Spinon waves in magnetized spin liquids
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 approximation. Short-range interaction leads to a new collective mode: a “spinon 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 naïvely forbidden for two-particle excitations and giving the spinon wave a finite lifetime, which, however, remains a well-defined excitation at small momentum. We also study the effect of Dzyaloshinskii-Moriya anisotropy on the zero momentum susceptibility, which is measured in electron spin resonance (ESR), and obtain a resonance linewidth linear in temperature and varying as with magnetic field 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 quantum spin liquid state.
The search for the enigmatic spin liquid state has switched into high gear in recent years. Dramatic theoretical (Kitaev model Kitaev (2006); Savary and Balents (2017) and spin liquid in triangular lattice antiferromagnet Zhu and White (2015)) and experimental (YbMgGaO Paddison et al. (2017); Shen et al. (2018) and -RuCl Banerjee et al. (2016)) developments leave no doubt of the eventual success of this enterprise. To 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 paper, we focus on the two dimensional quantum spin liquid (QSL) with a spinon Fermi surface. This is a priori the most exotic two dimensional QSL state, and yet one which has repeatedly been advocated for in both theory Ioffe and Larkin (1989); Motrunich (2005); Lee and Lee (2005); Motrunich (2006); Nave and Lee (2007) and experiment Yamashita et al. (2008); Shen et al. (2016); Fåk et al. (2017); Shen et al. (2018). Specifically, we study the dynamical susceptibility of the -component of the spin operator
which is an extremely information-rich quantity, and is accessible through inelastic neutron scattering Banerjee et al. (2017), ESR Smirnov et al. (2015); Ponomaryov et al. (2017), and RIXS Halász et al. (2016). The fractionalization of triplet excitations into pairs of spinons is a fundamental aspect of a QSL, and is expected to manifest in as two-particle continuum spectral weight Norman (2016); Savary and Balents (2017); Zhou et al. (2017), a surprising feature which appears more characteristic of a weakly correlated metal than a strongly correlated Mott insulator.
In a mean-field treatment in which the spinons are approximated as non-interacting fermions, this continuum has a characteristic shape at small frequency and wavevector in the presence of an applied Zeeman magnetic field, as discussed in Li and Chen (2017). In particular, there is non-zero spectral weight in a wedge-shaped region which terminates at a single point along the energy axis at zero momentum. Our analysis reveals the full structure in this regime beyond the mean field approximation. Notably, we find that interactions between spinons qualitatively modify the result from the mean-field form, introducing a new collective mode – a “spinon wave” – and modifying the spectral weight significantly.
We recapitulate the derivation of the theory of the spinon Fermi surface phase Nagaosa (1999); Lee et al. (2006). One introduces Abrikosov fermions by rewriting the spin operator , where are canonical fermionic spinors on site with spin-1/2 index (repeated spin indices are summed). This is a faithful representation provided the constraint is imposed – this constraint induces a gauge symmetry. In a path integral representation, the constraint is enforced by a Lagrange multiplier , which takes the role of the time-component of a gauge field, i.e. scalar potential. Microscopic exchange interactions, which are quadratic in spins, and are therefore quartic in fermions, are decoupled to introduce new link fields whose phases act as the spatial components of the corresponding gauge fields , i.e. the vector potential.
To describe the universal low energy physics, it is appropriate to consider “coarse-grained” fields descending from the microscopic ones, and include the symmetry-allowed Maxwell terms for the U(1) gauge field. Furthermore, due to the finite density of states at the spinon Fermi surface, the longitudinal scalar potential is screened and the time component can then be integrated out to mediate a short-range repulsive interaction between like charges. Therefore we consider the Euclidean action , where Kim et al. (1994); Nagaosa (1999); Lee et al. (2006)
Here is the space-time coordinate, is the three-momentum, is a two-component spinor, with spin indices that are suppressed when possible, describes static magnetic field and includes the g-factor as well as the Bohr magneton. The gauge dynamics is derived in the Coulomb gauge with . The gauge action is generated by spinons and and represent Landau damping and diamagnetic susceptibility of non-interacting spinon gas, correspondingly ( is the spinon mass, is the spinon density and is the Fermi momentum of non-magnetized system).
We proceed with the assumption of SU(2) symmetry, a good first approximation for many spin liquid materials, and address the effect of its violations in the latter part of the paper. Previous investigations focused on the transverse vector potential , which is not screened but Landau damped, and hence induces exotic non-Fermi-liquid physics. For example, one finds a self-energy varying with frequency as , and a singular contribution to the specific heat Nagaosa (1999); Lee et al. (2006). However, notably, the transverse gauge field has negligible effects on the hydrodynamic long-wavelength collective response Kim et al. (1994). Here, we instead focus on the short-range repulsion, which produces an exchange field that dramatically alters the behavior in the presence of an external Zeeman magnetic field/finite magnetization. Gauge fluctuations play a subsidiary role which we also include.
An important constraint follows purely from symmetry. Provided the Hamiltonian in zero magnetic field has SU(2) symmetry, a Zeeman magnetic field leads to a fully determined structure factor at zero momentum. Specifically, the Larmor/Kohn theorem Oshikawa and Affleck (2002) dictates that the only response at , , where is the magnetization and is the spinon Zeeman energy. For free fermions, the delta function is precisely at the corner of the spinon particle-hole continuum (also known as the two-spinon continuum). However, the contact exchange interaction shifts up the particle-hole continuum, at small momentum , away from the Zeeman energy to . This is seen by the trivial Hartree self-energy
where we use a zig-zag line to diagrammatically represent the local interaction, and , and is the expectation value of spin- spinon density in the presence of magnetic field. Consequently, for the Larmor theorem to be obeyed, there must be a collective transverse spin mode at small momenta.
This collective spin mode is most conveniently described by the Random Phase Approximation (RPA), which corresponds to a standard resummation of particle-hole ladder diagrams Aronov (1977). For the particular case of a momentum-independent contact interaction, one has
where the fermion lines correspond to the spinon Green’s functions including the Hartree shift (3), and in this approximation is the bare susceptibility bubble, calculated using these functions. We will however use the second line in Eq. (Spinon waves in magnetized spin liquids) to later define the RPA approximation even when gauge field corrections (but not the local interaction ) are included in . For the moment, we simply evaluate the bare susceptibility,
Here are bosonic and fermionic Matsubara frequencies, respectively. A simple calculation, followed by analytical continuation , gives
where square-roots are defined when their arguments are positive. The real/imaginary spin susceptibility describes domains outside/inside two-spinon continuum in the plane, correspondingly. At
and therefore : the position of the two-spinon continuum is renormalized by the interaction shift. However, inserting (7) in the RPA formula (Spinon waves in magnetized spin liquids) one finds that the RPA successfully recovers Larmor theorem at zero momentum for the interacting -invariant system,
Therefore the contribution at is solely from the collective mode, with no spectral weight from the continuum at . Dispersion of the collective spin mode is obtained with the help of (Spinon waves in magnetized spin liquids) and ,
For small the collective mode is dispersing downward quadratically , while in the opposite limit it approaches the low boundary of the two-spinon continuum, . Retaining quadratic in terms in (5) will lead to the termination of the collective mode at some at which the spin wave enters the two-spinon continuum.
This physics is not unique to spin liquids but applies to paramagnetic metals. Historically, this spin wave mode was predicted by Silin for non-ferromagnetic metals in 1958 within Landau Fermi liquid theory Silin (1958, 1959); Platzman and Wolff (1967); Leggett (1970), and observed via conduction electron spin resonance (CESR) in 1967 Schultz and Dunifer (1967). At the time, this observation was considered to be one of the first proofs of the validity of the Landau theory of Fermi-liquids Platzman and Wolff (1973). Unlike the more well-known zero sound Abel et al. (1966), an external magnetic field is required in order to shift the particle-hole continuum up along the energy axis to allow for the undamped collective spin wave to appear outside the particle-hole continuum, in the triangle-shaped window below it. Second order in the interaction corrections (beyond the ladder series) do cause damping of this spin mode Meyerovich and Musaelian (1994); Golosov and Ruckenstein (1995).
However, in the spin liquid, there is an additional branch of low energy excitations due to the gauge field, dispersing as . The very flat dispersion of the gauge excitations suggests it may act as a momentum sink, so that, for example, an excitation consisting of a particle-hole pair plus a gauge quantum may exist in the “forbidden” region where the bare particle-hole continuum vanishes and the spin wave mode lives. It is therefore critical to understand the effect of the gauge interactions upon the dynamical susceptibility. To this end, we consider the dressing of the particle-hole bubble by gauge propagators. Guided by the above thinking, we expect that it is sufficient to consider all diagrams with a single gauge propagator (denoted by wavy line). We obtain , with the correction term (see SM ())