Collective spinon wave in a magnetized U(1) spin liquid

Leon Balents1 and Oleg A. Starykh2 1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA 2Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA (Dated: January 8, 2020)

We study the transverse dynamical spin susceptibility of the two dimensional U(1) spinon Fermi surface spin liquid in a small applied Zeeman field. We show that both short-range interactions, present in a generic Fermi liquid, as well as gauge fluctuations, characteristic of the U(1) spin liquid, qualitatively change the result based on the frequently assumed non-interacting spinon approxima- tion. The short-range part of the interaction leads to a new collective mode: a “spinon spin wave” which splits off from the two-spinon continuum at small momentum and disperses downward. Gauge fluctuations renormalize the susceptibility, providing non-zero power law weight in the region outside the spinon continuum and giving the spin wave a finite lifetime, which scales as momentum squared. We also study the effect of Dzyaloshinskii-Moriya anisotropy on the zero momentum susceptibility, which is measured in spin resonance (ESR), and obtain a resonance linewidth linear in tem- perature and varying as B2/3 with magnetic field B at low temperature. Our results form the basis for a theory of inelastic neutrons scattering, ESR, and resonant inelastic x-ray scattering (RIXS) studies of this state.

The search for the enigmatic spin liquid state has ω switched into high gear in recent years. Dramatic the- oretical (Kitaev model [1,2] and spin liquid in tri- angular lattice antiferromagnet [3]) and experimental ��������-���� ��������� (YbMgGaO [4,5] and α-RuCl [6]) developments leave 2ωB+2uM 4 3 2ω no doubt of the eventual success of this enterprise. To B push this to the next stage, it is incumbent upon the community to identify specific experimental signatures ����� ����� �������� ����� that evince the unique aspects of these states. In this q paper, we focus on the two dimensional U(1) quantum 0 spin liquid (QSL) with a spinon Fermi surface. This is a priori the most exotic two dimensional QSL state, and FIG. 1. Magnetic excitation spectrum of an interacting U(1) yet one which has repeatedly been advocated for in both spin liquid with spinon Fermi surface. Wedge-shaped (blue) region, bounded by dashed lines, denotes spinon particle-hole theory [7–11] and experiment [5, 12–14]. Specifically, we 0 continuum inside which =χ±(q, ω) 6= 0. Collective spinon study the dynamical susceptibility of the q-component of spin wave, which is promoted by short-ranged repulsive inter- a the spin operator Sq (a = x, y, z) action u, is shown by the bold black line. The linewidth of this transverse spin wave is determined by the gauge fluctu- Z ∞ 1 + − iωt ations, which produce finite =χ±(q, ω) 6= 0 outside the non- X±(q, ω) = −i dth[Sq (t),S−q(0)]ie , (1) interacting spinon continuum, see (11). 0 which is an extremely information-rich quantity, and is accessible through inelastic neutron scattering [15], ESR momentum. Our analysis reveals the full structure in this [16, 17], and RIXS [18]. The of triplet regime beyond the mean field approximation. Notably, excitations into pairs of spinons is a fundamental aspect we find that interactions between spinons qualitatively of a QSL, and is expected to manifest in X± as two- modify the result from the mean-field form, introducing particle continuum spectral weight [2, 19, 20], a surpris- a new collective mode – a “spinon spin wave” – and mod- arXiv:1904.02117v2 [cond-mat.str-el] 7 Jan 2020 ing feature which appears more characteristic of a weakly ifying the spectral weight significantly. correlated metal than a strongly correlated Mott insula- We recapitulate the derivation of the theory of the tor. spinon Fermi surface phase [22, 23]. One introduces In a mean-field treatment in which the spinons are Abrikosov fermions by rewriting the spin operator 1 † † approximated as non-interacting fermions, this contin- Si = 2 ciασαβciβ, where ciα, ciα are canonical fermionic uum has a characteristic shape at small frequency and spinors on site i with spin-1/2 index α (repeated spin wavevector in the presence of an applied Zeeman mag- indices are summed). This is a faithful representation † netic field, as discussed in [21]. In particular, there is provided the constraint ciαciα = 1 is imposed – this con- non-zero spectral weight in a wedge-shaped region which straint induces a gauge symmetry. In a path integral rep- terminates at a single point along the energy axis at zero resentation, the constraint is enforced by a Lagrange mul- 2 tiplier Ai0, which takes the role of the time-component of determined structure factor at zero momentum. Specif- a gauge field, i.e. scalar potential. Microscopic exchange ically, the Larmor/Kohn theorem [25] dictates that the 00 interactions, which are quadratic in spins, and are there- only response at q = 0, X± = −2Mδ(ω − 2ωB), where fore quartic in fermions, are decoupled to introduce new M = (¯n↑ − n¯↓)/2 is the magnetization and ωB is the link fields whose phases act as the spatial components of spinon Zeeman energy. For free fermions, the delta func- the corresponding gauge fields A, i.e. the vector poten- tion is precisely at the corner of the spinon particle-hole tial. continuum (also known as the two-spinon continuum). To describe the universal low energy physics, it is ap- However, the contact exchange interaction shifts up the † particle-hole continuum, at small momentum q, away propriate to consider “coarse-grained” fields ψα, ψα de- scending from the microscopic ones, and include the from the Zeeman energy 2ωB to 2ωB + 2uM. This is symmetry-allowed Maxwell terms for the U(1) gauge seen by the trivial Hartree self-energy field. Furthermore, due to the finite density of states at the spinon Fermi surface, the longitudinal scalar po- −σ tential is screened and the time component A can then 0 Σ = = un¯ = −uMσ + un/¯ 2, (3) be integrated out to mediate a short-range repulsive in- σ −σ teraction u between like charges. Therefore we consider where we use a zig-zag line to diagrammatically repre- the Euclidean action S = S + S + S , where [22–24] ψ A u sent the local u interaction, σ =↑= 1 and σ =↓= −1, Z  1  andn ¯ is the expectation value of spin-σ spinon density S = d3x ψ† ∂ − µ − (∇ − iA)2 − ω σ3 ψ, σ ψ τ 2m r B in the presence of magnetic field. Consequently, for the Z 3 Larmor theorem to be obeyed, there must be a collective d q 1 2 2 SA = γ|ωn|/q + χq |A(q)| , transverse spin mode at small momenta. (2π)3 2 Z This collective spin mode is most conveniently de- 3 † † Su = d x u ψ↑ψ↑ψ↓ψ↓. (2) scribed by the Random Phase Approximation (RPA), which corresponds to a standard resummation of particle- Here x = (τ, x) is the space-time coordinate, q = (ωn, q) hole ladder diagrams [26]. For the particular case of a is the three-momentum, ψα is a two-component spinor, momentum-independent contact interaction, one has with spin indices α, β =↑, ↓ that are suppressed when possible, ωB describes static magnetic field B = Bzˆ ↑ ↑ ↑ and includes the g-factor as well as the Bohr magneton. X±(q, iωn) = + + + ··· The gauge dynamics is derived in the Coulomb gauge ↓ ↓ ↓ ∇·A = 0 with A(q) = izˆ×qˆA(q). The gauge action SA is χ±(q, iωn) generated by spinons and γ = 2¯n/kF and χ = 1/(24πm) = , (4) represent Landau damping and diamagnetic susceptibil- 1 + uχ±(q, iωn) ity of non-interacting spinon gas, correspondingly (m is where the fermion lines correspond to the spinon Green’s the spinon mass,n ¯ is the spinon density and kF is the functions including the Hartree shift (3), and in this ap- Fermi momentum of non-magnetized system). 0 proximation χ±(q, iωn) = χ±(q, iωn) is the bare suscep- We proceed with the assumption of SU(2) symmetry, tibility bubble, calculated using these functions. We will a good first approximation for many spin liquid materi- however use the second line in Eq. (4) to later define the als, and address the effect of its violations in the latter RPA approximation even when gauge field corrections part of the paper. Previous investigations focused on the (but not the local interaction u) are included in χ±. For transverse vector potential A, which is not screened but the moment, we simply evaluate the bare susceptibility, Landau damped, and hence induces exotic non-Fermi- liquid physics. For example, one finds a self-energy vary- 0 1 X 1 χ±(q, iωn) = ing with frequency as ω2/3, and a singular contribution βV ikn − k + ωB − un¯↓ kn,k 2/3 to the specific heat cv ∼ T [22, 23]. However, notably, 1 the transverse gauge field has negligible effects on the × . (5) ik + iω −  − ω − un¯ hydrodynamic long-wavelength collective response [24]. n n k+q B ↑ Here, we instead focus on the short-range repulsion u, Here ωn, kn are bosonic and fermionic Matsubara fre- which produces an exchange field that dramatically al- quencies, respectively. A simple calculation, followed by ters the behavior in the presence of an external Zeeman analytical continuation iωn → ω + i0, gives magnetic field giving rise to finite magnetization. Gauge 2Msign(ω − 2ω − 2uM) fluctuations play a subsidiary role which we also include. <χ0 (q, ω) = B , ± p 2 2 2 An important constraint follows purely from symme- (ω − 2ωB − 2uM) − vF q −2M try. Provided the Hamiltonian in zero magnetic field has =χ0 (q, ω) = , (6) ± p 2 2 2 SU(2) symmetry, a Zeeman magnetic field leads to a fully vF q − (ω − 2ωB − 2uM) 3 where square-roots are defined when their arguments therefore critical to understand the effect of the gauge are positive. The real/imaginary spin susceptibility de- interactions upon the dynamical susceptibility. To this scribes domains outside/inside two-spinon continuum in end, we consider the dressing of the particle-hole bubble the (q, ω) plane, Fig.1, correspondingly. At q = 0 χ0 by gauge propagators. Guided by the above thinking, we expect that it is sufficient to consider all diagrams 0 2M with a single gauge propagator (denoted by wavy line). χ±(q = 0, ω) = , (7) ω − 2ωB − 2uM + i0 ↑ ↑ ↑ and therefore =χ0 (q = 0, ω) ∼ δ(ω − 2ω − 2uM): the ± B χ1 (q,iω ) = + + (10) position of the two-spinon continuum is renormalized by ± n the interaction shift. However, inserting (7) in the RPA ↓ ↓ ↓ formula (4) one finds that the RPA successfully recovers Larmor theorem at zero momentum for the interacting Calculations described in [37] lead to SU(2)-invariant system, √ 3γ1/3k q2ω7/3 = χ1(q, ω) = − F . (11) 2 4/3 4 2M 56π χ (ω − 2˜ωB) X (q = 0, ω) = . (8) ± ω − 2ω + i0 B That is, the dressed susceptibility has a non-zero imagi- Therefore the contribution at q = 0 is solely from the nary part in the previously kinematically forbidden re- collective mode, with no spectral weight from the con- gion outside the spinon particle-hole continuum, see tinuum at 2ωB + 2uM. Dispersion of the collective Fig.1. This is a new continuum weight. However, spin mode is obtained with the help of (6) and =X± = the weight in this new continuum contribution vanishes 0 0 2 0 2 =χ±/[(1 + u<χ±) + (u=χ±) ], quadratically in momentum as q = 0 is approached. This is an important check on the calculations, since the Lar- q ω (q) = 2ω + 2uM − 4u2M 2 + v2 q2. (9) mor theorem still applies to the full theory (2) with the swave B F gauge field, which implies that precisely at zero momen- tum, there can be no new contributions. Similar to Kim For small q  uM/vF the collective mode is dispersing 2 et al. [24], who considered diagrams for the density cor- downward quadratically ω ≈ 2ωB −(vF q) /(4uM), while in the opposite limit q  uM/v it approaches the low relations and optical conductivity, this result relies on F important cancellations between self energy (first two di- boundary of the spinon continuum, ω ≈ 2ωB + 2uM − v q. Retaining quadratic in q terms in (5) will lead to agrams) and vertex corrections (last diagram), which are F needed to obtain this proper behavior of χ1. the termination of the collective mode at some qmax at which the spin wave enters the two-spinon continuum. Within the RPA approximation of Eq. (4), but now 0 1 2 with χ± = χ± + χ±, we see that the q dependence of This physics is not unique to spin liquids but applies to 1 =χ± is sufficient to ensure that the width (in energy) paramagnetic metals. Historically, this spin wave mode of the collective spin mode becomes narrow compared was predicted by Silin in 1958 for non-ferromagnetic met- to its frequency at small momentum: this is the stan- als within Landau Fermi liquid theory [27–30], and ob- dard criteria for sharpness and observability of a collec- served via conduction electron spin resonance (CESR) in tive excitation. The real part of χ1, derived in Eq. (S53), 1967 [31]. At the time, this observation was considered modifies the dispersion of the collective mode too, but to be one of the first proofs of the validity of the Landau preserves its downward q2 character within the 1/N theory of Fermi-liquids [32]. Unlike the more well-known approximation[37]. The final result for the dynamical zero sound [33], an external magnetic field is required in susceptibility is summarized in Fig.1. Away from the order to shift the particle-hole continuum up along the zero momentum axis =X±(q, ω) is always non-zero, and energy axis to allow for the undamped collective spin is the sum of several distinct contributions. Inside this wave to appear outside the particle-hole continuum, in spinon-gauge continuum, the spinon spin wave appears as the triangle-shaped window below it. Second order in a resonance which is asymptotically sharp at small mo- the interaction u corrections (beyond the ladder series) mentum. We note that, while our calculations are done in do cause damping of this spin mode [34–36]. two dimensions, a spinon spin wave with the same qual- However, in the U(1) spin liquid, there is an additional itative features is also present in the three dimensional branch of low energy excitations due to the gauge field U(1) QSL. A, dispersing as ω ∼ q3. The very flat dispersion of the For observation of the spinon spin wave via inelas- gauge excitations suggests it may act as a momentum tic neutron or RIXS experiment, the mode should be sink, so that, for example, an excitation consisting of a present over a range of momenta which is not too nar- particle-hole pair plus a gauge quantum may exist in the row. Because the extent of the ‘decay-free” triangu- “forbidden” region where the bare particle-hole contin- lar shaped region in Fig.1 is determined by 2ωB/vF ∼ p uum vanishes and the collective spin mode lives. It is mωB(ωB/EF ), this requires that the Zeeman energy 4 should be a substantial fraction of the exchange integral a splitting of the ESR line into a doublet [46]. (effective Fermi energy). This makes spin liquid materi- The gauge field part of Eq. (12) consists mathemati- als with J of order 10 K ideal candidates for observation, cally of coupling of A to the spin-non-conserving bilinear in constrast with usual metals for which ωB/EF is van- operators of spinons: ishingly small. 0 iαso X † + † − The above results apply to the case in which SU(2) spin H = [ψ ψp,↑A (q)−ψ ψp,↓A (q)], (13) A 2 p+q,↓ p+q,↑ rotation symmetry is broken only by the applied Zeeman p,q

field. Breaking of the SU(2) invariance by anisotropies in- ± validates the Larmor theorem and causes a shift and more where A = Ax ± iAy. This term has no simple mean importantly a broadening of the spin collective mode even field description, and is responsible for the magnetic field at zero momentum [38]. This is of particular importance and temperature dependence of the dynamical spin sus- for electron spin resonance, which has high energy resolu- ceptibility, and in particular the ESR line width η. tion but measures directly at zero momentum only [25]. Instead of a technically-involved diagrammatic cal- The way in which the resonance is broadened depends culation (which is also possible and confirms the re- in detail on the nature of the anisotropy, the orientation sults otherwise obtained) we employ an elegant short- of the applied magnetic field, etc., so it is not possible cut which is based on the modern reincarnation [25, 47] to give a single general result. Instead, we provide one of the classic ESR formulation by Mori and Kawasaki example of this physics and consider the influence of a [48]. We are interested in the retarded Green’s function of the transverse spin fluctuations GR (q = 0, ω) = Dzyaloshinskii-Moriya (DM) interaction, which is typi- S+S− cally the dominant form of anisotropy for weakly spin- 2M/(ω − 2ωB − Σ(ω)), which defines the zero momen- orbit coupled systems, provided it is symmetry allowed tum self-energy Σ(ω). The ESR theory shows (see [37]) by the lattice. See for example [39, 40]. that this self-energy is related to the correlations of the perturbation operator R = [H0 ,S+], according to Guided by the arguments of symmetry and simplicity A we next suppose that when projected to the spin liquid 1 R η = =Σ(ω) = − =G † (ω). (14) ground state manifold of the two-dimensional spin model, 2M RR the DM interaction appears as a familiar spin-orbit in- teraction of the Rashba kind. A momentum-dependent Eq. (14) directly expresses the ESR linewidth in terms of spin splitting, of which this is the simplest example, is the retarded Green’s function of the perturbing operator expected to appear in a model without spatial inversion R. Observe that R ∝ αso and hence to the second order symmetry because in the spinon Fermi surface state the in the spin-orbit coupling, Eq. (14) may be calculated spinons transform under lattice point group symmetries with respect to the isotropic Hamiltonian of the ideal in the same way as usual [41]. Then the term spin liquid subject to the Zeeman magnetic field ωB. 0 in the Hamiltonian breaking SU(2) spin invariance, H , F reads 60 Z 0 2 † y x 40 H = d x αsoψ ((ˆpx + Ax)σ − (ˆpy + Ay)σ )ψ. (12)

20

Herep ˆµ is the i-th component of the momentum operator, 2ω B 5 10 15 20 25 30 T and αso is the strength of the Rashba coupling. The dependence on the minimal combination pˆ+A is required -20 by the gauge invariance of the action in Eq. (2). Note that the magnetic field continues to couple to σz. FIG. 2. Scaling function F (x). In the fixed Coulomb gauge, the momentum and gauge For the Rashba coupling, one obtains R = terms within the Rashba anisotropy of Eq. (12) have dis- −i αso P ψ† σzψ A−(q), so that the calculation of η tinct effects. The former, momentum term, may be con- 2 p,q p+q p reduces to a convolution-type integral over energy and sidered at the mean field level, as an intrinsic spin split- momentum of the spectral functions of the spinon mag- ting in the spinon dispersion. Taking this into account, netization density Sz(q) and the gauge field A(q). This the ESR signal arises from vertical inter-band transitions instructive calculation is described in [37] and results in [42]. The variation of these transitions with momentum the full scaling function prediction for the ESR linewidth leads to an intrinsic lineshape, from which useful infor- mation about van Hove and other special points of the spinon bands may be extracted by a detailed analysis[43]. α2 mT 2ω η(ω ,T ) = so ( + const T 5/3F ( B )). (15) In Fermi liquids, this physics is responsible for chiral spin B 2M 8πχ T resonance [44, 45]. In one dimensional spin chains with uniform DM interaction, the same basic physics leads to The scaling function F (x) is plotted in Figure2 and 5 is characterized by these limits: F (x  1) = −4.4x Kelley, A. Banerjee, J.-Q. Yan, C. A. Bridges, D. G. Man- and F (x  1) = 0.75x5/3. Consequently, in the low- drus, S. E. Nagler, A. K. Kolezhuk, and S. A. Zvyagin, temperature limit the linewidth follows a ‘fractional’ scal- Phys. Rev. B 96, 241107 (2017). 2 5/3 [18] G. B. Hal´asz,N. B. Perkins, and J. van den Brink, Phys. ing with the magnetic field, η → αso(2ωB) /M ∼ 2 2/3 Rev. Lett. 117, 127203 (2016). αsoB . Also notable is the non-monotonic dependence [19] M. R. Norman, Rev. Mod. Phys. 88, 041002 (2016). of the scaling function F on its argument. The full scaling [20] Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. function represents a non-trivial quantitative prediction 89, 025003 (2017). for the the present model of magnetic anisotropy. [21] Y.-D. Li and G. Chen, Phys. Rev. B 96, 075105 (2017). However, while all isotropic magnets are alike, all [22] N. Nagaosa, Quantum Field Theory in Strongly Corre- anisotropic magnets are anisotropic in their own way. lated Electronic Systems (Springer, Berlin, Heidelberg, We leave an exhaustive study of different mechanisms 1999). [23] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. of anisotropy on ESR in spin liquids for future work. 78, 17 (2006). We thank A. 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Supplemental Materials for “Spinon waves in magnetized spin liquids” Leon Balents and Oleg A. Starykh

GAUGE FIELD CORRECTIONS TO THE TRANSVERSE SPIN SUSCEPTIBILITY

In the main text, we show that coupling to the gauge field induces spectral weight outside the region of the particle- hole continuum of the free fermion theory. On physical grounds, this is expected because in addition to the fermionic “” (we use quotes because they are not gauge invariant and have a non-Fermi liquid self-energy) the system possesses collective gauge excitations with a very soft dispersion relation ω ∼ k3. By creating a particle-hole pair and a , one may shunt enough of the total momentum of the excitation in to the photon to bring the remainder into the kinematically allowed region for particle-hole pairs, and because the energy of the photon is so small, this should be possible at just about any energy. With this picture in mind, we seek contributions to the dynamical spin structure factor with a single gauge field propagator, because “cutting” this line corresponds to a single excited photon excitation. With a single gauge propagator, there are three diagrams, as shown in Eq.(10). In 1 1 the first and second contributions, χ1(q, ωn) and χ2(q, ωn), the gauge line does not cross the particle-hole bubble, so the gauge field acts here as a self-energy correction to one of the two fermion lines. In the third diagram, the gauge line crosses the bubble, which represents a vertex correction. Care must be taken to combine all three terms because, as shown by Kim et al.[24], there are important cancellations between them which are required to maintain gauge invariance and avoid unphysical results at low frequency and momentum. The formal expressions for these three contributions are Z 2k + p 2k + p χ1(q, ω ) = (dk)(dp)G2(k, k )G (k + p, k + p )G (k + q, k + ω ) µ µ ν ν D (p, p ), 1 n ↑ n ↑ n n ↓ n n 2m 2m µν n Z 1 2 χ2(q, ωn) = (dk)(dp)G↑(k, kn)G↓(k + q, kn + ωn)G↓(k + p + q, kn + pn + ωn) 2k + 2q + p 2k + 2q + p × µ µ µ ν ν ν D (p, p ), 2m 2m µν n Z 1 χ3(q, ωn) = (dk)(dp)G↑(k, kn)G↑(k + p, kn + pn)G↓(k + q, kn + ωn)G↓(k + p + q, kn + pn + ωn) 2k + p 2k + 2q + p × µ µ ν ν ν D (p, p ), (S1) 2m 2m µν n where we introduced bold face for spatial vectors, kn and pn as Matsubara frequencies, and the notation (dk) = 2 3 d kdkn/(2π) . The gauge propagator is

 p p  D (p, p ) = δ − µ ν D(p, p ), (S2) µν n µν p2 n with 1 D(p, pn) = 2 . (S3) γ|pn|/p + χp

Here in a strict 1/N expansion the γ coefficient, which reflects Landau damping, itself originates from a fermion bubble. We will treat it however as just a bare kinetic term for the gauge field, following numerous previous works. Consider the first two diagrams. In their expressions, the integral over p defines the self-energy Σ, and these terms can be rewritten in terms of Σ. Specifically, Z 1 2 χ1(q, ωn) = (dk)Σ↑(k, kn)G↑(k, kn)G↓(k + q, kn + ωn), Z 1 2 χ2(q, ωn) = (dk)Σ↓(k + q, kn + ωn)G↑(k, kn)G↓(k + q, kn + ωn), (S4) 2 where the self-energy is Z 2k + p 2k + p Σ (k, k ) = (dp)G (k + p, k + p ) µ µ ν ν D (p, p ) α n α n n 2m 2m µν n Z k × pˆ2 = (dp) D(p, p )G (k + p, k + p ). (S5) m n α n n

The self-energy is a standard calculation in the spinon gauge theory. Writing the Green’s function explicitly, we have Z  2 k × pˆ D(p, pn) Σα(k, kn) = (dp) , (S6) m ikn + ipn − k+p,α with

k,α = ξk − αωB + Un−α, (S7)

2 2 with α =↑= +1 and α =↓= −1, and ξk = (k − kF )/2m. Owing to the singular nature of the gauge propagator, the ˆ ˆ integral for the self-energy is dominated by small p. Choosing coordinates p = kpk + zˆ × kp⊥, we have

p2  ≈  + v p + ⊥ . (S8) k+p,α k,α F k 2m

Here k,α includes a spin-dependent Zeeman shift and a Hartree correction, which are both constant. From this form 2 we expect the scaling pk ∼ p⊥  p⊥, which means that p is approximately normal to k. This means we can replace 2 2 2 2 2 p by p⊥ inside the gauge propagator, and that |k × pˆ| = k p⊥ ≈ k ≈ kF , for momentum k near the Fermi surface. Consequently, we have

2 Z   kF dpn dp⊥ dpk |p⊥| 1 Σα(k, kn) ≈ 2 3 2 . (S9) m 2π 2π 2π γ|pn| + χ|p⊥| ikn + ipn − k,α − p⊥/2m − vF pk

The integral over pk can be done immediately – there is a small subtlety in the real part of the integral is conditionally convergent and dependent upon the cutoff, but this is anyway absorbed in a simple Fermi energy shift and can be set to zero. The imaginary part is well-defined and we obtain

2 Z −ikF dpn dp⊥ |p⊥|sign(kn + pn) Σα(k, kn) = 2 3 . (S10) 2m vF 2π 2π γ|pn| + χ|p⊥| Note that the dependence on momentum k has dropped out, so the self-energy is purely frequency dependent – what is called a “local” self-energy. The actual function can be calculated by performing the pn integration first: the regions at large positive and negative frequency cancel one another and the full result is just obtained from the integral between −|kn| and |kn|. We find

Z 3 −ivF dp⊥ γ|kn| + χ|p⊥| Σα(k, kn) = sign(kn) |p⊥| ln 3 . (S11) 2πγ 2π χ|p⊥|

Then carrying out the p⊥ integration gives finally

2/3 −ivF sign(kn)|kn| Σα(k, kn) = √ . (S12) 2 3π γ1/3χ2/3 The 2/3 power law dependence on frequency of the self-energy is a famous result for the spinon Fermi surface. Now we will rewrite Eqs. (S4) in a form more amenable to seeing the partial cancellations of the three diagrams. To do so, we use the partial fractions rewriting

2 G↑(k, kn)[G↑(k, kn) − G↓(k + q, kn + ωm)] G↑(k, kn)G↓(k + q, kn + ωn) = , (S13) iωn − k+q,↓ + k,↑

2 G↓(k + q, kn + ωn)[G↓(k + q, kn + ωm) − G↑(k, kn)] G↓(k + q, kn + ωn) G↑(k, kn) = , −iωn + k+q,↓ − k,↑ 3 which is straightforward to show using of the explicit forms for the Green’s functions. Using these forms, we obtain 1 1 1 the sum of the two self-energy contributions as χ12 = χ1 + χ2 Z 1 Σ↓(k + q, kn + ωn) − Σ↑(k, kn) χ12(q, ωn) = (dk) G↑(k, kn)G↓(k + q, kn + ωn) iωn + k↑ − k+q,↓ 2 2 Z Σ↑(k, kn)G (k, kn) − Σ↓(k + q, kn + ωn)G (k + q, kn + ωn) + (dk) ↑ ↓ . (S14) iωn + k↑ − k+q,↓

We aim to massage the susceptibility corrections into a form which exposes the small q dependence more clearly and makes physical interpretation easier. First we split (S14) into two distinct parts,

1 1 1 χ12 = χ12,A + χ12,B, (S15) with Z 1 Σ↓(k + q, kn + ωn) − Σ↑(k, kn) χ12,A(q, ωn) = (dk) G↑(k, kn)G↓(k + q, kn + ωn) (S16) iωn + k↑ − k+q,↓ Z 2 2 1 Σ↑(k, kn)G↑(k, kn) − Σ↓(k + q, kn + ωn)G↓(k + q, kn + ωn) χ12,B(q, ωn) = (dk) . (S17) iωn + k↑ − k+q,↓

1,B We show below that χ12 does not contribute to the imaginary part of the susceptibility and for a moment just neglect it. Next we use the identity

G↑(k) − G↓(k + q) G↑(k)G↓(k + q) = (S18) iωn + ↑(k) − ↓(k + q) to write Z 1 Σ↓(k + q, kn + ωn) − Σ↑(k, kn) χ12,A(q, ωn) = (dk) 2 [G↑(k) − G↓(k + q)] . (S19) (iωn + k↑ − k+q,↓)

Next we use the expression for the self-energy, (S5), and obtain Z 1 1 D(p, pn) h 2 2 i χ12,A(q, ωn) = 2 (dk)(dp) 2 ((k + q) × pˆ) G↓(k + q + p) − (k × pˆ) G↑(k + p) [G↑(k) − G↓(k + q)] . m (iωn + k↑ − k+q,↓) (S20)

Similar manipulations of the vertex diagram give Z 1 1 D(p, pn) χ3(q, ωn) = 2 (dk)(dp) (k × pˆ) · ((k + q) × pˆ) m (iωn + k↑ − k+q,↓)(iωn + k+p,↑ − k+p+q,↓)

× [G↑(k + p) − G↓(k + q + p)] [G↑(k) − G↓(k + q)] . (S21)

IMAGINARY PART

Let us consider the imaginary part of the (real frequency) susceptibility outside the particle hole (PH) continuum. This means that the explicit denominators in Eqs. (S20,S21) can be trivially analytically continued and considered purely real. We will however have to do the pn and kn integrals. Start with (S20) and observe that if the product of the two square brackets is multiplied out, two of the four resulting terms, those involving the integral of G↓(k + p + q)G↓(k + q) and G↑(k + p)G↑(k), do not have imaginary part outside the PH continuum simply because the integral over k gives a result which is ωn-independent (in the first of these we can make a shift k → k − q and then integrate over k). That is, the only source of the imaginary part in these contributions is provided by the denominator in 1/(iωn + k↑ − k+q,↓) – and its imaginary part is restricted 1 to the non-interacting PH continuum. The same argument actually applies to χ12,B in (S17), one just needs to shift k → k − q in the 2nd term on the right hand side there. 4

1 The two terms in χ12,A that contribute outside the PH continuum are therefore given by Z 1 1 D(p, pn) h 2 2 i χ12,A(q, ωn) = 2 (dk)(dp) 2 ((k + q) × pˆ) G↓(k + q + p)G↑(k) + (k × pˆ) G↑(k + p)G↓(k + q) . m (iωn + k↑ − k+q,↓) (S22)

Since the gauge propagator (S3) is even in p = (p, pn), we can transform p → −p, followed by k → k + p, in the first term in the square brackets above to obtain

Z  2 2  1 1 (k × pˆ) ((k + q) × pˆ) χ12,A(q, ωn) = 2 (dk)(dp)D(p, pn) 2 + 2 G↑(k + p)G↓(k + q). m (iωn + k,↑ − k+q,↓) (iωn + k+p,↑ − k+q+p,↓) (S23)

Now insert the spectral representation,

Z d(p, ν) D(p, pn) = dν . (S24) ipn − ν Here we want 1 D(p, pn) = 2 . (S25) γ|pn|/p + χp In the usual way, we extract the spectral function via 1 1 γωp d(p, ω) = − ImD(p, ip → ω + i0+) = − . (S26) π n π γ2ω2 + χ2p6

Eq.(S23) factorizes into

Z  2 2  1 1 (k × pˆ) ((k + q) × pˆ) χ12,A(q, ωn) = 2 (dk)(dp) 2 + 2 I12, (S27) m (iωn + k,↑ − k+q,↓) (iωn + k+p,↑ − k+q+p,↓) where Z Z dkndpn 1 1 1 I12 = dνd(p, ν) 2 . (S28) (2π) ipn − ν ikn + ipn − k+p,↑ ikn + iωn − k+q,↓ Carrying out the contour integrations we obtain

Z d(p, ν) I12 = − dν [θ(ν)θ(−k+p,↑)θ(k+q,↓) + θ(−ν)θ(k+p,↑)θ(−k+q,↓)] . (S29) iωn + k+p,↑ − k+q,↓ − ν

Now we can analytically continue iωn → ω + i0, with ω > 0, and obtain the imaginary part, which constraints ν = ω + k+p,↑ − k+q,↓ and collapses the ν-integration

ImI12 = πd(p, ω + k+p,↑ − k+q,↓)θ(k+q,↓)θ(−k+p,↑)θ(ω + k+p,↑ − k+q,↓) for ω > 0. (S30)

We observe that the second set of step-functions in (S29) is always zero for ω > 0.

Now consider the vertex part, Eq. (S21). Expanding the product in the second line, we seek terms that have some ωn = qn dependence. There are just two such parts Z 1 1 D(p, pn) χ3(q, ωn) = − 2 (dk)(dp) (k × pˆ) · ((k + q) × pˆ) m (iωn + k↑ − k+q,↓)(iωn + k+p,↑ − k+p+q,↓)

× [G↑(k + p)G↓(k + q) + G↓(k + q + p)G↑(k)] . (S31)

We again observe that the set of transformations p → −p, followed by k → k + p, make the 2nd term in the square 5 brackets above equal to the 1st, so that Z 1 2 (k × pˆ) · ((k + q) × pˆ) χ3(q, ωn) = − 2 (dk)(dp) D(p, pn)G↑(k + p)G↓(k + q). (S32) m (iωn + k↑ − k+q,↓)(iωn + k+p,↑ − k+p+q,↓)

Evidently it reduces to a form very similar to (S27). Namely, Z 1 2 (k × pˆ) · ((k + q) × pˆ) χ3(q, ωn) = − 2 (dk)(dp) I12. (S33) m (iωn + k↑ − k+q,↓)(iωn + k+p,↑ − k+p+q,↓)

1 1 1 Therefore, the imaginary part of χ = χ12 + χ3 is determined by the imaginary part of I12 (S30),

Z 2 2 1 1 h (k × pˆ) ((k + q) × pˆ) Imχ (q, ω) = 2 (dk)(dp) 2 + 2 m (ω + k,↑ − k+q,↓) (ω + k+p,↑ − k+q+p,↓) 2(k × pˆ) · ((k + q) × pˆ) i − ImI12. (S34) (ω + k↑ − k+q,↓)(ω + k+p,↑ − k+p+q,↓)

At this point, we can see that the quantity in the square brackets in Eq. (S34) vanishes at q = 0. Hence we may expect a quadratic dependence on q (although this quantity has linear terms in q, they vanish on integration or pair off with another linear part from I12).

Let us see how to make the integration explicit. We choose polar coordinates according to

q = (q, 0),

k = kF + δk = (kF + δk)(cos θ, sin θ),

p = pk + p⊥ = pk(cos θ, sin θ) + p⊥(− sin θ, cos θ). (S35)

We assume δk, q, pk, p⊥  kF . The vector vertices in (S34) can be simplified as follows:

k + δk k + δk k × pˆ =z ˆ(k p − k p )/p =z ˆ F [cos θ(p sin θ + p cos θ) − sin θ(p cos θ − p sin θ)] =z ˆ F p x y y x p k ⊥ k ⊥ p ⊥ p ≈ zkˆ ⊥ . (S36) F p

2 p2 p⊥ k 2 At this point we need to compare pk and p⊥. From k+p − k = vF pk + 2m + 2m we deduce that pk ∼ p⊥/(mvF ) = 2 2 pk q 2 2 p⊥/kF  p⊥. Therefore 2m can be neglected in comparison with the two first terms and, moreover, p = p⊥ + pk ≈ p⊥. Hence (S36) reduces to justzk ˆ F .

At the same time we see that q × pˆ ≈ zqˆ sin θ  zkˆ F . Therefore all vertices in (S34) can be safely approximated byzk ˆ F . This strongly simplifies (S34),

2 Z 1 kF h 1 1 2 i Imχ (q, ω) = 2 (dk)(dp) 2 + 2 − m (ω + k,↑ − k+q,↓) (ω + k+p,↑ − k+q+p,↓) (ω + k↑ − k+q,↓)(ω + k+p,↑ − k+p+q,↓) Z 2 2 h 1 1 i × ImI12 = vF (dk)(dp) − ImI12. (S37) ω + k,↑ − k+q,↓ ω + k+p,↑ − k+q+p,↓

Next we observe that k+p,↑ + k+q,↓ − k,↑ − k+q+p,↓ = −2q · p ≈ 2qp⊥ sin θ. This combination appears when combining denominators in the squared factor in Eq. (S37), and explicitly demonstrates the q2 dependence of the result.

Next we can write

k+q,↓ ≈ vF (δk + q cos θ) +ω ˜B, (S38) 2 k+p,↑ ≈ vF (δk + pk) + p⊥/(2m) − ω˜B, (S39) p2 p q sin θ  ≈ v (δk + p + q cos θ) + ⊥ − ⊥ +ω ˜ . (S40) k+q+p,↓ F k 2m m B 6

The sign constraints on the energy in (S30) therefore are

k+q,↓ > 0 → vF (δk + q cos θ) +ω ˜B > 0, (S41) 2 k+p,↑ < 0 → vF (δk + pk) + p⊥/(2m) − ω˜B < 0,

The frequency argument of the photon spectral function is

2 ν = ω + k+p,↑ − k+q,↓ = ω − 2˜ωB − vF q cos θ + vF pk + p⊥/(2m). (S42)

Eq.(S41) shows that ν is bounded by ω from above, ν ≤ ω. At the same time ν > 0. Let us use Eq. (S42) as a definition to trade the pk integration for one over ν. We have dpk = dν/vF . In particular we see that

2 vF pk + p⊥/(2m) = ν + 2˜ωB − ω + vF q cos θ. (S43)

Introduce ε

ε = vF δk + vF q cos θ +ω ˜B, (S44) so that d(δk) = dε/vF . Hence the second line in Eq. (S41) becomes ε < ω − ν while the first line just reads ε > 0. Hence we obtain

0 < ε < ω − ν. (S45)

Now we can express all the energies in these variables.

k+q,↓ ≈ ε, (S46)

k+p,↑ ≈ ν − ω + ε, (S47)

k↑ ≈ vF δk − ω˜B ≈  − 2h − vF q cos θ, (S48) 2 k+p+q,↓ ≈ vF (δk + q cos θ + pk) + p⊥/2m +ω ˜B ≈ ε + ν − ω + 2˜ωB + vF q cos θ. (S49)

Putting everything together and using p ≈ p⊥ and (S26) for d(p⊥, ν), we obtain for (S37)

Z 2 2 1 2 2 p⊥ sin θ Imχ (q, ω) = 4vF q (dk)(dp) 2 2 ImI12 (ω − 2˜ωB − vF q cos θ) (ω − 2˜ωB − vF q cos θ + qp⊥ sin θ/m) 2 2 Z ∞ Z ω Z 2π Z ω−ν 2 2 4vF q dν d p⊥ sin θ γν|p⊥| ≈ − 4 dp⊥ dθ kF 4 2 2 2 6 . (S50) (2π) −∞ 0 vF 0 0 vF (ω − 2˜ωB − vF q cos θ) γ ν + χ p⊥ √ R ∞ 3 Now we can finally complete the evaluation. We integrate over p⊥ first, using 0 duu/(1 + u ) = 2π/3 3, and 1 R ω 1/3 obtain that Imχ ∼ 0 dν(ω − ν)ν . Then we obtain the final result √ 3γ1/3k q2ω7/3 Im χ1(q, ω) ≈ − F . (S51) 2 4/3 4 56π χ (ω − 2˜ωB)

The characteristic scaling q2ω7/3 is now explicitly shown.

REAL PART

The calculations of the real part are more difficult. In accordance with the common wisdom, see for example Chap.37 in the book by Abrikosov, Gor’kov, Dzyaloshinskii, the outcome depends on the order of the integration over Matsubara frequencies kn, pn and momenta k, p. Generally the first integrals carried out can be calculated simply and exactly in the absence of any ultraviolet cut-off. For a zero temperature quantum system, there is indeed no frequency cut-off, so integrating over all frequencies is correct. Integrating over all momenta is often not correct, because there is some lattice or bandwidth scale. Nevertheless, it can be tempting and simpler to try. In our case, carrying out the 7 momentum integration first, similar to the procedure used by Kim et al.[24], gives the incorrect result

2 2 2 1 1 vF q ω Re χ (q, ω) k integral first ' 3 4 , (incorrect), (S52) 8π χ (ω − 2˜ωB) which holds in the limit |ν| = |ω − 2˜ωB| > vF q. Importantly, the “result” is seemingly independent of a momentum cut-off even though the intermediate steps do require an explicit cut-off of the order kF when integrating over gauge momentum p⊥. Carrying out the frequency integration first, similar to the calculations in Maslov et al. [36], produces quite a different answer 2 2   1 kF q (ω + 2˜ωB) Λχ Re χ (q, ω) ' 2 3 ln , (correct). (S53) 16π χ(2˜ωB − ω) γ

Here Λ is a high-momentum cut-off. Observe that <χ1 diverges in the γ → 0 limit, which represents the static limit of gauge fluctuations, see (S3). The technicalities of integrations leading to Eq. (S53) are tedious and require a number of simplifi- cations performed at the proper stages in the calculation. These steps are carried out in the Math- ematica Notebook “realchi1.nb” which is included in the Supplementary Materials, see the following link https://journals.aps.org/prb/supplemental/10.1103/PhysRevB.101.020401 . There we find that the most divergent part of the Reχ1(q, ω) is given by

3 2 2 Z 2π Z Λ 7 2 1 χ kF q (ω + 2˜ωB) p cos φ Reχ (q, ω) = 3 3 dφ dp 2 2 2 4 2 . (S54) 16π (2˜ωB − ω) 0 0 (γ (p − 2kF sin φ) + χ p )

We rescale by pulling out χ from the denominator, introducing α = γ/(χkF ) and letting p = kF y. Then

2 2 Z 2π Z ΛF 7 1 kF q (ω + 2˜ωB) 2 y Reχ (q, ω) = 3 3 dφ cos φ dy 2 2 4 2 , (S55) 16π χ(2˜ωB − ω) 0 0 (α (y − 2 sin φ) + y ) where ΛF = Λ/kF is the dimensionless upper cut-off of the momentum integration. We observe that for large ΛF  1 R ΛF 3 2 2 2 integration variable y ∼ ΛF is large as well, and therefore the y-integral is approximated as 0 dyy /(α + y ) ≈ ln(ΛF /α). This gives us the quoted result, Eq.(S53). In addition to the leading q2 term in (S53), our calculation also produces a finite zero momentum independent contribution, which represents a correction δM to the magnetization M

2 1 2δM 1 ωBm γ χ (0, ω) = = 4 2 I(γ/χkF ). (S56) ω − 2ωB ω − 2ωB 2π χ kF Note that the prefactor is a simple pole, which reflects the Larmor theorem. Here I(α) is a momentum integral, which we calculate numerically. It parametrically depends on its argument α = γ/χkF . For example, for α = 1 we obtain I = 0.76. Note that (S56) represents a quantum correction to M (and hence the uniform susceptibility itself since M is proportional to the appleid field), and vanishes in the limit γ → 0. Interestingly, this correction comes from 1 the real part of χ12,B,(1), and is entirely absent when one integrates over the momenta before integrating over the frequencies. We conclude by noting that imaginary part of χ1,(S51), is not sensitive to the order of integrations discussed here. Carrying out calculations following [24] and doing the momentum integration before the frequency one precisely reproduces Eq. (S51) (we do not show those redundant calculations here).

RPA FORM

In the RPA approximation

0 1 χ±(q, iωn) + χ±(q, iωn) X±(q, iωn) = 0 1 (S57) 1 + u[χ±(q, iωn) + χ±(q, iωn)] 8 so that after the continuation to real frequency iωn → ω + i0 we obtain for the imaginary part of the susceptibility outside the non-interacting particle-hole continuum

1 Im[χ±(q, ω)] Im[X±(q, ω)] = 0 1 2 1 2 . (S58) (1 + u Re[χ±(q, ω) + χ±(q, ω)]) + (u Im[χ±(q, ω)])

0 Damping of the collective spin mode is determined by (S51), since Im[χ±(q, ω)] = 0 in this frequency range, while its dispersion is given by

0 1 1 + u Re[χ±(q, ω) + χ±(q, ω)] = 0. (S59)

Considering (S53) as a small correction to the dispersion of the collective mode, which is justified by the 1/N expansion 2 ∗ assumption of the gauge theory [24], we parameterize the dispersion as ωq = 2ωB −ζq and solve (S59) for ζ = 1/(2m ) by iterations. We find that the effective mass m∗ increases,

Λχ  1 v2 u(2ωB + uM) ln = F − γ , (S60) m∗ 2uM 4π2χ(2uM)3 and the dispersion of the collective mode flattens, correspondingly. But it retains its q2 character.

ESR IN THE U(1) FERMI SURFACE SPIN LIQUID

Mori-Kawasaki formula

In the following we use Mori-Kawasaki (MK) formula, as derived in the Appendix of [25]. A calculation with Matsubara Green’s functions is also possible, and was in fact carried out – the end structure of result is the same in the two approaches. This Matsubara calculation is instructive for understanding which diagrams contribute to the result. It shows that the ESR response is determined, in the leading α2 order, only by the diagrams with self-energy insertions inside the spinon bubble. The vertex diagram is not present in that order. However, the calculation at finite temperature T requires an analytical continuation to real frequency which is not an easy task. The MK approach is formulated directly in terms of retarded Green’s functions which is more convenient. 0 Let us parameterize H = H0 + H + HZ , where the first term is SU(2) invariant, the second contains symmetry- z breaking perturbations and the third describes interaction with static external field HZ = −hStot. The total spin of the system S+ obeys the equations of motion

dS+ dS− = −ihS+ + iR, = ihS− − iR+, (S61) dt dt where R = [H0,S+]. R R ∞ iωt + − Then for the retarded Green’s function GS+S− (ω) = −i 0 dte h[S (t),S (0)]i one obtains, using the identity iωt 1 d iωt e = iω dt e and integration by parts,

z R 2hS i h R 1 R G + − (ω) = + G + − (ω) − G − (ω). (S62) S S ω ω S S ω RS

R Repeating these steps for GRS− (ω) one finds

z − R R 2hS i h[R,S ]i GRR+ (ω) G + − (ω) = − + . (S63) S S ω − h + i0 (ω − h + i0)2 (ω − h + i0)2

z R 2hS i Assuming next that GS+S− (ω) = (ω−h−Σ(ω)) and expanding to second order in the perturbation R ∼ α, we obtain explicit the expression for the self-energy Σ(ω)

 − R  z Σ(ω) = h[R,S ]i − GRR+ (ω) /(2hS i) (S64)

The real part of this determines the shift of the resonance frequency from the ideal ω = h = 2ωB value, while the 9 imaginary part determines all-important linewidth, resulting in Eq. (14) of the main text. Note that the first term in Eq. (S64) proportional to the commutator is frequency independent and hence does not contribute to the imaginary part, because the latter must be odd in frequency (as shown by the spectral representation). Direct calculation gives the result shown in the main text

αso X R = −i ψ† σzψ A−(q). (S65) 2 p+q p p,q

z P † σz This is a composite operator consisting of a convolution in momentum space of S (q) = p ψp+q 2 ψp and the gauge − P z − field A (q), R = −iαso q S (q)A (q).

Convolution formula

2 Since R ∼ αso, the retarded Green’s function of the composite operator R can be approximated by αso multiplied z − by the simple convolution of independent Green’s functions of Sq and Aq . This well known result is worked out below as follows. Let O = AB, where A and B stand for two arbitrary independent (commuting) operators, [A, B] = 0. Then

R † > < GO(t) = −iΘ(t)h[Ot, O0]i = Θ(t)(GO(t) − GO(t)), (S66) where the greater and lesser Green’s functions are defined by

> † < † GO(t) = −ihOtO0i,GO(t) = ihO0Oti. (S67)

Note that here the lower index of the operators denotes time dependence, Ot = O(t). Using this representation and commutativity of A and B it is easy to show that

R  > > < <  GO(t) = iΘ(t) GA(t)GB(t) − GA(t)GB(t) . (S68)

iHt −iHt Next, the spectral decomposition and At = e A0e lead to

−i X Z n−i X o G>(t) = |hm|A|ni|2e−βEm e−it(En−Em) = de−it |hm|A|ni|2e−βEm δ( − (E − E )) (S69) A Z Z n m n,m n,m and −i X Z n−i X o G<(t) = |hm|A|ni|2e−βEn e−it(En−Em) = de−ite−β |hm|A|ni|2e−βEm δ( − (E − E )) . (S70) A Z Z n m n,m n,m

R As a result, spectral density (which determines GA(ω)) can be written as 1 X ρ () = |hm|A|ni|2(e−βEm − e−βEn )δ( − (E − E )). (S71) A Z n m n,m

It satisfies sign[ρ()] = sign[]. It is easy now to obtain Z Z > −it < −it GA(t) = (−i) de ρA()[1 + n()],GA(t) = (−i) de ρA()n(). (S72)

This allows us to write (S68) as Z R −i(1+2)t GO(t) = −iΘ(t) d1d2e ρA(1)ρB(2){(1 + n(1))(1 + n(2)) − n(1)n(2)}. (S73) 10

R R ∞ i(ω+i0)t R Using G (ω) = 0 dte G (t) we obtain Z R ρA(1)ρB(2) GO(ω) = d1d2{1 + n(1) + n(2)} , (S74) ω − 1 − 2 + i0 so that Z R ImGO(ω) = −π d1d2{1 + n(1) + n(2)}ρA(1)ρB(2)δ(ω − 1 − 2). (S75)

We can express this back in terms of the Green’s function, because for the individual retarded Green’s function one has Z ρ () GR(ω) = d A , (S76) A ω −  + i0

M † while the corresponding Matsubara Green’s function GA (τ) = −hTτ Aτ A0i is written as Z M ρA() GA (ωn) = d (S77) iωn − 

Therefore 1 Z ImGR (ω) = − d d {1 + n( ) + n( )}ImGR( )ImGR( )δ(ω −  −  ). (S78) O π 1 2 1 2 A 1 B 2 1 2

Calculation

Spectral densities

z − And now we consider the relevant case here with O = AB and A = Sq and B = Aq . Note that we pulled out the M q R q factor of αso so R = αsoO. Ref.24 tells us that GB (ν) = γ|ν|+χq3 so that GB() = −iγ+χq3 . Here |ν| → −i + 0 is used for ν > 0. Hence

R γq2 ImGB(2) = 2 2 2 6 . (S79) γ 2 + χ q

M GA is the polarization bubble of two spinon lines, Z Z M 2 f(ξp) − f(ξp+q) GA (ωn) = d (d p) δ( + ξp − ξp+q) (S80) iωn −  Standard angular integration gives

R m 1 ImG (1) = Θ(vq − |1|) (S81) A 2π p 2 2 2 v q − 1

Thus we need to evaluate Z Z R 1 2 m  γq(ω − ) ImGO(ω) = − d (d q){1 + n() + n(ω − )} Θ(vq − ||) π 2π pv2q2 − 2 γ2(ω − )2 + χ2q6 mγ Z Z ∞ dqq2 (ω − ) = − d{1 + n() + n(ω − )} (S82) 4π3v3 p 2 2 2 2 χ2q6 || q −  γ (ω − ) + v6 11

T = 0

At zero temperature 1 + n() + n(ω − ) → Θ() − Θ( − ω) = Θ()Θ(ω − ) and (S82) simplifies to mγ Z ω Z ∞ dqq2 (ω − ) ImGR (ω) = − d = O 4π3v3 p 2 2 2 2 χ2q6 0 || q −  γ (ω − ) + v6 m Z ω 3 Z ∞ dxx2 1 = − d √ . (S83) 3 3 2 χ2 3 2 4π v γ 0 ω −  1 x − 1  6 1 + v6 ω− x

The integration√ over x contains no singularities and, for ω (hence also ) small, it is governed by large x, so we can approximate x2 − 1 → x which allows one to evaluate it as

Z dxx v2 ω − 2/3 ∼ λ−1/3 = . (S84) 1 + λx6 χ2/3 3 The -integration simplifies immediately as well

Z ω  3 1/3 Z 1 duu d = ω5/3 . (S85) 1/3 0 ω −  0 (1 − u)

R 2/3 5/3 5/3 2/3 So we find ImGO(ω) ∼ −(m/(vγχ ))ω . Therefore, from Eq. (14) of the main text, η ∼ ω /M ∼ ωB since z hS i = χuniformωB and near the resonance ω = 2ωB. If no magnetic field is present, absorption is still possible because spin-orbit interaction breaks spin-rotational invariance. In that case, at T = 0, one has η(ω) ∼ ω2/3. Such fractional dependence is quite familiar in U(1) gauge theory [24].

T > 0

This case is more complicated. Eq.(S85) suggests that the -integration is dominated by the region where  ≈ ω. Let  = ω + ν and approximate n(−ν) ≈ −T/ν, then the leading part of (S82) reduces to mγ Z −T Z ∞ dqq2 ω(−ν) mT ω Z ∞ dq ImGR (ω)= − dν = − O 4π3v3 ν p 2 2 2 2 χ2q6 4π2χ p 2 2 |ω| q − ω γ ν + v6 |ω| q q − ω mT sign(ω) Z ∞ dx = − √ . (S86) 2 2 4π χ 1 x x − 1 Here we approximated  → ω in all places which do not cause any divergence. The ω-dependence has dropped out! The x-integral is actually π/2. This important observation suggests the following manipulation of (S82): add and subtract T/(ω − ) to separate the linear in T piece obtained above from the expected T 5/3f(ω/T ) scaling part. So, {1 + n() + n(ω − )} = T T [1 + n() + n(ω − ) − ω− ] + ω− . The last term on the right gives (S86). We now look at the first one, coming from the expression inside square brackets. Denote this contribution to R ImGO(ω) as −C. Then mγ Z T Z ∞ dqq2 (ω − ) C = d[1 + n() + n(ω − ) − ] (S87) 4π3v3 ω −  p 2 2 2 2 χ2q6 || q −  γ (ω − ) + v6 p Now we change to scaling variables: ω = T x,  = T y, q = T 1/3p. Then pq2 − 2 = T 2/3p2 − T 2y2 → T 1/3p. The p-integral reduces to

2/3 ∞ 2/3 3 2/3 ∞ T Z y(x − y) T γv 2/3 |x − y| Z dzz dpp =  y(x − y) . (S88) γ2 2 χ2 6 γ2 χ (x − y)2 1 + z6 0 (x − y) + γ2v6 p 0 12 √ The z-integral is just π/(3 3). This turns (S87) into

mT 5/3 Z ∞ ey 1 1 y(x − y) C = √ dy[ + − ] (S89) 2 1/3 2/3 y x−y 4/3 12 3π vγ χ −∞ e − 1 e − 1 x − y |x − y|

We split the y-integral into two, according to the sign of |x − y|, C = Ca + Cb. Hence Z x y Z ∞ y C = c dy[...] ,C = (−c ) dy[...] , (S90) a 0 1/3 b 0 1/3 −∞ (x − y) x (y − x)

5/3 where c0 =c ˜0T is an overall constant in (S89), i.e. m c˜0 = √ . (S91) 12 3π2vγ1/3χ2/3

0 0 In the Ca, we change y = 2x − y so that x − y = y − x, and find

Z ∞ e2x−y 1 1 2x − y C = c dy[ + + ] (S92) a 0 2x−y y−x 1/3 x e − 1 e − 1 x − y (y − x)

0 where we instantly renamed y → y again. We now add Cb and simplify to find Z ∞ dy  2x − y y 1 1  C = c − + 2x[ − ] . (S93) 0 1/3 y−2x y y−x x (y − x) 1 − e e − 1 e − 1 y − x This expression is free from divergences. Finally, setting y = z + x we obtain

Z ∞ dz  x − z z + x 1 1  ImGR (ω) = −c˜ T 5/3f(x) = −c˜ T 5/3 − + 2x[ − ] . (S94) O 0 0 1/3 z−x z+x z 0 z 1 − e e − 1 e − 1 z This represents the desired scaling function, with x = ω/T . The plot of the scaling function f(x) is shown in Figure 2. Its limits are as follows:

f(x) → −4.37746x for x  1; f(x) → 0.75x5/3 for x  1. (S95)

The small-x limit works for x ≤ 5 while the large-x limit turns on only for really large x ∼ 100. Altogether we have

2 2 αso  mT 5/3 2ωB  αso  m T 2/3 ˜ 2ωB  η = +c ˜0T f( ) = +c ˜0T f( ) , (S96) 2χuniformωB 8πχ T 2χuniform 8πχ ωB T

˜ 2/3 where we used the resonance condition ω = 2ωB. Also here f(x) = f(x)/x. The interesting T behavior of η is clearly subleading.