Spin liquid mediated RKKY interaction

We propose an RKKY-type interaction that is mediated by a spin liquid. If a spin liquid exists such an interaction could leave a fingerprint by ordering underlying localised moments such as nuclear spins. This interaction has a unique phenomenology that is distinct from the RKKY interaction found in fermionic systems; most notably the lack of a Fermi surface and absence of the requirement for itinerant electrons, since most spin liquids are insulators. We demonstrate that the interaction is predominately shaped by the lattice symmetries of the underlying spin liquid. As a working example we investigate the possible ordering of nuclear spins that interact through an underlying lattice of the two-dimensional spin-1/2 kagome antiferromagnet (KHAF), although the treatment remains general and can be extended to other spin liquids and dimensions. We find that several different nuclear spin orderings minimise the RKKY-type energy induced by the KHAF but are unstable due to a zero-energy flat magnon band in linear spin-wave theory. Despite this we show that a small magnetic field is able to gap out this magnon spectrum resulting in an intricate nuclear magnetism.

In the main text we motivate the use of a Luttinger-Tisza expansion of the spins reading, I a (q) = ∑ ν W ν (q)U a,ν (q), where the W ν (q) are (orthogonal) directions in spin-space. This allows us to find the minimum possible energy of the slRKKY interaction subject on average spin-length within the unit cell, which is also know as the "weak constraint" on spin-length.
Inserting this Luttinger-Tisza expansion of spins in terms of the eigenvectors of the susceptibility matrix χ ab (q) into the slRKKY Hamiltonian gives Using the fact that U a,ν q is an eigenvector of the static susceptibility matrix with eigenvalue λ ν (q) gives Because the eigenvectors U a,ν q are orthonormal this then reduces to The weak constraint on average spin length in Fourier space reads ∑ q,a I a q · I a −q = NI 2 means that we require W ν · W ν = I 2 and so the minimum ground state energy with this weak constraint applied is It is important to note the key difference between the slRKKY interaction and nearest neighbour classical KAFM is that the former is extended and therefore sensitive to the alignment of the spins on the same sublattice because the susceptibility matrix includes diagonal elements coupling these spins, whereas the latter has fewer spin constraints and therefore a much larger ground state degeneracy. 6 In particular, for slRKKY on the KAFM, once the spins on one plaquette are fixed the entire nuclear magnetic order is itself fixed since the Luttinger-Tisza energy minimum can only be achieved by an ordering over the whole lattice.

Multiple-Q ordering
In addition to the minimum energy configurations discussed in the main text characterised bya single Q Q Q ordering vector, a state mixing different Q Q Q values is also possible. Due to the symmetry of the susceptibility and the fact that the Q Q Q = 0 ordering is doubly degenerate the slRKKY energy is also minimised for the multiple-Q Q Q state: Such an ordering is non-coplanar and contains to two Q Q Qs, namely: Q Q Q = 0 in the spinẑ z z direction and Q Q Q y = Q yŷ y y in thex x x −ŷ y y plane, where the x, y, z directions in spin space can be chosen arbitrarily.
To see why such an state is allowed to exist it is easiest to write this multiple-Q Q Q state in Fourier space We see that this is a true ground state of the slRKKY interaction on kagome because: 1) The state satisfies the hard-constraint on spin-length, that is the constraint |I i | = I for each spin -in contrast to the weak spin constraint where spin length is only imposed on average; 2) (1, −1/2, −1/2) T is an eigenvector of the susceptibility at Q Q Q = 0 with minimum eigenvalue; 3) (0, 1, −1) T is an eigenvector of the susceptibility at Q Q Q = Q yŷ y y with minimum eigenvalue. Point 3) is the key to understanding why such a state is allowed. The susceptibility of the KAFM is doubly degenerate at Q Q Q = 0 with an eigen-space spanned by (1, −1/2, −1/2) T and (0, 1, −1) T . This means we have the possibility to use two

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Luttinger-Tisza vectors to make a ground state. However, by symmetry, the vectors (0, 1, −1) T , (1, 0, −1) T , and (1, −1, 0) T which are in the minimum eigen-basis at Q Q Q = 0 are the eigenvectors along the direction ±ŷ y y, the direction ± √ 3/2x x x ∓ 1/2ŷ y y, and the direction ∓ √ 3/2x x x ∓ 1/2ŷ y y, respectively. This means that we can start with a pure Q Q Q = 0 state and then move the component corresponding to one of the vectors (0, 1, −1) T , (1, 0, −1) T , or (1, −1, 0) T along Q Q Q parallel to the direction in which it remains a lowest eigenvector and the state will remain a ground state.
This multiple-Q Q Q state is therefore only possible due to a combination of features of the kagome lattice and the KAFM susceptibility: 1) One of the vectors at Q Q Q = 0 does not affect one of the spins in the unit cell, 2) the susceptibility at Q Q Q = 0 is degenerate, 3) Q Q Q = 0, on each sublattice, only uses up one of three available orthogonal directions in spin space, and 4) the symmetry of kagome means these vectors remain the minimum eigenvalue vector along the high symmetry directions listed above. Since √ 3 × √ 3 order does not satisfy 1, 2, or 3, Q Q Q = 0 order possess no further Luttinger-Tisza vectors, and there are no further Q Q Q vectors that minimise the slRKKY energy we see that it is likely the multi-Q state and symmetry partners are the only possible multiple-Q Q Q states of slRKKY.

Holstein-Primakoff
As outlined in the main text: We can chose a basis within the unit cell with respect to the directionẑ z z defined by a central spin and with the other two spins defined within this basis as is the rotation matrix about spin y-axis and θ i is the angle between the first spin and the spin on site i.
Since the nuclear spins are large we can take a 1/I Holstein-Primakoff expansion to estimate the nuclear spin-wave spectrum, 7 which in Fourier space reads where a †a q , a a q are the bosonic creation and annihilation operators, respectively, for magnons at momentum q on sublattice a about the mean field equilibrium ordering vector Q n .
Inserting the Holstein-Primakoff expansion into the slRKKY Hamiltonian, to zeroth-order in bosonic operators -order I 2gives E min the minimal ground state energy where the second equality follows from the fact that we chose the angles θ a for our equilibrium ordering such that the matrix NI 2 cos(θ a − θ b ) = I a (Q n ) · I b (−Q n ) outputs the minimum eigenvalue of χ ab (Q n , 0). The next order of the expansion comes from mixed x and z spin components. These will equal zero if The construction of cos(θ a − θ b ) to minimise the energy of the slRKKY Hamiltonian means that Eq. (24) must hold or else one could rotate and achieve a lower minimum than that set by θ a . Conversely the the condition at the minima that the exchange interaction has equal magnitude on each site and the above condition in Eq. (24) -that the effective magnetic field on each site vanishes -set limits on the possible regular periodic orderings we could find. Given these two conditions the mutual angles between the three spins in the Fourier-space unit cell -that is, before translation to real space -must either be collinear, which automatically satisfies Eq. (24), or have angles of 60 • or 120 • between them such that the terms from the susceptibility matrix cancel. As such the only orderings that can satisfy both conditions are those orderings found in the main text.
In the presence of a magnetic field, unlike in the free field case, the linear terms no longer cancel because the additional Zeeman term -after rotation on each site, as above -contains components of I x 2 , I x 3 from the spins that are not aligned with the field. This causes linear terms in the spin-wave operators,

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These terms represent the fact that the effective magnetic field from the RKKY and the external field now compete and the nuclear spins will cant upwards slightly from the true 120 • from the spin parallel to the field, further increasing magnetic field would eventually form a ferromagnetic arrangement. For our purposes we assume that fields are sufficiently small that the order can be approximated by the original mean field RKKY result.

Spin-wave diagonalisation
We wish to solve the general problem of finding the transformation of N species of bosons that diagonalises a block-diagonal spin-wave Hamiltonian in terms of new operators d(q) and d † (q) to a Hamiltonian of the form where [∆ ∆ ∆] ab = ω a δ ab is a diagonal matrix of the spin wave spectra and the transformation matrix D between operators with N × N matrices u and v. To preserve bosonic commutation relations these must satisfy and where 1 N is the N × N identity matrix. Eq. (28) and (29) require that which enables us to write the diagonalization problem as an eigenvalue problem of the form which means that to find the spin-wave spectra contained within ∆ ∆ ∆ we must find the eigenvalues of the left hand side of Eq. (31).

Zero-energy spin-wave band
To see that the the spin-waves of the spin liquid RKKY interaction on kagome have a zero-energy flat band for all q we use the fact that the determinant of the matrix in Eq. (31) is the product of the eigenvalues and so the matrix must have zero determinant -and hence be non-invertible -if it has a zero eigenvalue. Since a zero determinant is unchanged by addition of rows and columns; subtracting the second row from the first and then subtracting the first column from the second leads to, which is now in the form of a block triangular matrix. We can therefore write the determinant as From the definitions of Γ Γ Γ and Λ Λ Λ the first block matrix in terms of the susceptibility reads where the λ min in the final line follows from the choice of the spin orientations θ i such that we achieve the mean field minimum energy. The matrix in Eq. (33) therefore has a zero-eigenvalue from the cancellation of the diagonal of the first term and the λ min flat band of the susceptibility, χ ab (Q n + q). From this it follows that det {Γ Γ Γ − Λ Λ Λ} = 0 and hence the original Bogoliubov matrix of spin-wave excitations in Eq. (32) as has a zero-energy spectrum for all q.

5/7 7 Flat linear order spin wave band from hard constraint on spin length
In order to understand the origin of the flat-band it is instructive to consider the full classical equation of motion of the spins such that we treat them as classical vectors I i,m (where i labels the unit cell and m the sublattice). The dynamics of these spins are set by the slRKKY Hamiltonian and they have the (classical) equation of motion Which in momentum space reads: If the classical nuclear spins are ordered then we can expand them about the mean field ordering direction such that I i,m = I i,m + δ I i,m or in Fourier space I q,m = I Q ord ,m δ q,Q ord + δ I q,m , which at linear order in the expansion will be a fully classical equivalent to the Holstein-Primakoff expansion. Inserting this into Eq. (35) we obtain to linear order the estimate of the classical equation of motion for the spin in a unit cell where R R R(θ j ) are defined in the main text. From this we see that the matrix governing the classical motion of the spins will also have a zero eigenvalue since the constant eigenvalue of the two matrices will cancel. To understand why in a system where the energy minima result entirely from the constraint on spin length necessarily has a flat band at linear order (but can be dispersive at higher orders) in spin length, consider the hard constraint for classical spins in real space I i,m · I i,m = I. This constraint can be written in terms of spins fluctuating around some equilibrium position I and requires | I + δ I| 2 = I 2 . After expanding this reads 2δ I · I + |δ I| 2 = I 2 − | I | 2 ≥ 0, where in the last line we obtain the inequality because the component of spin in the direction of position | I | must be smaller or the same size as I for a fixed spin length. Spin waves are transverse and so 2δ I · I = 0. We see therefore that for finite fluctuations the inequality can only be satisfied -and hence the hard constraint on spin length satisfied -by high orders in δ I. As such only higher than linear order fluctuations can enforce the true constraint on spin length and, if the system solely achieves its energy minima because of this constraint, only these higher order terms will capture the increase in energy resulting from the ability of the spins to fluctuate. The above argument holds not only for regular periodic orders, but also for the multiple-Q Q Q states.

Effect of nuclear (Overhauser) field on the KHAF
As mentioned in the main text, after ordering of the nuclear spins there exists an Overhauser field of the nuclear moments on the spin liquid with the nuclear spins form an effective magnetic field. In principle this field could have a destabilizing effect on the spin liquid. Here we discuss why this is unlikely to impact our conclusions relating to nuclear orderings. Although such a back action may be of interest for a discussion of the KHAF spin-liquid state in real materials at very low temepratures, the focus of this work was on orderings of nuclear moments.
There are two effects from the reciprocal action that could qualitatively affect the physics of the nuclear moments: 1) a significant change in the susceptibility or 2) the loss of the spin liquid's dynamics.
1) As was shown in detail in the previous sections and main text, the coincidence of the symmetries of the lattice of the electron spins and nuclear spins allows only certain high symmetry positions within reciprocal space to minimise the slRKKY energy. The coincidence also means that any feedback between the two species of spins must maintain the symmetries of the susceptibility. Since the interactions leading to the spin liquid itself are constrained to the same symmetries both J and A act on the same scales. But since A I J the effect from the Overhauser field is weak. 2) Other than the susceptibility, the other key consideration in the calculation of nuclear orders is that the spin liquid remains dynamic -at least down to temperatures at which the nuclear spins order. This consideration is in fact related to the question of the precise nature of the spin liquid, which for instance in the case of KHAF is still undecided despite extensive research over many years. However, as the slRKKY physics is dictated by its symmetries, only the still unanswered question whether or not the ground state of the S = 1/2 KAFM is gapped or gapless can be of importance for the effect of the Overhauser field. In addition the question has two facets, whether the Overhauser field plays a role in principle or whether the reciprocal action matters in practice at the relevant temperatures.
We address first the question in principle: For a gapped spin liquid, the Overhauser field could enforce or compete against the gap. But even a small gap such as ∆ ∼ 0.05 − 0.1J 8,9 is still very large in comparison to the Overhauser field's back action 6/7 as A I ∆. This makes the back action highly unlikely to destabilise the spin liquid. For a gapless spin liquid 10,11 the question is if the Overhauser field could freeze the dynamics by opening a gap itself. It is not possible to conclusively say whether the Overhauser field will open a gap, since even the nature of the kagome spin liquid without any back action is not fully resolved, a gap could open up that is capable of modifying the temperature scales discussed in the remainder of the manuscript. However, as argued in 1) even if a gap is opened up it is unlikely to freeze the dynamics of the spin liquid since the Overhauser field acts on the same underlying lattice as the kagome spin liquid so that J and A act with the same weight and, since A I J, the feedback is not likely to open a sizeable gap capable of destroying the spin liquid.
Practically, however, it is important only that the spin liquid remains dynamic at temperatures corresponding to those which the nuclear spins have already ordered, T N , even if a small gap does open that causes a loss of the spin liquid dynamics. If the freezing temperature of the spin liquid is smaller than T N , then any opening of a gap would have no significant impact on the nuclear order. As no freezing of the spin liquid has been observed in kagome systems down to 40 − 50 mK 12,13 it is quite safe to assume that realistically sufficient dynamics can be maintained at the nuclear ordering temperatures required.