hydrodynamics in the rest frame and comment the results in moving frames.We summarize this work in Sec.VI.

II. FIRST ORDER SPIN HYDRODYNAMICS
In this section, let us briefly review the first order relativistic spin hydrodynamics.In spin hydrodynamics, we have the conservation equations for energy, momentum, total angular momentum, and particle number, i.e., [45,48,50,58,59,67,163] where Θ µν is the energy momentum tensor, J λµν is the total angular momentum current, and j µ is the current for particle number.Different with conventional relativistic hydrodynamics, the total angular momentum conservation equation in Eq. ( 1) plays a crucial role to describe the evolution of spin.The total angular momentum current can be written as [45,48] where the first two terms corresponds to the conventional orbital angular momentum, and Σ λµν is the rank-3 spin tensor.Using Eq. ( 2), the conservation equation ∂ λ J λµν = 0 can be rewritten as the spin evolution equation, Eq. ( 3) implies that the anti-symmetric part of energy momentum tensor Θ [µν] is the source for spin, and the spin can be viewed as a conserved quantity if and only if Θ [µν] = 0.
The spin density is defined as with the fluid velocity u µ .Analogy to the relationship between µ and n, here we introduce the anti-symmetric spin chemical potential ω µν as the conjugate of S µν .
Before decomposing the Θ µν and Σ λµν , we emphasize that there exist different choices for them.For example, by applying the Nöther theorem to two equivalent Lagrangian density for Dirac field, where , two distinct sets of energy momentum tensors and spin tensors emerge, Here, Σ λµν 1 is antisymmetric only with respect to µ and ν indices, while Σ λµν 2 is totally antisymmetric.In principle, one can derive the spin hydrodynamics from the microscopic theories, as demonstrated in Refs.[43,65,66,69,164] for kinetic theories and Refs.[41,165,166] for statistical methods.An alternative method to derive the spin hydrodynamics is to map the tensor structure of hydrodynamic variables to operators mentioned above, e.g.see Refs.[45,48,59].In this work, we follow Refs.[45,48] and adopt the energy momentum tensor and spin tensor sharing the similar tensor structure with Θ µν 1 and Σ λµν 1 , respectively.For other choices, one can refer to Refs.[59,147,155] and references therein.
Following Refs.[45,48], the energy momentum tensor and particle current can be decomposed as where h µ , ν µ , Π, and π µν stand for heat current, particle diffusion, bulk viscous pressure, and shear stress tensor, respectively, and the antisymmetric parts 2q [µ u ν] and ϕ µν are related to the spin effects.As for the rank-3 spin tensor Σ λµν , we have [45,48] where the spin density S µν defined in Eq. ( 6) has six independent degrees of freedom.
In this work, we follow the power counting scheme in Refs.[48,62,64], The spin density S µν is chosen as the leading order in the gradient expansion.It corresponds to the case in which the most of particles in the system are polarized, i.e. the order of S µν is considered as the same as the one for the number density n.While in Refs.[45,59], the authors have chosen a different power counting scheme, S µν ∼ O(∂), ω µν ∼ O(∂), Following [45,48], it is straightforward to get the entropy production rate, where S µ can is the entropy density current.The second law of thermodynamics ∂ µ S µ can ≥ 0 can give us the first order constitutive relations [45,48], where the heat conductivity coefficient κ, shear viscosity coefficient η, and bulk viscosity ζ also exist in conventional hydrodynamics, while λ and γ s are new coefficients corresponding to the interchange of spin and orbital angular momentum.The entropy principle also requires that the transport coefficients are positive.As the system approaches global equilibrium, the entropy production rate in Eq.( 15) tends to zero.It yields the well-known killing condition [41,165], which causes the right hand sides of Eqs.(16,17,18,19,20) to vanish.Especially, we have q µ , ϕ µν = 0 such that the energy momentum tensor Θ µν is symmetric in global equilibrium state.Note that, pointed out by Refs.[67,167], some cross terms between the different dissipative currents may also exist due to the Onsager relation, but here we neglect them for simplicity.
Before ending this section, we would like to comment on the heat flow h µ .Interestingly, when we set ν µ = 0 and n = 0, we find that one cannot fix the expression for heat current h µ in the first order of gradient expansion.By using ∆ να ∂ µ Θ µν = 0 and Eqs.(4,5), we ) when ν µ = 0 and n = 0.In that case, the term , will be neglected in the entropy production rate (15), i.e., we cannot determine the expression of h µ by the entropy principle there.The similar behavior was also observed in conventional hydrodynamics [168,169].

DYNAMICS
In this section, we analyze the causality and stability for the first order spin hydrodynamics.It is well-known that the conventional relativistic hydrodynamics in Landau frame up to the first order in gradient expansion are always acausal, e.g., see Refs.[156,157] as the early pioneer works.
In linear modes analysis, one can consider the perturbations δX to the hydrodynamical quantities X in the equilibrium.By assuming the δX ∼ δ Xe iωt−ikx with δ X being constant in space-time, one can solve the dispersion relation ω = ω(k) from the conservation equations.In the conventional hydrodynamics, the causality condition is usually given by [159,168,[170][171][172] where the condition (22) can also be written as lim k→∞ Re ∂ω ∂k ≤ 1 in some literature [159,171,172].However, the above condition is insufficient to guarantee the causality.We need an extra condition that, [173] As pointed out by the early pioneer work [173], the unbounded lim k→∞ ω k gives the infinite propagating speed of the perturbation, even if the ω is pure imaginary.One simple example is the non-relativistic diffusion equation, ∂ t n − D n ∂ 2 x n = 0 with D n being the diffusion constant.It is easy to check that its dispersion relation gives ω = iD n k 2 , which satisfies condition (22) but does not obey condition (23).Therefore, the perturbation in the nonrelativistic diffusion equations has the unlimited propagating speed, i.e. with any compact initial value for n(t 0 , x), the n(t 0 + ∆t, x) at x → ∞ can still get the influence [174].We emphasize that the conditions (22,23) are necessary but not sufficient to guarantee that the theory is casual [175][176][177].One example is the transverse perturbations of an Eckart fluid with shear viscous tensor, whose dispersion relation satisfies the conditions (22,23), but the velocity can exceed the speed of light (see Eqs. (47) and (48) in Ref. [157] for the perturbation equations and the propagating velocity).
The stability means that the the imaginary part of ω = ω(k) must be positive for k ̸ = 0, i.e.
Im ω(k) > 0. ( Note that the case of Im ω = 0 corresponds to the neutral equilibrium, which means the equilibrium state is not unique.In this work, we will not consider such special cases, and we only consider the condition (24) to study the stability of spin hydrodynamics as in Ref. [70].
It is necessary to study the causality and stability for the relativistic spin hydrodynamics in the first order.To see whether the first order spin hydrodynamics can be casual or not, we consider the linear modes analysis to the system, i.e. we take the small perturbations on top of static equilibrium.Following Refs.[156,157], the static equilibrium background is assumed to be irrotational global equilibrium state.We label the quantities with subscript (0) as those at the global equilibrium state, while we use "δX" to denote the small perturbations of the quantity X, e.g., e (0) and δe stand for the energy density at the global equilibrium and the small perturbations of energy density, respectively.
From now on, unless specified otherwise, we adopt the Landau frame, and neglect the conserved charge current j µ .
We now consider the small perturbations on top of static equilibrium.Not all of the perturbations are independent of each other, and we can choose as independent variables.
The variation of pressure δp and spin chemical potential δω µν can be expressed as functions of δe and δS µν through where the speed of sound c s , and χ b , χ s ,χ µν e are in general the functions of thermodynamic variables.For simplicity, we take c s , χ b , χ s ,χ µν e as constants in the linear modes analysis.Note that χ µν e comes from the anisotropy of the system.Under the assumption of an irrotational global equilibrium, from Eq. ( 19) the spin chemical potential vanishes ω µν (0) = 0.For simplicity, we further choose S µν (0) = 0.The variation of the temperature δT can be obtained by the thermodynamics relations, with the help of Eqs.(4,5), Next, we consider the variation of the conservation equations ∂ µ δΘ µν = 0 and ∂ λ δJ λµν = 0, where the perturbations δΘ µν and δJ λµν can be derived from the constitutive relations in Eqs.(2,11,(16)(17)(18)(19)(20).It is straightforward to obtain the linearized equations for the independent perturbations δe, δϑ i , δS µν , Here we introduce the following shorthand notations, In linear modes analysis, the perturbations are assumed along the x direction only, where δẽ, δ θi , and δ Sµν are independent of space and time.
Inserting the perturbations in Eq. ( 33) into Eqs.(28-31) yields, where and with The off-diagonal blocks A 1 , A 2 , A 3 in the matrix M 1 , whose expressions are shown in Appendix A, and are irrelevant to the following discussions.The non-trivial solutions in Eq.
In the k → ∞ limit, the dispersion relations become, where first four modes come from det M 1 = 0, and others can be derived by det M 2 = 0.
On the other hand, we also find that in Eqs.(48)(49)(50) |ω/k| is unbounded, which violates the causality condition (23).We also notice that Ref. [71] has also analyzed the causality for the first order spin hydrodynamics in small k limit.
We find that the first order spin hydrodynamics is acausal and unstable similar as the conventional relativistic hydrodynamics in Landau frame.
We now conclude that the first order spin hydrodynamics at static equilibrium state are unstable and acausal in the rest frame.We do not need to discuss the stability and causality of the first order spin hydrodynamics in moving frames again.

IV. MINIMAL CAUSAL SPIN HYDRODYNAMICS
In the previous section, we have shown that the first order spin hydrodynamics in Landau frame are acausal and unstable.The acausal and unstable theory is not physical, we therefore need to consider the second order spin hydrodynamics in gradient expansion.In this section we follow the idea of minimal causal extension in conventional hydrodynamics and implement it to the spin hydrodynamics.
Up to now, there are two ways to establish causal hydrodynamics.The first way is to add the second order corrections to the dissipative terms, such as the Müller-Israel-Stewart (MIS) theory [160,161] or other related second order hydrodynamics.The MIS theory is a famous causal conventional hydrodynamic theory up to O(∂ 2 ) in gradient expansion.Here, we consider a relativistic dissipative hydrodynamics with the bulk viscous pressure Π only as an example to explain why the MIS theory can be casual.The entropy current in MIS theory is assumed to be [161,179,180] where the coefficient β 0 > 0 and the ellipsis stands for other possible O(∂ 2 ) terms.Then the second law of thermodynamics ∂ µ S µ ≥ 0 leads to, where d/dτ ≡ u µ ∂ µ , and τ Π = ζβ 0 > 0 is defined as the relaxation time for the bulk viscous pressure.If the τ Π → 0, the hydrodynamic equations reduce to parabolic equations and become acausal.With a finite τ Π , the hydrodynamic equations are hyperbolic and can be causal under certain conditions [158,159,171,181,182].In linear modes analysis, the dispersion relations from Eq. ( 58) satisfy causal conditions (22,23) when the relaxation time τ Π is sufficiently large.The second order constitutive equations for shear viscous tensor π µν , heat flow h µ and heat current ν µ can be obtained in a similar way.These equations represent evolution equations that incorporate the respective relaxation time [161,179,180].Apart from the MIS theory, many other second order causal conventional hydrodynamic theories, e.g., Baier-Romatschke-Son-Starinets-Stephanov (BRSSS) theory [183] and the Denicol-Niemi-Molnar-Rischke (DNMR) theory [184], have been established.All of them contain the terms proportional to the relaxation times and can be causal and stable under certain conditions [183,185,186].Following these discussion, we can say that the key to recover the causality of the theory is to introduce the terms proportional to relaxation time.
It roughly says that one can choose some preferred frames to satisfy the causality and stability conditions.Unfortunately, the commonly used Landau or Eckart frame are not the preferred fluid frames in the BDNK theory.Therefore, we will not discuss the spin hydrodynamics in the BDNK theory in this work.We also notice that recent studies in Ref. [190] discuss the casual spin hydrodynamics in the first order similar to BDNK theory.
In this work, we follow the basic idea in MIS, BRSSS, and DNMR theories to construct a simplified causal spin hydrodynamics.Instead of considering the complete second order spin hydrodynamics, we only analyze the called "minimal" extended second order spin hydrodynamics.Here, the word "minimal" means that we concentrate on the essential terms in the second order of gradient expansion to get a causal theory and neglect the other terms which do not contribute to the dispersion relations in the linear modes analysis.As mentioned below Eq. ( 58), the key to get the causal theory is to add the terms proportional to the relaxation times similar to τ Π dΠ/dτ , in the left hand side of Eq. (58).Following this idea, the constitutive equations (16)(17)(18)(19)(20) in the minimal extended causal spin hydrodynamics can be rewritten as, where τ q , τ ϕ , τ π and τ Π are positive relaxation times for q µ , ϕ µν , π µν , Π, respectively.Eqs.(61,62) are the same as those in the conventional hydrodynamics 1 [158, 159, 171].Recently, the second order spin hydrodynamics similar to MIS theory has been introduced in Ref. [74] by using the entropy principle.Our minimal causal spin hydrodynamics can be regarded as a simplified version of it.We also notice that in the Refs.[60], the authors have proposed the same expressions for q µ and ϕ µν as presented in Eqs.(59,60) for minimal causal spin hydrodynamics.
Let us give some physical interpretation for Eqs.(59)(60)(61)(62).The nonzero relaxation times imply that the system requires time to transition from a non-equilibrium state to an equilibrium state.In other words, the dissipative fluxes Π, π µν , q µ , and ϕ µν do not undergo sudden transitions from nonzero to zero [161,179].As an example, we consider the general where Π 0 is constant, and the Green's function is defined as follows: The general solutions for π µν , q µ , and ϕ µν in Eqs.(59)(60)(61) share a structure similar to that of Eq. ( 63).Now, we assume that ζ∂ µ u µ jumps from nonzero to zero at time τ 0 .Due to the nonzero relaxation time τ Π , the solution (63) indicates that Π cannot instantaneously switch from a nonzero (non-equilibrium) value to zero (equilibrium).However, if τ Π = 0, the solution (63) reduces to Π = −ζ∂ µ u µ , and then Π undergoes sudden change from nonzero to zero and it thus causes acausality.Therefore, to obtain a physical theory, we introduce the nonzero relaxation times and treat Π, π µν , q µ , and ϕ µν as dynamical variables in Eqs.(59)(60)(61)(62).In principle, we can also consider the nonzero Σ λµν (1) in Eq. ( 13), which might involve corrections similar to the relaxation terms for q µ and ϕ µν .In this work, we concentrate on the simplest extension of the second-order terms and leave the more general discussion for future research.

HYDRODYNAMICS
In this section we analyze the causality and stability of the minimal causal spin hydrodynamics.We use the similar notations in Sec.III, i.e., for a physical quantity X, we use X (0) and δX to denote the X at the global equilibrium state and the small perturbations of the quantity X, respectively.We adopt the independent perturbations as δe, δu i , δS µν , δΠ, δπ ij , where δπ i i = 0 and δπ ij = δπ ji .
We first start from the spin hydrodynamics in the rest frame, i.e., u µ (0) = (1, 0).The conservation equations ∂ µ δΘ µν = 0 and ∂ λ δJ λµν = 0 with the constitutive equations (59 -62) read, where χ b , χ µν e , χ s , D s , D b , δϑ i , λ ′ , γ ′ , γ ⊥ , γ ∥ are defined in Eqs.(26,32) and we have used the spin evolution equation (3) to replace δq i and δϕ ij by δS µν , A. Zero modes for the spin hydrodynamics with zero viscous effects Following the conventional hydrodynamics, we consider a fluid with the dissipative terms q µ and ϕ µν only for simplicity, i.e., we remove Eqs.(67,68) and take δΠ = 0 and δπ ij = 0 in Eqs.(65,66,69,70).The detail of the calculation is shown in Appendix C 1.The causality condition requires The stability conditions give, The above conditions are derived from the small k and large k limits only.We can implement the Routh-Hurwitz criterion [168-170, 188, 192, 193] to prove that the condition ( 73) is sufficient and necessary for stability.More discussion can be found in Appendix C 2.
Interestingly, there exist zero modes, i.e., ω = 0 for all k, coming from Eq. ( 70) with vanishing δΠ, δπ ij .Generally, the zero modes in the linear mode analysis do not mean the perturbations are not decaying with time.It indicates that the nonlinear modes should be included in Eq. ( 70) if δΠ = δπ ij = 0. To continue our linear mode analysis, we need to set non-vanishing δΠ, δπ ij .
The det M 6 = 0 gives, which are non-propagating modes or non-hydrodynamic modes.
In k → 0 limit, the det M 4 = 0 and det M 5 = 0 give where Eq. ( 84) and Eq. ( 87) are doubly degenerate.In large k limit, we have where the expressions of these k-independent coefficients c 1,2,3,4,5 are shown in Appendix B.
Now, let us analyze the causality conditions.From Eqs. (90-95), we find that all modes in minimal causal spin hydrodynamics correspond to finite propagation speed since |ω/k| is bounded as k → +∞.Imposing Eq. ( 22) on the propagating modes in Eqs.(92)(93), the causality requires, where b 1,2 are defined in Appendix B. The causality conditions imply that the relaxation times τ q , τ π , τ Π , τ ϕ cannot be arbitrarily small, which is consistent with the discussion in Sec.
IV.We also notice that the Eq. ( 96) reduces to Eq. (C17) when we take a smooth limit

C. Non-trivial stability conditions in rest frame
The requirement of stability is non-trivial.Inserting Eq. ( 24) into Eqs.(82-95) yields, The stability condition λ ′ < 0 in Eq. ( 46) for the first order spin hydrodynamics becomes λ ′ < 2τ q in Eq. (97).When the relaxation time τ q is sufficiently large, the inequality λ ′ < 2τ q is satisfied, and then the previous unstable modes are removed.We also notice that the conditions (97, 98) agree with Eq. (C18) except the χ 0x e = 0.The strong constraint χ 0x e = 0 is released in this case.The satisfaction of the stability condition (98) relies on the specific equation of state governing S µν and ω µν .In Ref. [70], it was found that the stability condition (98) cannot be satisfied if δS µν ∼ T 2 δω µν [62,64].In more general cases, we can have where χ 1,2 are susceptibility corresponding to the S 0i and S ij in the rest frame.In this case, according to the definitions in Eqs.(26,32), the stability condition ( 98) is satisfied if Details can be found in Appendix D. Note that the parameters χ 1 and χ 2 strongly depend on the equation of state for S µν and ω µν .To determine the equation of state, we need the microscopic theories, and we will leave it for the future studies.
Another remarkable observation for the stability conditions is that there exist unstable modes at finite k.Eqs.(97,98,99) are the stability conditions in small k and large k limits only.We still need to study the Im ω in finite k region.One analytic method, named the Routh-Hurwitz criterion [168-170, 188, 192, 193], are usually implemented to study the sign of Im ω in finite k region.Unfortunately, det M 2 cannot be reduced to the form that Routh-Hurwitz criterion applies, thus, we analyze the behavior of Im ω numerically instead of the Routh-Hurwitz criterion.For a finite k, we find that Im ω can be negative, even if all the conditions (97, 98, 99) are satisfied.In Fig. 1

, we present an example to show that
Im ω can be negative for finite k.We choose the parameters as, It is straightforward to verify that the parameters in Eq. ( 102) satisfy the stability and causality constraints (22,23,24).We pick up three modes derived from det M 4 = 0. We observe that the Im ω at both small and large k limits are positive, while it becomes negative when kτ Π ∼ 0.5 and kτ Π ∼ 10.0, i.e., the modes are unstable in finite k region.
We comment on the unstable modes at finite k.The unstable modes in the minimal causal spin hydrodynamics are significantly different with those in the conventional hydrodynamics.
As discussed in Refs.[158,159,168,170,171,188], the stability conditions obtained in k → 0 and k → +∞ limits are sufficient to ensure the stability at any real k.However, it looks failed in minimal causal spin hydrodynamics.It implies that the conditions (97,98,99) are necessary but may not be sufficient.At last, it is still unclear whether the unstable modes at finite k indicate the fluid becomes unstable or not.
Figure 1.We plot the imaginary parts of ωτ Π as a function of kτ Π in three modes derived from det M 4 = 0.The parameters are chosen as in Eq. ( 102), which satisfy the causality and stability conditions Eqs. (22,23,24).The soild, dashed and dotted lines stand for three unstable modes.

D. Causality and stability analysis for extended q µ and ϕ µν
In principle, we can introduce the coupling terms for Π, π µν , q µ , and ϕ µν on the righthand side of Eqs.(59)(60).These terms will alter the linearized hydrodynamic equations and then the causality and stability conditions can be changed.In the current work, as the first step, we focus on the simplest coupling between q µ and ϕ µν , where g 1,2 are new transport coefficients describing the coupling between q µ and ϕ µν .For more general coupling terms, one can refer to Ref. [74].
Following the same method, Eqs.(65,66) become, We first consider the cases without viscous effects.The causality condition (72) becomes where m is defined in Eq. (B12).While the stability condition ( 73) is changed to Details can be found in Appendix C 3. We implement the Routh-Hurwitz criterion [168-170, 188, 192, 193] again to prove that these conditions ( 108) are sufficient necessary for stability.Details for the proof can be found in Appendix C 4. Similar to Sec.V A, we still find the zero modes coming from Eq. ( 70).
Therefore, we need to consider the non-vanishing viscous effects.Now, the sub-matrix M 5 shown in Eq. ( 79) is replaced with while other sub-matrices are unaffected by g 1 and g 2 .The Eqs. (93)(94)(95) become, where the definitions of f, f ′ , c 6 , and c 7 are defined in Appendix B.
As a brief summary, the extended q µ and ϕ µν can modify the causality and stability conditions, but cannot remove the zero modes when we turn off other dissipative effects.
The unstable modes at finite k cannot be cured by the extended q µ and ϕ µν .

E. Causality and stability in moving frames
Let us briefly discuss the causality and stability of the minimal causal spin hydrodynamics in moving frames.
For the causality in a moving frame, we refer to the studies in Refs.[168,176,177].
The authors in Refs.[168,176,177] have studied the dispersion relations at large k limit in moving frames and demonstrate that the system is causal in moving frames if it is causal in the rest frame.Thus, the minimal causal spin hydrodynamics is causal in moving frames when the causality condition (96) in the rest frame are satisfied.
For the stability, it has also been proved that if a causal theory is unstable in the rest frame, then it is also unstable in moving frames (see Theorem 2 of Ref. [194]).We now apply this theorem to the minimal causal spin hydrodynamics.If the equation of state gives δω µν = χ 1 δS µν with constant χ 1 , the minimal causal spin hydrodynamics will be unstable The green solid, red dashed, blue dash-dotted and brown dotted lines stand for the results with (g 1 /τ Π , g 2 /τ Π ) = (0.0, 0.0), (2.0, 0.1), (6.0, 0.1), (6.0, 0.05).Other parameters are also chosen as in Eq. (102).
in moving frames since it has unstable modes in the rest frame.For more general cases, the stability of the theory in both moving frames and the rest frame depends on the equation of state for S µν and ω µν .In summary, the minimal causal spin hydrodynamics is causal in any reference frame when Eq. ( 96) is fulfilled.Hence, we have solved the problem of acausality by introducing the minimal causal spin hydrodynamics.However, the stability of minimal causal spin hydrodynamics remains unclear.Our findings indicate that the validity of the stability condition ( 98) is highly contingent upon the equation of state governing spin density and spin chemical potential.Moreover, we also find that the stability conditions (97,98,99) obtained at k → 0 and k → +∞ are necessary but not sufficient.

VI. CONCLUSION
In this work, we investigate the linear causality and stability of the spin hydrodynamics proposed in Refs.[45,48].
In linear modes analysis, we consider perturbations to the spin hydrodynamics near the static equilibrium.We obtain the dispersion relations ω = ω(k) and analyze the all possible modes.The results show the stability condition ( 46) cannot be fulfilled.Moreover, the value of |ω/k| in Eqs.(48)(49)(50) is unbounded, which violates the causality condition (23).
In Refs.[45,70,71], the expression of q µ are modified by using the equation of motion for the fluid.We emphasize that the first order spin hydrodynamics in Refs.[45,70,71] are still acausal since one mode shown in Eq. ( 56) breaks the causality condition (23).We conclude that the spin hydrodynamics in the first order of gradient expansion are acausal and unstable.
We then follow the basic idea in MIS, BRSSS, and DNMR theories and consider the minimal causal spin hydrodynamics.The constitutive equations (16)(17)(18)(19)(20) in a minimal extended causal spin hydrodynamics are replaced by Eqs.(59)(60)(61)(62).One can view it as a natural extension of the first order spin hydrodynamics or a simplified version of the complete second order spin hydrodynamics [74].We investigate the causality and stability for this minimal causal spin hydrodynamics.We analyze the causality and stability for dissipative fluids with q µ and ϕ µν only and find the zero modes in the linear modes analysis.This suggests that linear mode analysis is inadequate in this case.Therefore, we consider dissipative spin fluids with shear viscous tensor and bulk viscous pressure.
For causality, we find that the modes with infinite speed disappear and all modes are causal in the rest frame if the conditions in Eq. ( 96) are fulfilled.Following the statement in Refs.[168,176,177], we comment that the minimal causal spin hydrodynamics are causal in any reference frame when the conditions (96) are fulfilled.
For the stability, although we obtain the stability conditions in Eqs.(97,98,99) from the constraints in the k → 0 and k → +∞ limits, the stability of the theory in both moving frames and the rest frame remains unclear.Two kinds of problems can lead to instabilities.The first one is related to stability condition (98).Interestingly, we prove that the coefficients D s , D b do not obey the stability condition (98) if the equation of state S µν ∼ T 2 ω µν is adopted.In more general cases, the fulfillment of the stability condition (98) hinges on the specific equations of state.One has to assess the condition (98) on a caseby-case basis.Surprisingly, different with the conventional hydrodynamics, we find that the stability condition (24) breaks at finite k as shown in Fig. 1.It implies that the conditions (97,98,99) are necessary but may not be sufficient.
We also considered the extended q µ and ϕ µν , in which the q µ and ϕ µν are coupled in the second order constitutive equations.The causality and stability conditions are modified in this case.However, in dissipative fluids with q µ and ϕ µν only the zero modes cannot be removed.The unstable modes at finite wavelength are still there.
We conclude that the spin hydrodynamics in the first order of gradient expansion, proposed in Refs.[45,48], are always acausal and unstable.The minimal causal extension of it makes the theory be causal in the sense of Eqs.(22,23).However, the linear stability of the minimal causal spin hydrodynamics remains unclear.The studies beyond the linear modes analysis may provide us a better and clear answer to the problem of stability.The coefficients introduced in Eqs.(90)(91)(92)(93)(94)(95) are defined as follows,

−4i(ωλχ
where The coefficients used in Eqs.(110)(111)(112) are given by f = mτ π + 8γ ⊥ τ q τ ϕ , (B8) where Appendix C: Causality and stability of the minimal causal spin hydrodynamics with q µ and ϕ µν only In this appendix, we study the causality and stability of minimal causal spin hydrodynamics, considering only q µ and ϕ µν .We will discuss two different cases.We name the system in which q µ and ϕ µν are not coupled, as depicted in Eqs.(59,60) as case I. Conversely, we name the system q µ and ϕ µν are coupled, as described by Eqs.(103,104) as case II.
1. Analysis for the case I We let δΠ = δπ ij = 0 in Eqs.(65,66,69,70) and remove Eqs.(67,68).Then we substitute the plane wave solutions Eq. ( 33) into Eqs.(65,66,69,70) and derive where δ X′ 2 and M ′ 2 are given by and with The off-diagonal matrices A ′ 4,5,6 are given by The dispersion relations ω = ω(k) are derived from We find that there exist two zero modes coming from the equation det M ′ 5 = 0. Now, let us focus on the nonzero modes.The det M ′ 6 = 0 gives two non-hydrodynamic modes From det M ′ 4 = 0 and det M ′ 5 = 0, we obtain the dispersion relation in small k limit, and, in large k limit, The causality conditions (22,23) require, which implies that the relaxation times τ q , τ ϕ cannot be arbitrarily small.It is consistent with the discussion in Sec.IV.
The stability condition (24) leads to where χ 0x e = 0 comes from the stability of the sound mode (C10).Although the conditions in Eq. (C18) are derived from the small k and large k limits only, we can implement the Routh-Hurwitz criterion [168-170, 188, 192, 193] to prove that the conditions (C18) are sufficient and necessary for stability, i.e., if (C18) are satisfied, then Im ω > 0 for all k ̸ = 0.
Details for the proof are given below.

Condition (C18) is sufficient and necessary for the stability
As mentioned, we have derived the stability condition (C18) from the linear modes analysis in small and large k limits only.Now, we implement the Routh-Hurwitz criterion [168-170, 188, 192, 193] to prove that the condition (C18) guarantees stability for all real nonzero k.
We only need to prove that the nonzero modes derived from det M ′ 4 = 0 and det M ′ 5 = 0 satisfy Im ω > 0 for all k.First, we discuss the modes coming from the det with We redefine ω = −i∆ and rewrite Eq. (C19) as, Notice that the coefficients a 0,1,2,3,4 are pure real.According to the Routh-Hurwitz criterion [168-170, 188, 192, 193], the stability condition (24), i.e., Im ω > 0 or Re ∆ < 0, is fulfilled for all nonzero k if and only if When the conditions in Eq. (C18) are fulfilled, the first inequality a i > 0 are automatically satisfied.The second inequality can be expressed as λ ′ = 2λ/[e (0) + p (0) ] > 0, which has already been guaranteed by entropy principle (21).Thus the modes derived from det M ′ 4 = 0 are stable for all k if condition (C18) is satisfied.
Second, we consider the nonzero modes derived from det M where Similarly, the Routh-Hurwitz criterion provides the necessary and sufficient conditions for Im ω > 0 in Eq. (C23), Each a ′ i > 0 does not give new constraints for stability.We now show that the second inequality holds for all k if the conditions in Eq. (C18) are fulfilled.Define a new function Since τ q > λ ′ /2 in Eq. (C18), we have H(D s , k) > 0 for any k and any D s > 0.

Analysis for case II
We now consider a more general case where q µ and ϕ µν are coupled as shown in Eqs.(103,104).Here we consider the q µ and ϕ µν only and neglect other dissipative terms for simplicity.In this case, M ′ 5 in Eq. (C5) should be replaced with while the matrix M ′ 4 is the same as before.The dispersion relations in Eq. (C15-C16) become where m = 2g 1 g 2 + 8g 1 γ ′ + g 2 λ ′ + 8γ ′ τ q .We also notice that the zero modes mentioned before cannot be solved by introducing the coupling between q µ and ϕ µν .

Condition (C39) is sufficient and necessary for the stability
Let us now prove that the condition (C39) ensures Im ω > 0 for all nonzero real k.
Consider the nonzero modes derived from det M ′ 5 = 0.The det M ′ 5 = 0 gives where The necessary and sufficient conditions for Im ω > 0 in Eq. (C40) are The first conditions are automatically satisfied when we have the constraints for stability.
Then we need to analyze whether Eq. ( C43) is satisfied under the existing constraints.
there exist unstable modes, although the analytic solutions in Refs.[62,64] do not rely on it.For general cases where χ 2 ̸ = 0, whether the stability condition (98) D s > 0, D b < 0 is satisfied depends on χ 1 , χ 2 , which relates with the equation of state for S µν and ω µν .To determine the value of χ 1 , χ 2 , further investigations should be done from the microscopic theory.

Figure 2 .
Figure 2. Imaginary parts of ωτ Π as a function of kτ Π in one mode derived from det M 5 = 0.