# Low-dimensional antiferromagnetic fluctuations in the heavy-fermion
paramagnetic ladder UTe2

W. Knafo  LNCMI-EMFL, CNRS UPR3228, Univ. Grenoble Alpes, Univ. Toulouse,
Univ. Toulouse 3, INSA-T, 143 Avenue de Rangueil, 31400 Toulouse, France  G.
Knebel  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000,
Grenoble, France  P. Steffens  Institut Laue Langevin, 6 rue Jules Horowitz,
BP 156, 38042 Grenoble, France  K. Kaneko  Materials Sciences Research Center,
Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan  A. Rosuel  Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000, Grenoble, France
J.-P. Brison  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, 38000,
Grenoble, France  J. Flouquet  Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG,
PHELIQS, 38000, Grenoble, France  D. Aoki  Institute for Materials Research,
Tohoku University, Ibaraki 311-1313, Japan  G. Lapertot  Univ. Grenoble Alpes,
CEA, Grenoble INP, IRIG, PHELIQS, 38000, Grenoble, France  S. Raymond  Univ.
Grenoble Alpes, CEA, IRIG, MEM, MDN, 38000 Grenoble, France

###### Abstract

Inelastic-neutron-scattering measurements were performed on a single crystal
of the heavy-fermion paramagnet UTe2 above its superconducting temperature. We
confirm the presence of antiferromagnetic fluctuations with the incommensurate
wavevector $\mathbf{k}_{1}=(0,0.57,0)$. A quasielastic signal is found, whose
momentum-transfer dependence is compatible with fluctuations of magnetic
moments $\mu\parallel\mathbf{a}$, with a sine-wave modulation of wavevector
$\mathbf{k}_{1}$ and in-phase moments on the nearest U atoms. Low
dimensionality of the magnetic fluctuations, consequence of the ladder
structure, is indicated by weak correlations along the direction $\mathbf{c}$.
These fluctuations saturate below the temperature $T_{1}^{*}\simeq 15$ K, in
possible relation with anomalies observed in thermodynamic, electrical-
transport and nuclear-magnetic-resonance measurements. The absence or weakness
of ferromagnetic fluctuations, in our data collected at temperatures down to
2.1 K and energy transfers from 0.6 to 7.5 meV, is emphasized. These results
constitute constraints for models of magnetically-mediated superconductivity
in UTe2.

###### pacs:

71.27.+a,74.70.Tx,75.30.Mb,75.40.Gb

The discovery of superconductivity at temperatures below $T_{sc}\simeq 1.6$ K
in the heavy-fermion paramagnet UTe2 Ran  _et al._ (2019a); Aoki  _et al._
(2019) opened a breach in correlated-electron physics. UTe2 was presented as a
candidate for topological superconductivity, whose triplet and chiral
characters have been proposed Ishizuka  _et al._ (2019); Ran  _et al._
(2019a); Aoki  _et al._ (2019); Nakamine  _et al._ (2019); Metz  _et al._
(2019); Kittaka  _et al._ (2020); Jiao  _et al._ (2020); Shishidou  _et al._
(2021); Hayes  _et al._ ; Bae  _et al._ (2021); Sato and Ando (2017). It
turned rapidly out that this system is a unique model to study the electronic
correlations and their feedback on magnetism and superconductivity Shick  _et
al._ (2021); Ishizuka and Yanase (2021); Miao  _et al._ (2020); Xu  _et al._
(2019); Machida (2020). A competition between different superconducting
pairing mechanisms was indicated from the observation of multiple
superconducting phases stabilized near quantum magnetic instabilities under
pressure and magnetic field Braithwaite  _et al._ (2019); Knafo  _et al._
(2019); Miyake  _et al._ (2019); Knebel  _et al._ (2019); Ran  _et al._
(2019b); Aoki  _et al._ (2020); Ran  _et al._ (2020); Lin  _et al._ (2020);
Knebel  _et al._ (2020); Knafo  _et al._ (2021); Aoki  _et al._ (2021). UTe2
was first suspected to be a nearly-ferromagnet, in which ferromagnetic
fluctuations were thought to lead to triplet superconductivity Ishizuka and
Yanase (2021). Longitudinal magnetic fluctuations were evidenced by NMR
Tokunaga  _et al._ (2019) and muon-spin relaxation measurements Sundar  _et
al._ (2019), but these studies could not unambiguously distinguish
ferromagnetic and antiferromagnetic fluctuations. The nature of magnetic order
induced under pressure Braithwaite  _et al._ (2019) also constitutes an open
question: ferromagnetism was first suggested Ran  _et al._ (2020); Lin  _et
al._ (2020), but antiferromagnetism was proposed from more recent studies
Thomas  _et al._ (2020); Aoki  _et al._ (2020, 2021); Li  _et al._ (2021). In
addition, UTe2 crystallizes in an Immm orthorhombic structure, where the U
atoms form a two-leg ladder structure with legs along $\mathbf{a}$ and rungs
along $\mathbf{c}$ [see Fig. 1(a-b)] Hutanu  _et al._ (2020); Xu  _et al._
(2019); Knafo  _et al._ (2021). Therefore, one could suspect that low
dimensionality may impact superconducting pairing too.

Inelastic neutron scattering is wavevector-resolved and allows directly
determining whether magnetic fluctuations are ferromagnetic or
antiferromagnetic. A first neutron-scattering study of UTe2, by time-of-flight
spectroscopy on an assembly of $61$ crystals offering a total mass of 700 mg
and mosaicity of $\simeq 15^{\circ}$, led to the identification of
antiferromagnetic fluctuations with the incommensurate wavevectors
$\mathbf{k}_{1}=(0,0.57,0)$ and $\mathbf{k}_{2}=(0,0.43,0)$ [expressed in
reciprocal lattice units (rlu) and defined within the first Brillouin zone]
Duan  _et al._ (2020). Here, we present a neutron-scattering study using the
triple-axis spectrometer Thales, at the Institut Laue Langevin in Grenoble. A
large single crystal of UTe2 with a mass of 241 mg and a mosaicity of
$\lesssim 2^{\circ}$ allowed a fine experimental resolution (see Supplementary
Materials SM ). We confirm the presence of magnetic fluctuations with
wavevector $\mathbf{k}_{1}$ and do not see clear signatures of ferromagnetic
fluctuations with wavevector $\mathbf{k}=0$. A low-dimensional character
related with the ladder structure of the U atoms is emphasized, and the
temperature-evolution of the antiferromagnetic fluctuations with wavevector
$\mathbf{k}_{1}$ is carefully investigated.

Figure 1: (a) Orthorhombic and Wigner-Seitz primitive cells, and
identification of the four smallest distances (from neutron diffraction at
$T=2.7$ K Hutanu  _et al._ (2020)), (b) extended structure emphasizing the
ladder structure of U atoms, and (c) Brillouin zone of UTe2. A
phenomenological magnetic-exchange scheme implied in the magnetic-fluctuations
mode with wavevector $\mathbf{k}_{1}$ is indicated in (b) (details are given
in the text).

Figure 2 presents energy scans measured at different momentum transfers
$\mathbf{Q}$ and $T=2.1$ K, for energy transfers $0.6\leq E\leq 7.5$ meV. A
large signal at the momentum transfer $\mathbf{Q}_{1}=(0,1.43,0)$ indicates
the presence of strong antiferromagnetic fluctuations at the incommensurate
wavevector $\mathbf{k}_{1}=(0,0.57,0)$ [from
$\mathbf{Q}_{1}=\mathbf{\tau}-\mathbf{k}_{1}$, where $\mathbf{\tau}=(0,2,0)$
is a nuclear Bragg position]. $\mathbf{k}_{1}$ is close to the Brillouin-zone
boundary [see Fig. 1(c)]. The momentum transfer $\mathbf{Q}=(0,1.16,1.9)$,
chosen far from $\mathbf{Q}_{1}$ (and with the same modulus), is
characteristic of a background without magnetic fluctuations. A few points
collected at $T=60$ K indicate a nearly-temperature independent background.
Spectra at three momentum transfers $\mathbf{Q}=(0,2.1,0)$, $(0,1.07,1.07)$,
and $(0,1.07,3.07)$, expected to characterize ferromagnetic fluctuations with
wavevector $\mathbf{k}\simeq 0$, present intensities near to the background
level 111The momentum transfers $\mathbf{Q}=(0,2.1,0)$, $(0,1.07,1.07)$, and
$(0,1.07,3.07)$ were chosen near, but not exactly at, the nuclear Bragg
positions $\mathbf{\tau}=(0,2,0)$, $(0,1,1)$, and $(0,1,3)$ corresponding to
magnetic wavevector $\mathbf{k}=0$ to avoid contamination by the nuclear Bragg
peaks.. At $\mathbf{Q}=(0,2.1,0)$, the intensity is slightly higher than at
the two other ferromagnetic positions, but similar intensities measured at
$T=2.1$ and 60 K indicate a presumably non-magnetic signal. Within the
experimental window investigated here ($T\geq 2.1$ K and $0.6\leq E\leq 7.5$
meV), we do not identify, thus, any clear signature of ferromagnetic
fluctuations in UTe2.

Figure 2: Energy scans measured at the temperature $T=2.1$ K and the momentum
transfers $\mathbf{Q}_{1}=(0,1.43,0)$ characteristic of antiferromagnetic
fluctuations (red squares), $\mathbf{Q}=(0,2.1,0)$,
$\mathbf{Q}=(0,1.07,1.07)$, and $\mathbf{Q}=(0,1.07,3.07)$ characteristic of
ferromagnetic fluctuations (circles), and $\mathbf{Q}=(0,1.16,1.9)$
characteristic of the background (gray triangles). A few points are shown at
$T=60$ K and the momentum transfers $\mathbf{Q}=(0,2.1,0)$ and
$\mathbf{Q}=(0,1.16,1.9)$ (open symbols). Lines correspond to a fit to the
data by a quasielastic Lorentzian shape at the momentum transfer
$\mathbf{Q}_{1}$, to a fit to the data by a linear shape at the momentum
transfer $\mathbf{Q}=(0,1.16,1.9)$, and to guides to the eyes at the other
momentum transfers.

$\mathbf{Q}$-dependences of the antiferromagnetic signal centered at
$\mathbf{Q}_{1}$, measured at the energy transfer $E=3$ meV and at the
temperature $T=2.1$ K, are presented in Fig. 3. The $(0,Q_{k},0)$ scan shown
in Fig. 3(a) indicates an asymmetric anomaly of maximal intensity near
$\mathbf{Q}_{1}=(0,1.43,0)$, with a shoulder for $Q_{k}>1.43$. It can be
fitted by two Gaussian contributions of full width at half maximum
$\kappa=0.24$ rlu, corresponding to a correlation length $\xi_{b}=3.75b$,
where $b=6.09$ Å Hutanu  _et al._ (2020), along the direction $\mathbf{b}$
(see Supplementary Materials SM ). The contribution centered at the momentum
transfer $\mathbf{Q}_{1}=(0,1.43,0)$, corresponding to the wavevector
$\mathbf{k}_{1}=(0,0.57,0)$, is twice more intense than that centered at the
momentum transfer $\mathbf{Q}_{2}=(0,1.58,0)$, corresponding to the wavevector
$\mathbf{k}_{2}=(0,0.42,0)$. In the following, focus will be given to the
dominant magnetic-fluctuations mode at wavevector $\mathbf{k}_{1}$.
$(0,Q_{k},0)$ scans at different energy transfers $E$, shown in the
Supplementary Materials SM , further indicate that the magnetic fluctuations
signal at the wavevector $\mathbf{k}_{1}=(0,0.57,0)$ is weakly or not
dispersive. In Fig. 3(b), a $(0,1.43,Q_{l})$ scan presents a damped sine-wave
evolution of the scattered intensity. Best fit to the data is made assuming i)
fluctuating magnetic moments $\mathbf{\mu}\parallel\mathbf{a}$ and ii) in-
phase magnetic moments on the two U of the primitive cell [see Fig. 1(a) and
Supplementary Materials SM ]. The sine-wave modulation indicates that, within
first approximation, the correlations along $\mathbf{c}$ can be neglected,
i.e., that the inter-ladder magnetic coupling along $\mathbf{c}$ is weak. To
our knowledge, magnetic correlations along $\mathbf{a}$ have not been
characterized by neutron scattering so far. However, a first component
$k_{1,h}=0$ of the propagation vector $\mathbf{k}_{1}$ is implicitly assumed
here and in Duan  _et al._ (2020). Indeed, strong correlations along the
ladder direction $\mathbf{a}$ are expected from the short distance
$d_{2}=4.12$ Å between U atoms along $\mathbf{a}$. For these reasons, one can
safely expect that the fluctuations with wavevector $\mathbf{k}_{1}$ imply
correlations with in-phase fluctuating moments along $\mathbf{a}$. We can
further conclude that the antiferromagnetic fluctuations investigated here are
low-dimensional, and that they are characterized by a set of equivalent, or
nearly-equivalent, lines of wavevectors $\mathbf{k_{L}}=(0,0.43,\delta)$ and
$(0,0.57,\delta)$, which include $\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ (see
Supplementary Materials SM ).

Figure 3: (a) $(0,Q_{k},0)$ and (b) $(0,1.43,Q_{l})$ scans at the energy
transfer $E=3$ meV and the temperature $T=2.1$ K. Gray dashed lines correspond
to the estimated background. Red lines correspond to fit to the data as
described in SM .

Figure 4(a) shows spectra measured at the momentum transfer
$\mathbf{Q}_{1}=(0,1.43,0)$ for a large set of temperatures $T$ from $2.1$ to
60 K. The increase of $T$ leads to a progressive decrease and broadening of
the antiferromagnetic signal. We fitted data shown in Fig. 4(a) assuming a
quasielastic Lorentzian variation of the imaginary part of the dynamical
susceptibility $\chi^{\prime\prime}(\mathbf{Q}_{1},E)$ (see Supplementary
Materials SM ). The temperature variations of the relaxation rate
$\Gamma(\mathbf{k}_{1})$ and of the real part of the static susceptibility
$\chi^{\prime}(\mathbf{k}_{1})$ extracted from these fits are plotted within
log-log scales in Fig. 4(b-c). When the temperature is lowered, the
strengthening of antiferromagnetic fluctuations is evidenced by the decrease
of $\Gamma(\mathbf{k}_{1})$ and the increase of
$\chi^{\prime}(\mathbf{k}_{1})$, which both saturate at temperatures below
$T_{1}^{*}\simeq 15$ K. $T_{1}^{*}$ and the low-temperature value
$\Gamma(\mathbf{k}_{1})\simeq 2.5$ meV $\simeq 2~{}k_{B}T_{1}^{*}$ are
characteristic of the antiferromagnetic fluctuations with wavevector
$\mathbf{k}_{1}$.

Anomalies are observed at temperatures near $T_{1}^{*}$ in various physical
properties. The bulk magnetic susceptibility $\chi_{a}=M/H$, where $M$ is the
magnetization and $\mu_{0}H=0.1$ T a magnetic field applied along the easy
magnetic axis $\mathbf{a}$, is compared to $\chi^{\prime}(\mathbf{k}_{1})$ in
Fig. 4(c). Knowing that $\chi_{a}=\chi_{a}^{\prime}(\mathbf{k}=0)$ and
$\chi^{\prime}(\mathbf{k}_{1})=\chi_{a}^{\prime}(\mathbf{k}_{1})$ (since the
magnetic fluctuations with wavevector $\mathbf{k}_{1}$ were attributed to
magnetic moments $\mu\parallel\mathbf{a}$), both quantities are expected to
converge at temperatures at which intersite magnetic correlations have
vanished, i.e., presumably $T\gtrsim 100$ K. A broad kink in $\chi_{a}$ is
observed at the temperature $T_{\chi-a}^{kink}\simeq 15$ K $\simeq T_{1}^{*}$.
Maxima in the electronic heat capacity Willa  _et al._ and electrical
resistivity measured with a current $\mathbf{I}\parallel\mathbf{c}$ Eo  _et
al._ and minima in the thermal expansion measured with lengths
$\mathbf{L}\parallel\mathbf{b},\mathbf{c}$ Thomas  _et al._ ; Willa  _et al._
and thermoelectric power measured with a current
$\mathbf{I}\parallel\mathbf{a}$ Niu  _et al._ (2020) are also observed at a
temperature of $\simeq 15$ K. These anomalies may result from the development
of antiferromagnetic fluctuations with wavevector $\mathbf{k}_{1}$ and their
possible feedbacks on the Fermi surface. Figure 4(d) presents a comparison of
$\chi^{\prime}(\mathbf{k}_{1})/\Gamma(\mathbf{k}_{1})$ extracted here and the
NMR relaxation rate $1/T_{1}T$, measured in a magnetic field
$\mathbf{H}\parallel\mathbf{b}$ by Tokunaga et al (for the two sites Te1 and
Te2) Tokunaga  _et al._ (2019). These two quantities are dominated by
fluctuations of magnetic moments $\mathbf{\mu}\parallel\mathbf{a}$ and vary
similarly, increasing with decreasing temperatures before saturating at low
temperature. $1/T_{1}T$ saturates below a temperature of $\simeq 20$ K near
$T_{1}^{*}$. Knowing that $1/T_{1}T$ consists in a sum of
$\chi^{\prime}(\mathbf{k})/\Gamma(\mathbf{k})$ over the reciprocal space (see
Supplementary Materials SM ), the magnetic fluctuations at $\mathbf{k}_{1}$
(and its equivalent positions $\mathbf{k}_{L}$) may contribute significantly
to $1/T_{1}T$. The slower $T$-variation of $1/T_{1}T$, in comparison with that
of $\chi^{\prime}(\mathbf{k}_{1})/\Gamma(\mathbf{k}_{1})$, may be due to the
contribution of fluctuations at wavevectors far from $\mathbf{k}_{1}$. The
question of low-energy ferromagnetic fluctuations (not observed here) and,
thus, of their possible contribution to $1/T_{1}T$, which is sensitive to
energies $E\rightarrow 0$, remains open.

Similar relationship between anomalies in bulk properties and magnetic
fluctuations were observed in other heavy-fermion paramagnets Aoki  _et al._
(2013). In the prototypical heavy-fermion paramagnet CeRu2Si2, the
characteristic temperature $T_{1}^{*}\simeq 10$ K of longitudinal
antiferromagnetic fluctuations Knafo  _et al._ (2009) is comparable to those
of anomalies in the magnetic susceptibility Haen  _et al._ (1987), thermal
expansion Paulsen  _et al._ (1990), and NMR relaxation rate Ishida  _et al._
(1998). The situation in UTe2 is more complex. In addition to the temperature
scale of $\simeq 15$ K considered earlier, a second temperature scale of
$\simeq 35$ K can be identified from maxima in the magnetic susceptibility
$\chi_{b}$ measured with $\mathbf{H}\parallel\mathbf{b}$ (hard magnetic axis)
Ikeda  _et al._ (2006), in the Hall coefficient Niu  _et al._ (2020) and
electrical resistivity Knafo  _et al._ (2019) measured with a current
$\mathbf{I}\parallel\mathbf{a}$. Further experiments are needed to determine
the relationship between the 15-K and 35-K anomalies, and to test if a second
and higher-energy magnetic-fluctuations mode could be present.

The possible coexistence of ferromagnetic and antiferromagnetic couplings in
the ladder structure of UTe2 was theoretically discussed in Xu  _et al._
(2019), and a first report of antiferromagnetic fluctuations was made in Duan
_et al._ (2020). Thanks to a careful investigation of this antiferromagnetic-
fluctuations mode, we emphasize here its quasielastic and low-dimensional
characteristics. It can be visualized as fluctuations of magnetic moments
$\mathbf{\mu}\parallel\mathbf{a}$, with the following phenomenological scheme:
i) an inter-ladder antiferromagnetic coupling $J^{b}_{inter}(AF)$ ending in
sine-wave modulation with wavevector $\mathbf{k}_{1}$ of the moments, ii)
intra-ladder ferromagnetic couplings $J^{c}_{1}(FM)$ along $\mathbf{c}$ and
$J^{a}_{2}(FM)$ along $\mathbf{a}$, and iii) a weak inter-ladder magnetic
coupling $J^{c}_{inter}$ along $\mathbf{c}$ [see Fig. 1(a)] 222The
phenomenological inter-ladder magnetic couplings $J^{b}_{inter}$ and
$J^{c}_{inter}$ introduced here may result from two successive ’diagonal’
inter-ladder exchanges [along third shortest U-U distance $d_{3}$, see Fig.
1(a-b)].. $\mathbf{Q}$-modulations of a magnetic-fluctuations signal similar
to those reported here for UTe2 were observed by inelastic neutron scattering
in other low-dimensional magnets whose primitive cell contains several
magnetic ions, as the magnetic ladder Sr14Cu24O41 Regnault  _et al._ (1999);
Boullier (2005), the layered paramagnet Sr3Ru2O7 Capogna  _et al._ (2003),
antiferromagnet YBa2Cu3O6.2 Sato  _et al._ (1988), and superconductors
YBa2Cu3O6.85 Pailhès  _et al._ (2004) and CaKFe4As4 Xie  _et al._ (2018). It
has been argued that a more robust superconducting pairing is expected for
two-dimensional, rather than three-dimensional, itinerant magnets Monthoux and
Lonzarich (2001). A low-dimensional character of the magnetic fluctuations
may, thus, be of importance for the development of superconductivity in UTe2.
The question of the role of ferromagnetic fluctuations for the development of
a superconducting phase, often suspected to be triplet, needs to be clarified.
Ferromagnetic fluctuations were not observed here, within the investigated
experimental window $0.6\leq E\leq 7.5$ meV, at $T=2.1$ K, but Duan et al
reported a small signal at the ferromagnetic momentum transfer
$\mathbf{Q}=(0,1,1)$, at $E\simeq 0.4$ meV and $T=300$ mK Duan  _et al._
(2020). An unambiguous experimental evidence for ferromagnetic fluctuations is
now needed. We note that in the paramagnet Sr2RuO4, a former candidate for
triplet superconductivity, intense incommensurate antiferromagnetic
fluctuations were found to coexist with weak, and broad in $\mathbf{Q}$-space,
quasi-ferromagnetic fluctuations Pustogow  _et al._ (2019); Steffens  _et al._
(2019). Conversely, triplet superconductivity was proposed to occur in UPt3,
where only antiferromagnetic fluctuations were observed Aeppli  _et al._
(1988); Tou  _et al._ (1998). Our finding that the antiferromagnetic
fluctuations imply ferromagnetically-coupled U atoms on the ladders may be of
importance. Indeed, it is compatible with the proposition of pseudo-triplet
superconductivity induced by a ferromagnetic interaction between U atoms of
the ladder rungs, without necessarily implying ferromagnetic fluctuations Xu
_et al._ (2019); Shishidou  _et al._ (2021); Anderson (1985). In the future,
challenges will be to determine precisely the respective roles of
antiferromagnetic and ferromagnetic fluctuations and low dimensionality for
the stabilization of different superconducting phases in UTe2. The topology
and possible low-dimensionality of some Fermi-surface sheets, which set in at
low temperatures, also play a role for superconductivity Xu  _et al._ (2019);
Miao  _et al._ (2020); Shishidou  _et al._ (2021) and would merit further
consideration too.

Figure 4: (a) Energy scans measured at $\mathbf{Q}_{1}=(0,1.43,0)$ and
temperatures $T$ from $2.1$ to 60 K. Full lines are fits to the data assuming
a Lorentzian shape, and the dashed line indicates the estimated background.
Temperature dependence, in log-log scales, (b) of the relaxation rate
$\Gamma(\mathbf{k}_{1})$, (c) of the real part of the static susceptibility
$\chi^{\prime}(\mathbf{k}_{1})$ and the bulk magnetic susceptibility
$\chi_{a}=M/H$ measured in a magnetic field
$\mu_{0}\mathbf{H}\parallel\mathbf{a}$ of 0.1 T, and (d) of the ratio
$\chi^{\prime}(\mathbf{k}_{1})/\Gamma(\mathbf{k}_{1})$ and of the NMR
relaxation rate $1/T_{1}T$ measured on the Te1 and Te2 sites in a magnetic
field $\mathbf{H}\parallel\mathbf{b}$ (from Tokunaga  _et al._ (2019)). In
panels (c-d), left and right scales have the same number of decades to allow
direct comparison between plotted data.

## Acknowledgements

We acknowledge useful discussions with Y. Tokunaga, K. Ishida, and C. Simon.

This work was supported by the ANR FRESCO.

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Supplementary Materials:
Low-dimensional antiferromagnetic fluctuations in the heavy-fermion
paramagnetic ladder UTe2

In these Supplementary Materials, we present complementary graphs and details
concerning the analysis of our data obtained by inelastic neutron scattering
on a single crystal of UTe2.

##  I Sample growth and characterization

The 241-mg single-crystalline sample of UTe2 used in this neutron-scattering
study was grown using chemical vapor transport method. Figure S1(a) presents a
photo of the crystal. Its heat-capacity $C_{p}$, measured at the CEA-Grenoble,
is presented as $C_{p}/T$ versus $T$ graph in Fig. S1(b). The onset of
superconductivity is visible at the temperature $T_{sc}\simeq 1.6$ K. The
transition is sharp, with a total width $\Delta T\simeq 0.05$ K. This
indicates a high quality and small degree of distortion of the crystal. A
large increase of $C_{p}/T$ is also observed at temperatures $T<250$ mK.

$A3$-scans, where the angle $A3$ characterizes the rotation of the sample
relatively to the vertical axis, around the Bragg positions
$\mathbf{Q}=(0,2,0)$ and $\mathbf{Q}=(0,0,4)$ and measured by elastic neutron
diffraction at the temperature $T=2$ K, are presented in Fig. S2. These scans
show that the sample has a mosaicity of $\lesssim 2~{}^{\circ}$.

Figure S1: (a) Photo and (b) heat capacity $C_{p}$, plotted as $C_{p}/T$
versus $T$, of the single crystal used in the present neutron scattering
study. Figure S2: $A3$-scans around the Bragg positions $\mathbf{Q}=(0,2,0)$
and $\mathbf{Q}=(0,0,4)$ measured by neutron diffraction at the energy $E=0$
and the temperature $T=2$ K.

##  II $\mathbf{Q}$-scans at different energies

Complementarily to Fig. 3 of the paper, Fig. S3 presents $\mathbf{Q}$-scans
around the antiferromagnetic signal centered at $\mathbf{Q}_{1}=(0,1.43,0)$,
measured at energy transfers $E=1,3,~{}\rm{and}~{}5$ meV and at the
temperature $T=2.1$ K. The $(0,Q_{k},0)$ scans shown in Fig. S3(a) present an
asymmetric anomaly of maximal intensity near $\mathbf{Q}_{1}=(0,1.43,0)$, with
a shoulder for $Q_{k}>1.43$. The $(0,Q_{k},Q_{l})$ scans at constant $Q_{k}$
shown in Fig. S3(b) show a damped sine-wave variation of intensity. These
$\mathbf{Q}$ scans performed at different energies indicate that the
excitations peaked at $\mathbf{Q}_{1}$ form a rod as a function of energy, and
is therefore weakly or not dispersive in the energy range investigated here.

$Q_{k}$ scans were fitted to Gaussian contributions expressed as:

$\displaystyle I(Q_{k})=I_{0}\exp[-4\ln 2(Q_{k}/\kappa)^{2}],$ (S1)

where $\kappa$ corresponds to the full-width at half maximum (FWHM) of the
contribution. A Fourier transformation of Eq. (S1) leads to the real-space
distribution of intensity:

$\displaystyle I^{\prime}(y)=I_{0}^{\prime}\exp[-4\ln 2(y/\xi_{b})^{2}],$ (S2)

where $\xi_{b}=4b\ln 2/(\pi\kappa)$ is defined as the correlation length in
the direction $\mathbf{b}$ of the magnetic fluctuations considered here.

Figure S3: (a) $(0,Q_{k},0)$ scans at the energy transfers $E=1,3,5$ meV and
the temperature $T=2.1$ K. Full lines are guides to the eyes. (b)
$(0,Q_{k},Q_{l})$ scans, where $Q_{k}$ is fixed to a value $1.4\leq Q_{k}\leq
1.5$, at the energy transfers $E=1,3,5$ meV and the temperature $T=2.1$ K.
Full lines correspond to fits to the data. Figure S4: $(0,1.43,Q_{l})$ scan at
the energy transfer $E=3$ meV and the temperature $T=2.1$ K and its fits
assuming magnetic moments (a) $\mathbf{\mu}\parallel\mathbf{a}$, (b)
$\mathbf{\mu}\parallel\mathbf{b}$, and (c) $\mathbf{\mu}\parallel\mathbf{c}$.
Dashed gray lines correspond to the estimated background, dashed colored lines
to the envelope $f_{m}(\mathbf{Q})^{2}\sin^{2}[\alpha(\mathbf{Q})]$, and full
colored lines correspond to fit by $F_{m}(\mathbf{Q})^{2}$ to the data. (d)
Variation of magnetic form factor $f_{m}$ with $\sin\theta/\lambda$, where
$\theta$ is the scattering angle and $\lambda$ the neutron wavelength,
expected for U3+, U4+, and U5+ [Brown, 2002], and comparison with scaled
intensities measured on UTe2 at the momentum transfers $(0,1.43,0)$ and
$(0,1.43,4)$, for $E=3$ meV and $T=2.1$ K. Figure S5: $Q_{l}$ scans expected
for different strengths (from strong to weak) of correlations along the
direction $\mathbf{c}$, assuming (a) one U atom per primitive cell (b) in
phase magnetic moments on the two U atoms of the primitive cell of UTe2.

##  III Fit to the $(0,1.43,Q_{l})$ scan by a damped sine-wave function

Figure S4 presents details concerning the fit to the damped sine-wave
variation of intensity from the $(0,1.43,Q_{l})$ scan made at $T=2.1$ K and
shown in Fig. 3(b) of the paper. Data were fitted by a function proportional
to the square $F_{m}(\mathbf{Q})^{2}$ of the magnetic structure factor:

$\displaystyle
F_{m}(\mathbf{Q})^{2}=f_{m}(\mathbf{Q})^{2}\sin^{2}[\alpha(\mathbf{Q})]\cos^{2}(\pi
Q_{l}d_{1}/c),$ (S3)

where $f_{m}(\mathbf{Q})$ is the magnetic form factor of individual U atoms,
$\alpha(\mathbf{Q})$ is the angle between $\mathbf{Q}$ and the magnetic moment
$\mu$, and the cosine pre-factor is controlled by the magnetic pattern in the
primitive cell, with $d_{1}=3.72$ Å [see Fig. 1(a) of the paper]. The envelope
$f_{m}(\mathbf{Q})^{2}\rm{sin}^{2}[\alpha(\mathbf{Q})]$ and the full fitted
function $F_{m}(\mathbf{Q})^{2}$ are plotted assuming the three hypotheses
$\mathbf{\mu}\parallel\mathbf{a}$, $\mathbf{\mu}\parallel\mathbf{b}$, and
$\mathbf{\mu}\parallel\mathbf{c}$ in Fig. S4(a), (b) and (c), respectively.
While a good fit to the data is obtained assuming
$\mathbf{\mu}\parallel\mathbf{a}$, the hypotheses
$\mathbf{\mu}\parallel\mathbf{b}$ and $\mathbf{\mu}\parallel\mathbf{c}$ do not
permit fitting the data in the whole measured $Q_{l}$ range. In this fit, the
magnetic form factor $f_{m}(\mathbf{Q})$ of valence-3 U3+ ions was used
[Brown, 2002], following the conclusions from a photoelectron-spectroscopy
study [Fujimori  _et al._ , 2021]. We note that a valence 4 corresponding to
U4+ ions was also proposed from an x-ray-absorption-spectroscopy experiment
and the question of valence in UTe2 remains open [Thomas  _et al._ , 2020].
However, the fit done here is almost unaffected by the choice of valence,
since the magnetic form factors of U3+, U4+, but also U5+ ions are very
similar [Brown, 2002]. The pre-factor $\rm{cos}(\pi Q_{l}d_{1}/c)$, which
allows a good fit to the data, is expected for in-phase, i.e.,
ferromagnetically-coupled, magnetic moments on the two U of the primitive
cell. We note that the first zero of the pre-factor occurs for
$Q_{l}=c/(2d_{1})$, which is accidentally close to $\simeq 2$. Alternative
hypotheses can lead to different pre-factors, as $\rm{sin}(\pi Q_{l}d_{1}/c)$
expected for antiparallel magnetic moments, but do not reproduce the
experimental data.

For pedagogical purpose, Fig. S5 shows $Q_{l}$ scans expected for different
strengths of the correlations along the direction $\mathbf{c}$, assuming one U
atom per primitive cell [Fig. S5(a)] or assuming in phase magnetic moments on
the two U atoms of the primitive cell of UTe2 [Fig. S5(b)]. Well-defined peaks
in the case of strong correlations are replaced by broader peaks when the
correlations are weakened, ending asymptotically in the envelope given by Eq.
(S3) when the correlations vanish.

##  IV Magnetic fluctuations with wavevectors $\mathbf{k}_{1}$ and
$\mathbf{k}_{2}$

Figure S6 presents series of $\mathbf{Q}$-scans performed at the energy $E=3$
meV and the temperature $T=2.1$ K. These scans are indicated by arrows in the
scattering plane shown in Fig. S5(a). The $(0,1.43,Q_{l})$ and
$(0,1.58,Q_{l})$ scans shown in Fig. S6(b) both present a damped sine-wave
evolution of the intensity. Each of these scans presents an alternation of
$\mathbf{k}_{1}$ and $\simeq\mathbf{k}_{2}$ positions. Within first
approximation, their fit to Eq. S3 (see Sect. III) indicates that the
antiferromagnetic fluctuations are characterized by a set of wavevectors
$\mathbf{k_{L}}=(0,0.43,\delta)$ and $(0,0.57,\delta)$, which include
$\mathbf{k}_{1}$ and $\mathbf{k}_{2}$. These two lines of wavevectors are
equivalent for symmetry reasons. However, small deviations are indicated by
different scattered intensities at positions corresponding to $\mathbf{k}_{1}$
and $\mathbf{k}_{2}$. These deviations are clear in Fig. S6(c), which presents
$(0,Q_{k},Q_{l})$ scans at constant $Q_{l}=0,1,2,3,4$. Fits made using
Gaussian contributions of width $\kappa=0.19-0.24$ rlu and centered at
positions corresponding to the wavevectors $\mathbf{k}_{1}=(0,0.57,0)$ and
$\mathbf{k}_{2}=(0,0.43,0)$ are indicated by full lines in the graphs. These
$(0,Q_{k},Q_{l})$ scans (for constant $Q_{l}$ values) show a non-trivial
evolution of the two peak intensities:

  * •

the intensity at wavevector $\mathbf{k}_{2}$ probed at the momentum transfer
$\mathbf{Q}=(0,1.57,0)$ is smaller than that at the wavevector
$\mathbf{k}_{1}$ probed at the momentum transfer $\mathbf{Q}=(0,1.43,0)$ (they
should be nearly equal),

  * •

a small intensity remains at wavevector $\mathbf{k}_{2}$ probed at the
momentum transfer $\mathbf{Q}=(0,1.43,2)$.

The observed intensities are not compatible with simple phonon contaminations.
These deviations may be due to subtle magneto-elastic effects. For instance,
the acoustic phonon intensity is expected to be one or two orders of magnitude
higher for $Q_{k}=2$ than for $Q_{k}=1$, which may lead to observable coupling
with magnetism. A more complete set of experiments, i.e., energy scans at
$\mathbf{k}_{2}$ positions and momentum-transfer scans at different energies,
possibly at different temperatures, may be needed to study the origin of these
effects. Great care to the $\mathbf{Q}$-dependence of the experimental
background and of the experimental resolution may be mandatory for such works,
for which a larger crystal than the one used here may be needed.

We note that, in their work made on an assembly of 61 single crystals using a
time-of-flight spectrometer, Duan et al. observed a vanishing of magnetic
fluctuations signal in the line $(0,Q_{k},2)$, in agreement with the cosine
term in Eq. (S3) used to fit our data [Duan  _et al._ , 2020].

Figure S6: (a) Scattering plane of the momentum transfers $Q=(0,Q_{k},Q_{l})$
accessed in the neutron experiment. Points at the Brillouin zone centers
corresponding to wavevector $\mathbf{k}=0$ are indicated by gray open circles.
Points corresponding to the wavevectors $\mathbf{k}_{1}=(0,0.57,0)$ and
$\mathbf{k}_{2}=(0,0.42,0)$ are indicated by green and orange crosses,
respectively. $\mathbf{Q}$ scans shown in panels (b-c) are indicated by
arrows. (b) $(0,1.43,Q_{l})$ and $(0,1.58,Q_{l})$ scans and (c)
$(0,Q_{k},Q_{l})$ scans, with $Q_{l}=0,1,2,3,4$, performed at the energy $E=3$
meV and the temperature $T=2.1$ K. In panel (c), the lines correspond to fit
to the data by one or two Gaussian contributions.

##  V Fit to the energy scans by a quasielastic Lorentzian dynamical
susceptibility

The complex magnetic susceptibility, expressed as:

$\displaystyle\chi(\mathbf{k},E)=\chi^{\prime}(\mathbf{k},E)+i\chi^{\prime\prime}(\mathbf{k},E),$
(S4)

where $\chi^{\prime}(\mathbf{k},E)$ is its real part and
$\chi^{\prime\prime}(\mathbf{k},E)$ its imaginary part, is generally used to
described magnetic fluctuations in the reciprocal space.

Knowing that the neutron intensity $I(\mathbf{Q},E)$ is related to the
imaginary part of the dynamical susceptibility
$\chi^{\prime\prime}(\mathbf{Q},E)$ by:

$\displaystyle
I(\mathbf{Q},E)-I_{BG}(E)\propto\frac{F_{m}(\mathbf{Q})^{2}}{1-\exp(-E/k_{B}T)}\chi^{\prime\prime}(\mathbf{Q},E),$
(S5)

where $I_{BG}(E)$ is the background, data shown in Fig. 4(a) of the paper were
fitted assuming a quasielastic Lorentzian variation of
$\chi^{\prime\prime}(\mathbf{Q}_{1},E)$:

$\displaystyle\chi^{\prime\prime}(\mathbf{Q}_{1},E)=\chi^{\prime}(\mathbf{k}_{1})\frac{E/\Gamma(\mathbf{k}_{1})}{1+(E/\Gamma(\mathbf{k}_{1}))^{2}},$
(S6)

where $\Gamma(\mathbf{k}_{1})$ and $\chi^{\prime}(\mathbf{k}_{1})$ are the
relaxation rate and the real part of the static susceptibility, respectively,
at the wavevector $\mathbf{k}_{1}$ measured at the momentum transfer
$\mathbf{Q}_{1}$. We note that a quasielastic Lorentzian variation implies:

$\displaystyle\chi(\mathbf{k},E)=\frac{\chi^{\prime}(\mathbf{k})}{1-iE/\Gamma(\mathbf{k})},$
(S7)

and that the real part of static susceptibility is given by:

$\displaystyle\chi^{\prime}(\mathbf{k})=\chi^{\prime}(\mathbf{k},E=0)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\chi^{\prime\prime}(\mathbf{k})}{E}dE.$
(S8)

For a $\mathbf{k}$-dependent quasielastic-Lorentzian magnetic susceptibility,
the NMR spin-lattice relaxation rate $1/T_{1}T$ is related to
$\chi^{\prime}(\mathbf{k})$ and $\Gamma(\mathbf{k})$ by:

$\displaystyle\frac{1}{T_{1}T}\propto\sum_{\mathbf{k}}|A(\mathbf{k})|^{2}\frac{\chi^{\prime}(\mathbf{k})}{\Gamma(\mathbf{k})},$
(S9)

where $A(\mathbf{k})$ is a $\mathbf{k}$-dependent hyperfine coupling. A strong
$\mathbf{k}$ dependence of the weighting factor $|A(\mathbf{k})|^{2}$ can
enhance the contribution at given wavevectors.

## References

  * Brown (2002) P. J. Brown,  _Neutron Data Booklet, Chapter 2.5, Magnetic form factors_ (Institut laue Langevin, Grenoble, Edited by A.J. Dianoux and G. Lander, 2002).
  * Fujimori  _et al._ (2021) S.-i. Fujimori, I. Kawasaki, Y. Takeda, H. Yamagami, A. Nakamura, Y. Homma, and D. Aoki, J. Phys. Soc. Jpn. 90, 015002 (2021).
  * Thomas  _et al._ (2020) S. M. Thomas, F. B. Santos, M. H. Christensen, T. Asaba, F. Ronning, J. D. Thompson, E. D. Bauer, R. M. Fernandes, G. Fabbris, and P. F. S. Rosa, Sci. Adv. 6 (2020).
  * Duan  _et al._ (2020) C. Duan, K. Sasmal, M. B. Maple, A. Podlesnyak, J.-X. Zhu, Q. Si, and P. Dai, Phys. Rev. Lett. 125, 237003 (2020).


# Correlated electrons in a Crystalline Topological insulator

F. Manghi  Dipartimento di Scienze Fisiche, Informatiche e Matematiche,
Università di Modena e Reggio Emilia, Via Campi 213/A, I-41125 Modena, Italy
CNR - Institute of NanoSciences - S3

###### Abstract

We study the effects of e-e interaction in a 3D Crystalline Topological
Insulator by adding on-site repulsion to the single-particle Hamiltonian and
solving the many-body problem within Cluster Perturbation Theory. The goal is
to clarify how many body effects modify the topological phase that stems from
the crystal symmetries. Tuning the strength of the on-site interaction we show
that band inversion disappears in the bulk and surface states loose their
metallic character.

###### pacs:

71.10.-w,71.10.Fd ,73.20.-r ,73.43.-f

Crystalline Topological Insulators (CTIs) are a new entry in the fascinating
class of topological materials. In these systems, the basic ingredients that
give rise to topological insulators - time-reversal invariance and spin-orbit
interaction - are substituted by crystal symmetries. After the pioneering work
that introduced the very notion of CTIs Fu (2011), real materials that exhibit
this behaviour have been predicted by theory Hsieh  _et al._ (2012) and
revealed by experiments.Tanaka  _et al._ (2012); Dziawa  _et al._ (2012); Xu
_et al._ (2012); Wang  _et al._ (2013); Okada  _et al._ (2013); Zeljkovic  _et
al._ (2014) Most of the theoretical studies for both real and model CTIs Ando
and Fu (2015) have been based on a single particle description of the
electronic states and this in spite of the prediction of CTI behavior also for
materials where strong e-e correlations are expected.Hsieh  _et al._ (2014);
Weng  _et al._ (2014) Few works have addressed the effects of electron
interactions in CTIs investigating how electron correlations affect their
topological classifications Isobe and Fu (2015) or give rise to new
topological crystalline Mott insulator phases. Kargarian and Fiete (2013) The
combination of topological band structure and e-e interactions has been
extensively studied in standard topological insulators (see Refs. Hohenadler
and Assaad, 2013; Meng  _et al._ , 2014; Rachel, 2018 for recent reviews). In
this field, the two-dimensional honeycomb lattice with intrinsic spin-orbit
interaction and on-site e-e repulsion - the so-called Kane-Mele-Hubbard model
Rachel and Le Hur (2010)\- has been identified as a paradigmatic example. For
this system different many-body approaches have been applied, from Quantum
Montecarlo simulations Zheng  _et al._ (2011); Hohenadler  _et al._ (2011,
2012) to Quantum Cluster methods, Yu  _et al._ (2011); Grandi  _et al._
(2015); Gra  showing that the Hubbard interaction may drastically affect the
stability of the quantum spin Hall phase and supress topological protected
edge states.

In this paper we study the effects of Hubbard correlations on a CTI focusing
on quasi-particle states for both bulk and surface systems. We rely our
analysis on a solution of the many-body Hamiltonian by Cluster Perturbation
Theory(CPT), Sénéchal  _et al._ (2000); Sénéchal (2012) calculating one-
particle propagator and spectral functions. We investigate the effects of e-e
interaction in the prototypical CTI introduced by Fu Fu (2011) where a
tetragonal unit cell hosts two inequivalent atoms with two orbitals per site.
This simplified structure, containing different sites and different orbitals,
may exhibit non-trivial orbital textures that have been shown to be essential
in real CTIs. Zeljkovic  _et al._ (2014); Liu  _et al._ (2014); Huang  _et
al._ (2016); Walkup  _et al._ (2018) We will see that the many-body effects
may drastically modify the band orbital order and consequently affect the CTI
phase.

This paper is organized as follows: After a reminder of the single particle
properties and an outline of the method adopted to solve the many-body
Hamiltonian, the results are shown for both the 3D crystal and the surface-
terminated one, illustrating the effects of e-e correlations in terms of
orbital order in 3D and of metallic/non-metallic surface states.

_Single particle band structure_. As mentioned above, the model consists of a
tetragonal lattice with two atoms, A and B, per cell and two orbitals per
site. Intralayer first- and second- nearest neighbours hopping parameters
connect atoms of the same species with values
$t^{A}_{1}=-t^{B}_{1}=1,t^{A}_{2}=-t^{B}_{2}=0.5$; intralayer hopping
parameters connect sites A and B with values
$t_{1}^{\prime}=2.5,t^{\prime}_{2}=0.5$ , $t^{\prime}_{z}=2$ Fu (2011)(see
(Fig. 1 (a)). An orbital dependence is introduced in the intralayer hopping
parameters making them different from zero only when connecting identical
orbitals.

Figure 1:  (a) Tetragonal lattice containing two atomic species. Dashed lines
enclose the 4-site non-primitive unit cell that will be used as the cluster
for CPT implementation (see next section). (b)3D Brillouin Zone and high
symmetry points. (c)2D Brillouin Zone and high symmetry points.

The 3D crystal is insulating and this thanks to the hopping between sites A
and B: indeed, neglecting them, one would get the band structure reported in
panel (a) of Fig. 2 where bands localized on A sites intersect the
corresponding bands associated to B sites. The inclusion of interlayer hopping
opens the gap and gives rise to the peculiar orbital texture where states of
predominant A and B character interchange along the valence and conduction
band edges (panel (b) of Fig. 2). The site composition gives rise to a
particular mechanism of _band inversion_ essential for the occurrence of
topologically protected metallic surface states. Indeed band inversion is
ubiquitous in topological materials: it can be associated to orbital and/or
spin degrees of freedom Wojek  _et al._ (2014); Zhang  _et al._ (2013); Cao
_et al._ (2013); Bernevig  _et al._ (2006); Liu  _et al._ (2008); Zhang  _et
al._ (2013) or to atomic composition Zeljkovic  _et al._ (2014); Dziawa  _et
al._ (2012). The evolution of band inversion with e-e interaction will be the
main result of this paper.

Figure 2:  (a) Single-particle band structure calculated at k-points around
$A$ where the minimum energy separation between valence and conduction band
occurs. In blue (red) eigenvalues of predominant A- (B) character. Panel (a)
reports eigenvalues obtained with no intralayer hoppings
($t_{1}^{\prime}=t^{\prime}_{2}=t^{\prime}_{z}=0$); panel (b) eigenvalues
obtained with hopping parameters of ref. Fu, 2011.

In order to study the reduced dimensionality we have adopted a slab geometry
by stacking 20 A-B bilayers along the (001) axis; in this configuration we
have both A- and B-terminated surfaces and correspondingly surface states
localized either on A or B sites ( Fig. 3 (b) ). The same slab geometry can be
used as a non primitive unit cell to calculate bulk states, getting the so-
called Projected Bulk Band Structure (PBBS) (Fig. 3 (a) ). As currently done
in surface physics Manghi  _et al._ (1987) PBBS allows to identify
straightaway the energy regions that, prohibited in the bulk, can host
localized states at the surface. The orbital texture of PBBS whereby the A and
B bands alternate across the gap edges is also shown.

Figure 3:  a) Single-particle Projected Bulk Band Structure calculated along
high-symmetry directions of the 2D Brilluoine Zone of Fig. 1 (c) around
$\overline{M}$ point . In blue (red) eigenvalues with predominant A- (B)
character. (b) Eigenvalues obtained for a slab of 20 A-B bilayers. In blue
(red) eigenstates localized at the lower (upper) surface layer containing A
(B) sites.

_Interacting Crystalline Topological Insulator_

The interacting Hamiltonian reads:

$\hat{H}=\sum_{\alpha\beta}\sum_{ijll^{\prime}}t_{il\alpha,jl\beta}\hat{c}_{il\alpha}^{\dagger}\hat{c}_{jl^{\prime}\beta}+U\sum_{il\alpha\beta}\hat{n}_{il\alpha\uparrow}\hat{n}_{il\beta\downarrow}$
(1)

Here $\alpha$,$\beta$ are orbital indices, $i$,$j$ run over the atomic
positions within the unit cell and $l$,$l^{\prime}$ refer to lattice vectors.
$U$, the strength of the on-site Hubbard interaction, is assumed to be site
and orbital independent.

We solve the many-body problem by Cluster Perturbation Theory (CPT). Sénéchal
_et al._ (2000) CPT has been successfully used to study Mott-Hubbard physics
both in model systems Sénéchal  _et al._ (2000); Potthoff  _et al._ (2003) and
in real materials, Manghi (2014); Eder (2008, 2015) and more recently to
address correlated topological phases of matter. Yu  _et al._ (2011); Grandi
_et al._ (2015); Gra ; Puviani and Manghi (2016)

CPT belongs to the class of Quantum Cluster theories which solve the problem
of many interacting electrons in an extended lattice by approaching first the
many body problem in a subsystem of finite size - a cluster- and then
embedding it within the infinite medium. Maier  _et al._ (2005) The lattice is
seen as a periodic repetition of identical clusters each of them containing
$M$ atoms characterized by a set of $n_{i}^{orb}$ orbitals (K=
$\sum_{i}^{M}n_{i}^{orb}$ is the total number of sites/orbitals per cluster).
The Hamiltonian is then partitioned in two terms, an intra-cluster and an
inter-cluster one. Since the e-e Coulomb interaction is on-site, the inter-
cluster Hamiltonian contains only single particle hopping terms and the many
body part is present in the intra-cluster Hamiltonian only. The Green’s
function for the extended lattice can then be obtained by solving the equation

$G_{ij\alpha\beta}(k,\omega)=G^{c}_{i\alpha
j\beta}(\omega)+\sum_{i^{\prime}\delta}B_{i\alpha
i^{\prime}\delta}(k,\omega)G_{i^{\prime}\delta j\beta}(k,\omega).$ (2)

Here the $K\times K$ matrix $B_{i\alpha i^{\prime}\delta}(k,\omega)$ is given
by

$\displaystyle B_{i\alpha i^{\prime}\delta}(k,\omega)=\sum_{l}^{L}e^{ik\cdot
R_{l}}\sum_{i^{\prime\prime}}^{M}\sum_{\gamma}G^{c}_{i\alpha
i^{\prime\prime}\gamma}(\omega)t_{i^{\prime\prime}\gamma 0,i^{\prime}\beta l}$

with $R_{l}$ the lattice vectors and $t_{i^{\prime\prime}0,i^{\prime}l}$ the
hopping between site $i^{\prime}$ and $i^{\prime\prime}$ belonging to
different clusters. $G^{c}_{i\alpha j\beta}$ is the Green’s function obtained
by exact diagonalization of the interacting Hamiltonian for the finite
cluster; we separately solve the problem for N, N-1 and N+1 electrons and
express the cluster Green’s function in the Lehmann representation at real
frequencies. 111More details on CPT implementation for multi-orbital systems
can be found in https://www.cond-mat.de/events/correl16/manuscripts/manghi.pdf

For the tetragonal lattice, the 4-site cluster shown in Fig. 1 (a) has been
adopted. This choice is suggested by the strength of hopping parameters: in
the present model the largest hopping parameters are those connecting A-B
sites. They are included in the exact diagonalization while the smaller ones
are used, in the spirit of CPT, _perturbatively_ in the periodization of the
cluster Green’s function.

Once the cluster Green’s function in the local basis $G^{c}_{i\alpha
i^{\prime}\beta}(\omega)$ has been calculated by exact diagonalization, eq. 2
is solved by matrix inversion at each $k$ and $\omega$. The quasi-particle
band structure is then obtained in terms of spectral function $A(k\omega)$

$A(k\omega)=\frac{1}{\pi}\sum_{n}ImG(kn\omega).$ (3)

where

$\displaystyle
G(kn\omega)=\frac{1}{K}\sum_{ii^{\prime}}e^{-ik\cdot(r_{i}-r_{i^{\prime}})}\mathcal{C}^{n}_{i\alpha}(k)G_{ii^{\prime}}(k,\omega)$

Here $n$ is the band index and $\mathcal{C}^{n}_{i\alpha}(\textbf{k})$ are the
eigenstate coefficients obtained by the single-particle band
calculation.Manghi (2014)

_Results_. We compare now the spectral function obtained for the bulk crystal
with increasing values of $U$. We see that the minimum energy separation
between hole and particle excitations - the quasi-particle gap $\Delta$ \-
diminishes with $U$ and goes to zero at a critical value $U_{c}=2.5$. After
that, $\Delta$ keeps increasing linearly as in a standard Mott-Hubbard regime.
This behaviour resembles quite closely what happens in the Kane-Mele-Hubbard
model which describes graphene as a Quantum Spin Hall topological insulator in
the presence of on-site e-e interaction Yu  _et al._ (2011); Grandi  _et al._
(2015); Gra ; Wu  _et al._ (2012).

Figure 4:  Evolution of the minimum energy separation between hole and
particle excitations as a function of the strength of the $U$-parameter

Fig 5 shows the quasiparticle band structure of bulk CTI for different values
of $U$ around the critical value $U_{c}$. In order to elucidate the orbital
composition of the quasi-particle states we plot the _local_ spectral function
$A(k\alpha\omega)=\frac{1}{\pi}\sum_{n}ImG(kn\omega)|\mathcal{C}^{n}_{i\alpha}(k)^{2}|$
with $\alpha$ running over the two orbitals of site $A$ (right panel) or $B$
(left panel). We see that for $U<U_{c}$ states close to the Fermi level have a
predominat $A$-site or $B$-site character.

For $U\geq U_{c}$ this is no more the case and the states that are close to
the Fermi level exhibit a mixed $A$-$B$ composition. The orbital texture in
the bulk evolves then with $U$: below $U_{c}$ valence and conduction band
edges exhibit the same band inversion that has been identified in the non-
interacting system with the predominant contribution from $A$ and $B$ sites
alternating around the $A$ point (or the $\overline{M}$ point in the PBBS).
This band inversion fades away for $U\geq U_{c}$ where states close to the
Fermi level have a mixed $A$-$B$ character.

Figure 5:  Site selected bulk quasi-particle states at k-points where the
minimum energy separation between hole and particle states occurs. Panels (a)
,(b), (c) correspond to $U=1,2.5,3.5$ respectively. The left (right) panels in
blu (red) show the $A$-site ($B$-site) contribution.

Simultaneously, as band inversion disappears, surface states loose their
metallic character and a gap appears in their k-dispersion. This is shown in
Fig. 6 where the local spectral functions obtained for the 20-layer slab are
shown for the same $U$ values. In the non interacting CTI the existence of
gapless surface states has been associated to a topological phase; we see that
this phase persists only up to $U<U_{c}$.

Figure 6:  Surface quasi-particle states for increasing $U$ across $U_{c}$ .
Panels (a) ,(b), (c) correspond to $U=1,2.5,3.5$ respectively. The left
(right) panels in blu (red) show states localized at the $A$-site ($B$-site)
terminated surface.

In conclusion, we have demontrated that the Hubbard interaction modifies the
orbital texture of the quasi-particle band edges around the Fermi level; this
effect is remarkable and can be relevant also in real materials exhibiting CTI
behaviour. Above the critical value $U_{c}$ band inversion is removed and
surface states are gapped. This suggests the identification of this critical
value as a transition point from a topological to a trivial insulating phase.
An explicit confirmation of topological character and of its evolution with
$U$ would require the generalization to the interacting case of the new
$Z_{2}$ topological invariant that has been introduced to classify the
topological crystalline insulator phase in the non interacting system. Fu
(2011) The so-called _topological Hamiltonian_ Wang  _et al._ (2012); Wang and
Zhang (2012) \- a fictitious non-interacting Hamiltonian associated to the
inverse of the dressed Green’s function at zero frequency - would be the most
promising approach. It has been applied to identify the topological character
of heavy fermion mixed valence compounds Werner and Assaad (2013); Deng  _et
al._ (2013); Lu  _et al._ (2013); Yoshida  _et al._ (2013) and of the half-
filled honeycomb lattice Grandi  _et al._ (2015); Gra  in the presence of on-
site Hubbard interaction. The application of this method to CTI’s will be the
subject of a forthcoming paper.

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