of 29
Direct Measurement of the Soft Mode Driving a Quantum Phase
Transition
M. Libersky,
1
R.D. McKenzie,
2
D.M. Silevitch,
1
P.C.E. Stamp,
2, 3
and T.F. Rosenbaum
1,
1
Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, Pasadena California 91125, USA
2
Department of Physics and Astronomy,
University of British Columbia, Vancouver,
British Columbia V6T 1Z1, Canada
3
Pacific Institute of Theoretical Physics,
University of British Columbia, Vancouver,
British Columbia V6T 1Z1, Canada
(Dated: January 14, 2021)
1
arXiv:2101.05143v1 [cond-mat.str-el] 13 Jan 2021
Abstract
Quantum phase transitions in spin systems are supposed to be accompanied by a soft collective
mode, which has not been seen in experiments. Here, we directly measure the low energy excitation
modes of a well-known realization of the Ising model in transverse field, LiHoF
4
, using microwave
spectroscopy techniques to probe energies well below what is accessible via neutron scattering
experiments. Instead of the single excitation expected for a simple quantum Ising system, we find
and characterize a remarkable array of ‘electronuclear’ modes, arising from coupling of the spin-1/2
Ising electronic spins to a bath of spin-7/2 Ho nuclear spins. The lowest-lying electronuclear mode
softens at the approach to the quantum critical point from below and above, a softening that can be
quenched with the application of a longitudinal magnetic field. The electronuclear mode structure
has direct implications for the Ising systems that serve as the building blocks of adiabatic quantum
computers and quantum annealers.
POPULAR SUMMARY
Near a quantum phase transition, the mix of competing order parameters and dynamic
fluctuations reveals fundamental aspects of the development of long-range order in a quan-
tum environment. Moreover, many materials of technological import demonstrate unusual
electronic, optical, and magnetic properties that have been ascribed to the close proximity
of quantum critical points. The classic example of a quantum phase transition is the Ising
model in transverse magnetic field (a subject of inquiry across physics, chemistry, neural
nets, economics, and sociology). The mystery addressed here is that any quantum phase
transition in a Quantum Ising system must be accompanied by a “soft mode,” and yet this
soft mode has not been seen directly in any experiments.
We report the direct observation and characterization of the soft mode in a physical
realization of the Ising model in transverse field. The experiments show a remarkable array
of modes, hitherto inaccessible because they are at too low an energy to be seen in neutron
scattering, but reached here with microwave absorption spectroscopy. We argue that the
modes of interest arise from coupling between electronic and nuclear spins, creating a set
of hybridized electronuclear collective modes, the lowest of which indeed softens at the
Correspondence and requests for materials should be addressed to T.F.R.,tfr@caltech.edu
2
quantum critical point. We analyze these modes using a Random Phase Approximation and
find excellent agreement between theory and experiment. These results have implications
for many systems described by the Quantum Ising model – including the current generation
of quantum information processing systems, for which understanding of low-lying states is
crucial.
I. INTRODUCTION
Since they were first discussed nearly 50 years ago [1], quantum phase transitions (QPTs)
have come to occupy a central place in thermodynamics and statistical mechanics [2]. The
Ising model in a transverse field, the simplest example of a QPT, has been applied to systems
across all of physics [2–5], as well as in the theory of neural networks [6], chemistry [7],
and economics [8]. Copious current work focuses on the application to adiabatic quantum
computing [9] and quantum optimization [10–13]. The model finds its clearest natural
realization in the LiHoF
4
rare earth magnet [14], and experiments over the years in this
system (and in the associated random magnet LiHo
x
Y
1
x
F
4
) have probed QPTs in great
detail [15–17]. However, a key feature has been missing in experiments on this and other
systems. The low-energy excitation spectrum of the system must contain a ‘soft mode’ that
drives the QPT and whose energy goes to zero precisely at the QPT [1]. In this paper
we describe the observation of that soft mode in LiHoF
4
using microwave spectroscopy
at energies well below those achievable in neutron experiments. Instead of the single low-
energy excitation expected for a simple quantum Ising system, we find and characterize both
experimentally and theoretically a remarkable array of modes arising from coupling of the
spin-1/2 Ising electronic spins to a bath of spin-7/2 Ho nuclear spins. Additionally, we apply
a longitudinal magnetic field and map out the suppression of the mode softening, comparing
our results to RPA calculations. Our results show that in a real adiabatic quantum computer
with environmental spin bath modes, the low-energy states will be hybridized Ising/spin bath
modes.
The quantum Ising model [18, 19] describes a set of mutually interacting quantum spins in
a potential that orients each spin along an easy axis, with quantum fluctuations controlled
by an external magnetic field applied perpendicular to the easy axis. This is one of the
simplest systems in many-body physics. The spin-1/2 Ising electronic spins
S
i
have the
3
Hamiltonian
H
0
=
i<j
J
ij
S
z
i
S
z
j
Γ
i
S
x
i
,
(1)
where the longitudinal couplings
J
ij
compete with the transverse field term Γ. This com-
petition drives a zero-temperature QPT when
J
0
Γ, where
J
0
is the dominant longitudinal
coupling. Away from the QPT, the system has a single collective mode: a spin wave exci-
tation (‘magnon’) with energy smaller than either
zJ
0
or
z
Γ, where
z
is the coordination
number. The magnon energy goes to zero at the QPT. In real solids the Ising electronic
spins also couple to nuclear spins.
The magnetism in LiHoF
4
is carried by the Ho
3+
spins, with the Ising behavior arising
from the crystal fields of the surrounding lattice [20–23]. The detailed form of the Hamil-
tonian, along with numerical values for all the couplings, can be found in Appendix A. The
crystal field potential is shown in Fig. 1(a). At low energy, the electronic spin spectrum is an
effective Ising doublet with the spins oriented along the crystallographic c-axis. Hyperfine
coupling of each holmium electronic spin to the I=7/2 holmium nuclear spins yields the ef-
fective energy level structure for each Ho ion shown in Fig. 1(b). There are also transferred
hyperfine couplings to the Li and F nuclei which can be ignored here. Measurements of
the single Ho-spin dynamics, via replacement of 99.8% of the holmium with non-magnetic
yttrium, experimentally confirms this picture [24]. The measured ferromagnet-paramagnet
phase diagram is consistent with a mean-field calculation that incorporates an effective
rescaling of the electronic spins due to hyperfine interactions at temperatures below 0.4 K
(Fig. 1(c)) [14].
A low temperature truncation of the single ion Hamiltonian shows that the two lowest
electronic crystal field eigenstates split into two low-energy groups, each group containing
eight electronuclear eigenstates, with all 16 states well separated in energy from the higher-
lying crystal field excitations. These 16 eigenstates are used to calculate the low-energy
spectrum of the full lattice Hamiltonian from the poles of the longitudinal Green’s function
in the Random Phase Approximation (RPA). At zero temperature, the two groups of elec-
tronuclear states become the upper and lower bands in the extended crystal, in the form
of collective ‘electronuclear modes’ that are delineated in Fig. 2. The upper band contains
eight of these modes – notice that the lowest mode in this upper band splits away from
the others above a transverse field
H
x
= 3 T (where Γ
H
2
x
). This particular mode can
4
FIG. 1.
Low-Energy Electronuclear Spectroscopy in LiHoF
4
. (a) The energy of a single
electronic Ho spin in LiHoF
4
, plotted as a function of direction on the Bloch-sphere, and calculated
using a coherent state basis (see Appendix for details) (b) Energy-level diagram of the splitting by
hyperfine interactions of the two lowest electronic states for a single Ho ion in LiHoF
4
. (c) Exper-
imental ferromagnet/paramagnet phase diagram for LiHoF
4
, showing a classical phase transition
at
T
C
= 1
.
53 K and a transverse-field driven quantum phase transition at
H
C
= 4
.
93 T (adapted
from Ref. 14). (d) Multi-mode loop-gap resonator for generating ac magnetic fields at frequencies
ranging from 900 MHz to 4.5 GHz (3 to 19
μ
eV). Magnetic field lines join the circular loops to
the square sample chamber. The corresponding electric fields (not shown) are largely confined to
the narrow gaps, reducing dielectric heating of the sample. The lowest-frequency magnetic mode
is illustrated here. (e) Magnified view of the resonator sample chamber, showing directions of
ac and dc magnetic fields with respect to the crystal axes of the sample. The Ising axis is the
crystallographic
c
-axis.
5
0
1
2
3
4
5
6
H
x
(T)
0
50
100
150
Energy (GHz)
FIG. 2.
Electronuclear collective modes in LiHoF
4
, Random Phase Approximation (RPA)
calculation for a long cylinder of LiHoF
4
at zero temperature. The modes are plotted as a function
of transverse field
H
x
, using the parameters given in the text. The lowest mode goes soft at the
quantum critical field
H
C
5
.
3 T. The modes divide into upper and lower ‘bands’; at high fields
a mode splits off from the upper band, reaching a minimum energy when
H
x
=
H
C
.
be identified with an excitation measured via neutron scattering [25]. In the absence of a
nuclear spin bath, this excitation would have been the magnon soft mode noted above [25].
The seven collective mode excitations in the lower band are spaced below approximately
30 GHz, with energy differences of approximately 4 GHz and energies decreasing near the
quantum critical point (QCP). This makes them difficult to see in any neutron scattering
experiment. The RPA analysis (for details see Appendix) predicts that the lowest of these
electronuclear excitations should soften completely to zero energy at a QCP. If one goes
beyond the RPA to a mode-coupling analysis, the QCP is found to be stable up to fourth-
order fluctuations in the T = 0 theory [26]. Hence, according to the theory, a true QPT
should persist even in the presence of a local coupling between the electronic spins and a
nuclear spin bath, and this mode should still soften completely at the QCP. To establish the
6
existence of this mode softening at a QPT has long been a goal of experiments.
II. METHODS
To probe for excitations in the lower branch of modes, with energies too low to be seen by
inelastic neutron scattering, we turn to microwave spectroscopy, applying a small ac magnetic
field, with frequency ranging from 0.9 to 5.0 GHz, in addition to the large static transverse
field. We then measure the ac absorption as a function of frequency and applied transverse
magnetic field. To achieve the needed sensitivity, a resonator structure is required to amplify
the applied ac signal. The requirements of high quality factor
Q
and field homogeneity on the
one hand, and the physical size constraints imposed by the cryogenics and superconducting
magnetic solenoid on the other, rule out the use of planar waveguide and cavity resonator
structures. Instead, we adopt a loop-gap resonator (LGR) design [27, 28] (Figs. 1(d,e)) that
can be tuned over a wide range of frequencies while maintaining a high
Q
factor and an ac
magnetic field highly concentrated on the sample volume.
LGRs are effective 3D lumped-element structures, with a uniform magnetic field in the
loops and an electric field largely confined to the thin gaps. For measurements above 1.5
GHz, the 4-loop 3-gap design illustrated in Fig. 1(d) allows the simultaneous probe of two
resonant modes [28]; measurements between 900 MHz and 1.5 GHz used a 2-loop 1-gap de-
sign optimized for lower frequencies. Both designs can have their resonant frequencies tuned
by approximately a factor of four by varying the gap capacitance via partial or complete
filling with pieces of sapphire wafer, thereby increasing the dielectric constant.
Each resonator was fabricated from a single block of oxygen-free high-conductivity
(OFHC) copper using wire electrical-discharge machining (EDM) to eliminate potential
losses arising from seams in two-piece designs. The resonator, contained inside an OFHC
copper shield, was attached to the cold finger of a helium dilution refrigerator. Stub anten-
nas attached to stainless-steel semirigid coaxial cables provide transmit and receive ports.
The transmission coefficient,
s
21
, was measured using a vector network analyzer (Tektronix
TTR506A). The measurements presented were taken at an incident power level of approx-
imately 1
μ
W (-30 dBm) at the resonator, for which sample heating was negligible and
the sample was demonstrated—by varying power levels—to be well into the linear response
regime. Static magnetic fields were supplied using a 8 T superconducting solenoid and
7
a home-built superconducting Helmholtz pair oriented parallel to the Ising axis, allowing
alignment of the applied transverse field to within 0
.
5
of the crystalline
a
-axis. The LiHoF
4
sample used was a single crystal measuring 1
.
8
×
2
.
5
×
2
.
0 mm
3
.
Zero-temperature RPA calculations [26] found that the spectral weight of the soft mode
is expected to be strongest in the
χ
zz
configuration. This counter-intuitive result is a conse-
quence of the crystal fields, and is one reason why the mode has eluded previous identifica-
tion, including recent experiments that probed the magnetic resonance behavior in LiHoF
4
employing a
χ
yy
geometry, with the ac magnetic field mutually perpendicular to both the
Ising axis and the dc transverse magnetic field [29]. Thus, in the present experiments the
resonator and sample are oriented with the ac magnetic field applied along the Ising axis
of the sample, a solenoid along the transverse axis, and a split coil oriented along the Ising
axis. In this geometry, crystal field effects reduce the ac soft mode absorption along the
y
direction to zero at the QCP.
III. RESULTS AND DISCUSSION
We plot in Figs. 3 and 4 the measured transmission of single-crystal LiHoF
4
in loop
gap resonators tuned to different frequencies at
T
= 55 mK. The high
Q
of the resonator
and large filling factor of the LiHoF
4
crystal inside the resonator enhances the coupling
between the incident microwave field and the spins. With the resonator tuned to the lowest
experimentally accessible frequency of 930 MHz, we focus in Fig. 3 on the lowest energy
‘soft mode’ excitation in the sample. In this regime, the field-dependent evolution of the
cavity resonant frequency is driven primarily by the change in the static susceptibility of
the LiHoF
4
crystal. When the resonant frequency coincides with the energy of the soft
mode, the absorption is enhanced, resulting in a peak in the inverse quality factor 1
/Q
of
the resonator (insets). By varying the resonant frequency of the resonator via insertion of
sapphire wafers into the gap, we then can track the soft mode close to the QCP. As shown in
Fig. 4, probing the soft mode at higher frequencies results in two peaks bracketing the 4.8 T
quantum critical point, indicating that the mode does indeed persist into the paramagnet,
in accordance with predictions.
At frequencies above 2.8 GHz, the coupling between additional excitation modes in the
sample and the cavity becomes strong enough to be observable even well away from the
8
FIG. 3.
Resonant absorption probing a low-energy excitation mode.
Transmission mag-
nitude
|
s
21
|
2
vs. frequency and transverse magnetic field for a single-mode loop-gap resonator
with zero-field tuning of 1.0 GHz. As the static susceptibility of the LiHoF
4
sample increases with
increasing transverse field, the effective inductance of the resonator + sample circuit increases,
resulting in a decreasing resonant frequency, with a cusp at the quantum phase transition at 4.8
T. Lower inset: individual frequency spectrum (blue) and Lorentzian fit (orange); bar indicates
the full-width half-maximum point used to determine the quality factor
Q
. Upper inset: 1
/Q
vs.
transverse field, showing enhanced dissipation when the energy of the soft mode matches the 0.93
GHz circuit resonant frequency.
resonant modes of the cavity (Fig. 4(a,b)). We then can observe in principle transitions
between all the additional modes, at frequencies corresponding to the energy differences
between the modes. To track the evolution of these transitions, we use a linear combination
of absorptive and dispersive Lorentzian lineshapes to extract the field dependence of the
9
(a)
(b)
(c)
(d)
FIG. 4.
Resonant and broadband evolution of higher-energy excitation modes.
(a)
Transmission magnitude
|
s
21
|
2
vs. frequency and transverse magnetic field with bimodal resonator
tuned to 2.6 and 4.2 GHz. (b) Expanded view of the broadband transmission response. The field
evolution of the first excited state response appears as a well-defined continuous curve well away
from resonant modes of the LGR. Near the cavity tuning of 4.2 GHz and near an extraneous cavity
mode at 3.9 GHz, avoided level crossings can be ascribed to hybridization between the cavity
photons and the magnons. For enhanced contrast, the transmission between 2.7 and 3.8 GHz is
plotted relative to a zero-field frequency dependent background of approximate average value -70
dB. Inset: Magnified view of transmission in the low-field region where the soft mode and excited
states are expected to coincide. A few closely spaced modes are resolved, and the non-monotonic
shape is reproduced well by the RPA calculations. (c) Expanded view of the resonant response
between 2.58 and 2.63 GHz (region between horizontal dashed lines in panel a). (d) Transverse
field dependence of the inverse quality factor for the resonant response shown in (c). At 2.6 GHz,
peaks in 1
/Q
are observed both above (highlighted in red) and below the 4.8 T quantum critical
point , indicating that at higher frequencies, the soft mode is visible on both sides of the phase
transition.
frequency and linewidth. Near the cavity resonance at 4.2 GHz, the spectra were fit to a
coupled oscillator model to account for hybridization between the electronuclear mode and
resonator photons [30, 31]; the apparent avoided level crossing at 3.6 GHz is due to an
10
anti-resonance in the LGR response.
We show in Fig. 5 a summary of and comparison between the measured (top) and
theoretically expected (bottom) transition energies. The blue points in Fig. 5 are derived
from a series of on-resonance measurements such as those shown in Figs. 3 and 4(c,d); the
orange curve is the result of the broadband measurement shown in Fig. 4(b). We note that
it is essential to do a T = 55 mK finite-temperature RPA calculation since these transition
energies differ from their T = 0 values. In particular, the imaginary time longitudinal spin
correlation function is used to determine the finite temperature magnon modes in LiHoF
4
.
The correlation function is calculated in the random phase approximation by making use of a
basis of mean field electronuclear eigenstates for the effective low temperature Hamiltonian
of the LiHoF
4
system. In Matsubara frequency space, the denominator of the resulting
equation is a polynomial with zeros corresponding to the electronuclear collective modes
of the system. At finite temperature, these modes include all possible transitions between
thermally excited states of the system – they can be thought of as a generalization of the
magnon modes one finds in a simple electronic spin system. The spectral weights associated
with the longitudinal spin fluctuations follow from the residue associated with each of these
modes (see Appendix for additional details).
The close agreement of theory with experiment implies that the RPA collective modes
are weakly coupled and represent the true collective degrees of freedom unusually well. The
finite temperature of the measurement (
k
BT
= 1
.
15 GHz, dashed line in Fig. 5), allows
thermal population of excited states for the lowest frequencies. At low transverse field, the
three lowest excitation modes are essentially degenerate, resulting in a single
E
(
H
) curve
for both measurement and model. We plot in the insets to Fig. 5 this low transverse-
field behavior and note that the non-monotonic field dependence of the measured mode
is accurately reproduced by the model. The RPA calculations overestimate the critical
field, primarily due to the absence of mode-mode couplings in the theory (which, although
individually small, have a cumulative effect on the critical field).
Near the QPT, the RPA shows the soft mode to be sensitive to the effect of an applied
longitudinal dc field, consistent with the divergent longitudinal susceptibility of the material.
This prediction is compared to experimental results in Fig 6. With the possibility of domain
pinning and hysteresis in the ferromagnet, it is essential to follow a well-defined magnetic
field history in order to have a consistent initial state (although we note that in all of the
11
FIG. 5.
Measured and calculated excitation spectra.
Top: Measured field dependence of soft
mode (
E
21
) and excited state (
E
32
) spectra, as determined by on-resonance (blue points, derived
from Fig. 3 and Fig. 4(d)) and off-resonance (orange curve, derived from Fig. 4(b)) responses,
respectively. The dashed-line curve through the
E
21
points is a guide to the eye. Horizontal dashed
line indicates frequency corresponding to the 55 mK measurement temperature. Bottom: Three
lowest transition energies, calculated using a finite-temperature Random Phase Approximation
(RPA) method. The mode structure and energy scale of the measurement and the model are in
close agreement. The field scale for the QPT differs by approximately 8%. Insets: Measured and
calculated frequency evolution at low field, where the three lowest modes are effectively degenerate.
The energy scale for the measurement and model differ by approximately 4%.
12
experiments, hysteresis effects were small given that pure LiHoF
4
is a soft ferromagnet).
We start by applying a transverse field of 5.6 T, sufficient to ensure that the system is in
the paramagnetic state. In sequence, we then apply a longitudinal field of 70 mT, lower
the transverse field to the desired value, and collect the resonator spectra for a series of
longitudinal fields.
To obtain the curves displayed in Fig 6(a), this protocol is repeated for a series of trans-
verse fields, and the resultant mesh of absorptions 1
/Q
(
H
t
,H
l
) is plotted as a function of
transverse field for several longitudinal fields. The most visible feature of these curves is
the strong absorption at a critical value of the transverse field for which the lowest energy
excitation has a minimum. This is due to the system softening at the phase transition and
absorbing at all frequencies (similar to critical opalescence). The softening is cut off by the
longitudinal field, leading to a substantial suppression in the peak amplitude. Below the
critical value of the transverse field, we also see resonant absorption where the soft mode is
degenerate with the cavity mode. The longitudinal field lifts the minimum in the soft mode,
suppressing its absorption. This leads to a reduction in the cavity 1
/Q
as shown in Fig.
6(a).
In order to quantitatively model the dependence of the soft mode on the applied longitu-
dinal field, it is necessary to consider the domain structure and demagnetization field present
in the system. In this context, a crucial feature of the LiHoF
4
system is the overwhelm-
ing dominance of the interelectronic spin dipolar interaction over the small superexchange
interaction. As a consequence, the domain walls are expected to be very thin with low
surface energy. One expects a very large number of Ising domains, and an almost uniform
demagnetization field throughout the sample except very near the boundaries. This purely
theoretical expectation is justified by the direct observation of micron-sized domains in op-
tical Kerr and Faraday rotation experiments [32–34]. The precise structure of the domains
(which has in other contexts been argued to have either a branched, striped, or bubble
structure [35, 36], depending on the applied field, is then not crucial – what matters is the
value of the mean magnetization density in determining the demagnetization field.
It is then appropriate to simply model the system as a thick plate, with height along
the Ising axis much greater than the domain size, and of infinite extent in the directions
transverse to the easy axis – the neglect of the surfaces in the x and y directions being
justified by the large number of domains in the sample. At zero wavevector, the soft mode
13
0.0014
0.0012
0.0010
0.0008
1/Q
5.0
4.8
4.6
4.4
4.2
4.0
H
t
(T)
Longitudinal Field (Oe)
0
200
400
600
(a)
Soft mode
Phase transition
1.905
1.900
5.0
4.5
4.0
f
0
(GHz)
4.54
4.52
4.50
4.48
4.46
H
t
(T)
-15
-10
-5
0
5
10
H
l
(mT)
Experiment
RPA calculation
H
demag
= -H
l
H
demag
= -1.3 H
l
(b)
FIG. 6.
Tuning the soft mode with longitudinal field.
(a) Dissipation (1
/Q
) in the vicinity
of the phase transition and soft mode at 1.9 GHz, following the field-cooling protocol described
in the main text. The asymmetry in the longitudinal field range arises from the need to null out
geometrical misalignments between the LiHoF
4
crystal and the magnets. The phase transition is
marked by the large peak in dissipation at
4
.
7 T transverse field and the soft mode appears as a
satellite peak at
4
.
5 T. Longitudinal (Ising) magnetic fields suppress the dissipation in the main
peak, but do not significantly change the amplitude of the soft-mode satellite. Inset: Evolution of
resonant frequency for the same set of longitudinal fields. A small shift as a function of longitudinal
field is observed. (b) Location of the soft-mode peak at small longitudinal fields. RPA calculations
are for mode locations for two different scalings of internal demagnetization fields. An internal
demagnetization field equal and opposite to the applied field produces a sharper evolution of the
mode position than is observed in the experiment. A demagnetization field 30% larger than the
applied field, including a finite domain-wall energy, matches the observed behavior.
14
will only be affected by the average demagnetization field present in the sample, which may
be incorporated into the RPA calculation via the introduction of an effective demagnetizing
factor (see Appendix B). Provided that the domain walls have very small energy, the average
demagnetization field present in a thick plate divided into stripe domains is then known to
be equal and opposite to the applied longitudinal field [35].
We plot in Fig. 6(b) the transverse field location of the RPA soft mode at 1.9 GHz with
the average demagnetization field taken to be equal and opposite to the applied longitudinal
field. The location of the soft mode has a sharper longitudinal-field dependence than is
seen in experiment. If one takes into account the finite energy of the domain walls, the
average demagnetization field increases in magnitude [35]. Micron-sized stripe domains in a
thin sample of LiHoF
4
are consistent with a domain wall energy of about 0
.
1
erg/cm
2
(the
exact value will vary with the applied field and temperature). The actual structure of the
domains is likely to be more complicated (e.g. branching in thick samples [33, 36]), but will
still increase the value of the demagnetization field. Assuming a demagnetization field 30%
larger than the applied field yields a good match with the data plotted in Fig 6(b).
IV. CONCLUSIONS
The existence of a soft mode is a necessary feature of the quantum phase transition in
simple models of the quantum Ising system. The LiHoF
4
system has a long history as an
exemplar of the Ising model in transverse field. The reported lack of mode softening in
experiments has then raised questions of whether quantum criticality exists in real quantum
Ising systems, and whether the models are too idealized. We find here that even though
the low-energy spectrum of LiHoF
4
is very intricate, with a wealth of fascinating behavior,
there is indeed a low-energy electronuclear soft mode associated with quantum criticality and
appropriately sensitive to a longitudinal magnetic field. Moreover, the agreement with finite-
temperature RPA theory shows that it interacts very weakly with the 14 other electronuclear
modes.
These results suggest a picture of quantum annealing—and by implication adiabatic
quantum computation—in which hybridized collective modes play a key role. A field sweep
through a QPT, which is the key feature of adiabatic quantum computing, can no longer be
regarded as a simple 2 level-avoidance process. One also must consider all of the collective
15
modes, as well as the sensitivity of the zero mode to any longitudinal fields, whether these
be stray local fields or applied fields. Since materials with similar electronuclear hybridized
spins are promising candidates for solid-state qubit realizations [37–39], these collective
modes clearly will need to be considered in a broad range of quantum Ising materials and
models.
ACKNOWLEDGMENTS
The experimental work at Caltech was supported by US Department of Energy Basic
Energy Sciences Award DE-SC0014866. P.C.E.S. acknowledges support at Caltech from
Simons Foundation Award 568762 and National Science Foundation Award PHY-1733907.
The theoretical work at UBC was supported by the National Sciences and Engineering
Research Council of Canada.
Appendix A: Effective Hamiltonian and RPA for the LiHoF
4
System
In this Appendix we summarize the effective Hamiltonian used for the LiHoF
4
system, and
give details of the RPA used to derive the collective mode spectrum at finite temperature.
We work throughout in SI units.
1. Effective Hamiltonian
The Hamiltonian for a real quantum Ising system like LiHoF
4
is much more complex
than the toy model that is described early in the main text. If we neglect phonons, we can
describe the system accurately by the Hamiltonian [22, 26, 40]
H
=
i
V
C
(
J
i
)
g
L
μ
B
i
B
x
J
x
i
+
A
i
I
i
·
J
i
(A1)
1
2
J
D
i
6
=
j
D
μν
ij
J
μ
i
J
ν
j
+
1
2
J
nn
<ij>
J
i
·
J
j
,
where the
Ho
spin-8 ions
{
J
i
}
have a single ion crystal field energy
V
C
(
J
i
), given by
V
C
(
~
J
) =
B
0
2
O
0
2
+
B
0
4
O
0
4
+
B
0
6
O
0
6
+
B
4
4
(
C
)
O
4
4
(
C
)
(A2)
+
B
4
6
(
C
)
O
4
6
(
C
) +
B
4
4
(
S
)
O
4
4
(
S
) +
B
4
6
(
S
)
O
4
6
(
S
)
,
16