Main

Although of all the elements only liquid helium fails to crystallize at temperature T = 0, quantum liquids are quite common in condensed matter. Metals host electron Fermi liquids, and superconductors contain Bose liquids of Cooper pairs. Trapped ultracold atoms can also form quantum liquids, and some remarkable new examples were recently identified in magnetic crystals5,7,8,9,10. The organometallic material PHCC is an excellent physical realization of a quantum spin liquid (QSL) in a two-dimensional (2D) Heisenberg antiferromagnet (HAFM). Its Cu2+ spins are coupled through a complex network of orbital overlaps, and form an array of slightly skewed anisotropic spin-1/2 ladders10 in the crystalline a–c plane with highly frustrated super-exchange interactions5. The spin excitations in PHCC have a spectral gap Δs ≈ 1 meV and nearly isotropic 2D dispersion in the (h0l) plane with a bandwidth slightly larger than Δs. In the absence of a magnetic field, only the short-range dynamic spin correlations typical of a liquid exist: the spin gap precludes long-range magnetic order down to T = 0. Here we explore magnetic excitations in PHCC via inelastic neutron scattering and compare the results with similar measurements in the quantum fluid 4He, emphasizing the effects where quasiparticle dispersion reaches the threshold for two-particle decay and interferes destructively with the continuum.

The properties of superfluid 4He (ref. 4) can be explained by considering Bose quasiparticles with a finite-energy minimum (an energy gap) in their spectrum1,2. However, in a Bose quantum liquid, a spectral gap can produce an energy-momentum threshold where the quasiparticle description breaks down1,2,3. Beyond this threshold, single-particle states are no longer approximate eigenstates of the hamiltonian and the quasiparticle spectrum terminates. Neutron scattering experiments in 4He indicate that the spectrum of quasiparticles (quanta of longitudinal sound waves also called phonons) ends when the phonon is able to decay into two ‘rotons’11,12,13,14,15. These rotons are phonons with roughly quadratic dispersion that occur near the dispersion minimum, which is at energy Δ ≈ 0.74 meV and wavevector Q ≈ 2 Å-1 (compare Fig. 1, main panel). Spontaneous decays provide the only mechanism that destroys quasiparticles in 4He at T = 0. However, owing to the high density of two-roton states, this decay path is so effective that instead of acquiring a finite lifetime, the quasiparticles simply cease to exist. Specifically, the single-particle pole is absent in the Green's function of 4He atoms for Q > Qc (where Qc is a threshold wavevector), so that the quasiparticle spectrum does not continue beyond the threshold1,2,3.

Figure 1: Liquid helium excitation spectrum S(Q, ) from inelastic neutron scattering measurements.
figure 1

Main panel, excitation spectrum in 4He for 1.5 ≤ T ≤ 1.8 K. Data for wavevector Q ≥ 2.3 Å-1 are reproduced from ref. 13, data at smaller Q are from C.L.B. and S.-H. Lee, unpublished results. Solid black line, dispersion from ref. 13; red circle with cross, spectrum termination point at Q = Qc and ℏω = 2Δ. White line, Feynman–Cohen bare dispersion in absence of decays17; horizontal red line at ℏω = 2Δ, onset of two-roton states for ℏω ≥ 2Δ. Inset, excitations near termination point, at Q = 2.6 Å-1 ≈ Qc, for several temperatures13.

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The excitation spectrum of superfluid 4He as probed by neutron scattering is shown in Fig. 1, main panel. One can see the roton minimum in the dispersion and the spectrum termination point at Qc ≈ 2.6 Å-1. Near Qc the phonon hybridizes with two-roton excitations, its dispersion flattens, and spectral weight is transferred to the multiparticle continuum13,15. While a smeared maximum occurs at the leading edge of the continuum for Q > Qc and appears to continue the quasiparticle dispersion relation, it is instead ascribed to a two-roton bound state (resonance) resulting from roton–roton interactions15,16. Decays modify the ‘bare’ Feynman–Cohen quasiparticle dispersion in 4He (white line in Fig. 1, main panel)17. Instead of terminating where it reaches the energy 2Δ, the quasiparticle spectrum is suppressed to lower energies at Q ≤ Qc, approaching the threshold energy ℏω = 2Δ horizontally3 (black line in Fig. 1, main panel).

The generality of the physics underlying quasiparticle breakdown in 4He suggests that similar effects may occur in other quantum liquids. The quasiparticle instability in 4He relies on the isotropic nature of the fluid: since the spectrum only depends on |Q|, the roton minimum produces a strong singularity in the density of states (DOS). For QSLs on a crystalline lattice, the DOS available for quasiparticle decays is enhanced by the absence of dispersion in certain directions that occurs in low-dimensional systems (D < 3) and in systems with competing interactions (frustration). Quasiparticle breakdown effects should thus be strongest in one-dimensional (1D) QSLs, such as spin-1 chains with a spectral gap18. Though the term has not been used in this context, numerical work suggests that spectrum termination does occur in spin-1 HAFM spin chains19,20. Its observation through neutron scattering, however, is hindered by small scattering cross-sections at the appropriate wavevectors. In the spin-1 chain system Ni(C2H8N2)2NO2ClO4 (NENP), scattering becomes undetectable when the single-particle excitation meets the non-interacting two-particle continuum21, owing to either decays or a vanishing structure factor. While transformation of magnetic excitations from well-defined quasiparticles to a continuum was observed in the quasi-1D spin-1 HAFM CsNiCl3, it is only seen as an onset of damping beyond a certain momentum threshold, well before the dispersion crosses the lower bound of the projected non-interacting two-particle continuum22, which may be a result of inter-chain interactions.

In contrast to the HAFM spin-1 chain, the structure factor of PHCC is favourable for probing the interaction of magnon quasiparticles with their two-particle continuum. Its effects, however, could be less pronounced because the 2D DOS singularities are weaker. Prior measurements examined magnetic excitations in PHCC below ∼3 meV (ref. 5). Here we present data for energies ℏω ≤ 7 meV and for wavevectors along the (1/2, 0, l) and (h, 0, -1 - h) directions, elucidating both single- and multiparticle excitations in this 2D QSL. Data shown in Fig. 2a and selected scans shown in Fig. 3 demonstrate clear similarities to the spectrum of superfluid 4He. The one-magnon dispersion reaches the lower boundary of the two-magnon continuum, ℏω2m(Q) = minq{ℏω(q) + ℏω(Q - q)}, for Qc = (hc, 0, -1 - hc) with hc ≈ 0.15 near the magnetic Brillouin zone boundary. The first frequency moment23 integrated over different ranges of energy transfer shown in Fig. 2b reveals how oscillator strength is transferred from the quasiparticle excitation to the multiparticle continuum, in analogy to what is observed in 4He (ref. 13).

Figure 2: Magnetic excitation spectrum at T = 1.4 K in PHCC.
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a, Background-corrected intensity along the (1/2, 0, -1 - l) and (h, 0, -1 - h) directions. A δℏω = 0.25 meV running average was applied to each constant wavevector scan, retaining the actual point density of the acquired data. Black line, previously determined single-magnon dispersion5. White lines, bounds of two-magnon continuum calculated from this dispersion. Red circle with cross, the point where the single-particle dispersion relation intersects the lower bound of the two-particle continuum. b, First frequency moment of measured scattering intensity integrated over different energy ranges. Red squares (total), 0.8 ≤ ℏω ≤ 5.5 meV; black circles (quasiparticle), 0.8 ≤ ℏω ≤ 3 meV; blue diamonds (continuum), 3 ≤ ℏω ≤ 5.5 meV. c, Resolution-corrected half-width at half-maximum (HWHM) of the lower energy peak throughout the range of wavevector transfer for high resolution (solid points) and low resolution (open points) data. Error bars illustrate systematic error corresponding to 10% uncertainty in the neutron beam collimation used for resolution correction.

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Figure 3: Individual constant-wavevector scans of PHCC along the ( h , 0, -1 - h ) direction at T = 1.4 K.
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Identical vertical scales emphasize variation in lineshape in the vicinity of hc. a–c, Solid lines, fits to single resonant mode (yellow shaded region) plus a higher energy continuum excitation (blue shaded region) convolved with the instrumental resolution function. For wavevectors h ≥ 0.2, higher energy excitations are well represented by a two-particle continuum of the form where Θ is a Heaviside step function and ɛ2(Q) is defined by the calculated upper boundary of the two-particle continuum (white line in Fig. 2a); ɛ1(Q) and A were refined by the least squares fitting. d–f, For h ≤ 0.15 this description fails and the spectrum is fitted by two superimposed damped harmonic oscillator spectra, each in the form of a difference of two lorentzians whose HWHM Γ parameterizes damping, (green shaded regions). The gaussian representing elastic incoherent nuclear scattering is also included at all wavevectors. Dashed lines and solid symbols in a–c show data on a one-fifth intensity (I) scale. Error bars show statistical uncertainty estimated as the square root of total neutron count measured at each point.

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A change in the character of the excitation spectrum near hc is also apparent in Fig. 3, which shows the energy-dependent magnetic scattering for wavevectors along the (h, 0, -1 - h) direction at T ≈ 1.4 K≪Δs. For h ≥ 0.2, Fig. 3a–c, there are two distinct contributions, a resolution-limited quasiparticle peak at lower energy and a broad feature with a sharp onset at higher energy, which we associate with the two-particle continuum. This continuum is well described by a square-root singularity above an energy threshold, typical for two-particle scattering governed by a diverging spectral density19. The threshold obtained from such data analysis is slightly higher than the calculated lower boundary of the two-magnon continuum (white line in Fig. 2a), and is close to the lowest energy of two-particle states involving gap mode magnons with a diverging DOS. Alternatively, the shift could indicate magnon repulsion.

For h ≤ 0.15, the quasiparticle peak joins the continuum to form a complex spectral feature that extends from 2.5 to 4.5 meV (Fig. 3d–f). We parameterize this spectrum by the overlapping response of two damped harmonic oscillators. The onset of scattering occurs well above the lowest energy for two non-interacting magnons (dashed white line in Fig. 2a), which indicates significant interactions. While the lower energy peak that appears to continue the quasiparticle dispersion in PHCC carries more spectral weight than the corresponding resonance at Q > Qc in superfluid 4He, it also has a measurable energy width as quantified in Fig. 2c. This demonstrates that a decay mechanism abruptly becomes accessible to the low energy excitations for h ≤ 0.15. The width increases towards the Brillouin zone boundary, h = 0, where the peak at the leading edge can be described by a non-quasiparticle square root singularity as used for the continuum at h ≥ 0.2, or as an unstable non-dispersive resonance below the continuum.

The temperature dependence of scattering in 4He for Q between 2.4 and 2.6 Å-1, Fig. 1 inset13,14, provides additional evidence of quasiparticle spectrum breakdown. Data in PHCC for Q = (0.15, 0, -1.15) where the one and two magnon states converge (Fig. 4e–h) similarly indicate that proximity to the two-particle continuum enhances thermal damping: the peak whose energy is approximately 20 K is already severely broadened at T = 10 K (Fig. 4f). Its thermal broadening resembles that of the Q = (0.5, 0, -1.5) gap mode, which is shown in Fig. 4a–d. This differs from observations in copper nitrate, a 1D QSL with weak dispersion where the one-magnon band lies well below the two-magnon continuum and decays cannot occur24. Temperature-induced damping in that case is stronger for the lower-energy gap mode than for quasiparticles at the top of the dispersion curve; that is, heating mainly affects energy levels that become thermally populated. For PHCC, damping near the top and bottom of the band is governed by the same thermal population (Fig. 4a inset), consistent with the idea that high-energy excitations decay into gap-mode quasiparticles. As their thermal population increases, the probability of stimulated emission by the high-energy excitations also grows.

Figure 4: Temperature dependent energy spectra for PHCC.
figure 4

Data are shown at Q = (0.5, 0, -1.5) (a–d) and Q = (0.15, 0, -1.15) (e–h). Solid lines for T = 1.5 K in a and b are fits as described in Fig. 3. Solid lines for T ≥ 10 K are fits to the following response function satisfying a detailed balance constraint: S(Q,ω) = Γ/π(1/(Γ2 + (ℏω - ℏω0)2) - 1/(Γ2 + (ℏω + ℏω0)2))1/(1 - exp (- β ℏω)). The temperature dependence of the relaxation rate, Γ, for the lower energy peak at both wavevectors is shown in the inset to a. The line corresponds to exponentially activated behaviour with Δ = 2.0 meV. Coloured areas below peaks indicate the assignment of different contributions to the spectra. Dashed lines indicate incoherent elastic nuclear scattering. Solid (dashed) horizontal bars in d and h indicate resolution (width of the low energy peak). Error bars show statistical uncertainty estimated as the square root of total neutron count measured at each point.

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In summary, quasiparticle spectrum termination as seen in superfluid 4He can also occur in other condensed matter systems, and in quantum magnets in particular. The dramatic changes observed in the spectrum of magnetic excitations in PHCC provide compelling evidence for such a phenomenon in the 2D QSL. The termination point is marked by rapid transfer of intensity from the magnon peak to the continuum at higher energies and by an abrupt appearance of damping. Although in PHCC the damped peak at the leading edge of magnetic scattering carries more intensity than the analogous peak in superfluid 4He, the lineshape and temperature dependence of post-threshold excitations in these two very different quantum liquids are remarkably similar.

Quasiparticles are ubiquitous in nature, ranging from phonons, magnons, rotons1,2,3, magnetorotons25 and heavy electrons and holes in condensed matter physics to quasiparticles of the quark gluon plasma and various unstable particles and resonances in the standard model of particle physics26. Rarely, however, do experiments offer as detailed a view of quasiparticle decay as the present results in a 2D organometallic spin liquid. Our findings show that an analysis of excitations in terms of quasiparticles with a well-defined dispersion relation can fail beyond a certain energy-momentum threshold where the quasiparticles break down. This has important implications for a variety of condensed matter systems, in particular for other QSLs such as lamellar copper oxide superconductors, where spin excitations above a gap are considered as possible mediators of electron pairing and high-temperature superconductivity27.

Methods

Neutron scattering measurements of (C4H12N2)Cu2Cl6 (PHCC) were performed using the SPINS cold neutron triple axis spectrometer at the NIST Center for Neutron Research. Four deuterated PHCC crystals5 with a total mass of 7.5 g were co-aligned to within 1°. Energy scans were acquired by varying the incident beam energy for fixed monitor counts in a low-efficiency detector between the pyrolytic graphite (PG (002)) monochromator and the sample. A 138′ radial collimator was used between the sample and a horizontally focusing PG (002) analyser with an angular acceptance of 5° horizontally and 6° vertically, which was followed by a matching single-channel, high-efficiency cylindrical detector. A cooled Be filter was placed after the sample. Measurements in Fig. 4 employed an additional PG filter before the sample. Data in Figs 2 and 3 were acquired with 5 meV fixed final energy, and data in Fig. 4 at 3.7 meV fixed final energy. Projected full-width at half-maximum energy resolution of these configurations at ℏω = 0 is 0.18 meV and 0.11 meV, respectively. A wavevector independent fast-neutron background was measured by shielding the analyser entrance with cadmium. A wavevector-dependent thermal neutron background arising predominantly from incoherent phonon scattering was measured at T = 100 K and scaled using the thermal detailed balance factor for use as a low-temperature non-magnetic background. These backgrounds were subtracted from all data presented.