Quasiparticle breakdown in a quantum spin liquid.
ABSTRACT Much of modern condensed matter physics is understood in terms of elementary excitations, or quasiparticles--fundamental quanta of energy and momentum. Various strongly interacting atomic systems are successfully treated as a collection of quasiparticles with weak or no interactions. However, there are interesting limitations to this description: in some systems the very existence of quasiparticles cannot be taken for granted. Like unstable elementary particles, quasiparticles cannot survive beyond a threshold where certain decay channels become allowed by conservation laws; their spectrum terminates at this threshold. Such quasiparticle breakdown was first predicted for an exotic state of matter--super-fluid 4He at temperatures close to absolute zero, a quantum Bose liquid where zero-point atomic motion precludes crystallization. Here we show, using neutron scattering, that quasiparticle breakdown can also occur in a quantum magnet and, by implication, in other systems with Bose quasiparticles. We have measured spin excitations in a two-dimensional quantum magnet, piperazinium hexachlorodicuprate (PHCC), in which spin-1/2 copper ions form a non-magnetic quantum spin liquid, and find remarkable similarities with excitations in superfluid 4He. We observe a threshold momentum beyond which the quasiparticle peak merges with the two-quasiparticle continuum. It then acquires a finite energy width and becomes indistinguishable from a leading-edge singularity, so that excited states are no longer quasiparticles but occupy a wide band of energy. Our findings have important ramifications for understanding excitations with gapped spectra in many condensed matter systems, ranging from band insulators to high-transition-temperature superconductors.
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ABSTRACT: We determine the dynamical dimer correlation functions of quantum dimer models at the Rokhsar-Kivelson point on the bipartite square and cubic lattices and the non-bipartite triangular lattice. Based on an algorithmic idea by Henley, we simulate a stochastic process of classical dimer configurations in continuous time and perform a stochastic analytical continuation to obtain the dynamical correlations in momentum space and the frequency domain. This approach allows us to observe directly the dispersion relations and the evolution of the spectral intensity within the Brillouin zone beyond the single-mode approximation. On the square lattice, we confirm analytical predictions related to soft modes close to the wavevectors (pi,pi) and (pi,0) and further reveal the existence of shadow bands close to the wavevector (0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at (pi,pi,pi) is found, as predicted by the single mode approximation. The soft mode has a quadratic dispersion at very long wavelength, but crosses over to a linear behavior very rapidly. We believe this to be the remnant of the linearly dispersing "photon" of the Coulomb phase. Finally the triangular lattice is in a fully gapped liquid phase where the bottom of the dimer spectrum exhibits a rich structure. At the M point the gap is minimal and the spectral response is dominated by a sharp quasiparticle peak. On the other hand, at the X point the spectral function is much broader. We sketch a possible explanation based on the crossing of the coherent dimer excitations into the two-vison continuum.12/2007;
[show abstract] [hide abstract]
ABSTRACT: We determine the dynamical dimer correlation functions of quantum dimer models at the Rokhsar-Kivelson point on the bipartite square and cubic lattices and the non-bipartite triangular lattice. On the basis of an algorithmic idea by Henley, we simulate a stochastic process of classical dimer configurations in continuous time and perform a stochastic analytical continuation to obtain the dynamical correlations in momentum space and the frequency domain. This approach allows us to observe directly the dispersion relations and the evolution of the spectral intensity within the Brillouin zone beyond the single-mode approximation. On the square lattice, we confirm analytical predictions related to soft modes close to the wavevectors (pi, pi) and (pi, 0) and further reveal the existence of shadow bands close to the wavevector ( 0, 0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at (pi, pi, pi) is found, as predicted by the single-mode approximation. The soft mode has a quadratic dispersion at very long wavelength, but crosses over to a linear behavior very rapidly. We believe this to be the remnant of the linearly dispersing 'photon' of the Coulomb phase. Finally the triangular lattice is in a fully gapped liquid phase where the bottom of the dimer spectrum exhibits a rich structure. At the M point the gap is minimal and the spectral response is dominated by a sharp quasiparticle peak. On the other hand, at the X point the spectral function is much broader. We sketch a possible explanation based on the crossing of the coherent dimer excitations into the two-vison continuum.
Quasiparticle breakdown in a quantum spin liquid
Matthew B. Stone1,3, Igor A. Zaliznyak2, Tao Hong3, Collin L. Broholm3,4 and Daniel
1Condensed Matter Sciences Division, Oak Ridge National Laboratory, Oak Ridge, TN
2Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
3Department of Physics and Astronomy, The Johns Hopkins University, Baltimore,
Maryland 21218, USA
4National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
Much of modern condensed matter physics is understood in terms of elementary
excitations, or quasiparticles – fundamental quanta of energy and momentum1,2.
Various strongly-interacting atomic systems are successfully treated as a collection
of quasiparticles with weak or no interactions. However, there are interesting
limitations to this description: the very existence of quasiparticles cannot be taken
for granted in some systems. Like unstable elementary particles, quasiparticles
cannot survive beyond a threshold where certain decay channels become allowed
by conservation laws – their spectrum terminates at this threshold. This regime of
quasiparticle failure was first predicted for an exotic state of matter, super-fluid
helium-4 at temperatures close to absolute zero – a quantum Bose-liquid where
zero-point atomic motion precludes crystallization1-4. Using neutron scattering,
here we show that it can also occur in a quantum magnet and, by implication, in
other systems with Bose-quasiparticles. We have measured spin excitations in a
two dimensional (2D) quantum-magnet, piperazinium hexachlorodicuprate
(PHCC),5 in which spin-1/2 copper ions form a non-magnetic quantum spin liquid
(QSL), and find remarkable similarities with excitations measured in superfluid
4He. We find a threshold momentum beyond which the quasiparticle peak merges
with the two-quasiparticle continuum. It then acquires a finite energy width and
becomes indistinguishable from a leading-edge singularity, so that excited states
are not quasiparticles but occupy a wide band of energy. Our findings have
important ramifications for understanding phenomena involving excitations with
gapped spectra in many condensed matter systems, including high-transition-
Although of all the elements only liquid helium fails to crystallize at 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 among quantum spins in magnetic crystals5,7-10. The organo-metallic
material PHCC is an excellent physical realization of a QSL in a 2D Heisenberg
antiferromagnet (HAFM). Its Cu2+ spins are coupled through a complex super-exchange
network in the crystalline a-c plane forming an array of slightly skewed anisotropic
spin-½ ladders10 coupled by frustrated interactions5. The spin excitations in PHCC have
a spectral gap ∆s ≈ 1 meV and nearly isotropic 2D dispersion in the (h0l) plane with a
bandwidth slightly larger than ∆s. In the absence of a magnetic field, only 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 the quasi-particle dispersion reaches
the threshold for two-particle decay and interferes destructively with the continuum.
The properties of superfluid4
4He 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 down1-3. 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 (phonons) ends when the phonon is able to decay into
two “rotons”11-15. These rotons are phonons with roughly quadratic dispersion that occur
near the dispersion minimum, ∆ ≈ 0.74 meV and wavevector Q ≈ 2 Å-1, cf. Fig. 1a.
Spontaneous decays provide the only mechanism that destroys quasiparticles in 4He at T
= 0. However, due 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, so that the quasiparticle spectrum does not continue beyond the threshold1-3.
Figure 1 Liquid helium excitation spectrum. a Excitation spectrum in 4He for 1.5
≤ T ≤ 1.8 K from inelastic neutron scattering measurements13,16. Solid black line
is dispersion from Ref. 13; red circle with cross indicates spectrum termination
point at Q= Qc and ℏω = 2∆. White line is Feynman-Cohen bare dispersion in
absence of decays18, and horizontal red line at ℏω = 2∆ shows onset of two-
roton states for ℏω ≥ 2∆. b Inset depicts excitations near termination point, at Q
= 2.6 Å-1 ≈ Qc, for several temperatures13.
The excitation spectrum of superfluid 4He as probed by neutron scattering13,16 is
shown in Fig. 1a. 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
interactions15,17. Decays modify the “bare” Feynman-Cohen quasiparticle dispersion in
4He (white line in Fig 1a) 18. Instead of terminating where it reaches the energy 2∆, the
quasiparticle spectrum is suppressed to lower energies at Q ≤ Qc, approaching the
threshold E = 2∆ horizontally3 (black line in Fig 1a).
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 1D QSLs, such as spin-1 chains with a
spectral gap19. Though the term has not been used in this context, numerical work
suggests that spectrum termination does occur in spin-1 HAFM spin chains20,21. Its
observation through neutron scattering, however, is hindered by small scattering cross
sections at the appropriate wavevectors. In the spin-1 chain system NENP scattering
becomes undetectable when the single-particle excitation meets the non-interacting two-
particle continuum22, either due to decays or to 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 two particle continuum23, which may be a result of inter-
Figure 2 Magnetic excitation spectrum at T=1.4 K in PHCC. a Background
corrected intensity along the (½, 0, -1 - l) and (h, 0, -1-h) directions. A δℏω =
0.25 meV running average was applied to each constant wave-vector scan,
retaining the actual point density of the acquired data. Black line is previously
determined single-magnon dispersion5. White lines are bounds of two-magnon
continuum calculated from this dispersion. Red circle with cross indicates 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)
correspond to 0.8 ≤ ℏω ≤ 5.5 meV, black circles (quasiparticle) to 0.8 ≤ ℏω ≤ 3
meV, and blue diamonds (continuum) to 3 ≤ ℏω ≤ 5.5 meV. c resolution
corrected half width at half maximum of the lower energy peak throughout the
range of wavevector transfer for high resolution (solid points) and low resoution
(open points) data.
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 meV5.
Here we present data for energies ℏω ≤ 7 meV and for wavevectors along the (½, 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,
for Qc = (hc, 0, -1-hc) with hc ≈ 0.15 near the magnetic Brillouin zone (BZ) boundary.
The first frequency moment24 integrated over different ranges of energy transfer shown
in Fig. 2b reveals how oscillator strength is transferred from the quasi-particle excitation
to the multiparticle continuum, in analogy to what is observed in 4He13.
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 << ∆s. For h ≥ 0.2, Figs. 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
density20. The threshold obtained from such data analysis is slightly higher than the
calculated low-E 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.
Figure 3 Individual constant wavevector scans of PHCC along the (h, 0, -1-h)
direction at T = 1.4 K. Identical vertical scales emphasize variation in lineshape
in vicinity of hc. Solid lines are 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
with ε2(Q) defined by the
calculated upper boundary of the two-particle continuum (white line in Fig.
2a); ε1(Q) and A were refined by the least square fitting. For h ≤ 0.15 this
description fails and the spectrum is fitted by two superimposed DHO spectra,
(green shaded regions). The
Gaussian representing elastic incoherent nuclear scattering is also included at
all wavevectors. Dashed lines and solid symbols in panels a-c show data on a
one-fifth intensity scale.
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 (Figs. 3d-f). We parameterize this spectrum by
the overlapping response of two damped harmonic oscillators (DHO). 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 excitation for h ≤ 0.15. The
width increases towards the BZ 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. 1b,13,14 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 shown in
Figs. 4 e-h, similarly indicate that proximity to the two-particle continuum enhances
thermal damping: the peak whose energy is approximately 20 K is severely broadened
already at T=10 K (Fig. 4 f). Its thermal broadening resembles that of the Q = (0.5,0,-
1.5) gap mode, which is shown in Fig. 4 a-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 spectrum termination cannot occur25. Temperature-induced
damping in that case is stronger for the lower-energy gap mode than for quasiparticles at
the top of the dispersion curve, i.e. 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 (inset Fig. 4a), 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
Figure 4 Temperature dependent energy spectra for PHCC 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 detailed balance constraint
S Q ω
Γ +− Γ ++
dependence of the relaxation rate, Γ, for the lower energy peak at both
wavevectors is shown in inset to a. Line corresponds to exponentially activated
behaviour with ∆=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 frames d
and h indicate resolution (width of the low energy peak).
In summary, quasiparticle spectrum termination as seen in superfluid 4He can also
occur in other condensed matter systems, quantum magnets in particular. Dramatic
changes that we observe in the spectrum of magnetic excitations in PHCC provide
compelling evidence for its existence 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 line-shape 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, rotons1-3,
magnetorotons26 and heavy electrons and holes in condensed matter physics to the
quasiparticles of the quark gluon plasma and the various unstable particles and
resonances in the standard model of particle physics27. Rarely, however, do experiments
offer as detailed a view of quasiparticle decay as the present results in a 2D organo-
metallic spin liquid. Our findings show that an analysis of excitations in terms of
quasiparticles with a well-defined dispersion relation can be at fault 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
Neutron scattering measurements of 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 1º.
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))
monochrometer and the sample. A 138’ radial collimator was used between the sample
and a horizontally focusing PG (002) analyzer with an angular acceptance of 5o
horizontally and 6o vertically. A cooled Be filter was in place after the sample.
Measurements in Fig. 4 employed an additional PG filter before the sample. Data in
Figs. 2 and 3 [Fig. 4] were acquired with 5 [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 analyzer 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.
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Acknowledgements We acknowledge discussions with L. Passell, J. Tranquada, A.
Abanov, M. Zhitomirsky, A. Tsvelik, A. Chitov, and M. Swartz. Work at BNL and
ORNL was supported by the Office of Science, U.S. Department of Energy, under
contracts DE-AC02-98CH10886 and DE-AC05-00OR2272. Work on SPINS and at
JHU was supported by the U.S. National Science Foundation. We are grateful to B. Fåk
for permission to reproduce the results of his measurements on 4He in Fig. 1.
Correspondence and requests for materials should be addressed to I.Z.