# Signature of a randomness-driven spin-liquid state in a frustrated magnet

J. Khatua  Department of Physics, Indian Institute of Technology Madras,
Chennai 600036, India  M. Gomilšek  Jožef Stefan Institute, Jamova c. 39, 1000
Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska u. 19, 1000 Ljubljana, Slovenia  J. C. Orain  Paul
Scherrer Institute, Bulk MUSR group, LMU 5232 Villigen PSI, Switzerland  A. M.
Strydom  Highly Correlated Matter Research Group, Department of Physics,
University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa  Max
Planck Institute for Chemical Physics of Solids, 40 Nöthnitzerstr., Dresden
D-01187, Germany.  Z. Jagličić  Faculty of Civil and Geodetic Engineering,
University of Ljubljana, 1000 Ljubljana, Slovenia  Institute of Mathematics,
Physics and Mechanics, 1000 Ljubljana, Slovenia  C. V. Colin  Institut Néel,
Université Grenoble Alpes, CNRS, Grenoble, 38042, France  S. Petit  LLB, CEA,
CNRS, Université Paris-Saclay, CEA Saclay, 91191 Gif-sur-Yvette, France  A.
Ozarowski  National High Magnetic Field Laboratory, Florida State University,
Tallahassee, Florida 32310, USA  L. Mangin-Thro  Institut Laue-Langevin, 38042
Grenoble, France  K. Sethupathi  Department of Physics, Indian Institute of
Technology Madras, Chennai 600036, India  Quantum Centre for Diamond and
Emergent Materials, Indian Institute of Technology Madras, Chennai 600036,
India.  M.S. Ramachandra Rao  Quantum Centre for Diamond and Emergent
Materials, Indian Institute of Technology Madras, Chennai 600036, India.
Department of Physics, Nano Functional Materials Technology Centre and
Materials Science Research Centre, Indian Institute of Technology Madras,
Chennai-600036, India  A. Zorko  Jožef Stefan Institute, Jamova c. 39, 1000
Ljubljana, Slovenia  Faculty of Mathematics and Physics, University of
Ljubljana, Jadranska u. 19, 1000 Ljubljana, Slovenia  P. Khuntia  Department
of Physics, Indian Institute of Technology Madras, Chennai 600036, India
Quantum Centre for Diamond and Emergent Materials, Indian Institute of
Technology Madras, Chennai 600036, India.  Functional Oxide Research Group,
Indian Institute of Technology Madras, Chennai 600036, India.

###### Abstract

Collective behaviour of electrons, frustration induced quantum fluctuations
and entanglement in quantum materials underlie some of the emergent quantum
phenomena with exotic quasi-particle excitations that are highly relevant for
technological applications. Herein, we present our thermodynamic and muon spin
relaxation measurements, complemented by ab initio density functional theory
and exact diagonalization results, on the recently synthesized frustrated
antiferromagnet Li4CuTeO6, in which Cu2+ ions ($S$ = 1/2) constitute
disordered spin chains and ladders along the crystallographic [101] direction
with weak random inter-chain couplings. Our thermodynamic experiments detect
neither long-range magnetic ordering nor spin freezing down to 45 mK despite
the presence of strong antiferromagnetic interaction between Cu2+ moments
leading to a large effective Curie-Weiss temperature of $–$154 K. Muon spin
relaxation results are consistent with thermodynamic results. The temperature
and magnetic field scaling of magnetization and specific heat reveal a data
collapse pointing towards the presence of random-singlets within a disorder-
driven correlated and dynamic ground-state in this frustrated antiferromagnet.

Geometrically frustrated magnets with incompatible magnetic interactions
between spins are characterized by strong quantum fluctuations and absence of
trivial order parameter related to broken symmetry. They are, therefore,
extremely sensitive to perturbations such as quenched disorder, next-nearest-
neighbor interactions, and magnetic anisotropy 1; 2. Strong quantum
fluctuations preclude symmetry breaking phase transitions and lead to novel
magnetically disordered ground states such as quantum spin liquids (QSLs) 1;
3; 4; 5; 6. QSLs are characterized by quantum fluctuations, absence of long-
range order even at T$\rightarrow{0}$, and exotic fractionalized excitations
in contrast to spin-wave excitations found in conventional magnets 7; 8. These
properties of QSLs are fingerprints of strong quantum entanglement.
Triangular-lattice compound $\kappa$-(ET)2Cu2(CN)3 9 and 1-TaS2 10, kagome-
lattice compound ZnCu3(OH)6Cl2 11, and hyperkagome-lattice compound PbCuTe2O6
12; 13 are some of the most investigated candidate spin-liquid materials with
unusual ground-state properties. Recently, enormous efforts have been devoted
towards experimental realizations of QSLs on the honeycomb-lattice ever since
Alexei Kitaev proposed an exactly solvable highly anisotropic Kitaev model for
$S$ = 1/2 spins on this two dimensional spin-lattice 14. In sharp contrast to
an isotropic nearest-neighbor exchange model on the honeycomb-lattice, which
exhibits long-range order 15, the ground-state of the Kitaev model is a QSL
with fractionalized Majorana fermions and gauge flux excitations 16; 17; 18;
19. The realization of this exotic state of matter is interesting not only
from a fundamental physics point of view but also holds immense promise for
potential applications in robust quantum computing technology 20.
Despite tremendous experimental efforts, the observed signatures of QSL are
far away from those of theoretically proposed QSLs states for a clean periodic
system 2; 21; 14. However, it has been suggested that some highly frustrated
magnets show quantum disordered ground-state due to randomness that might play
an important role in destabilizing conventional Néel order at low $T$. There
are several origins of randomness in magnetic insulators and the interplay
between quantum fluctuation and randomness may lead to novel quantum ground-
states. The triangular-lattice YbMgGaO4 with anti-site mixing between non-
magnetic Mg2+ and Ga3+ ions 22; 23 and square lattice Sr2CuTe1-xWxO6 with a
few percent of W on the Te site are some of examples of frustrated magnets
with a randomness-induced quantum disordered ground state 24; 25; 26.
Interestingly, while most of the reported Kitaev magnets including 5d iridates
A2IrO3 (where A = Na, Li) and ruthenate $\alpha$-RuCl3 with 4$d$ electrons
show long-range order at sufficiently low-temperature, the iridate H3LiIr2O6
and ruthenate $\alpha$-Ru1-xIrxCl3, which are derived from a parent ordered
magnet by substitution of non-magnetic ions, do not exhibit long-range
magnetic order 27; 28; 29; 30. Furthermore, some honeycomb-lattice magnets,
like A3LiIr2O6 (A = H, D, Ag etc.), show scaling behavior in the field
dependence of the specific heat 27; 31; 32, which is ascribed to the existence
of disorder, making these quantum materials interesting for studies of the
effect of disorder on a QSL state 32. A similar scenario is observed in three
dimensional spin-lattice where quantum fluctuation is less pronounced. For
instance, the oxynitride Lu2Mo2O5N2 with $S$ = 1/2 spins on a pyrochlore
lattice, which is derived from the spin-glass material Lu2Mo2O7 via random
substitution of O2- and N3- anions, shows a dynamic ground state 33; 34.
Disorder in quantum magnets is usually unavoidable, yet very often it can act
as a new prism in revealing many interesting quantum phenomena of the host
material 35; 36; 37; 38. Frustrated magnets with quenched disorder in the form
of material defects or a broad distribution of exchange interaction strengths
can exhibit a randomness induced spin-liquid state. Understanding the impact
of quenched disorder on the quantum fluctuations that drive quantum disordered
states in antiferromagnets is an interesting area of study 39; 22; 40; 41. In
the so-called random-singlet phase, singlets form due to a random distribution
of antiferromagnetic exchange interaction between spin-1/2 defect sites,
leading to a power-law distribution of exchange energies and density of
states, which lead to unconventional scaling behavior in magnetic
susceptibility and specific heat at low-temperatures 22; 32; 42. The signature
of such novel random singlet state was initially observed in one-dimensional
spin-chain organic compounds due to presence of random antiferromagnetic
Heisenberg exchange interaction 43; 44. Furthermore, there is also
experimental evidence of random singlet state in inorganic spin-chain compound
BaCu2(Si1-xGex)2O7 45 wherein the random substitution of Ge in place of Si
introduce the bond disorder in antiferromagnetic long-range ordered magnet
BaCu2Si2O7 46. Theoretically, it has been proposed that such exchange
randomness in spin-chain compound is one of the key ingredients for quantum
information processor which deals with properties of quantum entanglement in
quantum materials 47; 48; 49; 50. In recent years, higher dimensional quantum
materials with intrinsic disorder either from spin vacancies, anti-site
disorder or non-magnetic impurities and dislocations or grain boundaries have
drawn enormous attention to realize novel low-temperature phases including
random singlet that is expected to be unique in disordered quantum many-body
systems 33; 34; 25. In frustrated magnets with quenched disorder, the majority
of intrinsic $S$ = 1/2 spins may constitute a dynamic liquid-like state while
the remaining $S$ = 1/2 moments act as defects and form a random network with
exotic low-lying excitations that destabilize a glassy state and instead lead
to a dynamic low-energy ground-state 32. Theoretically, it has been proposed
that randomness in the frustrated lattice can induce such spin-liquid like
states in a wide region of parameter space 51. Experimentally, this phenomenon
is much less explored due to a lack of suitable frustrated magnets, which
leads one to look for candidate materials wherein the degree of randomness can
be tuned in a controlled manner. In this context, 3$d$ transition-metal
frustrated S = 1/2 spin lattices are promising potential hosts of a randomness
driven quantum spin-liquid ground state.

Figure 1: Crystal structure and high temperature spin model of Li4CuTeO6. (a)
Visualization of one unit cell of LCTO where edge-sharing TeO6 and CuO6
octahedra connect Cu2+ ions on $2d$ (Cu2) sites with an exchange $J$ through a
Cu–O${\cdots}$O–Cu super-superexchange (SSE) bridge around Te6+. Additionally,
corner-sharing CuO6 octahedra connect Cu2+ ions on $2d$ (Cu2) and $4g$ (Cu3)
sites with an exchange $J^{\prime}$ through a nearly-linear Cu–O–Cu
superexchange (SE) bridge. (b) Resulting spin model of randomly-depleted 1D
spin chains of Cu2 sites with antiferromagnetic exchange $J$ running along the
[101] direction, to which randomly-occupied Cu3 sites couple via an
antiferromagnetic exchange $J^{\prime}$ and introduce strong frustration. (c)
Rietveld refinement of neutron diffraction data of LCTO at room temperature
for the incident wavelength 1.28 Å. The solid circles represent the observed
intensity (Obs.) whereas the black solid line is the calculated intensity
(Cal.). The residual signal is denoted by a green solid line and is shifted
vertically for clarity. (d) The ESR spectrum of LCTO measured at 250 K and
256.3 GHz (points). The fit (solid line) includes two components with uniaxial
symmetry and a Lorentzian line shape.

Herein, we report crystal structure, neutron diffraction, magnetic properties,
specific heat, and muon spectroscopy, as well as complementary density
functional theory (DFT) and exact diagonalization (ED) results of the
antiferromagnet Li4CuTeO6 (henceforth LCTO), where Cu2+ ($S$ = 1/2) ions are
arranged on a frustrated spin-lattice with unavoidable anti-site disorder
between Li+ and Cu2+. In LCTO, the majority of Cu2+ ions (84 %) at the 2$d$
crystallographic site form random-length strongly antiferromagnetic spin
chains while a minority of Cu2+ ions ($\sim$ 7 %) at defect 4$g$ sites
strongly antiferromagnetically couple to these spin chains, leading to
significant frustration. The large and negative value of Curie-Weiss
temperature reflects the presence of strong antiferromagnetic interaction
between Cu2+ spins. Furthermore, we find that weaker inter-chain interactions
effectively couple the chains in a random, disordered network at low $T$. As a
consequence, this compound neither undergoes a phase transition to long-range
magnetic order nor spin freezing down to at least 45 mK. Furthermore, specific
heat and magnetization results reveal data collapse behavior, which suggests
the presence of a random-singlet state 32. Muon spin relaxation measurements
corroborate a dynamic ground-state in this frustrated magnet. This is
attributed to the presence of subdominant inter-chain interactions that couple
the chains in a disordered network. Our results thus establish that LCTO hosts
a randomness induced spin-liquid-like state in a frustrated magnet.

## Results

### Rietveld refinement and crystal structure

The room-temperature powder X-ray diffraction data were taken to confirm the
phase purity of the polycrystalline samples of LCTO. The Rietveld refinement
of the X-ray diffraction data was performed using GSAS software 52, which
confirms that LCTO crystallizes in a monoclicnic structure with the space
group $C2/m$ (No. 12) 53. To deepen the analysis of the crystalline structure
and in particular to determine the occupation of the mixed sites by the light
elements a neutron diffractogram was measured at 300 K at an incident
wavelength 1.28 Å. Fig. 1 (c) depicts the Rietveld refinement of neutron
diffraction data performed using fullprof which yields lattice parameters $a$
= 5.2752(5) Å, $b$ = 8.8163(8) Å, $c$ = 5.2457(5) Å and $\alpha$ = $\gamma$ =
90∘, $\beta$ = 113.172(8)∘, in agreement with powder X-ray diffraction data
(see SI table I) and previously reported values 53. A unit cell of LCTO is
shown in Fig. 1 (a). We found that among three crystallographic sites of Li,
the sites 2d and 4g are partially shared with Cu2 and Cu3 sites, respectively,
with Cu2 site occupancy $p_{2}$ = 84 % and Cu3 site occupancy $p_{3}$ = 7 %.
This is likely due to the similar ionic radii of Li+ and Cu2+ ions. Cu2+ ions
form distorted CuO6 octahedra with nearest-neighbor oxygen ions with Cu2 and
Cu3 sites connecting to each other directly by corner- and edge-sharing, and
indirectly via corner- and edge-sharing TeO6 octahedra in an a priori 3D
lattice. However, a critical look at the crystal structure reveals only two
plausibly strong exchange pathways: (i) a nearly-linear Cu–O–Cu superexchange
(SE) bridge between Cu2 and Cu3 sites with bond angle 173.3∘ and bond distance
4.12 Å, which is expected to yield a strong antiferromagnetic exchange
$J^{\prime}$ by Goodenough-Kanamori rules 54; 55, and (ii) a double
Cu–O${\cdots}$O–Cu super-superexchange (SSE) bridge between two Cu2 sites
going around a Te6+ ion with a large symmetric Cu–O${\cdots}$O bond angle of
138.6∘ and a small O${\cdots}$O distance 2.87 Å (well below the van der Waals
radii sum $\sim$ 3.04 Å), which is expected to yield a strong
antiferromagnetic exchange $J$ 56; 57; 58; 59; 60 [Fig. 1(a)]. The resulting
1D spin model is shown in Fig. 1(b) and consists of random-length linear spin
chains of Cu2 (occupation probability $p_{2}$) with exchange $J$ running along
the [101] direction, with randomly-occupied Cu3 defect sites (occupation
probability $p_{3}$) displaced along $\pm b$ from each main-chain bond $J$
introducing strong frustration by coupling to both neighboring Cu2 main-chain
sites via a $J^{\prime}$ exchange. All other SE and SSE pathways, which, if
present, introduce 3D inter-chain couplings in general, are expected to have
much smaller strength as they either proceed via very nearly 90∘ Cu–O–Cu and
Cu–O${\cdots}$O angles, or involve O${\cdots}$O distances well above the O2-
van der Waals radii sum.

###  Ab initio and exact diagonalization calculations

The expected interaction between Cu2+ moments are fully born out by ab initio
DFT exchange-coupling calculations on a $2\times 1\times 2$ supercell of LCTO
using the local (spin) density approximation (LSDA) in an LSDA+$U$ scheme with
an effective Hubbard $U_{\mathrm{eff}}$ = 9 meV (see SI for details). By first
assuming full occupancy of Cu2 sites and zero occupancy of Cu3 sites the main-
chain Cu2–Cu2 antiferromagnetic exchange strength $J_{\mathrm{DFT}}$ = 177 K
was extracted, and then by introducing a single occupied Cu3 site the
frustrating Cu2–Cu3 antiferromagnetic exchange strength
$J^{\prime}_{\mathrm{DFT}}$ = 824 K was obtained. As expected, all other
exchange interactions, which introduce 3D couplings between the chains, were
found to have strengths below $\sim$ 6 K, making them difficult to resolve by
DFT. For $T\gg$ 6 K the frustrated 1D random-occupancy model of Fig. 1(b) is
thus expected to hold, while at low $T$ further frustrating inter-chain
exchanges might be expected. Temperature dependent thermodynamic quantities of
the high-$T$ random model obtained from DFT were calculated using ED by
probability weighing ED results for different random-length chains of Cu2
spins with a different distribution of occupied Cu3 sites around them (see SI
for details). This way, ED predictions for magnetic susceptibility
$\chi_{\mathrm{ED}}$, magnetic specific heat $C_{\mathrm{mag,ED}}$ and
magnetic entropy $S_{\mathrm{mag,ED}}$ as a function of $T$, applied magnetic
field $\mu_{0}H$, $g$ factor, and $J$ and $J^{\prime}$ exchange strengths at
any given occupation probabilities $p_{2}$ and $p_{3}$ could be calculated to
high accuracy.

### Magnetic susceptibility

Fig. 2 (a) presents the experimental temperature dependence of the magnetic
susceptibility ($\chi(T)$) in an applied magnetic field $\mu_{0}H$ = 1 T in
the temperature range 1.9 K $\leq$ $T$ $\leq$ 300 K. The magnetic
susceptibility data do not indicate the presence of a phase transition down to
1.9 K. In order to estimate dominant magnetic interactions, the high-
temperature 1/$\chi(T)$ data were fitted with a Curie-Weiss law (right $y$\-
axis; Fig. 2 (a))

$\chi=\frac{C}{\textit{T}-\theta_{\rm CW}}.$ (1)

In Eq. (1), $\theta_{\rm CW}$ and C are the Curie-Weiss temperature and the
Curie constant, respectively. The Curie-Weiss fitting in the temperature range
180 K $\leq$ T $\leq$ 300 K yields $\theta_{\rm CW}$ = $-$154 K and C = 0.39
cm3 K/mol. The estimated effective magnetic moment $\mu_{\rm eff}$ =
$\sqrt{8C}$ = 1.76 $\mu_{\rm B}$ is close to the expected value for a Cu2+ ion
with $S$ = 1/2 spin. The large negative $\theta_{\rm CW}$ confirms the
presence of dominant antiferromagnetic interactions between Cu2+ spins.

Figure 2: Magnetization results of Li4CuTeO6. (a) The temperature dependence
of the magnetic susceptibility (left y-axis) and inverse magnetic
susceptibility (right y-axis) in a magnetic field $\mu_{\rm 0}H$ = 1 T. The
solid line represents the Curie-Weiss fit. (b) The temperature dependence of
magnetic susceptibility at 1 T with a high-$T$ fit using ED results, and a
low-$T$ fit using a power-law. The inset shows the zero-field cooled (ZFC) and
filed cooled (FC) susceptibility in a magnetic field of 100 Oe. (c) Data
collapse for $MT^{-0.15}$ as a function of $\mu_{0}H/T$. (d) The temperature
dependence of the real part of ac susceptibility in different frequencies down
to 45 mK.

To obtain a more quantitative estimate of the $J$ and $J^{\prime}$ exchange
strengths, the $\chi(T)$ data were next fitted with the results of ED
calculations on the random spin model of Fig. 1(b). Fixing the ratio of
experimental $J/J^{\prime}$ to be the same as that found in DFT,
$J_{\mathrm{DFT}}/J^{\prime}_{\mathrm{DFT}}$, which is expected to be a more
robust DFT prediction than raw exchange strengths, and fixing the main-chain
Cu2 site occupancy to the experimental value $p_{2}$ = 84 %, a good fit at
$T>$ 20 K is obtained [Fig. 2(b)] with only 3 free, sensible parameters:
$g=2.28$, $J/J_{\mathrm{DFT}}=J^{\prime}/J^{\prime}_{\mathrm{DFT}}=1.38$, and
$p_{3}=$ 20 %. We note that while fit parameters $g$ and
$J/J_{\mathrm{DFT}}=J^{\prime}/J^{\prime}_{\mathrm{DFT}}$ are quite robust to
perturbations like magnetic anisotropies or unwanted extrinsic contributions
due to any impurities, the fitted value of $p_{3}$ is substantially less so
since the experimental data is limited to $T\ll J^{\prime}$. Thus, the
experimental value of Cu3 occupancy from neutron diffraction should be taken
as definitive. The final estimates for the dominant exchanges are thus $J$ =
244 K and $J^{\prime}$ = 1140 K, with the discrepancy from DFT likely arising
from the choice of Hubbard $U_{\mathrm{eff}}$, which acts to rescale all
exchanges. Importantly, for $T<$ 20 K the high-$T$ model of Fig. 1(b) starts
to systematically deviate from experiment, indicating the presence of further,
most likely inter-chain, exchanges $J_{\mathrm{inter}}\lesssim$ 10 K in
strength acting between the random spin-chain fragments of the high-$T$ model
and dominating the physical response of LCTO at low $T<20\ \rm K\ll$
$J,J^{\prime}$. In this regime most spin degrees of freedom of spin-chain
fragments are expected to freeze out, leaving only their ground state degrees
of freedom to couple via $J_{\mathrm{inter}}$ exchanges in an emergent $\geq
2$ dimensional (likely 3D) random lattice. Remarkably, though LCTO exhibits
such quenched disorder, ultimately arising from partial Cu2+ site occupancy,
that could lead to a spin-glass state at low $T$, the absence of glass-like
freezing behavior is witnessed by the absence of any separation between zero-
field-cooled and field-cooled susceptibility measurements in 100 Oe (inset of
Fig. 2 (b)). Furthermore, to confirm the absence of spin-freezing, we have
performed ac susceptibility measurement down to 45 mK at different
frequencies. It is observed that there is no peak or frequency dependence of
real part of the ac susceptibility (Fig. 2 (d) and SI Fig. 3) in the
temperature range 45 mK $\leq$ $T$ $\leq$ 50 K, which strongly rules out the
presence of spin glass transition in Li4CuTeO6 61. Quenched Li/Cu anti-site
disorder in the host lattice could, alternatively, also lead to unconventional
scaling behavior of thermodynamic observables. Such a scenario has been
proposed in several frustrated quantum materials 25; 22; 32; 62.
As shown in Fig. 2 (c), a data collapse of $MT^{-0.15}$ as a function of
$\mu_{\rm 0}H/T$ is found in LCTO in a broad temperature range between 2 and
50 K, which could be a sign of the presence of random-singlets 62.

Figure 3: Specific heat measurements reveal a randomness induced spin-liquid
state in Li4CuTeO6 at low $T$. (a) The temperature dependence of the specific
heat ($C_{\rm p}$) in zero magnetic field in a broad temperature range 0.052 K
$\leq$ T $\leq$ 250 K. (b) The temperature-dependent specific heat in
different magnetic fields upto 7 T. (c) The temperature dependence of magnetic
specific heat divided by temperature in zero-field in a log-log scale where
the red line is the power-law fit. Inset shows the temperature dependence of
associated entropy change. (d) The scaling behavior of the specific heat with
$C_{\rm mag}(\mu_{0}H)^{0.85}/T$ as a function of the scaled temperature
T/$\mu_{0}H$ for several fields.

Furthermore, theoretically, it has been suggested that in the absence of the
spin-orbit interaction, low-temperature magnetic susceptibility data of
random-singlet states show a power-law $\chi\propto T^{-\gamma}$ behavior with
$\gamma$ $<1$ 24. The susceptibility data were fitted with a power-law
$\chi\propto T^{-\gamma}$ in the low-temperature region, which yielded
$\gamma$ = 0.82 $\pm$ 0.01 (Fig. 2 (b)). A similar power-law fit was also
observed in the quasi-two dimensional magnet Sr2CuTexW1-xO6, which is a
promising candidate for realizing a random-singlet state on a square lattice
25. A power-law fit of low-temperature magnetic susceptibility thus suggests
the realization of a random-singlet state attributed to the presence of
isolated magnetic moments in the host lattice 25; 32. The data collapse is a
remarkable feature found also in other QSL candidates with quenched disorder
such as the honeycomb-lattice H3LiIr2O6 27 and the triangular-lattice Y2CuTiO6
62. The data collapse behavior in the low-temperature region suggests the
presence of random-singlets in LCTO induced by quenched disorder as predicted
theoretically 32. The powder ESR spectrum consists of two components, both
featuring uniaxial symmetry, in agreement with the crystal symmetry. A fit of
the spectrum with a powder-averaged Lorentzian shape (see Fig. 1 (d)) yields
principal $g$ factors $g_{x,y}^{1}$ = 2.06 and $g_{z}^{1}$ = 2.31 for the
component with larger intensity and $g_{x,y}^{2}$ = 2.06 and $g_{z}^{2}$ =
2.20 for the component with smaller intensity. The ratio of the intensities is
85:15, which is close to the ratio $p_{2}:2p_{3}$ of the occupied Cu2 and Cu3
sites as confirmed from the structural refinement.

### Specific heat

Fig. 3 (a) depicts the temperature dependence of specific heat in zero
magnetic field. The absence of any anomaly in the entire temperature range of
investigation rules out any phase transition down to 52 mK, which is much
below both the Curie-Weiss temperature and the dominant $J$ and $J^{\prime}$
exchange interactions and thus indicates the presence of strong spin
frustration that promotes a dynamic ground-state in LCTO. The specific heat
results in several magnetic fields up to 7 T are shown in Fig. 3 (b). It is
observed that upon increasing the magnetic field the broad maximum shifts
toward higher temperatures. Though similar to a Schottky contribution, which
could arise from quasi-free spins, it, in fact, describes intrinsic spin
physics due to both spins that appear isolated in the high-$T$ spin model
[Fig. 1(b)], and due to finite-length spin-chain fragments with an odd total
number of spins, as the total spin of those cannot be lower than $1/2$ by the
rules for quantum addition of spin. In both cases, an applied field Zeeman
splits the spin-$1/2$ ground-state introducing a local-field-dependent energy
gap that results in a specific heat peak whose position is proportional to the
applied field. However, all spins, both those in finite-length spin-chain
fragments, as well as nominally isolated spins, in fact couple together to
form a frustrated 3D random spin-lattice via weaker $J_{\mathrm{inter}}$
inter-chain exchanges, together shaping the intriguing, liquid-like low-$T$
behaviour of LCTO at $T\lesssim J_{\mathrm{inter}}$.

Figure 4: Muon spin relaxation results reveal a spin-liquid state in
Li4CuTeO6.(a) The time evolution of zero-field muon spin asymmetry curves for
several temperatures where solid lines are the fits to Eq. (3). (b) The time
evolution of muon spin asymmetry in zero-field and in a longitudinal field
(LF) of 50 G at 1.55 K with fits as discussed in the text. The simultaneous
fit of zero-field and longitudinal field data to the Eq. (3) yields $\sigma=$
3.66 G and $\lambda$ = 0.029 $\mu$s-1. (c) The temperature dependence of
Knight shift where red line is the fit of low-temperature Knight shift i.e.,
$K\sim$ $T^{-\gamma}$ with $\gamma$ = 0.75 in the applied field of 4900 G.
Inset shows the scaling of the muon Knight shift ($K$) with bulk
susceptibility with temperature as an implicit parameter. The solid line is
the linear fit in two regions (see text). (d) The temperature dependence of
the longitudinal muon spin-relaxation rate in zero-field. (e) The temperature
dependence of the muon relaxation rate in the presence of a transverse field
(TF) of 4900 G where red line is the power-law $T^{-0.2}$ fit in the low-
temperature region.

In order to gain further insights into the magnetic properties relevant to
this frustrated magnet and understand the ground-state degeneracy, it is
important to extract the magnetic specific heat. Therefore, we have considered
the zero-field specific heat data ($C_{\rm p}$) as a sum of intrinsic magnetic
specific heat ($C_{\rm mag}$) due to exchange-coupled Cu2+ ions, and lattice
specific heat ($C_{\rm lattice}$) due to phonons. After subtracting the
lattice contribution, the obtained magnetic specific heat exhibits a clear
broad maximum (SI Fig. 6) around 0.4 K releasing $\sim 13{\text{\%}}$ of
entropy [inset in Fig. 3(c)], which is a typical signature of short-range spin
correlations in spin liquid candidates 62; 27. Comparing this with ED results
on the high-$T$ model with no inter-chain coupling, which shows no such peak
in zero field, and instead exhibits residual entropy at absolute zero due to
unavoidable Kramers ground-state degeneracy of individual spin-chain fragments
(see SI), this clearly shows the dominant impact of frustrated inter-chain
couplings $J_{\mathrm{inter}}$ at low $T$. Furthermore, it is observed that
the temperature dependence of $C_{\rm mag}/T$ shows a $T^{-0.5}$ power-law
behaviour (Fig. 3 (c)) at low temperature, which suggest the presence of a
random singlet state with strong low-lying excitations 27; 63. Next, the
entropy was obtained by integrating $C_{\rm mag}/T$ in the temperature range
0.052 K $\leq$ $T$ $\leq$ 2 K and is shown in the inset of Fig. 3 (c), with
the contribution to entropy from specific heat below at $T$ below the lowest
measured $T$ = 52 mK, estimated as $S({52}\ {\rm mK})=C_{\mathrm{mag}}({52}\
{\rm mK})/(0.5)$. The saturation entropy amounts to just 13 % of expected
total entropy for a spin-1/2 system, which could suggest the presence of a
highly degenerate ground state, but also naturally arises just from the fact
that many spin degrees of freedom inside individual spin-chain fragments
already freeze out at these low $T\ll J,J^{\prime}$, reducing the total number
of available spin degrees of freedom and making individual spin-chain
fragments behave as single effective spins in a frustrated random 3D spin-
lattice formed by weaker inter-chain interactions $J_{\mathrm{inter}}$. The
presence of these frustrated interactions is also seen from the shift of
specific heat to lower $T$ than predicted by theory if they were not present
(see SI for a detailed comparison with ED calculations). The power-law
behaviour and the absence of any long range ordering down to 52 mK in zero
field data suggest that the antiferromagnet Li4CuTeO6 is a quantum spin liquid
candidate.
Recently, Kimchi et al. suggested that quenched disorder in spin-liquid
candidates can form a network of local moments in a random-singlet phase 64;
39 and the low-temperature specific heat data recorded in various magnetic
fields should collapse to a universal curve characterized by a parameter
$\gamma$, where $\gamma$ is defined by the relation
$C_{p}/T\propto\frac{1}{(\mu_{0}H)^{\gamma}}F(\frac{T}{\mu_{0}H})$ and $F$ is
a scaling function 32. We scaled the magnetic specific heat data of LCTO
accordingly (see Fig. 3 (d)) and we indeed find data collapse for $\gamma$ =
0.85, which could be attributed to the presence of random-singlets in spin-
liquid ground-state. A similar data collapse is reported in a triangular-
lattice Y2CuTiO6 with $\gamma$ = 0.70, where 50% of Cu2+-sites are diluted by
Ti4+ ions that yields quenched disorder 62.

### Muon spin relaxation

In order to gain microscopic insights into the intrinsic susceptibility and
the ground-state spin dynamics, we performed zero-field (ZF) and transverse-
field (TF) muon spin relaxation ($\mu$SR) measurements on polycrystalline
samples. $\mu$SR is a unique local-probe technique with a broad time window to
track electron spin fluctuations. Therefore, ZF-$\mu$SR is one of the best
tools to gain comprehensive microscopic information on magnetic ordering and
spin dynamics of frustrated quantum materials. In Fig. 4 (a), we present
ZF-$\mu$SR spectra at a few representative temperatures showing that muon
asymmetry remains more or less unchanged for all temperatures, which confirms
the absence of a phase transition down to 1.55 K 65. The time dependence of
the muon asymmetry provides the information concerning internal field
distributions at the muon stopping site. In multi-domain or polycrystalline
samples, static local-field of nuclear origin $B_{\rm loc}$ is randomly
oriented with respect to the initial muon spin direction. The corresponding
local-field distribution can be approximated by a Gaussian shape. In this case
the muon relaxation function $P_{z}$($t$) (also known as the Gaussian Kubo-
Toyabe function (KT) is of the form

$P_{z}(t)=\mathrm{KT}(t)=\frac{1}{3}+\frac{2}{3}(1-\sigma^{2}t^{2})e^{-\frac{1}{2}\sigma^{2}t^{2}},$
(2)

where $\sigma$ gives the local-field distribution width.
In LCTO, the presence of competing magnetic fields arising from electron spins
provides dynamic muon spin relaxation in addition to relaxation due to quasi-
static nuclear fields. Namely, a damping factor exp($-\lambda(T)t$)
characteristic of dynamical fields has to be added to the KT function to
account for the experimental results. Therefore, the zero-field asymmetry
spectra were fitted by the damped Gaussian Kubo-Toyabe relaxation function

$A(t)=A_{0}\mathrm{KT}(t)e^{-\lambda(\textit{T})t},$ (3)

where $\lambda(T)$ is the dynamical muon spin relaxation rate due to
electronic spins. The fitting of the observed ZF asymmetry data (Fig. 4 (a))
by Eq. (3) yields the nuclear field distribution width of approximately 3.66 G
and the dynamical relaxation rate $\lambda$ = 0.023 $\mu$s-1. In addition, the
strong change of asymmetry spectra between zero-field and a small longitudinal
field of 50 G as shown in Fig 4 (b) also supports this model, as the applied
field decouples the relaxation due to small static nuclear fields while it
does not affect the relaxation due to larger but dynamical fields of
electronic origin. The obtained temperature dependence of the muon spin
relaxation rate due to dynamical fields in ZF remains constant down to 1.55 K
(Fig. 4 (d)) without any signature of magnetic ordering, which suggests a
dynamic ground-state. We note that the presence of a disordered static local
field at the muon site generally leads to the “1/3” tail in zero-field muon
decay asymmetry of powder samples that is a hallmark of static magnetism.
However, zero-field muon asymmetries do not show the “1/3” tail, which
confirms the absence of spin freezing 66. In LCTO, the absence of any
oscillation in ZF spectra (see Fig. 4 (a)) also corroborates the absence of
long-range magnetic ordering down to 1.55 K.
Next, we carried out muon rotation measurements in the presence of large
transverse field (TF) $B_{T}$ = 4900 G, which is much greater than the local
fields. The observed experimental TF spectra (not shown) were fitted using an
oscillatory Gaussian decaying function

$A(t)=A_{0}cos(2\pi\nu(\textit{T})t-\phi)e^{(-\lambda_{T}(\textit{T})t)^{2}/2}.$
(4)

Here $\phi\sim\pi/3$ is the initial phase of the oscillation, $\nu$(T) is the
frequency of muon precession originating from the local magnetic field and
external transverse magnetic field, whereas $\lambda_{T}(T)$ is the transverse
muon spin relaxation rate. The temperature dependence of
$\lambda_{T}(\textit{T})$ is shown in Fig. 4 (e). Upon decreasing the
temperature, the relaxation rate increases and a rather strong change of
$\lambda_{T}$ is observed in the temperature range 1.55 K $<$ $T$ $<$ 10 K,
which suggests the presence of a crossover region 12. Above $T$ $>$ 20 K,
$\lambda_{T}$ varies very slowly with temperature as would be expected for a
paramagnetic region. In the presence of an external transverse field
($B_{T}$), the total magnetic field experienced by muon can be defined as the
sum of $B_{T}$ and $B_{\rm loc}$. So, the mean precession frequency can be
defined as $\nu(T)=\gamma_{\mu}(B_{T}+B_{\rm loc}(T))/2\pi$, where only the
local magnetic field $B_{\rm loc}(T)$ changes with temperature
($\gamma_{\mu}=2\pi\times 135.53$ MHz/T is the muon gyromagnetic ratio). The
muon Knight shift ($K$) that is directly proportional to magnetic
susceptibility ($\chi$) via $K$ = ($\nu$($T$) $-$ $\nu_{\rm 0}$)/$\nu_{\rm 0}$
= $A\chi$, where $\nu_{\rm 0}$ = ($\gamma_{\mu}/2\pi)$$B_{T}$ is the reference
frequency and $A$ is the coupling constant between the muon magnetic moment
and the electron magnetic moments, thus captures intrinsic susceptibility
relevant to the spin-lattice. Inset of Fig. 4 (c) depicts the muon Knight
shift as a function of bulk magnetic susceptibility ($\chi$) with temperature
as an implicit parameter. It is seen that $K$ as a function of $\chi$ exhibits
two linear regions that are consistent with the observed crossover region in
temperature dependence of $\lambda_{T}$. Below $T<$ 75 K, the linear fit
(orange line) yields a coupling constant $A$ = $-$114 Oe/$\mu_{\rm B}$ whereas
we find $A$ = $-$211 Oe/$\mu_{\rm B}$ from the linear fit (olive line) above
75 K. Furthermore, it is observed that the Knight shift data follows $K\sim$
$T^{-0.75}$ power-law behavior at low $T$ (see Fig. 4 (c)), which is
consistent with bulk magnetic susceptibility and confirms that such
susceptibility is intrinsic to the system. This behavior further corroborates
the presence of “random-singlets” in LCTO. $\mu$SR results do not indicate any
static local fields of electronic origin down to 1.55 K. The measured muon
asymmetry decay in zero field is, in fact, dominated by static nuclear fields,
while relaxation due to dynamical fields of electronic origin remains hindered
behind the nuclear relaxation due to exchange narrowing down to 1.55 K. This
reveals spin fluctuations are fast even at $T/\theta_{\rm CW}\approx 0.01$,
which again suggests a dynamic ground-state in LCTO. We note that the zero-
field muon spin relaxation rate due to electronic fields (Fig. 4 (d)) is much
lower than the transverse muon spin relaxation rate (Fig. 4 (e)) and its
temperature dependence is different. As the former relaxation is only due to
field fluctuations, while the latter is affected also by static local-field
distributions induced by the applied transverse field, it is the induced local
fields that are responsible for the increase of the transverse relaxation rate
with decreasing temperature. This increase is characterized by a power-law
$T^{-0.2}$ at the lowest temperatures (Fig. 4 (e)).

## Discussion

Our investigation reveals the presence of a strong antiferromagnetic exchange
interactions between spins of Cu2+ ions constituting a random frustrated spin-
chain model with weaker inter-chain exchanges couplings forming an effective
frustrated random 3D lattice of spin chain fragments at low $T$. Remarkably,
the material doesn’t show any signature of a phase transition or spin-glass
freezing down to at least 45 mK, which suggests a dynamical liquid-like
ground-state in this antiferromagnet. The scaling behavior of magnetization
and specific heat data at low $T$ reflects the presence of “random-singlets”
in a quantum disordered ground-state of LCTO. Magnetic materials with a dilute
random network of $S$ = 1/2 sites can be considered as a combination of two
sub-systems where the majority of constituent spins develop spin-liquid or
valence bond crystal, and the minority forms random-singlets, and leaves out a
random network of $S$ = 1/2 sites. For such a system, in the ground-state,
spins do not undergo any long-range antiferromagnetic order. The spins that
are strongly coupled rather constitute singlet pairs and the remaining spins
are almost free due to the development of effective antiferromagnetic exchange
interaction ($J$) with a power-law probability distribution,
$\mathcal{[}P](J)\sim J^{-\gamma}$, between them in the low-$T$
renormalization group flow of the random-singlet phase, where $\gamma$ is a
positive quantity related to the spin correlation length 22. The small sub-
system of spin-1/2 moments in the low-energy limit leads to unconventional
scaling behavior in specific heat $C[T]\sim T^{1-\gamma}$ and spin
susceptibility $\chi[T]$ $\sim T^{-\gamma}$ 32. In the presence of an applied
magnetic field, specific heat captures the modified distribution of energy
through a field-dependent coefficient, i.e., $C\sim T/(\mu_{0}H)^{\gamma}$ at
low-temperatures. It is worth to mention that the temperature and field
dependence of magnetic specific heat displays $T/\mu_{\rm 0}H$ data collapse
for $\gamma$ = 0.85 in LCTO, which is a bit higher than the observed $\gamma$
= 0.5 for the honeycomb-lattice H3LiIr2O6 and triangular-lattice LiZn2Mo3O8
but it is close to the observed $\gamma$ = 0.7 for triangular-lattice
Y2CuTiO6. The absence of strong spin-orbit interaction in Y2CuTiO6 and LCTO
possibly plays a vital role resulting in data collapse with higher value of
$\gamma$ compared to H3LiIr2O6 and Li2Zn2Mo3O8. In addition, we also found a
divergence in the low-temperature susceptibility of LCTO, which scales as
$T^{-\gamma}$ with $\gamma$ = 0.82 for bulk measurements and $\gamma$ = 0.75
for local-probe muon Knight-shift measurements. This value of $\gamma$ is
comparable with the reported $\gamma$ = 0.68 and 0.76 for Y2CuTiO6 and
Sr2CuTe1-xWxO6, respectively 24; 62; 25. Further theoretical and experimental
studies are desired to shed additional insights into the magnetism and spin
dynamics of this promising spin-liquid candidate and to clarify the degree to
which the random-singlet picture might be modified due to the presence of an
underlying spin-chain fragment structure with random connectivity. The chain
fragments could conceivably contribute further ingredients to low temperature
physics of LCTO, leading to novel phenomena outside the paradigm of the
usually considered random-singlet states.

## Conclusion

We have investigated the structure, thermodynamic and local magnetic
properties of Li4CuTeO6 by employing X-ray, neutron diffraction,
magnetization, specific heat, ESR, and muon spin relaxation measurements.
These were supported by state-of-the-art ab initio density functional theory
(DFT) and exact diagonalization (ED) calculations to elucidate the high-$T$
spin model and its thermodynamics. Li4CuTeO6 crystallizes in the monoclinic
space group $C$2/$m$, wherein the Cu2+ site constitute spin chains with
exchange $J$ along the [101] direction of random length due to partial
occupancy of the Cu2 site, with random additional Cu2+ ions on partially
occupied Cu3 sites surrounding the main chains at positions displaced along
$\pm b$ from the main-chain bond centers and coupling to two Cu2 sites on the
main chain through an exchange $J^{\prime}$. The large and negative Curie-
Weiss temperature, $\theta_{\rm CW}$ = $-$ 154 K, suggests the strong
antiferromagnetic nature of the $J$ and $J^{\prime}$ exchange interactions
between Cu2+ moments, which is confirmed by DFT and fits of ED calculations to
experimental data for $T>$ 20 K. These gradual deviation from these fits at
low $T$ reveals further frustrating 3D interactions between the random spin-
chain fragments, which dominate the low-$T$ response. Specific heat
experiments reveal the absence of long-range magnetic ordering down to 52 mK
despite the presence of a large Curie-Weiss temperature, implying a highly
frustrated spin-lattice. The absence of spin-freezing despite the Li/Cu anti-
site disorder reflects that Cu2+ moments remain fluctuating down to 45 mK. The
zero-field and transverse-field $\mu$SR measurements support a dynamic ground-
state down to 1.55 K. Moreover, the existence of $T/\mu_{\rm 0}H$ scaling as
well as magnetization and specific heat data collapse indicate the presence of
a spin-1/2 network of random-singlets due to unavoidable anti-site disorder in
the host lattice and frustrated weak 3D couplings. Our results demonstrate a
randomness induced spin-liquid state in a frustrated magnet. These results
offer an exciting ground to explore novel quantum states with unconventional
low-energy excitations in novel frustrated quantum materials with quenched
disorder.

## Methods

Polycrystalline samples of Li4CuTeO6 were synthesized by the conventional
solid-state reaction route as outlined in ref. 53. To obtain a single phase of
LCTO, stoichiometric amounts of Li2CO3 (Alfa Aesar, 99.0 %), CuO (Alfa Aesar,
99.995 %), and TeO2 (Alfa Aesar, 99.9995 %) were mixed and the mixture was
finely ground and pressed into a pellet, and after several intermediate steps,
finally sintered at 850∘C for 30 h. The X-ray diffraction measurements were
performed at 300 K by employing a Rigaku smart LAB X-ray diffractometer with
Cu K$\alpha$ radiation ($\lambda$ = 1.54 Å). Due to the light mass of Li and O
atoms, the analysis of X-ray powder diffraction data is not sufficient to
determine the structural disorder and the mixed site occupancy. Therefore,
neutron diffraction measurements were performed at room temperature for
incident wavelength of 1.28 Å using crg-D1B Two-Axis Powder Diffractometer at
the Institut of Laue-Langevin, France. Our neutron diffraction studies detects
neither spurious phases nor non-stoichiometry of oxygen. The neutron
diffraction results at 1.28 Å are shown in Fig. 1 (c), while the neutron
diffraction pattern, XRD pattern and resulting refinements parameters are
presented in the Supplementary Information (SI). This suggests that the
polycrystalline samples used in this study are of very high quality. The
refined occupancy of all the atomic sites are within the experimental
uncertainty of about 1 %, thus ruling out the presence of non-stoichiometric
oxygen.
Magnetization measurements were carried out using a Quantum Design, SQUID
(MPMS) in the temperature range 1.9 K $\leq\textit{T}\leq$ 300 K under
magnetic fields up to 5 T. The electron spin resonance (ESR) spectrum was
measured at 250 K at the irradiation frequency of 256.3 GHz on a custom built
spectrometer with homodyne detection at the NHMFL, Tallahassee, USA.
Thermoremanent measurements were performed using a Quantum Design, SQUID
(MPMS) and the sample was cooled down from 100 to 2 K in either 1000 or 8 Oe,
and turned down at 2 K. After 1h, the signal was measured again (at $t$ = 0).
These results rule out the presence of a spin-glass transition and are shown
in SI Fig. 4.
Specific heat measurements were performed in a Quantum Design, Physical
Properties Measurement System (PPMS) by the thermal relaxation method, in the
temperature range 1.9 K $\leq\textit{T}\leq$ 240 K and in magnetic fields up
to 7 T. Furthermore, specific heat measurements were carried out separately in
the temperature range 0.052 K $\leq$ $T$ $\leq$ 4 K in zero-field and 3 T
using a dilution refrigerator which was also used to measure ac susceptibility
in the temperature range 0.045 K $\leq$ $T$ $\leq$ 4 K at four different
frequencies using a Dynacool PPMS instrument from Quantum Design, San Diego
(USA).
$\mu$SR measurements were performed using the GPS spectrometer at the Paul
Scherrer Institute, Villigen, Switzerland, on a polycrystalline sample with
mass of approximately 1 g down to 1.55 K in zero field and in a transverse
field of 4900 G.
DFT+$U$ calculations on a $2\times 1\times 2$ supercell of LCTO were carried
out using the plane-wave DFT code 67 using a local (spin) density
approximation (LSDA) exchange-correlation functional in a LSDA+$U$ scheme 68
with an effective Hubbard $U_{\mathrm{eff}}$ = 9 eV, a 2000 plane-wave energy
cutoff, and a $2\times 3\times 2$ Monkhorst-Pack grid reciprocal-space
sampling 69. To extract exchanges total energies of ${\geq}120$ random
collinear spin configurations were calculated and fitted to a symmetry-adapted
spin model 70. See SI for further details. ED calculations with up to $N=18$
spins were carried out on $>13,000$ distinct configurations of occupied Cu2
and Cu3 sites around finite-length spin-chain fragments in the high-$T$ random
1D model of Fig. 1(b), achieving a total probability between 60.2 % and more
than 98.8 %, depending on parameters, that a randomly-chosen Cu2+ spin in LCTO
is part of one of the ED calculated spin-chain fragments. Details of these ED
calculations are presented in the SI.

## Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request. The neutron diffraction data can
be found at https://doi.org/10.5291/ILL-DATA.EASY-1043.

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## Acknowledgments

We thank DST, India for the PPMS facility at IIT Madras. PK acknowledges the
funding by the Science and Engineering Research Board, and Department of
Science and Technology, India through Research Grants. AZ and MG acknowledge
the financial support of the Slovenian Research Agency through Programme No.
P1-0125 and Projects No. N1-0148 and J1-2461. A portion of this work was
performed at the National High Magnetic Field Laboratory, which is supported
by National Science Foundation Cooperative Agreement No. DMR-1644779 and the
State of Florida. AMS thanks the NRF (93549) and the URC and FRC of the
University of Johannesburg. Experiments at ILL was sponsored by the French
Neutron Federation (2FDN), with data references from Easy Proposal N° 1043.

## Author contributions

All authors contributed substantially to this work.

## Competing interests

The authors declare no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to P.K. (email:
<EMAIL_ADDRESS>


1468241 [Physics]Fisica Scuola di dottorato Vito Volterra XXXIV 2022 2022
Prof. Paolo Pani<EMAIL_ADDRESS>22 February 2022 Prof. Alfredo
Urbano Prof. Massimo Bianchi Prof. Enrico Barausse

# Probing new physics on the horizon of black holes with gravitational waves

Elisa Maggio

###### Abstract

Black holes are the most compact objects in the Universe. According to general
relativity, black holes have a horizon that hides a singularity where
Einstein’s theory breaks down. Recently, gravitational waves opened the
possibility to probe the existence of horizons and investigate the nature of
compact objects. This is of particular interest given some quantum-gravity
models which predict the presence of horizonless and singularity-free compact
objects. Such exotic compact objects can emit a different gravitational-wave
signal relative to the black hole case. In this thesis, we analyze the
stability of horizonless compact objects, and derive a generic framework to
compute their characteristic oscillation frequencies. We provide an
analytical, physically-motivated template to search for the gravitational-wave
echoes emitted by these objects in the late-time postmerger signal. Finally,
we infer how extreme mass-ratio inspirals observable by future gravitational-
wave detectors will allow for model-independent tests of the black hole
paradigm.

###### Contents

  1. 0 Preface
  2. 1 Introduction

##  Chapter 0 Preface

The work presented in this thesis has been carried out mainly at the Physics
Department of Sapienza University of Rome in the research group of gravity
theory and gravitational wave phenomenology. Part of this work was carried out
at the Consortium for Fundamental Physics, School of Mathematics and
Statistics, University of Sheffield, United Kingdom, and at the Centro de
Astrofísica e Gravitação (CENTRA), Instituto Superior Técnico, Universidade de
Lisboa, Portugal. I thank these institutions for their kind hospitality.

### List of publications

The work in this thesis was accomplished with different scientific
collaborations, whose members I kindly acknowledge.

  * •

Chapter LABEL:chapter2 is the outcome of a collaboration with Paolo Pani and
Guilherme Raposo based on:

    * E. Maggio, P. Pani, G. Raposo, “Testing the nature of dark compact objects with gravitational waves,” Invited chapter for C. Bambi, S. Katsanevas, K.D. Kokkotas (editors), Handbook of Gravitational Wave Astronomy, Springer, Singapore (2021), arXiv:2105.06410, https://doi.org/10.1007/978-981-15-4702-7$\\_$29-1.

  * •

Chapter LABEL:chapter3 is the outcome of a collaboration with Luca
Buoninfante, Anupam Mazumdar and Paolo Pani based on:

    * E. Maggio, L. Buoninfante, A. Mazumdar, P. Pani, “How does a dark compact object ringdown?,” Phys. Rev. D 102, 064053 (2020),
arXiv:2006.14628.

  * •

Chapter LABEL:chapter4 is the outcome of a collaboration with Vitor Cardoso,
Sam Dolan, Valeria Ferrari and Paolo Pani based on:

    * E. Maggio, P. Pani, and V. Ferrari, “Exotic compact objects and how to quench their ergoregion instability,” Phys. Rev. D 96, 104047 (2017), arXiv:1703.03696.

    * E. Maggio, V. Cardoso, S. Dolan, and P. Pani, “Ergoregion instability of exotic compact objects: electromagnetic and gravitational perturbations and the role of absorption,” Phys. Rev. D 99, 064007 (2019), arXiv:1807.08840;

  * •

Chapter LABEL:chapter5 is the outcome of a collaboration with Swetha Bhagwat,
Paolo Pani and Adriano Testa based on:

    * E. Maggio, A. Testa, S. Bhagwat, and P. Pani, “Analytical model for gravitational-wave echoes from spinning remnants,” Phys. Rev. D 100, 064056 (2019), arXiv:1907.03091.

  * •

Chapter LABEL:chapter6 is the outcome of a collaboration with Maarten van de
Meent and Paolo Pani based on:

    * E. Maggio, M. van de Meent, P. Pani, “Extreme mass-ratio inspirals around a spinning horizonless compact object,” Phys. Rev. D in press (2021), arXiv:2106.07195.

As a part of the activities during my PhD, I served as a member of the LISA
Consortium, being involved in the writing of the LISA Fundamental Physics and
the LISA Waveform White Papers, the LISA Figure of Merit analysis, and the
LISA Early Career Scientists (LECS) group. Part of the outcome of these
activities is in preparation or has been submitted for publication and is not
included in this thesis.

### Conventions

In this work, geometrized units, $G=c=1$, are adopted where $G$ is the
gravitational constant, and $c$ is the speed of light.

The signature of the metric adopts the $(-,+,+,+)$ convention. The Greek
letters run over the four-dimensional spacetime indices. The comma stands for
an ordinary derivative, and the semi-colon stands for a covariant derivative.

$\mathbf{M}^{*}$ is the complex conjugate of a matrix, and $\mathbf{M}^{\top}$
is the transpose of a matrix. $\mathfrak{R}(n)$ and $\mathfrak{I}(n)$ are the
real and the imaginary part of a number, respectively.

### Abbreviations

BH | Black Hole
---|---
ECO | Exotic Compact Object
EMRI | Extreme Mass-Ratio Inspiral
ISCO | Innermost Stable Circular Orbit
LIGO | Large Interferomenter Gravitational-wave Observatory
LISA | Laser Interferometer Space Antenna
GR | General Relativity
GW | Gravitational Wave
NS | Neutron Star
PN | Post-Newtonian
QNM | Quasi-Normal Mode
SNR | Signal-to-Noise Ratio
TH | Tidal Heating
ZAMO | Zero Angular Momentum Observer

##  Chapter 1 Introduction

The landmark detection of gravitational waves (GWs) provides the unique
opportunity to test gravity in the strong-field regime and infer the nature of
astrophysical sources. So far, the ground-based detectors LIGO and Virgo have
detected ninety GW events from the coalescence of compact binaries
[LIGOScientific:2018mvr, LIGOScientific:2020ibl, LIGOScientific:2021usb,
LIGOScientific:2021djp]. These detections allowed us to observe for the first
time the coalescence of two black holes (BHs) and revealed that their masses
can be heavier than the ones observed in the electromagnetic spectrum
[LIGOScientific:2016lio, LIGOScientific:2020ibl]. Recent important discoveries
include the first multi-messenger observation of a binary neutron star (NS)
merger [LIGOScientific:2017vwq, LIGOScientific:2017ync] and the observation of
the formation of an intermediate-mass BH [LIGOScientific:2020iuh].

Furthermore, GWs provide a new channel for probing Einstein’s theory of
gravity in a regime inaccessible to traditional astronomical observations,
namely the strong-field and highly dynamical one. Several consistency tests of
the GW data with the predictions of general relativity (GR) have been
performed. No evidence for new physics has been reported within current
measurement accuracies [LIGOScientific:2016lio, LIGOScientific:2019fpa,
LIGOScientific:2020tif, LIGOScientific:2021sio].

The GW signal emitted by compact binary coalescences is characterized by three
main stages: the _inspiral_ phase, when the two bodies orbit around each other
and the emission of GWs makes the orbit shrink, the _merger_ phase, when the
two bodies coalesce, and the _ringdown_ when the final remnant relaxes to an
equilibrium solution. The study of the different stages of the GW signal
allows us to infer the properties of the compact objects and understand their
nature.

Several extensions of GR predict the existence of regular and horizonless
compact objects, also known as exotic compact objects (ECOs) [Giudice:2016zpa,
Cardoso:2019rvt]. Indeed, the presence of the horizon poses some theoretical
problems, the most notable ones being the existence of a singularity in the
black-hole interior and the Hawking information loss paradox [Mathur:2009hf].

ECOs can mimic the features of BHs through electromagnetic observations since
they can be as compact as BHs [Abramowicz:2002vt]. Indeed, the supermassive
object at the center of the M87 galaxy observed by the Event Horizon Telescope
poorly constrains few models of ECOs [EventHorizonTelescope:2019pgp].
Furthermore, current GW observations do not exclude ECOs that could
potentially explain events in the mass gap between NSs and BHs and due to
pair-instability supernova processes [LIGOScientific:2020iuh,
Bustillo:2020syj, LIGOScientific:2020zkf].

One way to distinguish ECOs from BHs is by analyzing the ringdown stage of a
compact binary coalescence. The ringdown is dominated by the complex
characteristic frequencies – the so-called quasi-normal modes (QNMs) – of the
remnant, that differ dramatically if the latter is a BH or an ECO
[Cardoso:2016rao]. By inferring the QNMs of the remnant, we can test whether
they are compatible with the predicted spectrum for a BH.

Current observations of the fundamental QNM in the ringdown of binary
coalescences are compatible with remnant BHs as predicted by GR
[LIGOScientific:2019fpa, LIGOScientific:2020tif, LIGOScientific:2021sio];
however, the characterization of the remnant is still an open problem. The no-
hair theorems establish that BHs in GR are determined uniquely by two
parameters, i.e., their mass and angular momentum [Carter:1971zc,
Robinson:1975bv]. Therefore, the measurement of one complex QNM allows us only
to estimate the parameters of the BH. A test of the BH paradigm would require
the identification of at least two complex QNM frequencies. Louder GW events,
to be collected as detector sensitivity improves, and more sophisticated
parametrized waveforms will allow us to extract more information about the
remnant.

If the remnant of a merger is an ECO that is almost as compact as a BH, the
prompt ringdown signal would be nearly indistinguishable from that of a BH
[Cardoso:2016rao]. A characteristic fingerprint of ECOs would be the
appearance of a modulated train of GW echoes at late times due to the absence
of the horizon [Cardoso:2016rao, Maggio:2019zyv]. Tentative evidence for GW
echoes in LIGO/Virgo data has been reported in the last few years
[Abedi:2016hgu], but recent independent searches argued that the statistical
significance for GW echoes is consistent with noise [Westerweck:2017hus,
LIGOScientific:2020tif, LIGOScientific:2021sio].

Besides ECO fingerprints in the GW emission, in this thesis we analyze the
astrophysical viability of ECOs as BH alternatives. Indeed, spinning
horizonless compact objects are prone to the so-called ergoregion instability
when spinning sufficiently fast [Friedman:1978wla, 10.1093/mnras/282.2.580,
Kokkotas:2002sf]. The endpoint of the instability could be a slowly-spinning
ECO [Cardoso:2014sna, Brito:2015oca] or dissipation within the object could
lead to a stable remnant [Maggio:2017ivp, Maggio:2018ivz, Wang:2019rcf]. If
confirmed, the ergoregion instability could provide a strong theoretical
argument in favor of the BH paradigm for which rapidly spinning compact
objects must have a horizon.

The prospect for detectability of new physics will improve in the future with
the next-generation detectors like the ground-based observatories Einstein
Telescope [Punturo:2010zz] and Cosmic Explorer [Reitze:2019iox], and the
space-based Laser Interferometer Space Antenna (LISA) [LISA:2017pwj]. In
particular, LISA is an extremely promising observatory of fundamental physics.
Planned for launch in 2034, LISA will detect GWs in a lower frequency band
than ground-based detectors. LISA will observe a plethora of astrophysical
sources, particularly extreme mass-ratio inspirals (EMRIs) in which a stellar-
mass object orbits around the supermassive object at the center of a galaxy
[Gair:2017ynp].

EMRIs are unique probes of the nature of supermassive compact objects. Since
LISA will observe inspirals that can last for years, the phase shift of the
waveform will be tracked with high precision and deviations from GR will be
measured accurately. During the inspiral around a horizonless supermassive
object, extra resonances would be excited leaving a characteristic imprint in
the waveform [Cardoso:2019nis, Maggio:2021uge]. Any evidence of partial
reflectivity at the surface of the object would also indicate a departure from
the classical BH picture [Datta:2019epe, Maggio:2021uge].

Within this broad and flourishing context, this thesis aims to investigate the
nature of compact objects and test the existence of horizons with GWs. This
work is organized as follows.

Chapter LABEL:chapter1 is dedicated to the tests of the BH paradigm that have
been currently performed using GWs. A review of the recent GW observations is
presented, and the stages of the gravitational waveform are analyzed. In
particular, the consistency tests of GR and the parametrized tests of
deviations from GR are described. Finally, the prospects of detecting
deviations from GR with next-generation detectors are assessed.

Chapter LABEL:chapter2 illustrates the theoretical motivations for studying
horizonless compact objects. A parametrized classification of horizonless
compact objects is presented depending on their deviations from the standard
BH picture. Some remarkable models of ECOs are reviewed, and their
phenomenology is compared to the BH case.

Chapter LABEL:chapter3 derives the GW signatures of static horizonless compact
objects, particularly their QNM spectrum in the ringdown. The system is
described by perturbation theory. The imposition of the boundary conditions
that describe the response of the compact object to perturbations requires
careful analysis. A numerical procedure for the derivation of the QNMs is
illustrated. The QNMs of horizonless compact objects are derived both for
remnants almost as compact as BHs and with smaller compactness. In the former
case, the presence of characteristic low frequencies in the QNM spectrum is
highlighted. In the latter case, an extended version of the BH membrane
paradigm is applied to derive model-independent deviations from the BH QNM
spectrum. Finally, current constraints on horizonless compact objects and
prospects of detectability are assessed.

Chapter LABEL:chapter4 analyzes spinning horizonless compact objects that are
prone to the ergoregion instability above a critical threshold of the spin.
The QNMs of spinning horizonless compact objects are derived. From the
analysis of the imaginary part of the QNMs, the conditions for which
horizonless compact objects are unstable are identified. An analytical
description of the QNMs in terms of superradiance is detailed, and ways of
quenching the ergoregion instability are provided.

Chapter LABEL:chapter5 describes the GW echoes that would be emitted after the
prompt ringdown in the case of a horizonless merger remnant. An analytical,
physically-motivated template for GW echoes is provided that can be
implemented to perform a matched-filter-based search for echoes. Finally, the
properties of GW echoes are analyzed, and the prospects of detection with
current and future detectors are assessed.

Chapter LABEL:chapter6 is devoted to the analysis of EMRIs with a central
horizonless compact object. During the inspiral, extra resonances are excited
when the orbital frequency matches the characteristic frequencies of the
central ECO. The impact of the resonances in the GW dephasing with respect to
the BH case is assessed. This analysis shows that LISA will be able to probe
the reflectivity of compact objects with unprecedented accuracy.


$\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\rho_{P}^{\mathrm{ext}})$ have the
same terminal point for each $P\in S$, then
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\rho)$ and
$\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\rho)$ have the same terminal
point.

###### Proof.

Let $(M,\phi)$ be the unit-root $\sigma$-module over $A_{\pi}^{\infty}$
corresponding to $\rho_{U}$. From the equality (14), the Newton polygon
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\phi)$ is the concatenation of
$\operatorname{\mathrm{NP}}_{\pi_{q}}^{<r}(\rho)$ with
$r_{0}+r_{1}+r_{\infty}-|S|$ segments of slope $0$. By definition, the Hodge
polygon $\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\phi)$ is a concatenation
of $\operatorname{\mathrm{HP}}_{\pi_{q}}^{<r}(\rho)$ with
$r_{0}+r_{1}+r_{\infty}-|S|$ segments of slope $0$. The result follows by
canceling out these extra slope-$0$ segments. ∎

######  Corollary 7.15.

Theorem 1.1 holds.

###### Proof.

By Theorem 7.10 we know that any finite character of order $p^{n}$ is
$\bm{\delta}$-Hodge, where $\delta_{P}$ is the Swan conductor at $P$ divided
by $p^{n-1}$. The result is then an immediate consequence of Corollary 7.14. ∎

######  Corollary 7.16.

Theorem 1.3 holds for finite characters of order $p$.

###### Proof.

By Remark 1.4 we only need to show the “if” direction. By our assumption on
the order of $\rho$ we have $d_{P}\in\mathbb{Z}$ and $p\equiv 1\mod d_{P}$. By
[22] (see also [2, Remark 4.1]) we know
$\operatorname{\mathrm{NP}}_{q}(\rho_{P}^{\mathrm{ext}})=\operatorname{\mathrm{HP}}_{q}(\rho_{P}^{\mathrm{ext}})$
for each $P\in S$. The result follows from Theorem 7.10. ∎

##  8 Glossary of notation

The spaces studied in this paper are numerous and intricate. For this reason,
we thought the reader would benefit from a notation glossary.

###  8.1 Basic definitions

  1. $R$

A complete discrete valuation ring with maximal ideal $\mathfrak{m}$ and
residue field $\mathbb{F}_{p}$.

  2. $\pi$

A non-zero topologically nilpotent element of $R$.

  3. $K$

The fraction field of $R$, i.e. $K=R[\frac{1}{\pi}]$.

  4. $q$

A power of $p$.

  5. $R_{q}$

The unramified extension of $R$ whose residue field is $\mathbb{F}_{q}$.

  6. $K_{q}$

The fraction field of $R_{q}$, i.e. $K_{q}=R_{q}[\frac{1}{\pi}]$.

  7. $\pi_{q}$

The $v_{p}(q)$-th power of $\pi$: i.e. $\pi_{q}=\pi^{v_{p}(q)}$.

  8. $X$

A smooth affine curve over $\mathbb{F}_{q}$.

  9. $\overline{X}$

The smooth compactification of $X$.

  10. $S$

The points of $\overline{X}$ at infinity, i.e. $S=\overline{X}\backslash X$.

  11. $\eta$

A tame morphism $\overline{X}\to\mathbb{P}_{\mathbb{F}_{q}}^{1}$ that is only
ramified over $\\{0,1,\infty\\}$. See §4.1 for more details.

  12. $S_{\eta}$

The branch points of $\eta$, i.e. $S_{\eta}=\eta^{-1}(\\{0,1,\infty\\})$.

  13. $\mu$

A function from $S_{\eta}$ to $\mathbb{Z}$. More precisely, if
$\eta(P)\in\\{0,\infty\\}$ we have $\mu(P)=0$ and if $\eta(P)=1$ we have
$\mu(P)=p-1$.

  14. $A$

A smooth $\mathbb{Z}_{q}$-algebra whose special fiber is the coordinate ring
of $\overline{X}-S_{\eta}$.

###  8.2 The space $A$, its modifications, and some associated notation

In this article, we frequently work with modifications to the space of
functions $A$. These modified spaces are notated using combinations of
superscripts, subscripts, and font changes. These modifiers are occasionally
applied to spaces other than $A$. However, for expository purposes, it seems
prudent to use $A$ as our ‘running example’. We will now explain these
notation modifiers, as well as explain some other notation for objects
associated to $A$.

  1. $A_{\pi}$

The ring $R\otimes_{\mathbb{Z}_{p}}A$.

  2. $A_{\pi}^{\infty}$

The $\pi$-adic completion of $A_{\pi}$.

  3. $A_{\pi}^{m}$

The elements of $A_{\pi}^{\infty}$ that ‘overconverge’ $\pi$-adically with a
radius of $m$. See §2.1 for more details.

  4. $A_{\pi}^{\dagger}$

The elements of $A_{\pi}^{\infty}$ that ‘overconverge’ $\pi$-adically. In
particular, it is the union of all $A_{\pi}^{m}$.

  5. $V_{\pi}^{\dagger}$

The $K$-vector space $K\otimes_{R}A_{\pi}^{\dagger}$. More generally, when $A$
is replaced with $V$, this signifies tensoring with $K$.

  6. $\tilde{A}_{\pi}^{\dagger}$

The semi-local ‘twist’ of $A_{\pi}^{\dagger}$.

  7. $\mathcal{A}_{\pi,P}$

The completion of $A_{\pi}$ along the ideal defining $P$. We use the
superscripts $m$, $\dagger$, and $\infty$ for the appropriate $\pi$-adic
growth conditions.

  8. $\mathbf{m}$

A tuple $(m_{P})_{P\in S_{\eta}}$ of rational numbers indexed by $S_{\eta}$.

  9. $\mathcal{A}_{\pi,P}^{\mathbf{m}}$

This is a subspace of $\mathcal{A}_{\pi,P}^{\dagger}$ with very precise growth
conditions depending on $\eta(P)$ and $\mathbf{m}$. See §6.1.1 and §6.1.2 for
the precise definitions.

  10. $\mathcal{A}_{\pi,P}^{\mathrm{tr}}$

The subspace of elements in $\mathcal{A}_{P}$ of ‘truncated’ power series.
More precisely, elements that can be written as
$\sum\limits_{k=\mu(P)}^{\infty}a_{k}t_{P}^{-k}$. This will frequently be
combined with another ‘growth’ superscript, such as $\dagger$, $\infty$, or
$\mathbf{m}$.

  11. $\mathcal{A}_{\pi}$

The product $\prod\limits_{P\in S_{\eta}}\mathcal{A}_{\pi,P}$. We use the
superscripts $\mathbf{m}$, $\dagger$, and $\infty$ for the appropriate
$\pi$-adic growth conditions. We also use the $\mathrm{tr}$ superscript to
denote the ‘truncated’ power series in each summand.

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