WITNESSING QUANTUM CRITICALITY AND ENTANGLEMENT IN THE TRIANGULAR ANTIFERROMAGNET KYBSE2

Witnessing quantum criticality and entanglement in the triangular antiferromagnet KYbSe2

 A. O. Scheie,1, ∗ E. A. Ghioldi,2, 3 J. Xing,4 J. A. M. Paddison,4 N. E. Sherman,5, 6 M. Dupont,5, 6 D.
 Abernathy,1 D. M. Pajerowski,1 Shang-Shun Zhang,7 L. O. Manuel,3 A. E. Trumper,3 C. D. Pemmaraju,8 A.
 S. Sefat,4 D. S. Parker,4 T. P. Devereaux,8, 9 J. E. Moore,5, 6, 10 C. D. Batista,2, 11, † and D. A. Tennant1, 10, 11
 1 Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
 2 Department of Physics and Astronomy, University of Tennessee, Knoxville, TN, USA
 3 Instituto de Física Rosario (CONICET) and Universidad Nacional de Rosario,
 Boulevard 27 de Febrero 210 bis, (2000) Rosario, Argentina
 4 Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
 5 Department of Physics, University of California, Berkeley, California 94720, USA
 6 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
arXiv:2109.11527v2 [cond-mat.str-el] 27 Sep 2021

 7 School of Physics and Astronomy and William I. Fine Theoretical Physics Institute,
 University of Minnesota, Minneapolis, MN 55455, USA
 8 Stanford Institute for Materials and Energy Sciences,
 SLAC National Accelerator Laboratory, Stanford, CA 94025, USA
 9 Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, USA
 10 Quantum Science Center, Oak Ridge National Laboratory, TN 37831, USA
 11 Shull Wollan Center - A Joint Institute for Neutron Sciences, Oak Ridge National Laboratory, TN 37831. USA

 The Heisenberg triangular lattice quantum spin liquid and the phase transitions to nearby magnetic orders
 have received much theoretical attention, but clear experimental manifestations of these states are rare. This
 work investigates a new spin-half Yb3+ delafossite material, KYbSe2 , whose inelastic neutron scattering spectra
 reveal a diffuse continuum with a sharp lower bound. Applying entanglement witnesses to the data reveals
 multipartite entanglement spread between its neighbors, and analysis of its magnetic exchange couplings shows
 close proximity to the triangular lattice Heisenberg quantum spin liquid. Key features of the data are reproduced
 by Schwinger-boson theory and tensor network calculations with a significant second-neighbor coupling 2 . The
 strength of the dynamical structure factor at the point shows a scaling collapse in ~ / B down to 0.3 K,
 indicating a second-order quantum phase transition. Comparing this to previous theoretical work suggests that
 the proximate phase at larger 2 is a gapped Z2 spin liquid, resolving a long-debated issue. We thus show that
 KYbSe2 is close to a spin liquid phase, which in turn sheds light on the theoretical phase diagram itself.

 INTRODUCTION that case, it has been found that a realistic strength as small as
 ≈ 10% of the main interaction is enough to destroy magnetic
 A quantum spin liquid (QSL) is an elusive state of matter order and bring the system into a QSL phase [8–14], which
 where magnetic degrees of freedom on a lattice are in a highly is continuously connected to a QSL phase driven by nearest
 entangled, fluctuating ground state with exotic quasiparticle neighbor anisotropic exchange [15]. Determining the nature
 excitations [1–4]. The quasiparticles are of singular interest of the QSL phase is a theoretical challenge, with proposals
 for, e.g., quantum information applications [2, 5] but have ranging from gapped Z2 and gapless (1) Dirac to chiral [8–
 been, together with the extended entanglement, frustratingly 14], with no clear consensus within the community. In order
 difficult to identify experimentally. to discern among possible QSL states, experiments are called
 for.
 Despite tremendous effort over the last decades, no two or
 three dimensional materials have unambiguously been shown In the last decade, Yb3+ based materials have become pop-
 to realize a genuine QSL. This is partly because many studies ular as QSL candidates because of the Yb3+ effective = 1/2
 focus on “negative evidence” such as lack of magnetic order, state. Most recently, a class of delafossite materials have been
 lack of coherent excitations, etc., which are not unique to proposed as relatively disorder-free QSL candidates, includ-
 QSL states. Instead, to conclusively identify an experimental ing NaYbO2 [16–18], NaYbS2 [19, 20], NaYbSe2 [5, 22] and
 QSL, “positive evidence” is needed: experimental evidence CsYbSe2 [23]. Each of these materials shows diffuse exci-
 of either (i) a highly entangled ground state, or (ii) exotic tations and no long-range magnetic order down to 0.4 K or
 quasiparticles—both key properties of a QSL. lower, but because neither are unique to QSL states (both are
 Beginning with Anderson’s resonating valence bond also caused by spin glass [24], random singlet phases [25], or
 state [6], the two-dimensional (2D) triangular geometry has 2D magnetic order only in the zero temperature limit), they
 long been studied as a platform for QSLs. Although the sim- remain QSL candidates only.
 plest spin-1/2 model with nearest-neighbor antiferromagnetic Here we investigate a new member of the Yb3+ delafossite
 Heisenberg interactions orders magnetically in a 120◦ phase, family: KYbSe2 which forms a layered triangular lattice of
 the magnetic frustration makes the order weak [7]. The mag- magnetic Yb3+ ions, see Fig. 1(a). This material shows no
 netic order can be further destabilized by additional interac- long-range order above 400 mK [1], and finite-field ordered
 tions such as a next-nearest-neighbor exchange coupling. In phases similar to NaYbO2 [17] and NaYbS2 [20]. Thus it

2

 a KYbSe
 b
 Yb

 Quantum critical
 regime

 KYbSe
 /
 Se
 b

 c
 K c a
 stripe
 a b order QSL order
 . .
 /

FIG. 1. Crystal structure and phase diagram of KYbSe2 . Panel a shows the crystal structure with a side view of the stacked triangular layers
and a top view showing the Yb3+ triangular lattice mediated by Se2− ions. Panel b shows a schematic phase diagram of the triangular lattice
Heisenberg antiferromagnet as a function of second neighbor exchange strength 2 . This includes a zero temperature 120◦ ordered phase for
 2 / 1 . 0.06, a zero temperature stripe ordered phase for 2 / 1 & 0.16, and an intermediate QSL phase [8–14]. Near the quantum critical
points we expect quantum critical regime extending at finite temperature.

appears promising as a quantum spin liquid candidate. We of 47 ± 10 Å at 0.3 K (≈ 10 unit cells in the plane).
successfully apply entanglement witnesses one-tangle, two- In the inelastic channel, two features stand out in the low-
tangle, and quantum Fisher information (QFI) to KYbSe2 [4], temperature KYbSe2 spectrum: a diffuse continuum of ex-
and detect the presence of quantum entanglement at low tem- citations, and a pronounced 0.2 meV energy minimum at
peratures. Using a combination of density-functional theory, = (1/2, 0, 0). Both of these features are seen in the triangu-
Onsager reaction field theory, Schwinger bosons, and ten- lar lattice compound Ba3 CoSb2 O9 [30–33]. The “roton-like”
sor network approaches to model KYbSe2 , we find that its minimum at is a generic feature of the 2D quantum triangu-
physics is well-captured by a microscopic spin-1/2 Hamilto- lar lattice Heisenberg antiferromagnet and is a nonlinear effect
nian with nearest and next-nearest neighbor Heisenberg inter- (i.e., not captured by linear spin wave theory) [34–36]. Fits
actions on the triangular lattice in proximity to the QSL phase to the KYbSe2 roton mode [see Supplementary Information]
[see Fig. 1(b)]. Finally, the neutron spectrum displays signa- show a mode maximum of 0.288(12) meV, and a roton mini-
tures of quantum criticality and fractionalized spinon quasipar- mum 0.200(13) meV at . This indicates that strong quantum
ticles. Together, these results show KYbSe2 to be proximate effects are at work in KYbSe2 .
to a spin liquid with positive evidence for the two key features: The continuum, meanwhile, extends up to 1.6 meV, over five
quantum entanglement and exotic quasiparticles. times the roton mode bandwidth. This is far too high in energy
 to be a two-magnon continuum, which is limited to twice the
 single-magnon bandwidth. Integrating the scattering intensity
 EXPERIMENTS over the entire Brillouin zone shows that ∼ 60% of the mag-
 netic scattering intensity is found above 0.4 meV, compared
 Cold Neutron Chopper Spectrometer (CNCS) to only ∼ 29% between 0.05 meV and 0.4 meV, showing that
 the continuum scattering carries twice the spectral weight of
 We measured the low-energy KYbSe2 single crystal neutron the “single-magnon” intensity. Perhaps most interestingly, the
spectrum on the CNCS spectrometer [28] at Oak Ridge Na- continuum in KYbSe2 comes all the way down to the sharp
tional Laboratory’s Spallation Neutron Source [29] between low-energy modes [Fig. 2(a)]. The KYbSe2 diffuse contin-
0.3 K and 2 K using a 3 He refrigerator (for details, see the uum with a sharp lower bound is reminiscent of the Van Hove
methods section). The data are shown in Fig. 2. singularity observed in 1D spin chains—which are known to
 In the elastic channel, quasi-Bragg intensities appear be- have highly entangled ground states with fractionalized spinon
tween 1 K and 0.3 K which look like (1/3, 1/3) Bragg peaks excitations [4, 37, 38]. This well-defined lower bound to the
signaling 120◦ correlations. They have no dependence upon continuum distinguishes KYbSe2 from other QSL candidates,
ℓ [Fig. 2(p)] which evidences truly 2D static correlations and such as NaCaNi2 F7 [39], YbMgGaO4 [40, 41], and herbert-
weak inter-plane exchange. As an aside, this weak inter-plane smithite [42] which are diffuse everywhere.
exchange is expected given the fragility of the crystal inter-
plane bonds: KYbSe2 planes readily flake off when the crys-
tals are not handled gently. Fitting the in-plane scattering to Wide Angular-Range Chopper Spectrometer (ARCS)
extract the correlation length using the (101) peak to define the
resolution width, we find the magnetic peaks are much broader In order to understand how “quantum” the KYbSe2 spins
than the nuclear Bragg peaks with a fitted correlation length are, we measured the the crystal electric field (CEF) excita-

3

 ( , ) (meV ) ( , ) (meV )
 ( , ) (meV ) 0.0 0.1 0.2 0.3 0.4 0.5 0 10
 10 1 10 0 101
 . meV . meV . meV . meV elastic
 a 0.3 K d e f g 1p
 0.5
 (meV)

 (/, , )
 1 0.0 0

 )
 0.5

 (
 / / 1 0.3 K
 0 1q
 b 1K h i j k
 0.5
 (meV)

 (/, , )
 1 0.0 0

 )
 0.5

 (
 / 1 1K
 0 1r
 c 2K l m n o
 0.5
 (meV)

 (/, , )
 1 0.0 0

 )
 = . meV
 = ± . meV
 0.5 12 K subtracted

 (
 / 1 2K
 0 1.0 0.5 0.0 0.5 1.0 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 1.0 0.5 0.0
 ( , , ) (, , ) (, , ) (, , ) (, , ) (/, , )

 FIG. 2. Neutron spectrum of KYbSe2 at 0.3 K (top row), 1 K (middle row) and 2 K (bottom row). The left panels show energy-dependent
 scattering along (− /2 − 1/2, , 0) which includes where the dispersion touches zero energy. These plots comprise data with = 1.55 meV
 below ~ = 0.5 meV, and = 3.32 meV above ~ = 0.5 meV. Note the roton-like mode at 0.3 K and the diffuse high energy spectrum.
 The center panels show constant energy slices measured with = 1.55 meV. Panel d shows elastic intensity associated with (1/3, 1/3) static
 magnetism which disappears at higher temperatures. The right panels plot this elastic intensity as a function of ℓ, which reveals almost no
 dependence on ℓ, and thus 2D correlations.

 7K 100 K 200 K 3 The best fit crystal field Hamiltonian shows a ground state
 40 (a) (b) (c) doublet
(meV)

 5 1 7
 20 2
 | ± i = 0.78(3) ∓
 2
 ∓ 0.44(4) ±
 2
 − 0.44(3) ±
 2
 (1)
 (arb. u.)

 0 with a first excited state at 17.1(3) meV. This ground state dou-
 40 (d) (e) (f) blet gives a weak easy plane tensor = = 3.0(2), and
 1
(meV)

 = 1.8(6). As the large and indicate, the ground
 20 state doublet allows for significant quantum tunnelling from
 bkg subtracted effective spin operator ± . Thus, the Yb3+ spins in KYbSe2
 0 4 8 4 8 4 8 0 can be treated like a spin-1/2 system.
 (Å ) (Å ) (Å )
 4 (g) (h) (i) CEF fit ENTANGLEMENT WITNESSES
 Å < < Å data
 (arb. u.)

 2 Diffuse neutron excitations suggest but do not prove QSL
 behavior, which makes their observation ambiguous by them-
 0 20 40 20 40 20 40
 selves. Fortunately, entanglement witnesses provide a way
 (meV) (meV) (meV) out of this quandary: by quantifying entanglement in KYbSe2
 we can show that its excitations are associated with a highly-
 FIG. 3. Crystal field spectrum of KYbSe2 . The top row shows entangled ground state.
 the measured intensity at = 50 meV at 7 K, 100 K, and 200 K. We apply three entanglement witnesses to the KYbSe2 data
 The middle row shows the same data with the self-consistent 300 K (same as in Refs. [4, 37]): one-tangle 1 , which quantifies
 background subtracted. The bottom row shows a cut through the data entanglement of a spin with the entire system [45, 46]; the
 between 2 and 3 Å−1 compared to the fitted CEF Hamiltonian two-tangle 2 , which quantifies the total bipartite entanglement
 derived from quantum concurrence [47, 48]; and QFI which
 gives a lower bound on multi-partite entanglement [3]. For
 tions using the ARCS spectrometer [43] at Oak Ridge National details of these calculations, see the methods section.
 Lab’s Spallation Neutron Source. We fitted a single-ion CEF One-tangle is calculated from the static spin susceptibil-
 Hamiltonian to the excitations using PyCrystalField [9] soft- ity at zero temperature and ranges between zero (unentangled
 ware; data and fits are shown in Fig. 3 [Details on the CEF state) and one (maximally entangled state). The ratio be-
 fitting procedure are given in the Supplemental Information]. tween elastic and inelastic scattering (to within our energy

4

 a b K=(1/3,1/3,0) stead, they point to many sites entangled together at the lowest
 0.3 K 0.3 K
 0.75

 )
 1K temperatures—as one would expect for a QSL.
 1K

 ( ) (meV
 2K
 2K 100
 0.50 MICROSCOPIC MODELING
 nonzero two tangle 10 2
 0 1 2
 0.25 zero two (meV)
 tangle c To better understand the features observed in KYbSe2 , and

 1 partite entangled find a microscopic model for the compound, we use a com-
 0.00 bination of theoretical techniques such as density-functional

 nQFI
 theory (which showed a magnetic insulating state, discussed
 0.25 in the Methods section), the Onsager reaction field, Schwinger

 0 1 2 3 4 00 1 2
 bosons, and tensor networks.
 neighbor (K)
 Onsager reaction field:
FIG. 4. KYbSe2 entanglement witnesses. Panel a shows the real- estimating the exchange ratios
space spin spin correlations for the first four neighbors used to cal-
culate two-tangle: a measure of bipartite entanglement [47]. None
of the neighbor spin-spin correlators exceed the classical threshold, First, we employ the Onsager reaction field (ORF) [15] to fit
which means two-tangle is zero at all temperatures. Panel b shows the energy-integrated paramagnetic scattering shown in Fig. 5.
the intensity as a function of energy at = (1/3, 1/3, 0) (integrated This approach neglects quantum fluctuations, but in the para-
over ±0.025 RLU), with the normalized quantum Fisher informa- magnetic regime it is accurate up to a temperature-dependent
tion (nQFI) shown below in panel c. At 0.3 K, nQFI just exceeds energy scale normalization [53] which in our case is unknown.
the threshold for at least bipartite entanglement. Thus no classical
 Despite this limitation, ORF does give relative anisotropy and
arrangement of spins could produce the observed KYbSe2 spectrum.
 ratios between exchanges. Using the -tensor derived from
 crystal electric field fits and allowing for first and second neigh-
resolution of 40 eV at ~ = 0) at 0.3 K gives a one-tangle bor exchange, we find the off-diagonal anisotropic exchange is
 1 = 0.62(8). This is of course not at zero temperature, but small and the nearest neighbor exchange is isotropic to within
if the zero temperature value is anywhere close (which mag- uncertainty [see Methods] making KYbSe2 a very good ap-
netic entropy measurements indicate is so [1]) it evidences proximation to a triangular lattice Heisenberg antiferromagnet
substantial spin entanglement in KYbSe2 . described by the microscopic 1 − 2 Hamiltonian,
 Two tangle is calculated from the Fourier transform to real ∑︁ ∑︁
 Ĥ = 1 ˆ · ˆ + 2 ˆ · ˆ . (2)
space of the frequency integrated ( , ) and is shown in
 h , i hh , ii
Fig. 4. We find that none of the neighboring spin correlators
exceed the classical h · i threshold, and thus two-tangle is What is more, the fitted 2 / 1 = 0.047(7). This is extremely
zero for all temperatures in KYbSe2 . This makes sense given close to the the predicted phase boundary between 120◦ mag-
quantum monogamy [50] and six equivalent nearest neighbors netic order and a QSL phase on the triangular lattice Heisen-
for every site to distribute its entanglement. The significance berg antiferromagnet: 2 / 1 ≈ 0.06 [8–14]. Thus, ORF fits
of this will become apparent shortly. show KYbSe2 has nearly isotropic Heisenberg exchange and
 The third entanglement witness, QFI, is calculated from an is very close to a quantum spin liquid phase.
energy integral at a specific point in [3]. For KYbSe2 we
evaluate QFI at (1/3, 1/3), the wavevector associated with the
strongest correlations. The scattering and nQFI are shown in Schwinger bosons: comparing the neutron spectrum
Fig. 4. At 1 K and 2 K, nQFI = 0.42(2) and 0.238(10)
respectively, which is too small to witness entanglement. At To understand the inelastic neutron spectrum, we turn to a
0.3 K, nQFI = 1.03(4), which indicates at least bipartite en- Schwinger Boson (SB) approach [54–56]. This is a parton
tanglement in a highly correlated ground state. formulation where the Heisenberg model is expressed in terms
 Clearly, these entanglement witnesses reveal appreciable of interacting spin-1/2 bosons or spinons, whose condensation
spin entanglement in KYbSe2 , but the combination of two- leads to long-range magnetic ordering [54, 55]. For details,
tangle and QFI is particularly revealing. The zero two- see the Methods section.
tangle shows that the entanglement is spread out over near- The dynamical spin structure factor ( , ) at = 0 using
est neighbors rather than pairing with a particular neighbor SB [56] for 2 / 1 = 0.05 is shown in Fig. 5(d). On a qualitative
in singlets. This is what one expects for a highly-entangled level, this result captures the features seen in the experimental
ground state (c.f. vanishing two-tangle for the Kitaev spin data: the strong dispersive cone emanating from , the contin-
liquid [51]). Meanwhile the QFI shows at least bipartite en- uum scattering at higher energies, the diffuse high-energy fea-
tanglement within the (1/3, 1/3) correlations. Both of these ture at , and the pronounced low-energy “roton-like” mode
rule out classical glassiness or random singlet formation. In- at . We note that the downturn of the roton-like mode is

5

 1K 2K 1.5
 1a b
 Tensor networks: full spectrum model

 1.0 The third technique we use to model the diffuse inelastic
 (/, , )

 0 neutron scattering is based on tensor networks [see the Meth-

 ()
 0.5 ods section]. A related approach was recently used to interpret
 and describe the scattering of CsYbSe2 [23], and provides a
 1 data 0.5 SCGA data SCGA full quantum picture of the neutron spectrum. The downside
 0.0 0.5 0.5 0.0 0.5 0.0 to this technique is finite size effects, which cause broadened
 (, , ) (, , )
 modes and gaps in the low energy spectrum. Nevertheless,
 101
 1.5 c 0.3 K qualitative comparisons can be made.
(meV)

 The simulated data along high symmetry directions of the
 1.0 Brillouin zone for 2 / 1 = 0.05 is shown in Fig. 5(e). The
 0.5 overall features of the experimental data are reproduced in
 0
 the simulations: the asymmetric dispersive modes emanating
 1.5 d Schwinger boson theory from , the diffuse continuum extending to high energies, and

 ( , ) (mev
 100
(meV)

 even the broad 1 meV feature at . This shows that the trian-
 1.0 gular lattice Heisenberg 1 - 2 model is indeed an appropriate
 0.5 model for KYbSe2 . Further microscopic simulations show
 0
 )
 that most of the high energy scattering remains unchanged as
 1.5 e DMRG 2 is increased and the system enters the QSL phase, showing
(meV)

 that the high-energy scattering can be interpreted as bound
 1.0 spinons of a proximate spin liquid.
 0.5 10 1

 0
 CRITICAL SCALING
 (, , ) (, , ) (, , ) (, , )

 FIG. 5. Comparison between experimental KYbSe2 scattering
 So far, the entanglement witnesses and theoretical compar-
 and theoretical simulations. Panels a and b show Onsager reaction isons indicate that KYbSe2 is close to the 1 / 2 QSL quantum
 field (ORF) fits to energy-integrated paramagnetic KYbSe2 scattering critical point. If this is true, we should see quantum criti-
 at 1 K and 2 K. In each panel, the data is on the left and the fit is on cal scaling in the finite temperature neutron spectrum [58–
 the right. Panels c-e show neutron scattering along high-symmetry 61]. Plotting scattered intensity times ( ) versus ~ / B ,
 directions. c shows the experimental data for KYbSe2 and d shows shown in Fig. 6, we see a critical exponent = 1.73(12) over
 the zero-temperature simulated spectrum from Schwinger boson cal- more than a decade in / . Theoretically, the semiclassical
 culations with 1 = 0.56(3) meV and 2 / 1 = 0.05. Panel e shows
 tensor network simulations with the same 1 and 2 . On a qualita-
 spin wave scattering from an ordered Heisenberg triangular
 tive level, the theory captures the continuum excitations observed in lattice predicts an exponent = 1. The observed scattering
 experiment. is unquestionably inconsistent with this [Fig. 6(a)]. Thus this
 scaling suggests that, at least down to 300 mK, KYbSe2 is
 within the critical fan of a quantum phase transition and is
 much less pronounced in the SB result because of the lack of extremely close to the quantum spin liquid phase.
 1/ corrections to the internal vertices and the single-spinon The large magnetic correlation length in the elastic channel
 propagator [56]. However, the most remarkable aspect of suggests that KYbSe2 may be on the 120◦ side of the phase
 this comparison is that the SB approach captures the intensity boundary. It is not clear experimentally whether KYbSe2 is on
 modulation of the the continuum scattering at higher energies, the QSL side of the critical point—measures below 300 mK
 which is determined by the two-spinon continuum of the SB may reveal a transition to long-range magnetic order—but the
 theory. This correspondence points to the continuum scatter- critical scaling is strong evidence that KYbSe2 is within the
 ing in KYbSe2 originating from its proximity to a deconfined quantum critical regime at finite .
 spin liquid state with fractionalized spinon excitations. The proximity to a quantum critical point has important
 The measured continuum scattering extends up to higher en- implications regarding the nature of the QSL state. Indeed,
 ergies than SB predicts: ≈ 1.6 meV, approximately three times the gapped Z2 QSL state proposed by Sachdev [62] is the only
 the fitted 1 = 0.56(3) meV (see Supplemental Information). liquid which can be continuously connected with 120◦ Néel
 We attribute this discrepancy to the lack of 4-spinon contri- ordered state, as it does not break any symmetries and has
 butions arising from Feynman diagrams which have not been lowest energy modes at the -points [63]. (The low energy
 included in the SB calculation [56]. Note that the KYbSe2 con- excitations of the other possibility, a -flux state, are gapped at
 tinuum extent does match the predicted continuum extent near the -points and gapless at the -points, inconsistent with the
 the 2 / 1 ≈ 0.06 transition point as calculated by Gutzwiller observations.) The resulting quantum critical point is expected
 projected variational Monte Carlo [57]. to have a dynamically generated (4) symmetry [64, 65].

6

 102 SWT suggests pressure does more than just shift magnetic exchange
 constants.
 10 1
)

 The family of Yb3+ delafossites are a remarkable platform
 100 optimum
( = , ) (

 = . for 2D triangular lattice Heisenberg systems. By controlling
 2 / 1 , we are able to systematically approach a QSL from
 10 2
 10 2 the 120◦ ordered phase, which gives a clear pathway towards
 a = c =
 an experimentally verifiable QSL state. Scaling behavior in
 ~ / with a nontrivial exponent, i.e., a value inconsistent
 0.3 K 10
 1
 10 1
 0.3 K with gapless spin wave excitations, is observed in the spin
 1K 1K correlations down to the lowest temperature measured (0.3 K),
)

 10 2 2 K 10 1 2K with a correlation length of at least ten unit cells.
( = , ) (

 optimum While a weakly first-order transition with a long correla-
 10 3 = . ( ) 10 3 tion length is possible, the natural interpretation of the results
 in this work is that the phase transition from 120◦ to a QSL
 10 4 b = . d = . is second order, which combines with previous theoretical
 100 101 10 5
 100 101 102 work to constrain strongly the nature of the QSL. One of the
 / / frontiers in quantum condensed matter physics is to under-
 stand the possible phase transitions between topological and
 FIG. 6. Critical scaling in KYbSe2 , showing data at the point broken-symmetry phases, and the combined experimental and
 at three different temperatures scaled by ~ / B and ( =
 theoretical analysis of KYbSe2 helps clarify one piece of this
 , ) ( B ) . Measured KYbSe2 spectra are on the left column,
 and calculated spin wave theory (SWT) are on the right column. frontier.
 When = 1.73(12), the KYbSe2 data from the three temperatures
 follow the same curve, suggesting quantum critical scaling. SWT
 spectra, meanwhile, overlap when = 1.0. This suggests funda-
 mentally different behavior in KYbSe2 that cannot be captured by
 ∗ scheieao@ornl.gov
 non-interacting magnons.
 † cbatist2@utk.edu
 [1] J. Knolle and R. Moessner, Annual Review of Condensed Matter
 Physics 10, 451 (2019).
 CONCLUSION [2] C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R.
 Norman, and T. Senthil, Science 367 (2020), 10.1126/sci-
 These results show that KYbSe2 is within the quantum crit- ence.aay0668.
 [3] L. Savary and L. Balents, Reports on Progress in Physics 80,
 ical fan of a quantum spin liquid state. CEF fits show an 016502 (2016).
 isotropic = 1/2 doublet with strong quantum effects, and [4] Y. Zhou, K. Kanoda, and T.-K. Ng, Rev. Mod. Phys. 89, 025003
 ORF simulations show a 2 / 1 ratio very close to the QSL (2017).
 quantum critical point 2 / 1 ≈ 0.06. Entanglement witnesses [5] Y. Tokura, M. Kawasaki, and N. Nagaosa, Nature Physics 13,
 reveal an entangled ground state with distributed entangle- 1056 (2017).
 ment, just as a QSL should have. Finally, there are strong [6] P. Anderson, Materials Research Bulletin 8, 153 (1973).
 signs of quantum criticality in the neutron spectrum: (i) the [7] S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004
 (2007).
 majority spectral weight in the continuum, (ii) the sharp lower [8] Z. Zhu and S. R. White, Phys. Rev. B 92, 041105 (2015).
 continuum bound reminiscent of the 1D spinon spectrum, (iii) [9] W.-J. Hu, S.-S. Gong, W. Zhu, and D. N. Sheng, Phys. Rev. B
 strong correspondence to SB and tensor network simulations 92, 140403 (2015).
 near the transition to a spin liquid, and (iv) critical scaling in- [10] Y. Iqbal, W.-J. Hu, R. Thomale, D. Poilblanc, and F. Becca,
 compatible with semiclassical excitations all indicate that the Phys. Rev. B 93, 144411 (2016).
 KYbSe2 excitations are fractionalized spinons of a QSL phase. [11] S. N. Saadatmand and I. P. McCulloch, Phys. Rev. B 94, 121111
 (2016).
 These results have implications beyond just this material. [12] A. Wietek and A. M. Läuchli, Phys. Rev. B 95, 035141 (2017).
 As noted earlier, triangular lattice CsYbSe2 and NaYbSe2 also [13] S.-S. Gong, W. Zhu, J.-X. Zhu, D. N. Sheng, and K. Yang, Phys.
 show features of a QSL phase: with CsYbSe2 possibly more Rev. B 96, 075116 (2017).
 toward the 2 = 0 limit [23], and NaYbSe2 2 / 1 possibly [14] S. Hu, W. Zhu, S. Eggert, and Y.-C. He, Phys. Rev. Lett. 123,
 within the QSL phase [22]. This suggests that the periodic 207203 (2019).
 table can be used to “tune” 2 / 1 such that the delafossite lattice [15] Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Chernyshev,
 Phys. Rev. Lett. 120, 207203 (2018).
 can be brought into and out of a QSL phase depending on the A-
 [16] L. Ding, P. Manuel, S. Bachus, F. Grußler, P. Gegenwart, J. Sin-
 site element. This gives a remarkably controlled way to study gleton, R. D. Johnson, H. C. Walker, D. T. Adroja, A. D. Hillier,
 QSL materials. Another possible way to “tune” 2 / 1 could and A. A. Tsirlin, Phys. Rev. B 100, 144432 (2019).
 be through hydrostatic pressure—there are even reports of [17] M. M. Bordelon, E. Kenney, C. Liu, T. Hogan, L. Posthuma,
 superconductivity in NaYbSe2 under pressure [66, 67], which M. Kavand, Y. Lyu, M. Sherwin, N. P. Butch, C. Brown, M. J.

7

 Graf, L. Balents, and S. D. Wilson, Nature Physics 15, 1058 [39] K. Plumb, H. J. Changlani, A. Scheie, S. Zhang, J. Krizan,
 (2019). J. Rodriguez-Rivera, Y. Qiu, B. Winn, R. Cava, and C. L.
[18] K. M. Ranjith, D. Dmytriieva, S. Khim, J. Sichelschmidt, Broholm, Nature Physics 15, 54 (2019).
 S. Luther, D. Ehlers, H. Yasuoka, J. Wosnitza, A. A. Tsirlin, [40] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan, Q. Wang, H. C.
 H. Kühne, and M. Baenitz, Phys. Rev. B 99, 180401 (2019). Walker, P. Steffens, M. Boehm, Y. Hao, D. L. Quintero-Castro,
[19] M. Baenitz, P. Schlender, J. Sichelschmidt, Y. A. Onykiienko, L. W. Harriger, M. D. Frontzek, L. Hao, S. Meng, Q. Zhang,
 Z. Zangeneh, K. M. Ranjith, R. Sarkar, L. Hozoi, H. C. Walker, G. Chen, and J. Zhao, Nature 540, 559 (2016).
 J.-C. Orain, H. Yasuoka, J. van den Brink, H. H. Klauss, D. S. [41] J. A. M. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu,
 Inosov, and T. Doert, Phys. Rev. B 98, 220409 (2018). M. Stone, H. Zhou, and M. Mourigal, Nature Physics 13, 117
[20] R. Sarkar, P. Schlender, V. Grinenko, E. Haeussler, P. J. Baker, (2017).
 T. Doert, and H.-H. Klauss, Phys. Rev. B 100, 241116 (2019). [42] T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-
 [5] K. M. Ranjith, S. Luther, T. Reimann, B. Schmidt, P. Schlender, Rivera, C. Broholm, and Y. S. Lee, Nature 492, 406 (2012).
 J. Sichelschmidt, H. Yasuoka, A. M. Strydom, Y. Skourski, [43] D. L. Abernathy, M. B. Stone, M. J. Loguillo, M. S. Lucas,
 J. Wosnitza, H. Kühne, T. Doert, and M. Baenitz, Phys. Rev. B O. Delaire, X. Tang, J. Y. Y. Lin, and B. Fultz, Review of
 100, 224417 (2019). Scientific Instruments 83, 015114 (2012).
[22] P.-L. Dai, G. Zhang, Y. Xie, C. Duan, Y. Gao, Z. Zhu, E. Feng, [9] A. Scheie, Journal of Applied Crystallography 54 (2021).
 Z. Tao, C.-L. Huang, H. Cao, A. Podlesnyak, G. E. Granroth, [45] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
 M. S. Everett, J. C. Neuefeind, D. Voneshen, S. Wang, G. Tan, [46] V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A 61,
 E. Morosan, X. Wang, H.-Q. Lin, L. Shu, G. Chen, Y. Guo, 052306 (2000).
 X. Lu, and P. Dai, Phys. Rev. X 11, 021044 (2021). [47] T. Roscilde, P. Verrucchi, A. Fubini, S. Haas, and V. Tognetti,
[23] T. Xie, J. Xing, S. Nikitin, S. Nishimoto, M. Brando, P. Kha- Phys. Rev. Lett. 93, 167203 (2004).
 nenko, J. Sichelschmidt, L. Sanjeewa, A. S. Sefat, and [48] L. Amico, F. Baroni, A. Fubini, D. Patanè, V. Tognetti, and
 A. Podlesnyak, arXiv preprint arXiv:2106.12451 (2021). P. Verrucchi, Phys. Rev. A 74, 022322 (2006).
[24] S. Zhang, H. J. Changlani, K. W. Plumb, O. Tchernyshyov, and [3] P. Hauke, M. Heyl, L. Tagliacozzo, and P. Zoller, Nat. Phys. 12,
 R. Moessner, Phys. Rev. Lett. 122, 167203 (2019). 778 (2016).
[25] Z. Zhu, P. A. Maksimov, S. R. White, and A. L. Chernyshev, [50] T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503
 Phys. Rev. Lett. 119, 157201 (2017). (2006).
 [1] J. Xing, L. D. Sanjeewa, A. F. May, and A. S. Sefat, arXiv [51] G. Baskaran, S. Mandal, and R. Shankar, Phys. Rev. Lett. 98,
 preprint ArXiv:2109.00384 (2021). 247201 (2007).
 [4] A. Scheie, P. Laurell, A. M. Samarakoon, B. Lake, S. E. Nagler, [15] J. A. M. Paddison, Phys. Rev. Lett. 125, 247202 (2020).
 G. E. Granroth, S. Okamoto, G. Alvarez, and D. A. Tennant, [53] T. Huberman, D. A. Tennant, R. A. Cowley, R. Coldea, and C. D.
 Phys. Rev. B 103, 224434 (2021). Frost, Journal of Statistical Mechanics: Theory and Experiment
[28] G. Ehlers, A. A. Podlesnyak, J. L. Niedziela, E. B. Iverson, and 2008, P05017 (2008).
 P. E. Sokol, Review of Scientific Instruments 82, 085108 (2011). [54] D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988).
[29] T. E. Mason, D. Abernathy, I. Anderson, J. Ankner, T. Egami, [55] A. Auerbach, Interacting electrons and quantum magnetism
 G. Ehlers, A. Ekkebus, G. Granroth, M. Hagen, K. Herwig, (Springer-Verlag, New York, 1994).
 J. Hodges, C. Hoffmann, C. Horak, L. Horton, F. Klose, [56] E. A. Ghioldi, M. G. Gonzalez, S.-S. Zhang, Y. Kamiya, L. O.
 J. Larese, A. Mesecar, D. Myles, J. Neuefeind, M. Ohl, C. Tulk, Manuel, A. E. Trumper, and C. D. Batista, Phys. Rev. B 98,
 X.-L. Wang, and J. Zhao, Physica B: Condensed Matter 385, 184403 (2018).
 955 (2006). [57] F. Ferrari and F. Becca, Phys. Rev. X 9, 031026 (2019).
[30] D. Macdougal, S. Williams, D. Prabhakaran, R. I. Bewley, D. J. [58] B. Lake, D. A. Tennant, C. D. Frost, and S. E. Nagler, Nat.
 Voneshen, and R. Coldea, Phys. Rev. B 102, 064421 (2020). Mater. 4, 329 (2005).
[31] H. D. Zhou, C. Xu, A. M. Hallas, H. J. Silverstein, C. R. Wiebe, [59] A. Schröder, G. Aeppli, R. Coldea, M. Adams, O. Stockert,
 I. Umegaki, J. Q. Yan, T. P. Murphy, J.-H. Park, Y. Qiu, J. R. D. H. Löhneysen, E. Bucher, R. Ramazashvili, and P. Coleman,
 Copley, J. S. Gardner, and Y. Takano, Phys. Rev. Lett. 109, Nature 407, 351 (2000).
 267206 (2012). [60] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. B
[32] S. Ito, N. Kurita, H. Tanaka, S. Ohira-Kawamura, K. Nakajima, 39, 2344 (1989).
 S. Itoh, K. Kuwahara, and K. Kakurai, Nature Communications [61] S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
 8, 235 (2017). [62] S. Sachdev, Phys. Rev. B 45, 12377 (1992).
[33] J. Ma, Y. Kamiya, T. Hong, H. B. Cao, G. Ehlers, W. Tian, C. D. [63] F. Wang and A. Vishwanath, Phys. Rev. B 74, 174423 (2006).
 Batista, Z. L. Dun, H. D. Zhou, and M. Matsuda, Phys. Rev. [64] P. Azaria, B. Delamotte, and T. Jolicoeur, Phys. Rev. Lett. 64,
 Lett. 116, 087201 (2016). 3175 (1990).
[34] W. Zheng, J. O. Fjærestad, R. R. P. Singh, R. H. McKenzie, and [65] A. V. Chubukov, S. Sachdev, and T. Senthil, Nuclear Physics B
 R. Coldea, Phys. Rev. B 74, 224420 (2006). 426, 601 (1994).
[35] O. A. Starykh, A. V. Chubukov, and A. G. Abanov, Phys. Rev. [66] Y.-T. Jia, C.-S. Gong, Y.-X. Liu, J.-F. Zhao, C. Dong, G.-Y. Dai,
 B 74, 180403 (2006). X.-D. Li, H.-C. Lei, R.-Z. Yu, G.-M. Zhang, and C.-Q. Jin,
[36] A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B 79, Chinese Physics Letters 37, 097404 (2020).
 144416 (2009). [67] Z. Zhang, Y. Yin, X. Ma, W. Liu, J. Li, F. Jin, J. Ji, Y. Wang,
[37] P. Laurell, A. Scheie, C. J. Mukherjee, M. M. Koza, M. Enderle, X. Wang, X. Yu, et al., arXiv preprint arXiv:2003.11479 (2020).
 Z. Tylczynski, S. Okamoto, R. Coldea, D. A. Tennant, and [68] P. J. Brown, “Magnetic form factors,” The Cambridge Crystal-
 G. Alvarez, Phys. Rev. Lett. 127, 037201 (2021). lographic Subroutine Library (1998).
[38] B. Lake, D. A. Tennant, J.-S. Caux, T. Barthel, U. Schollwöck, [69] F. James and M. Roos, Comp. Phys. Commun. 10, 343 (1975).
 S. E. Nagler, and C. D. Frost, Phys. Rev. Lett. 111, 137205 [70] S.-S. Zhang, E. A. Ghioldi, Y. Kamiya, L. O. Manuel, A. E.
 (2013). Trumper, and C. D. Batista, Phys. Rev. B 100, 104431 (2019).

8

[71] A. Szasz, J. Motruk, M. P. Zaletel, and J. E. Moore, Phys. Rev.
 X 10, 021042 (2020).
[72] U. Schollwock, Annals of Physics 326, 96 (2011), january 2011
 Special Issue.
[73] L. Vanderstraeten, J. Haegeman, and F. Verstraete, SciPost Phys.
 Lect. Notes , 7 (2019).
[74] M. Fishman, S. R. White, and E. M. Stoudenmire, “The ITensor
 software library for tensor network calculations,” (2020),
 arXiv:2007.14822.
[75] P. Hohenberg, Physical Review 136, B864 (1964).
[76] W. Kohn and L. J. Sham, Physical Review 140, A1133 (1965).
[77] A. J. Cohen, P. Mori-Sánchez, and W. Yang, “Insights into
 current limitations of density functional theory,” (2008).
[78] X. Duan, F. Wu, J. Chen, P. Zhang, Y. Liu, H. Yuan, and C. Cao,
 Communications Physics 2018 1:1 1, 1 (2018).
[79] a. Seidl, a. Görling, P. Vogl, J. Majewski, and M. Levy, Physical
 FIG. 7. KYbSe2 sample used to measure the low-energy spin excita-
 review. B, Condensed matter 53, 3764 (1996).
 tions on CNCS. 20 crystals were coaligned and glued to two aluminum
[80] J. P. Perdew, W. Yang, K. Burke, Z. Yang, E. K. Gross, M. Schef-
 plates (top) which were then screwed to a copper rod (bottom).
 fler, G. E. Scuseria, T. M. Henderson, I. Y. Zhang, A. Ruzsin-
 szky, H. Peng, J. Sun, E. Trushin, and A. Görling, Proceedings
 of the National Academy of Sciences of the United States of
 America 114, 2801 (2017). Neutron Source. The sample for this experiment consisted of
[81] J. P. Perdew, Physical Review B 23, 5048 (1981). 20 coaligned plate-like crystals glued to aluminum discs (see
[82] J. Sun, A. Ruzsinszky, and J. Perdew, Physical Review Letters Fig. 7), for a total mass of 200 mg KYbSe2 in the (ℎ 0)
 115, 036402 (2015). scattering plane. The sample was mounted in a 3 He refriger-
[83] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, Journal ator and measured with double-disc chopper frequency 300.0
 of Physics: Condensed Matter 9, 767 (1997).
 Hz (high-flux mode, 9 degree opening on the double disk).
[84] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Journal of Chemical
 Physics 118, 8207 (2003). All CNCS data were corrected for the isotropic Yb3+ form
[85] N. Deilynazar, E. Khorasani, M. Alaei, and S. Javad Hashemifar, factor [68].
 Journal of Magnetism and Magnetic Materials 393, 127 (2015), The spectrum was measured over 180◦ rotation at =
 arXiv:1502.01814. 3.32 meV and = 1.55 meV at base temperature and at
[86] A. Payne, G. Avedaño-Franco, X. He, E. Bousquet, and 12 K. At 1 K and 2 K, we measured only over 60◦ and used
 A. H. Romero, Physical Chemistry Chemical Physics 21, 21932 −3 crystal symmetry to fold the scattering over and cover
 (2019).
 the full range of reciprocal space. In comparing intensity of
[87] M. Casadei, X. Ren, P. Rinke, A. Rubio, and M. Scheffler,
 Physical Review Letters 109, 1 (2012). nuclear Bragg peaks, we did find some degree of obverse-
[88] A. Payne, G. Avendaño-Franco, E. Bousquet, and A. H. Romero, reverse twinning of the crystal array, such that some crystal
 Journal of Chemical Theory and Computation 14, 4455 (2018). planes were rotated 60◦ from those below. This did not affect
[89] G. Kresse and J. Hafner, Physical Review B 47, 558 (1993). the in-plane scattering due to the lack of scattering dependence
[90] G. Kresse and D. Joubert, Physical Review B 59, 1758 (1999). upon ℓ. The sample thermometer at base temperature read
[91] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 270 mK, but because this thermometer was not exactly on the
 3865 (1996).
 sample we round up the effective base temperature to 300 mK.
[92] V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B
 44, 943 (1991). To probe a possible gap at , we also measured a rotation scan
[93] P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka, J. Luitz, over 15◦ at = 1.0 meV, for a resolution FWHM of 20 eV at
 et al., An augmented plane wave+ local orbitals program for ~ = 0. These data are shown in Fig. 8, and reveal a gapless
 calculating crystal properties 60 (2001). excitation spectrum at 0.3 K to within 40 eV.
[94] G. Pokharel, A. F. May, D. S. Parker, S. Calder, G. Ehlers,
 A. Huq, S. A. J. Kimber, H. S. Arachchige, L. Poudel, M. A.
 McGuire, D. Mandrus, and A. D. Christianson, Phys. Rev. B Background subtraction
 97, 134117 (2018).
[95] T. Pandey and D. S. Parker, Phys. Rev. Applied 10, 034038
 (2018). For the CNCS experiment, a phenomenological background
 was created and subtracted using the 12 K scattering data. At
 12 K, the spin excitations become totally diffuse paramagnetic
 excitations. To model and eliminate these, we took the median
 METHODS
 intensity at each constant energy slice to be the approximate
 value of paramagnetic intensity, and subtracted this value from
 CNCS experiment each pixel at that energy transfer. Then, we set any negative
 intensities to zero, and subtracted this background from the
 We measured the low-energy spin excitations with the CNCS data. This median-value subtraction was not done for elastic
spectrometer at Oak Ridge National Laboratory’s Spallation scattering because paramagnetic intensity has negligible elas-

9

 (a) = meV (b) . < < . sum of squared residuals, defined as
 0.3 0.6
 100
 !2
 0.2 ∑︁ ∑︁ data − calc − 

 (arb. u.)
(meV)

 0.4 2 = 
 , (3)
 0.1 
 d ∈d
 0.0 0.2 10 1
 where denotes a data set, data is the intensity of data point
 0.1 0.3 K
 , calc is the corresponding calculated intensity [see Supple-
 0.75 0.70 0.65 0.60 0.0 0.2 0.0 0.2 0.4 mental Information], is the corresponding uncertainty, and
 [HH0] (meV)
 and denote, respectively, fitted intensity scale and offset
 FIG. 8. High resolution KYbSe2 scattering at (1/3, 1/3, 0). Panel (a)
 factors determined at each iteration using linear-least-squares
 shows a slice through the data showing the gapless dispersion. Panel relations.
 (b) shows a 1D cut indicated by the faint vertical red bar in panel (a). The (Q) are elements of an interaction matrix given by
 Both plots show the dispersion to be gapless at 0.3 K to within 0.04
 meV. + 0
 J (Q) = − ­ − 0 ® ,
 © ª
 (4)
 « 0 0 ¬
 tic contributions. Thus, for elastic data the 12 K was directly
 in which
 subtracted from lower temperatures. We find that this proce-
 dure effectively eliminates artifacts in the data while leaving = 2[cos 2 (ℎ + ) + cos 2 ℎ + cos 2 ], (5)
 magnetic intensity unchanged, as shown in Fig. 9. Finally,
 = 2 cos 2 (ℎ + ) − cos 2 ℎ − cos 2 , (6)
 because entanglement witnesses require a total sum rule sat- √
 isfying ( + 1) = 0.75 for an effective = 1/2 system, we = 3(cos 2 − cos 2 ℎ), (7)
 normalized the background-subtracted 300 mK KYbSe2 scat-
 tering such that the total scattering is h 2 i = 0.75. where ℎ and are noninteger Miller indices. We find a best fit
 Hamiltonian

 = 2.33(10) K = 2.28(10) K
 Critical scaling fits = −0.018(8) K 2 = 0.11(2) K (8)

 where and are the -plane and -axis nearest neighbor
 To fit the critical exponent in Fig. 6, we used data at ~ / 
 exchange respectively, is off-diagonal exchange [15], 2 is
 above the “knee” where the power law behavior starts. Using
 second neighbor Heisenberg exchange, and spins have been
 this data range, we minimized the 2 of the scaled data fitted
 treated as classical vectors of unit length. These values show
 to a power law in (~ / ) , varying and rescaling the
 off-diagonal exchange being much smaller than the Heisen-
 data in each iteration. This resulted in a fitted = 1.73(12).
 berg terms and , showing that KYbSe2 can be effectively
 modeled by the 1 − 2 Heisenberg model of Eq. (S.3) in the
 main text.
 ARCS experiment To check the robustness of the results, we performed three
 checks. First, to check for the possibility of local 2 minima,
 The sample for the ARCS measurement was 3 g of plate- we performed 20 separate fits initialized with different parame-
 like crystals ground into a powder. We measured the inelastic ter values in the range [−1 : 1] K. No local minima were found
 neutron scattering at incident energies = 35 meV, 50 meV, to give acceptable agreement with the experimental data, and
 and 130 meV and at temperatures 7 K, 100 K, 200 K, and the parameters reported in the text correspond to the minimum
 300 K (for = 50 meV only). For details of the crystal field 2 we obtained. Second, we considered the effect of including
 fits, see see the Supplemental Information. an additional symmetry-allowed off-diagonal exchange inter-
 action, [15]. This parameter refined to a zero value within
 uncertainty, and has negligible effect on the results. Third,
 we considered the effect of the obverse-reverse twinning of
 Onsager reaction field fits the crystal array, and found that including this effect in the
 calculation had negligible effect on the fit quality or parameter
 The magnetic diffuse scattering from the CNCS experiment values.
 was analyzed using the Onsager reaction field approach of
 Ref. 15. Fits to the single-crystal diffuse-scattering data sets
 were performed using the Migrad algorithm in the Minuit Schwinger boson calculations
 program [69]. The fitted data sets comprised the (ℎ 0), (ℎ0 )
 and (ℎℎ ) scattering planes measured at 1 K and 2 K. All data Here we describe the main steps of the Schwinger boson cal-
 were energy-integrated over > 0.05 meV. We minimize the culations. The triangular antiferromagnetic Heisenberg model

10

 . meV . meV . meV . meV . meV . meV . meV
 (a) 0.3 K data (b) (c) (d) (e) (f) (g)
 0.5 0.5
(/, , )

 0.0
 0.5
 / / / 0.4
 (h) 12 K bkg (i) (j) (k) (l) (m) (n)
 0.5

 ( , ) (mev
 0.3
(/, , )

 0.0
 0.5
 / / / 0.2

 )
 (o) 0.3 K - bkg (p) (q) (r) (s) (t) (u)
 0.5
(/, , )

 0.0 0.1
 0.5
 / / /
 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 0.5 0.0 0.5 0.0
 (, , ) (, , ) (, , ) (, , ) (, , ) (, , ) (, , )

 FIG. 9. KYbSe2 background subtraction for CNCS data. The top row shows the raw data at 0.3 K. The middle row shows the phenomenological
 background generated from the 12 K scattering data. The bottom row shows the data with the background subtracted, eliminating artifacts near
 = 0 and ~ = 0.

 is given in Eq. (S.3). The spin operators can be expressed in sites in the triangular lattice to form a cylinder geometry, and
 terms of SB operators, ˆ = 12 † , where † = ( † ↑ , † ↓ ), this choice leads to different allowed momentum values in
 and ≡ ( , , ) is the vector of Pauli matrices. The the Brillouin zone. We use the XC6 boundary conditions ex-
 spin-1/2 representation of the spin operator is enforced by the plained in [71]. In Fig. 10(a), we show a sample of this lattice
 constraint † ↑ ↑ + † ↓ ↓ = 1. with = 6 and = 6. In Fig. 10(b) we show the allowed 
 The Heisenberg interaction can be expressed in terms of the values for the XC6 boundary conditions, as well as the path we
 bond operators = 21 ( ↑ ↓ − ↓ ↑ ) and = 21 ( † ↑ ↑ + take to generate Fig. 5(e). We note that this choice of boundary
 conditions yields the maximum number of allowed points in
 † ↓ ↓ ):
 the path through the high symmetry points of interest.
   To calculate the dynamical structure factor, ( , ), we first
 2 1
 · = 
 ˆ ˆ − − 2(1 − ) † + 2 : † :, (9) calculate the time-dependent spin-spin correlation function
 2
 given by
 where the real parameter fixes the decoupling scheme of a
 path integral formulation over coherent states (for KYbSe2 , ( , ) B ˆ ( ) · ˆ (0) , (10)
 we set = 0.45), where the bond fields and are in-
 troduced via a Hubbard-Stratonovich transformation. At the where h·i is the expectation value in the ground state, and ˆ 
 saddle-point level (uniform and static bond fields), the theory is the spin operator at the center site in the lattice. Due to
 describes non-interacting spin-1/2 spinons, whose conden- the rotational symmetry of this model, we only look at the
 sation leads to long-range magnetic ordering [54, 55]. The -component of the spin using the identity
 inclusion of fluctuations of the bond fields that mediate the
 spinon-spinon interaction drastically modifies the excitation
 ˆ ( ) · ˆ (0) = 3 ˆ ( ) ˆ (0) , (11)
 spectrum in the sense that the true collective modes (magnons)
 of the antiferromagnetically ordered phase emerge as two-
 spinon bound states and the two-spinon continuum is strongly and we drop the pre-factor of 3 in the calculations. In defining
 renormalized [56, 70]. ˆ , we subtract ˆ to remove potential disconnected contri-
 butions present from finite precision. The dynamical structure
 factor is then related to this quantity through the Fourier trans-
 Tensor network calculation form,
 ∫ ∞ 

 1 ∑︁
 e− · − 
 For the tensor network calculation we studied the spin- 

 , = √ d ( , ), (12)
 1/2 triangular antiferromagnetic Heisenberg model defined in 2 −∞ 

 Eq. (S.3). In these simulations we wrapped the triangular lat-
 tice into a cylinder with a circumference of = 6 and length where the sum is over all = lattice sites, and measures
 = 36 sites. There is some freedom in how one identifies the distance from the center site. The quantity ( , ) has the

11

a b M K
 300

 DOS (arb. units)
 Total
 K Yb 4f
 200

 100
FIG. 10. Illustration of the geometry used and corresponding
Brillouin zone for the tensor network simulations. Panel a is a
6 × 6 lattice that we make a cylinder by identifying the top and bottom
 0
rows shown in red. Panel b is the Brillouin zone for this geometry, -14 -12 -10 -8 -6 -4 -2 0 2 4
with the blue shaded region showing the allowed momenta, and the
arrows show the path we take to generate Fig. 5(e). Energy (eV)

 FIG. 11. DFT electronic density of states in KYbSe2 calculated
following properties, using the SCAN functional with an additional Hubbard-U correction
 of U=8 eV on the Yb 5f states. The Fermi energy is set to 0 eV.
 (− , ) = ( , ), (13)
 Re ( , − ) = Re ( , ), (14)
 Im ( , − ) = −Im ( , ). (15) tional Theory [75, 76] (DFT) owing to systematic self-
Meaning we only need positive times, and can write interaction and static-correlation errors [77] in semi-local ap-
 ∫ ∞ ∑︁ proximate exchange correlation (XC) functionals. When f-
 
 1 
 shell magnetism is not important, DFT simulations of rare-
 , = √ d cos · 
 0 earth compounds often relegate the f-electrons to a core-
  
 × cos( ) Re ( , ) − sin( ) Im ( , ) . (16) shell [78] within a pseudopotential approximation. Such a
 description yields satisfactory structural property predictions
One major advantage of performing the Fourier transform this especially in the Lathanides where the f-shell-ligand hybridiza-
way is this ensures ( , ) is real, even when the time integral tion is weak [78]. In KYbSe2 , magnetism is of primary rel-
is truncated to a finite upper limit. Due to the finite system size, evance and so f-electrons need to be treated explicitly as va-
the resulting spectral function will be a sum of delta-functions, lence electrons. While semi-local GGA XC functionals within
and not the desired analytic function in the thermodynamic the traditional Kohn-Sham scheme often describe open f-shell
limit. To remedy this, we broaden these delta functions with insulators as metals, modern XC functionals when deployed
a Gaussian distribution with a width . This is achieved by within a generalized Kohn-Sham (GKS) [79, 80] framework
scaling ( , ) by a Gaussian, are able to mitigate f-shell self-interaction [81] errors (SIE) and
 yield a qualitatively correct accounting of the transport gap.
 ( , ) −→ e− ( , ),
 2
 (17) In KYbSe2 , we find that a meta-GGA+U approach where the
before integrating. SCAN [82] meta-GGA functional is employed in conjunction
 To perform this calculation, we use the Density Matrix with an on-site Hubbard-U correction [83] of U=8 eV is able to
Renormalization Group (DMRG) algorithm [72] to find the describe the system as insulating with one unpaired f-electron
ground state, and then use the Time Dependent Variational per Yb site (see Fig 11). Non-local screened hybrid func-
Principle (TDVP) [73] for the time evolution. In this work, tionals in the HSE06 [84] family are similarly able to describe
we used a maximum bond-dimension = 512, a time step the band gap in this system. Once an insulating ground state
 = 0.1, a maximum time max = 60, and a Gaussian width of is obtained, the size of the gap can be tuned by varying the
 = 0.02 1 . The introduction of a finite max corresponds to a fraction of non-local Fock exchange in the XC functional.
frequency resolution ∼ 1/ max , for which lower frequencies Furthermore we find that the band gap in KYbSe2 is largely
are not reliable, and The finite system size introduces a gap insensitive to the specific magnetic ordering between differ-
in the spectrum that scales as Δ ∼ 1/ , even if the system is ent Yb sites. This is in line with the expectation that 4f-shell
gapless in the thermodynamic limit. For this study, we utilize hybridization is weak and the GKS band gap between a Se
the ITensor library [74]. ligand dominated valence band and a narrow Yb 4f conduc-
 tion band is almost purely determined by on-site Coulomb
 repulsion between Yb f-electrons. However, a quantitative
 Density Functional Theory simulation accounting of the energies of different low-energy magnetic
 orderings in KYbSe2 is complicated by the previously doc-
 Localized f-electron magnetism in Mott-Hubbard systems umented multiple-minima problem [85, 86] encountered in
has traditionally been a challenge for ab initio Density Func- orbital-dependent XC functional approaches to modeling d-

12

and f-electron systems. We find that in trying to stabilize of Energy, Office of Science, Basic Energy Sciences, Materials
specific in-plane magnetic orderings in KYbSe2 , the GKS Sciences and Engineering Division. J.X. and A.S. were sup-
self-consistency cycle can get trapped in any one of a plethora ported by U.S. Department of Energy, Basic Energy Sciences,
of local minima associated with an overall similar magnetiza- Materials Science and Engineering Division.
tion density. These stationary points are further found to be N.E.S., M.D., J.E.M., (C.D.P., T.P.D.) were supported by
separated by energies comparable to the inter-site J couplings the U.S. Department of Energy, Office of Science, Office of
(∼1K) that one wishes to extract, making unambiguous identi- Basic Energy Sciences, Materials Sciences and Engineering
fication of the lowest energy minimum associated with a given Division under Contract No. DE-AC02-05-CH11231 (DE-
magnetic ordering difficult. Meta-heuristic approaches [86– AC02-76SF00515) through the Theory Institute for Materials
88] that aim to mitigate the multiple-minima problem have and Energy Spectroscopy (TIMES). J.E.M. acknowledges ad-
been proposed and a systematic effort to explore the efficacy ditional support by a Simons Investigatorship.
of such methods in the context of specific 4f-electron systems This research used the Lawrencium computational cluster
such as KYbSe2 is worth pursuing in future. resource provided by the IT Division at the Lawrence Berkeley
 The above DFT simulations of electronic structure in this National Laboratory (supported by the Director, Office of Sci-
work were carried out using the Vienna Ab Initio Simula- ence, Office of Basic Energy Sciences, of the U.S. Department
tion Package (VASP) [89] version 6.2.1. which employs a of Energy under Contract No. DE-AC02-05CH11231). This
planewave basis set in conjunction with PAW potentials
 √ √[90]. research also used resources of the National Energy Research
A planewave cutoff of 400 eV was used. A 72 atom 3×2 3×1 Scientific Computing Center (NERSC), a U.S. Department of
magnetic supercell was considered to model various low en- Energy Office of Science User Facility operated under Contract
ergy spin ordered configurations and a 4x2x1 k-point mesh No. DE-AC02-05CH11231.
was used to sample the corresponding Brillouin zone. Spin
orbit coupling was included and an in-plane 120◦ ordered con-
figuration was used to calculate the electronic DOS shown in AUTHOR CONTRIBUTIONS
Fig 11. For ionic positions, the experimentally determined
geometry was used. A.O.S. and D.A.T. conceived and coordinated the project.
 We also carried out first principles calculations of KYbSe2 J.X. and A.S. synthesized and characterized single crystal
within the GGA+U+so approach [91, 92], as implemented in KYbSe2 samples for experiments. A.O.S., D.A., and D.M.P.
the all-electron planewave density functional code WIEN2K performed the neutron experiments, and A.O.S. analyzed the
[93]. We find a saturation moment on the Yb site of 1.63 , in neutron data and calculated entanglement witnesses. J.A.M.P.
good agreement with the approximate value of 1.5 found performed ORF fits. E.A.G., S-S.Z., L.O.M., A.E.T., and
from the experimental measurements. It is noteworthy that C.D.B. carried out Schwinger boson calculations. N.E.S.,
this former value is primarily orbital moment, with the Yb or- M.D., and J.E.M. carried out tensor network calculations
bital moment, at 1.055 , outstripping the Yb spin moment of dynamical structure factors. C.D.P., T.P.D., and D.S.P.
of 0.572 (there is an additional small component from the carried out DFT calculations. A.O.S., N.E.S., M.D., J.E.M.,
interstitial region, and Se spheres, of 0.12 ). The exact na- C.D.B., and D.A.T. wrote the manuscript with input from all
ture of exchange coupling, as well as the coupling to the lattice co-authors.
depicted by the substantial orbital moment, in such border-
line quantum magnets [94, 95] remains a matter of substantial
debate and controversy.
 COMPETING FINANCIAL INTERESTS

 DATA AVAILABILITY The authors declare no competing financial interests.

 All plotted experimental data will be made publicly avail-
 CORRESPONDING AUTHORS
able.

 Correspondence and requests for materials should be ad-
 ACKNOWLEDGMENTS dressed to A. O. Scheie and C. D. Batista.

 This research used resources at the Spallation Neutron
Source, a DOE Office of Science User Facility operated by
the Oak Ridge National Laboratory. The work by D.A.T. is
supported by the Quantum Science Center (QSC), a National
Quantum Information Science Research Center of the U.S. De-
partment of Energy (DOE). The work of J.A.M.P. (magnetic
diffuse scattering fits) was supported by the U.S. Department

S1

 0.3 K 1K 2K
 SUPPLEMENTAL INFORMATION FOR WITNESSING
 (a) (b) (c) 3
QUANTUM CRITICALITY AND ENTANGLEMENT IN THE
 TRIANGULAR ANTIFERROMAGNET KYbSe2 0.5
 2

 (/, , )
 0.0

 ()
 ENTANGLEMENT WITNESSES 1
 0.5
 One tangle 1.0 0
 0.0 0.5 0.0 0.5 0.0 0.5
 (, , ) (, , ) (, , )
 One-tangle is calculated from the static spin at zero tem- 5.0 (d) (e) (f) 0.5
perature and ranges between zero (unentangled state) and one 2.5
(maximally entangled state). We calculate this from the 0.3 K
data using the ratio of elastic (to within ±0.04 meV) to total 0.0 0.0
magnetic scattering: 1 = 1 − 4h i 2 = 1 − 3 , where 2.5
is the ratio of elastic to total scattering and h i 2 = 43 for
 5.0 5 0.5
effective = 1/2. Summing over all elastic intensity and 0 5 5 0 5 5 0 5
comparing to inelastic scattering, we find 13(2) % of the mag- + / + / + /
netic scattering is elastic at 0.3 K. This gives a one-tangle
 1 = 0.62(8). This is of course not at zero temperature, but FIG. S1. Energy-integrated KYbSe2 scattering (a)-(c) Fourier trans-
magnetic entropy measurements indicate that by 0.3 K nearly formed to obtain the static spin-spin correlation in real space (d)-(f).
all ln(2) is recovered [1], and so only small changes in the Red indicates ferromagnetic correlation, blue indicates antiferromag-
 netic correlation. Quasi-long-range order is visible at 0.3 K, but
magnetism are expected between 0.3 and 0.0 K.
 higher temperatures show a much shorter correlation length.

 Two tangle
 Quantum Fisher Information

 We calculate Two-tangle from the real-space spin corre-
 Quantum Fisher Information can be calculated from the
lations, which we obtain from the Fourier transform of the
 neutron spectrum by an integral over energy. When normalized
energy-integrated (ℎ 0) plane scattering. For an isotropic
 by spin length, QFI is defined as
 = 1/2 system (which KYbSe2 is to a good approximation—
see Onsager reaction field fits), the two-tangle is defined as ∫ ∞  
 ~ ~ 
 nQFI = d (~ ) tanh 00 (~ , ) (S.2)
   !2 3 2 0 2 
 ∑︁ 1
 2 = 8 max 0, 2| | − + , (S.1)
 4 [3, 4], where 00 (q, ) is obtained via the fluctuation dissi-
 ≠0
 pation theorem 00 (k, ) = ~1 tanh ( ~ /2 ) ( , ). In our
where = h + 
 
 i [2]. As eq. S.1 shows, must exceed case, because the correlations are 120◦ , we calculate the nQFI
the classical threshold of 1/4 for two-tangle to be nonzero. As at q = (1/3, 1/3) as shown in main text Fig. 4(b). The
is shown in main text Fig. 4(b), none of the first four neighbor magnetic excitations at q = (1/3, 1/3) are gapless to within
distances exceed this threshold, and thus two-tangle is zero for 40 eV and the QFI integral begins to diverge as temperature
all temperatures in KYbSe2 . decreases.
 We obtained the real space correlations in main text Fig. 4
for the two-tangle by taking Fourier transform of the energy-
integrated data in the (ℎ 0) plane. To do this, we cut out a FITTING THE ROTON MODE
section of reciprocal space from −0.5 < ℎ < 0.5 and −1 <
 < 1. Data were corrected for the Yb3+ form factor and To quantify the extent and the gap of the roton-like mode,
background subtracted as described above. Empty data near we fitted the intensity vs energy of many constant- cuts as
 = 0 was filled in with data from the next Brillouin zones, shown in Fig. S2. We used an asymmetric Gaussian to model
and then the data were integrated over all energies, yielding the mode, and with the exception of two data points near ,
the 2D slices shown in Fig. S1. it picks out the peak maximum very well. We then fitted these
 We then took the 2D Fourier transform of the data to obtain data points to a sinusoidal function sin( ) + sin(3 ) + 
the real-space spin-spin correlation function, which is shown to estimate the mode maximum and minimum. These fits
in the bottom row of Fig. S1. Note that the correlations show a mode maximum of 0.288(12) meV, a roton minimum
noticeably decrease as temperature increases, although they 0.200(13) meV, and a fitted gap of 0.059(7) meV. The fitted gap
retain an overall antiferromagnetic correlation to the nearest may be an artifact of the mode’s deviation from the idealized
neighbors. The average neighbor correlations for the two- sin function rather than an actual gap—the higher resolution
tangle calculation in main text Fig. 4 were extracted from scan in Fig. 8 of the main text do not reveal a clear gap up to
these plots. 40 eV.