††thanks: Corresponding author<EMAIL_ADDRESS>
# Electron correlation and spin-orbit coupling effects in US3 and USe3

Yu Yang  LCP, Institute of Applied Physics and Computational Mathematics, P.O.
Box 8009, Beijing 100088, People’s Republic of China  Ping Zhang  LCP,
Institute of Applied Physics and Computational Mathematics, P.O. Box 8009,
Beijing 100088, People’s Republic of China  Center for Applied Physics and
Technology, Peking University, Beijing 100871, People’s Republic of China

###### Abstract

A systematic density functional theory (DFT) +$U$ study is conducted to
investigate the electron correlation and spin-orbit coupling (SOC) effects in
US3 and USe3. Our calculations reveal that inclusion of the $U$ term is
essential to get energy band gaps for them, indicating the strong correlation
effects for uranium 5$f$ electrons. Taking consideration of the SOC effect
results in small reduction on the electronic band gaps of US3 and USe3, but
largely changes the energy band shapes around the Fermi energy. As a result,
US3 has a direct band gap while USe3 has an indirect one. Our calculations
predict that both US3 and USe3 are antiferromagnetic insulators, in agreement
with corresponding experimental results. Based on our DFT+$U$ calculations, we
systematically present the ground-state electronic, mechanical and Raman
properties for US3 and USe3.

##  Introduction

Actinide based materials possess interesting physical behaviors due to the
existence of strongly-correlated 5$f$ electrons and have attracted extensive
attentions Savrasov01 ; Albers01 ; Hecker04 ; Moore06 ; Prodan06 ; Prodan07 .
Different from actinide oxides which have been widely studied to reveal the
detailed configurations and correlation effects of the 5$f$ electrons Prodan06
; Prodan07 ; Dudarev97 ; Dudarev98 ; Dudarev00 ; Sun08JCP ; Sun08CPB ; Shi10 ;
WangBT10 ; Zhang10 ; Kern99 ; Petit03 ; Petit10 ; Wilkins06 ; Yin08 ; Jomard08
; Andersson09 ; Santini09 ; Sanati11 ; Dorado09 ; Dorado10 ; Nakamura10 ;
Meredig10 ; Geng10 ; Yasuoka12 ; Manley12 , none of the actinide chalcogenides
has ever received comparable concerns. Here we take US3 and USe3 as two
representatives to study the electron correlation and spin-orbit coupling
(SOC) effects of 5$f$ electrons in actinide chalcogenides. The other reason
for us to investigate their electronic structures is that they employ the
layered MX3 structure (with M to be a metal element, and X to be S, Se, or
Te), which can be used as models for studying electronic behaviors in
1-dimensional (1D) systems. The MX3 structure belongs to the space group of
P21/$m$, with its atomic organizations depicted in Fig. 1(a). In the
monoclinic lattice, the top two chalcogen (S , Se or Te) atoms form a tightly
bound pair suggesting that the MX3 compounds may be regarded as MX(X2)
Nouvel87 . The top X-X pair together with an underneath M and a more
underneath X atom form a triangular unit, which repeats along the $\vec{b}$
lattice direction forming a 1D chain, and two such triangle units form a unit
cell of the MX3 compounds in the ($\vec{a}$,$\vec{c}$) plane. In this way, the
MX3 compounds are always considered as 1D materials. For example, the metallic
ZrTe3 and NbSe3 exhibiting charge-density-wave transitions have been
thoroughly investigated in angle-resolved photoelectron spectroscopy (ARPES)
and optical experiments Schafer03 ; Perucchi041 ; Perucchi042 ; Yokoya05 ;
Pacile07 . In the present paper, by contrast, we address the electronic
structures of US3 and USe3, two semiconducting members of the MX3 family.

In studies of actinide based materials, one has to be careful for the electron
correlation and SOC effects of the actinide 5$f$ electrons. As an example, for
actinide dioxides conventional density functional theory (DFT) schemes that
apply the local density approximation (LDA) or the generalized gradient
approximation (GGA) underestimate the strong on-site Coulomb repulsion of the
5$f$ electrons and consequently fail to capture the insulating properties
Dudarev97 ; Zhang10 . Several approaches, the LDA/GGA+$U$, the hybrid density
functional of (Heyd, Scuseria, and Enzerhof) HSE, the self-interaction
corrected local spin-density (SIC-LSD), and the Dynamical Mean-Field Theory
(DMFT), have been developed to correct the pure LDA/GGA failures in
calculations of actinide materials. Among them the effective modification of
pure DFT by LDA/GGA+$U$ formalisms has been widely used in theoretical studies
of UO2 Dudarev97 ; Dudarev00 ; Sanati11  and PuO2 Sun08JCP ; Sun08CPB ;
Jomard08 ; Shi10 . The obtained structural parameters as well as the
electronic structure and phonon dispersion curves Zhang10 ; Manley12  accord
well with experiments. In our present work, the GGA+$U$ schemes due to Dudarev
et al. Dudarev97 ; Dudarev98 ; Dudarev00  are employed to study the electron
correlation and spin-orbit coupling effects in US3 and USe3, as well as the
two materials’ electronic, mechanical, and Raman properties.

The rest of the paper is organized as follows. In Sec. II the computational
method is briefly described. In Sec. III we present the results of the
physical properties of US3 and USe3, and discuss the electron correlation and
SOC effects of the uranium 5$f$ electrons. Finally in Sec. IV, we close our
paper with a summary of our main results.

##  Calculation method

Our total-energy calculations are carried out by employing the plane-wave
basis pseudopotential method as implemented in Vienna ab initio simulation
package (VASP) VASP . The exchange and correlation effects are described with
the GGA approximation in the Perdew-Burke-Ernzerhof (PBE) form PBE1 ; PBE2 .
The projected augmented wave (PAW) method of Blöchl PAW  is employed with the
frozen-core approximation. Electron wave function is expanded in plane waves
up to a cutoff energy of 400 eV and all atoms are fully relaxed until the
Hellmann-Feynman forces on them are less than 0.01 eV/Å. A 7$\times$9$\times$1
Monkhorst-Pack MPkpoints  k-point mesh is employed for integration over the
Brillouin zone of US3 and USe3. The uranium
6$s^{2}$6$p^{6}$5$f^{3}$6$d^{1}$7$s^{2}$, sulphur 3$s^{2}$3$p^{6}$, and
selenium 4$s^{2}$4$p^{6}$ electrons are treated as valence electrons.
Noncollinear calculations are used when considering the spin-orbital coupling
effects. The strong on-site Coulomb repulsions among the localized uranium
5$f$ electrons are described by using the formalism formulated by Dudarev et
al. Dudarev97 ; Dudarev98 ; Dudarev00 . In this scheme, the total GGA energy
functional is of the form

$E_{{\rm GGA}+U}=E_{\rm GGA}+\frac{U-J}{2}\sum_{\sigma}[{\rm
Tr}\rho^{\sigma}-{\rm Tr}(\rho^{\sigma}\rho^{\sigma})],$ (1)

where $\rho^{\sigma}$ is the density matrix of $f$ states with spin $\sigma$,
while $U$ and $J$ are the spherically averaged screened Coulomb energy and the
exchange energy, respectively.

In this paper, the Coulomb $U$ is treated as a variable, while the exchange
energy is set to be a constant $J$=0.51 eV. This value of $J$ is in the ball
park of the commonly accepted one for uranium compounds Sanati11  and close to
the theoretically predicted value of 0.54 eV in UO2 Kotani92 . Since only the
difference between $U$ and $J$ is significant, we will henceforth label them
as one single parameter $U_{\rm eff}$=$U$-$J$, while keeping in mind that the
nonzero $J$ has been used during calculations.

The phonon frequencies at the Gamma point for US3 and USe3 are calculated by
using the density functional perturbation theory (DFPT). And their Raman-
active frequencies are obtained through symmetry analysis on the corresponding
vibration modes. Before DFPT calculations, the lattice constants and atomic
positions of US3 and USe3 are further optimized using a denser $k$-point mesh
of 9$\times$13$\times$3, and a finer force convergence criteria of 0.001 eV/Å.
A 2$\times$2$\times$1 supercell is subsequently used for DFPT calculations.
Since both US3 and USe3 are stacked along the $\vec{c}$ direction through van
der Waals interactions, and with relatively large lattice constants, we do not
extend the supercell along the $\vec{c}$ direction. During DFPT calculations
on the 2$\times$2$\times$1 supercell, the $k$-point mesh is set to be
5$\times$7$\times$5, and the energy convergence criteria is set to be 1e-6 eV.

##  Results and discussion

Experimentally, the magnetic orderings of uranium trichalcogenides were
studied ever since 1961 Nouvel87 ; Trzebiatowski61 ; Suski76 ; Baran83 ;
Noel86 . The relatively large atomic distances between neighboring U atoms in
US3 and USe3 suggest that magnetic orderings may occur in these compounds
through super-exchange interactions of uranium 5$f$ electrons via the S or Se
ions Nouvel87 . And different measurements indicate that both US3 and USe3
crystals undergo AFM transitions at very low temperatures, with the magnetic
moments of the two uranium atoms within each unit cell opposite to each other
Nouvel87 ; Baran83 ; Noel86 . In the present GGA+$U$ study, we consider the
nonmagnetic (NM), ferromagnetic (FM), and antiferromagnetic (AFM) phases for
each choice of the $U_{\rm eff}$ value and then determine the lowest-energy
state by a subsequent total-energy comparison of these three phases. Our
calculated electronic energies for different magnetic phases of US3 and USe3
are all listed in Table I. One can see that within GGA formalism or the
GGA+$U$ formalism with a too small $U_{\rm eff}$ value, the FM state is more
stable for both US3 and USe3, in contradiction with experimental results.
Therefore, the electron correlation effect has to be accounted for to
correctly describe ground-state US3 and USe3. In the discussions that follow,
we will confine our reports to the AFM solutions for US3 and USe3.

The experimentally determined lattice parameters of US3 and USe3 are ($a$=5.37
Å, $b$=3.96 Å, $c$=9.94 Å) and ($a$=5.65 Å, $b$=4.06 Å, $c$=10.47 Å)
respectively Nouvel87 ; Gronvold68 ; Salem84 . Our calculated lattice
constants with different $U_{\rm eff}$ values are summarized in Table II,
together with the experimental results. We can see that different from what we
found for actinide dioxides Zhang10 ; Wang10 , the lattice constants of
actinide trichalcogenides do not change monotonically with the value of the
$U_{\rm eff}$ parameter. Besides, the lattice constants obtained in GGA
calculations are obviously too small in comparison with corresponding
experimental results, for both US3 and USe3. For US3, the value of $U_{\rm
eff}$ has to be as large as 6 eV to get reasonable lattice constants compared
with experimental results. Differently, a $U_{\rm eff}$ value of 4 eV is
enough to get reasonable lattice constants for USe3, and changing $U_{\rm
eff}$ from 4 to 6 eV has negligible effects on the obtained lattice constants.

Besides of the prominent changes in the atomic structure parameters, the most
dramatic improvement brought by the GGA+$U$ formalism when compared to the GGA
results is in the description of electronic structure properties. For this, we
have investigated the band structures in AFM phases for US3 and USe3 aiming at
revealing the fundamental influences of considering the on-site Coulomb
interaction. Figures 2(a) and 2(b) show the obtained local density of states
(LDOS) for the S1, S3, and U atoms in US3, and Se1, Se3, and U atoms in USe3
respectively. Without accounting for the on-site Coulomb repulsion, one can
see that the GGA calculations predict an incorrect metallic ground state by
nonzero occupation of uranium electrons at the Fermi energy ($E_{\rm f}$).
When switching on the $U_{\rm eff}$ parameter, as shown both in Figs. 2(a) and
2(b), the uranium electronic states in US3 and USe3 begin to split at $E_{\rm
f}$ and tend to form two peaks with the gap of $\Delta E$. The amplitude of
this gap increases with increasing $U_{\rm eff}$. Previous electric
resistivity measurements pointed out that USe3 did not conduct at room
temperatures Shlyk95 , thus a band gap at $E_{\rm f}$ should be contained in
its electronic structure. We can see from Fig. 2(b) that an energy band gap
appears only when $U_{\rm eff}$ is larger than or equal to 4 eV for USe3. For
US3, the value of $U_{\rm eff}$ has to be enlarged to 6 eV to open the band
gap at the Fermi energy. These results also imply that the electron
correlation strength might be different in US3 and USe3. From the last figure
in Figs. 2(a) and 2(b), we can see that the sulphur (selenium) and uranium
electronic states overlap with each other and contribute equally to the
valence band maximum (VBM) of US3 (USe3), while the conduction band minimum
(CBM) is composed of uranium electronic states. The orbital mixing between
sulphur (selenium) and uranium electronic states below the Fermi energy
indicates that there are covalent interactions between the sulphur (selenium)
and uranium atoms in US3 (USe3).

To further analyze the orbital-resolved electronic structures, we calculate
the projected density of states (PDOS) for the X-$np$ ($n$=3 (X=S) or 4
(X=Se)), U-5$f$ and U-6$p$ electronic states in UX3. Figures 3(a)-3(d) show
the obtained PDOS for US3 in different calculations, while the corresponding
PDOS results for USe3 are shown in Figs. 3(e)-3(h). The value of $U_{\rm eff}$
is chosen to be 6 eV in all GGA+$U$ and GGA+$U$+SOC calculations. One can see
that without considering the electron correlation effect, US3 and USe3 are
both wrongly predicted as metallic materials, which contradicts with
experimental results. After adding the $U$ parameter to describe the strong
on-site energy of U-5$f$ electrons, the energy gaps are opened for US3 and
USe3. We can see from Figs. 3(d) and 3(h) that further considering SOC has
little influences on the electronic states around the Fermi energy. For both
US3 and USe3, the SOC effect lies in the deep energy level, causing an energy
splitting of the U-6$p$ states. From the PDOS results, we can also see clear
difference between the X1- and X3-$np$ (n=3 (X=S) or 4(X=Se)) electronic
states. The PDOS for the electronic states of the X2 atom is very similar to
that of the X1 atom and thus is not shown here. The different electronic state
distribution of the X1(X2) and X3 atoms proves the theory of considering UX3
as UX(X2), and indicates the strong covalent bondings between the X1 and X2
atoms.

To analyze more carefully the SOC effects in US3 and USe3, we recalculate the
electronic structures of them with higher resolutions, with the reduced
Gaussian smearing width of only 0.02 eV and a PDOS resolution of 0.01 eV.
Figures 4(a) and 4(b) show the obtained PDOS results for uranium 5$f$
electrons of US3 and USe3, by using the GGA method, while Figs. 5(a) and 5(b)
show the obtained energy band structures of US3 and USe3 by using the GGA+$U$
method, respectively. All the GGA+SOC and GGA+$U$+SOC calculations are
noncollinear, with both SOC and orbital polarizations considered. And the
results before and after considering the SOC effects are shown in solid and
dotted lines respectively. From the PDOS results obtained by using the GGA
method, we can see that pure inclusion of SOC to the GGA method does not lead
to energy band gaps for US3 and USe3. This discovery is different from the
situation of CoO, where SOC can solely open a band gap Norman90 .

From the band structures of US3 shown in Fig. 5(a), we can see that both the
lowest unoccupied conduction and highest occupied valence bands distribute
along the $\Gamma$-Z direction. For the lowest conduction band, the SOC effect
causes an energy downshift of 0.05 eV along the $\Gamma$-Z direction.
Contrarily for its highest valence band, the SOC effect causes an energy
upshift of 0.03 eV at the $\Gamma$ point and an upshift of 0.07 eV at the Z
point. As a result, the energy band gap of US3 changes from indirect
($\Gamma\rightarrow$Z) into direct type (Z$\rightarrow$Z) after considering
the SOC effect, and the gap value reduces from 1.59 to be 1.48 eV. Different
from US3, the lowest conduction and highest valence bands of USe3 distributes
along the K1-B and B-$\Gamma$-Y directions respectively. For the lowest
conduction band along the K1-B direction, the SOC effect causes an energy
downshift of 0.04 eV at the K1 point and a downshift of 0.02 eV at the B
point. Correspondingly the CBM of USe3 moves from B to the K1 point. However,
since the VBM of USe3 is at some point along the B-$\Gamma$-Y direction
instead of at any high-symmetry $k$ points, the SOC effect does not change the
indirect band gap type. Overall the band gap of USe3 changes from 1.30 to be
1.23 eV after considering the SOC effect. Our studies reveal that the energy
band gap of USe3 is smaller than that of US3.

After systematically presenting the electronic structure results for US3 and
USe3, we now turn to their mechanical properties. The elastic constants are
obtained by solving the eigenvalues of their Hessian matrix, which is
calculated based on the Hooke’s law and small position changes on independent
S, Se and U atoms. The obtained elastic constants of US3 and USe3 in different
calculations by using the GGA and GGA+$U$ methods are both listed in Table
III. We can see that the elastic constants calculated by using the GGA
formalism for US3 and USe3 do not satisfy the stability criteria Nye85 ; Wu07
for monoclinic structures that the $C_{\rm 11}$, $C_{\rm 22}$, $C_{\rm 33}$,
$C_{\rm 44}$, $C_{\rm 55}$, $C_{\rm 66}$ should be all positive. This result
further proves that conventional GGA formalism fails for describing the
uranium trichalcogenides. Contrarily, the calculated elastic constants by
using the GGA+$U$ method for both US3 and USe3 satisfy the above stability
criteria, as well as the other six criteria for monoclinic structures Nye85 ;
Wu07 , proving the stable existence of US3 and USe3, and their strong electron
correlation effects.

Based on the elastic constants of US3 and USe3, we further calculate the Voigt
and Reuss bounds on their bulk ($B_{\rm V}$, $B_{\rm R}$) and shear moduli
($G_{\rm V}$, $G_{\rm R}$) Supple . And the bulk and shear moduli of US3 and
USe3 can be estimated by $B$=($B_{\rm V}$+$B_{\rm R}$)/2, and $G$=($G_{\rm
V}$+$G_{\rm R}$)/2. And the Poisson’s ratios can be calculated by
$\nu$=(3$B$-2$G$)/(6$B$+2$G$). The obtained mechanical properties for US3 and
USe3 are all listed in Table IV. We can see that the Voigt and Beuss bounds
obviously differ from each other for bulk and shear moduli. This result
reflects the structural anisotropy of US3 and USe3. Besides, the bulk and
shear moduli are both found to be larger for US3 than for USe3. The Poisson’s
ratio is calculated to be 0.19 and 0.22 for US3 and USe3 respectively.

Until now, there are no experimental reports on the mechanical properties of
US3 or USe3. Hence our theoretical values can be used as references for
further investigations or industrial applications of them. At another side,
the Raman properties of the MX3 group materials have been systematically
studied for a long time, and some of the characteristic peaks of US3 and USe3
were successfully observed in experiments Nouvel87 . Therefore, here we
further calculate the Raman properties of US3 and USe3. According to group
theory and the symmetry of them, both US3 and USe3 are predicted to have 12
Raman-active vibrational modes Nouvel87 ; Zwick79 : 8 Ag and 4 Bg modes
respectively. The symbols Ag and Bg represent for two different vibration
symmetries respectively: vibrations inside and out of the mirror plane.
Specifically, Ag corresponds to the vibrations where $dy$ is always 0 while Bg
corresponds to the vibrations where $dy$ is not 0.

Our Raman results of US3 and USe3 are obtained by symmetry analysis based on
the vibrational modes calculated within the GGA+$U$ method, where SOC effects
are considered. Figures 6(a) and 6(b) show our determined Raman frequencies
for US3 and USe3 respectively, together with the experimental results by G.
Nouvel et al. Nouvel87 . One can see that by using the GGA formalism for both
US3 and USe3, the Raman frequencies are totally confused compared with the
experimental results. At another side, the GGA+$U$ results on the Raman
frequencies are clearly reasonable in two obvious aspects: i) there exist a
highest single Raman frequency for both US3 and USe3, corresponding to the
molecular-like X2 mode vibrations; ii) the highest Raman frequency of US3 is
much larger than that of USe3, indicating that the S-S interaction is stronger
than Se-Se. When comparing all the Raman frequencies, there seems to be a
frequency shift of about 25$\sim$35 cm-1 between our GGA+$U$ and the
experimental results.

##  Conclusion

By using the GGA and GGA+$U$ methods, we have systematically studied the
electronic, mechanical, and Raman properties of US3 and USe3, aiming to reveal
the underlying electron correlation and SOC effects. By comparing the
calculated lattice constants of US3 and USe3 with corresponding experimental
results, and monitoring their electronic structures with different $U_{\rm
eff}$ parameters, we conclude that a $U_{\rm eff}$ value of 6 eV is needed to
get reasonable lattice constants, and the right insulating properties for both
US3 and USe3. After considering SOC effects, the deep U-6$p$ band will be
split into two separate energy bands in both US3 and USe3. The SOC effects
also changes the energy band shapes around the Fermi energies and lead to the
fact that US3 is a direct band gap while USe3 is an indirect band gap
insulator. Because of the SOC effects, the energy band gaps of US3 and USe3
reduce by 0.11 and 0.07 eV respectively. Based on the obtained
antiferromagnetic ground states of US3 and USe3, we then systematically give
out their electrical, mechanical, and Raman properties. The VBM and CBM of US3
(USe3) are found to be contributed by 3$p$-5$f$ (4$p$-5$f$) hybrid and 5$f$
(5$f$) electronic states respectively. The elastic constants of US3 and USe3
are found to satisfy the stability criteria, and their Poisson’s ratios are
calculated to be 0.19 and 0.22 respectively. For the Raman properties, we
theoretically repeat the two important experimental observations that the S-S
or Se-Se dimer vibrations have the largest frequencies in US3 and USe3, and
the frequency of the S-S dimer vibration is larger than that of the Se-Se
dimer vibration.

##  Acknowledgement

This work was supported by the National Natural Science Foundation of China
under Grants No. 10904004 and No. 90921003 and Foundations for Development of
Science and Technology of China Academy of Engineering Physics under Grants
No. 2011B0301060, No. 2011A0301016, and No. 2008A0301013.

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Table 1: Electronic energies per unit cell of US3 and USe3 in the nonmagnetic (NM), ferromagnetic (FM), and antioferromagnetic (AFM) states by using different calculation methods. All values are in units of eV. Methods | US3 | USe3
---|---|---
NM | FM | AFM | NM | FM | AFM
GGA | -55.33 | -55.55 | -55.47 | -50.48 | -50.87 | -50.79
GGA+$U$ ($U_{\rm eff}$=1 eV) | -53.30 | -53.82 | -53.78 | -48.46 | -49.30 | -49.31
GGA+$U$ ($U_{\rm eff}$=2 eV) | -51.34 | -52.50 | -52.24 | -46.57 | -48.14 | -48.17
GGA+$U$ ($U_{\rm eff}$=3 eV) | -49.68 | -51.53 | -51.53 | -45.07 | -47.72 | -47.72
GGA+$U$ ($U_{\rm eff}$=4 eV) | -48.42 | -50.85 | -50.85 | -44.50 | -47.26 | -47.35
GGA+$U$ ($U_{\rm eff}$=5 eV) | -46.55 | -50.74 | -50.74 | -43.95 | -46.79 | -47.26
GGA+$U$ ($U_{\rm eff}$=6 eV) | -48.39 | -51.68 | -51.69 | -43.48 | -46.52 | -46.85

Table 2: Lattice constants of US3 and USe3 obtained by using different
methods. All values are in units of Å.

Methods | US3 | USe3
---|---|---
$a$ | $b$ | $c$ | $a$ | $b$ | $c$
GGA | 5.11 | 3.83 | 9.25 | 5.50 | 3.99 | 10.22
GGA+$U$ ($U_{\rm eff}$=1 eV) | 5.12 | 3.82 | 9.26 | 5.52 | 3.95 | 10.30
GGA+$U$ ($U_{\rm eff}$=2 eV) | 5.13 | 3.82 | 9.28 | 5.52 | 3.94 | 10.42
GGA+$U$ ($U_{\rm eff}$=3 eV) | 5.09 | 3.74 | 9.61 | 5.50 | 3.87 | 10.98
GGA+$U$ ($U_{\rm eff}$=4 eV) | 5.09 | 3.75 | 9.65 | 5.71 | 4.02 | 10.90
GGA+$U$ ($U_{\rm eff}$=5 eV) | 5.06 | 3.82 | 9.61 | 5.73 | 4.04 | 10.52
GGA+$U$ ($U_{\rm eff}$=6 eV) | 5.44 | 3.92 | 9.76 | 5.73 | 4.04 | 10.60
Exp.a,b | 5.37 | 3.96 | 9.94 | 5.65 | 4.06 | 10.47

aReference Gronvold68,

bReference Salem84,

Table 3: Elastic constants of US3 and USe3 obtained in GGA and GGA+$U$+SOC calculations. The $U_{\rm eff}$ value is chosen to be 6 eV in both GGA+$U$+SOC calculations. All data are in units of GPa. | $C_{11}$ | $C_{22}$ | $C_{33}$ | $C_{44}$ | $C_{55}$ | $C_{66}$ | $C_{12}$ | $C_{13}$ | $C_{23}$ | $C_{45}$ | $C_{16}$ | $C_{26}$ | $C_{36}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
US3 (GGA) | -654 | -80 | -4132 | 452 | -332 | -273 | -923 | -2953 | -1766 | 32 | -511 | -640 | -522
US3 (GGA+$U$+SOC) | 963 | 1055 | 437 | 313 | 299 | 169 | 196 | 132 | 181 | -6 | -102 | -55 | -74
USe3 (GGA) | 946 | 583 | -107 | 363 | 309 | 7 | -37 | -349 | -65 | 3 | 171 | 24 | 198
USe3 (GGA+$U$+SOC) | 951 | 938 | 496 | 257 | 244 | 267 | 195 | 194 | 216 | -13 | -19 | -20 | -33

Table 4: Mechanical properties of US3 and USe3 including the bulk and shear moduli, their Voigt and Reuss bounds, and the Poisson’s ratio calculated by using the GGA+$U$ method. The $U_{\rm eff}$ value is chosen to be 6 eV. All moduli data are in units of GPa. | $B_{\rm V}$ | $B_{\rm R}$ | $G_{\rm V}$ | $G_{\rm R}$ | $B$ | $G$ | $\nu$
---|---|---|---|---|---|---|---
US3 | 386 | 299 | 286 | 245 | 342 | 266 | 0.19
USe3 | 400 | 365 | 272 | 257 | 382 | 265 | 0.22

List of captions

Fig.1 (Color online). (a) The atomic structures of MX3, where blue and dark
green balls representing for metal (M) and chalcogen (X) atoms respectively.
The monoclinic lattices are depicted by the dashed lines. (b) Depiction of the
Brillouin Zone for the monoclinic MX3 lattice.

Fig.2 (Color online). The local density of states (LDOS) for the S1, S3, and U
atoms in US3 (a) and Se1, Se2, and U atoms in USe3 (b) by using the GGA and
GGA+$U$ methods with $U_{\rm eff}$ ranging from 1 to 6 eV. The S1 (Se1) and S3
(Se3) atoms correspond to X1 and X3 atoms depicted in Fig. 1(a). The Fermi
energies are denoted by dashed lines.

Fig.3 (Color online). (a)-(d) The projected density of states (PDOS) for the
3$p$ electronic states of S1 and S3 atoms, and 5$f$, 6$p$ electronic states of
the U atom in US3. (e)-(h) The PDOS for the 4$p$ electronic states of Se1 and
Se3 atoms, and 5$f$, 6$p$ electronic states of the U atom in USe3. The S1
(Se1) and S3 (Se3) atoms correspond to X1 and X3 atoms depicted in Fig. 1(a).
The Fermi energies are denoted by dashed lines. The values of $U_{\rm eff}$
are chosen to be 6 eV in all GGA+$U$ and GGA+$U$+SOC calculations.

Fig.4 (Color online). The projected density of states for the uranium 5$f$
electronic states in US3 (a) and USe3 (b), by using the GGA method. The Fermi
energies are set to be zero and denoted by dashed lines. The results before
and after considering the spin-orbit coupling effects are shown in red solid
and blue dotted lines respectively.

Fig.5 (Color online). The electronic energy band structures for US3 (a) and
USe3 (b) by using the GGA+$U$ method. The values of $U_{\rm eff}$ are chosen
to be 6 eV. The Fermi energy and different $k$-points are denoted by the
dashed lines. The results before and after considering the spin-orbit coupling
effects are shown in solid and dotted lines respectively.

Fig.6 (Color online). The calculated frequencies at the $\Gamma$ point for
Raman-active vibrational modes of US3 (a) and USe3 (b) by using the GGA and
GGA+$U$ methods, together with the experimental results listed in Ref.
Nouvel87, . The values of $U_{\rm eff}$ are chosen to be 6 eV. The symmetry
for each vibration mode are also presented.

Figure 1:

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Figure 6:


Constraining the generalized uncertainty principle with the gravitational wave
event GW150914

Zhong-Wen Feng 1 ***E-mail<EMAIL_ADDRESS>Shu-Zheng Yang 2 †††E-mail:
<EMAIL_ADDRESS>Hui-Ling Li 1,3 ‡‡‡E-mail<EMAIL_ADDRESS>Xiao-Tao Zu 1
§§§E-mail<EMAIL_ADDRESS>
1\. School of Physical Electronics, University of Electronic Science and
Technology of China, Chengdu, 610054, China
2\. Department of Astronomy, China West Normal University, Nanchong, 637009,
China
3\. College of Physics Science and Technology, Shenyang Normal University,
Shenyang, 110034, China

Abstract: In this letter, we show that the dimensionless parameter in the
generalized uncertainty principle (GUP) can be constrained by the
gravitational wave event GW150914, which was discovered by the LIGO Scientific
and Virgo Collaborations. Firstly, according to the Heisenberg uncertainty
principle (HUP) and the data of gravitational wave event GW150914, we derive
the standard energy-momentum dispersion relation and calculate the difference
between the propagation speed of gravitons and the speed of light, i.e.,
$\Delta\upsilon$. Next, using two proposals regarding the GUP, we also
generalize our study to the quantum gravity case and obtain the modified speed
of gravitons. Finally, based on the modified speed of gravitons and
$\Delta\upsilon$, the improved upper bounds on the GUP parameters are
obtained. The results show that the upper limits of the GUP parameters
$\beta_{0}$ and $\alpha_{0}$ are $2.3\times 10^{60}$ and $1.8\times 10^{20}$.

##  1 Introduction

Several versions of quantum gravity models predict the existence of a
fundamental scale of length, which can be identified with the Planck scale.
This view is also supported by Gedanken experiments [1]. In this scenario, the
Heisenberg uncertainty principle (HUP) can be changed into the so-called
generalized uncertainty principle (GUP), which has various implications for a
wide range of physical systems [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. For example, the effects of the GUP
on the evolution of black holes were calculated in [5, 6, 7, 8, 9, 10, 11, 12,
13, 14]. The GUP-corrected quantum Hall effect was investigated in [15]. The
impact of the GUP on neutrino oscillations was studied in [16]. In [17, 18],
using the GUP, the authors discussed the thermodynamics of the Friedmann-
Robertson-Walker universe and the inflation preheating in cosmology. The
effect of the GUP has also been used to calculate the entropic force [19, 20].
According to the GUP, the critical temperature and Helmholtz free energy of
Bose-Einstein condensation in the relativistic ideal Bose gas were computed in
[21]. In [22, 23], Faizal and Majumder incorporated the GUP into Lifshitz
field theories and showed that the breaking of supersymmetry by a non-
anticommutative deformation can be used to generate the GUP. Moreover, it is
exciting that the effects of the GUP can be probed via quantum optics [24].
Based on this previous research, it is clear that two types of GUP have been
studied widely. The version of the proposed by Kempf, Mangano, and Mann (GUP
I) is

$\Delta x\Delta p\geq\frac{\hbar}{2}\left[{1+\beta\left({\Delta
p}\right)^{2}}\right],$ (1)

where $\Delta x$ and $\Delta p$ are the uncertainties for position and
momentum, respectively.
$\beta={{\beta_{0}\ell_{p}^{2}}\mathord{\left/{\vphantom{{\beta_{0}\ell_{p}^{2}}{\hbar^{2}}}}\right.\kern-1.2pt}{\hbar^{2}}}={{\beta_{0}}\mathord{\left/{\vphantom{{\beta_{0}}{M_{p}^{2}c^{2}}}}\right.\kern-1.2pt}{M_{p}^{2}c^{2}}}$,
$\beta_{0}$ is a positive dimensionless parameter, which is called the GUP
parameter, $\ell_{p}$ represents the Planck length, $M_{p}$ is the Planck
mass, and the Planck energy is $M_{p}c^{2}=1.2\times 10^{28}{\rm{eV}}$. Eq.
(1) implies a nonzero minimal uncertainty $\Delta
x_{\min}\approx\ell_{p}\sqrt{\beta_{0}}$. It should be noted that the
derivation of Eq. (1) relies on the modified fundamental commutation relation
$\left[{x_{i},p_{j}}\right]=i\hbar\delta_{ij}\left[{1+\beta p^{2}}\right]$
with the position operator $x_{i}$ and the momentum operator $p_{i}$ [25]. The
other version proposed in [26, 27] admits a minimal length and a maximal
momentum (GUP II)

$\displaystyle\Delta x\Delta p$ $\displaystyle\geq$
$\displaystyle\frac{\hbar}{2}\left[{1-2\alpha\left\langle
p\right\rangle+4\alpha^{2}\left\langle{p^{2}}\right\rangle}\right]$ (2)
$\displaystyle\geq$
$\displaystyle\frac{\hbar}{2}\left[{1+\left({\frac{\alpha}{{\sqrt{\left\langle{p^{2}}\right\rangle}}}+4\alpha^{2}}\right)\Delta
p^{2}}\right.+\left.{4\alpha^{2}\left\langle
p\right\rangle^{2}+4\alpha^{2}\left\langle
p\right\rangle^{2}-2\alpha\sqrt{\left\langle{p^{2}}\right\rangle}}\right],$

where
$\alpha={{\alpha_{0}}\mathord{\left/{\vphantom{{\alpha_{0}}{M_{p}c}}}\right.\kern-1.2pt}{M_{p}c}}={{\alpha_{0}\ell_{p}}\mathord{\left/{\vphantom{{\alpha_{0}l_{p}}\hbar}}\right.\kern-1.2pt}\hbar}$,
and $\alpha_{0}$ is the GUP parameter. Inequality (2) is equivalent to the
modified fundamental commutation relation
$[x_{i},p_{i}]=i\hbar[\delta_{ij}-\alpha(p\delta_{ij}+{{p_{i}p_{j}}\mathord{\left/{\vphantom{{p_{i}p_{j}}p}}\right.\kern-1.2pt}p})+\alpha^{2}(p^{2}\delta_{ij}+3p_{i}p_{j})]$.
Moreover, Eq. (1) implies a minimal length $\Delta
x_{\min}\approx\alpha_{0}\ell_{p}$ and a maximum momentum $\Delta
p_{\max}\approx{{\ell_{p}}\mathord{\left/{\vphantom{{\ell_{p}}{\alpha_{0}}}}\right.\kern-1.2pt}{\alpha_{0}}}$.

Theoretically, the GUP parameters $\beta_{0}$ and $\alpha_{0}$ are always
assumed to be of the order of unity, in which case corrections are negligible
unless the lengths or energies approach the Planck length $\ell_{p}$ or the
Planck energy $E_{p}$. However, if the assumptions regarding the GUP
parameters are not made a priori, the GUP parameters should be constrained by
previous experiments [28, 29, 30, 31, 32, 33, 34, 35, 37]. Recently, it was
argued that the modified dispersion relations motivated by quantum gravity can
affect the propagation of the observed gravitational wave signal [42].
Therefore, one can use the data of gravitational waves event GW150914 that
reported by the Laser Interferometer Gravitational-Wave Observatory (LIGO)
Scientific and Virgo Collaborations to constrain the possibility of Lorentz
violation during gravitational waves propagation. In [38], the authors showed
that the upper bound on the difference between the speed of light and
gravitational waves is $\left|{\Delta\upsilon}\right|\leq 10^{-17}$. According
to the gravitational wave event GW150914, Arzano and Calcagni showed that the
upper bound on the characteristic quantum-gravity mass scale is $M>4\times
10^{4}{\rm{eV}}$, which is much weaker than that coming from photon
propagation from gamma-ray bursts. They also found that the a phenomenological
dispersion relation $\omega^{2}=k^{2}\left({1+{{\alpha
k^{n}}\mathord{\left/{\vphantom{{\alpha
k^{n}}{M^{n}}}}\right.\kern-1.2pt}{M^{n}}}}\right)$ is compatible with
observations and it has a phenomenologically viable mass of $M>10{\rm{TeV}}$
only in the quite restrictive range of $0<n<0.68$ [39]. In order to explain
why the speed of gravitational waves is smaller than that of light, Gwak, Kim,
and Lee studied the speed of gravitons in the gravitational wave event
GW150914 by using gravity’s rainbow [40]. They noted that the upper bound of
the rainbow parameter $\eta$ is smaller than $4.6\times 10^{59}$ at a
frequency of $250{\rm{Hz}}$, which indicates that the effect of gravity’s
rainbow is very small in the gravitational waves event GW150914.

On the other hand, it is well known that the GUP is also motivated by the
quantum gravity. Thus, inspired by previous studies, it is beneficial to
investigate the possibility of using the gravitational wave event GW150914 to
constrain the GUP parameters $\beta_{0}$ and $\alpha_{0}$. In this study, we
first calculate the difference between the speed of gravitons and the speed of
light via the HUP, i.e., $\Delta\upsilon$. Next, according to Eq. (1) and Eq.
(2), we obtain the corresponding modified dispersion relations and the GUP-
corrected speed of gravitons. Finally, we find the improved upper bounds on
the GUP parameters by comparing $\Delta\upsilon$ with the GUP corrected speed
of gravitons.

The remainder of this letter is organized as follows. In Section 2, we derive
the standard energy-momentum dispersion relation that corresponding to the
HUP. Subsequently, following the definition of the group speed of the wave
front, the difference between the speed of gravitons and the speed of light is
obtained. In Section 3 and Section 4, we generalize our study to the quantum
gravity case and compute the effects of GUP on the speed of gravitons.
According to the calculated effect of the GUP on the speed of gravitons
measurements, we obtain the improved upper bounds on the GUP parameters
$\beta_{0}$ and $\alpha_{0}$. Finally, in Section 5 results are briefly
discussed.

##  2 The speed of gravitons in the Gravitational waves event GW150914

First, we consider the conventional Heisenberg uncertainty principle $\Delta
x\Delta p\geq{\hbar\mathord{\left/{\vphantom{\hbar 2}}\right.\kern-1.2pt}2}$.
This inequality is equivalent to the Heisenberg algebra
$\left[{x_{i},p_{j}}\right]=i\hbar\delta_{ij}$. The the position and momentum
operators in the Heisenberg algebra can be defined as follows

$\begin{array}[]{*{20}c}{x_{i}=x_{0i};\quad p_{i}=p_{0i}},\\\ \end{array}$ (3)

where $x_{0i}$ and $p_{0i}$ satisfying the canonical commutation relations
$\left[{x_{0i},p_{0j}}\right]=i\hbar\delta_{ij}$. In gravitational spacetime,
the background metric ansatz that we study is

$ds^{2}=g_{ab}dx^{a}dx^{b}=g_{00}c^{2}dt^{2}+g_{ij}dx^{i}dx^{j}.$ (4)

Eq. (4) leads to the square of the four-momentum
$p_{a}p^{a}=g_{ab}p^{a}p^{b}=g_{00}\left({p^{0}}\right)^{2}+g_{ij}p^{0i}p^{0j}$.
Considering that $p_{i}$ in this background satisfies the relation
$p^{2}=p_{i}p^{i}=g_{ij}p^{0i}p^{0j}$, then the square of the four-momentum
can be rewritten as

$p_{a}p^{a}=g_{00}\left({p^{0}}\right)^{2}+p^{2}.$ (5)

It should be mentioned that the right-hand side in RHS of Eq. (5) forms the
original dispersion relation
$g_{00}\left({p^{0}}\right)^{2}+p^{2}=-m^{2}c^{2}$. Therefore, the equation
above takes the form

$p_{a}p^{a}=-m^{2}c^{2},$ (6)

and the time component of the momentum is given by

$\left({p^{0}}\right)^{2}={{\left({-p^{2}-m^{2}c^{2}}\right)}\mathord{\left/{\vphantom{{\left({-p^{2}-m^{2}c^{2}}\right)}{g_{00}}}}\right.\kern-1.2pt}{g_{00}}}.$
(7)

It is well known that the energy of a particle can be defined in the following
form

${\omega\mathord{\left/{\vphantom{\omega
c}}\right.\kern-1.2pt}c}=-\xi_{a}p^{a}=-g_{ab}\xi^{a}p^{b},$ (8)

where $\xi^{a}=\left({1,0,0,\cdots}\right)$ is the Killing vector. Using Eq.
(8), the energy in the metric (4) can be defined as [41]

$\omega=-g_{00}cp^{0}.$ (9)

Now, by substituting Eq. (9) into Eq. (7), one can express the energy of a
particle in terms of the three spatial momentum momentum and the mass

$\omega^{2}=\left({-g_{00}cp^{0}}\right)^{2}=-g_{00}\left({p^{2}c^{2}+m^{2}c^{4}}\right).$
(10)

It is well known that the LIGO is in a weak gravitational spacetime, here we
only work in the Minkowski spacetime, i.e., $g_{00}=-1$. Thus, Eq. (10)
reduces to the standard energy-momentum dispersion relation
$\omega^{2}=p^{2}c^{2}+m^{2}c^{4}$. In addition, by assuming that the
gravitational waves propagate as free waves, the speed of gravitons can be
calculated by using the group speed of the wave front, i.e.,
$\upsilon:={{\partial\omega}\mathord{\left/{\vphantom{{\partial\omega}{\partial
p}}}\right.\kern-1.2pt}{\partial p}}$, where $\omega$ and $p$ represent the
energy and momentum, respectively [42, 43]. For the standard energy-momentum
dispersion relation, the speed of gravitons is given by

$\displaystyle\upsilon_{g}=\frac{{\partial\omega}}{{\partial
p}}=\frac{{c^{2}p}}{{\sqrt{c^{4}m^{2}+c^{2}p^{2}}}}=c\sqrt{1-\frac{{m^{2}c^{4}}}{{\omega_{g}^{2}}}}\approx
c\left({1-\frac{{m_{g}^{2}c^{4}}}{{2\omega_{g}^{2}}}}\right),$ (11)

where $\omega_{g}$ and $m_{g}$ are the energy and rest mass of gravitons,
respectively. Considering that $h=4.136\times 10^{-15}{\rm{eV}}\cdot{\rm{s}}$
and $c=3\times
10^{8}{{\rm{m}}\mathord{\left/{\vphantom{{\rm{m}}{\rm{s}}}}\right.\kern-1.2pt}{\rm{s}}}$,
then the difference between the speed of gravitons $\upsilon_{g}$ and the
speed of light $c$ can be expressed as

${\Delta\upsilon}=c-\upsilon_{g}={{m_{g}^{2}c^{5}}\mathord{\left/{\vphantom{{m_{g}^{2}c^{5}}{2\omega_{g}^{2}}}}\right.\kern-1.2pt}{2\omega_{g}^{2}}}.$
(12)

In [44, 45], the LIGO Scientific and Virgo Collaborations pointed out that the
signal of gravitational wave event GW150914 is peaked at $\nu=150{\rm{Hz}}$.
Thus, the maximum energy of gravitons is $\omega_{g}=h\nu\approx 6.024\times
10^{-13}{\rm{eV}}$. Moreover, they also found an upper bound for the mass of
gravitons is $m_{g}\leq 1.2\times
10^{-22}{{{\rm{eV}}}\mathord{\left/{\vphantom{{{\rm{eV}}}{c^{2}}}}\right.\kern-1.2pt}{c^{2}}}$.
Therefore, Eq. (12) turns out to be

${\Delta\upsilon}<5.6\times 10^{-12}{\rm{m/s}}.$ (13)

It is clear that the difference between the speed of gravitons and the speed
of light is very small. However, by using Eq. (13), one can investigate the
appropriate range of the parameter in GUP motivated by quantum gravity. In
Section 3 and Section 4, by considering the GUP corrected speed of gravitons
that obtained from the gravitational wave event GW150914 and Eq. (13), the
upper limit of GUP parameters will be calculated.

##  3 Bounds on the GUP parameters $\beta_{0}$

In this section, we set the upper bound of the GUP parameter $\beta_{0}$ by
considering the speed obtained from the gravitational wave event GW150914 and
Eq. (13). In Eq. (1), the operators for the position $x_{i}$ and momentum
$p_{i}$ can be defined as

$\displaystyle x_{i}=x_{0i};\quad p_{i}=p_{0i}\left({1+\beta p^{2}}\right),$
(14)

where $x_{0i}$, $p_{0i}$ satisfying the canonical commutation relations
$\left[{x_{0i},p_{0j}}\right]=i\hbar\delta_{ij}$, and
$p^{2}=g_{ij}p^{0i}p^{0j}$, leading to $p=\sqrt{g_{ij}p^{0i}p^{0j}}$.
Moreover, the $p_{0i}$ represents the the momentum at low energies, which
satisfies the standard representation in position space, i.e.,
$p_{0i}={{-i\hbar d}\mathord{\left/{\vphantom{{-i\hbar
d}{dx_{i}}}}\right.\kern-1.2pt}{dx_{i}}}$, and the $p_{i}$ as that at lower
energies. In the Minkowski spacetime, the modified square of the four-momentum
is given by

$\displaystyle
p_{a}p^{a}=g_{ab}p^{a}p^{b}=-g_{00}\left({p^{0}}\right)^{2}+g_{ij}p^{0i}p^{0j}\left({1+\beta
p^{2}}\right)^{2}.$ (15)

Then, by retaining the terms up to $\mathcal{O}\left(\beta^{2}\right)$, the
equation above can be rewritten as

$p_{a}p^{a}=-\left({p^{0}}\right)^{2}+p^{2}+2\beta p^{2}p^{2},$ (16)

It should be noted that the first two terms of Eq. (16) form the usual
dispersion relation, i.e., $-\left({p^{0}}\right)^{2}+p^{2}=-m^{2}c^{2}$.
Therefore, Eq. (16) can be rewritten as $p_{a}p^{a}=-m^{2}c^{2}+2\beta
p^{2}p^{2}$, and the time component of the momentum becomes

$\left({p^{0}}\right)^{2}=m^{2}c^{2}+p^{2}\left({1-2\beta p^{2}}\right).$ (17)

According to Eq. (8) and Eq. (9), the energy of a particle can be expressed in
terms of the three spatial momentum momentum and mass as follows

$\omega^{2}=m^{2}c^{4}+p^{2}c^{2}\left({1-2\beta p^{2}}\right).$ (18)

Note that as $\beta\to 0$, Eq. (18) goes to the standard energy-momentum
dispersion relation. Next, using Eq. (11), the group speed of massless
gravitons with a modified dispersion relation (18) is approximately equal to

$\upsilon_{massless}=\frac{{\partial\omega}}{{\partial
p}}=\frac{{c\left({1-4p^{2}\beta}\right)}}{{\sqrt{1-2p^{2}\beta}}}\approx
c\left({1-3\beta p^{2}}\right).$ (19)

In the infrared, one can make a correction to the massless dispersion relation
$\omega_{g}^{2}=p_{g}^{2}c^{2}$ [39]. Thus, the Eq. (19) can be rewritten as

$\upsilon_{massless}\approx
c\left({1-3\beta{{\omega_{g}^{2}}\mathord{\left/{\vphantom{{\omega^{2}}{c^{2}}}}\right.\kern-1.2pt}{c^{2}}}}\right),$
(20)

where $\omega_{g}$ is the energy of gravitons. The difference between the
modified speed of gravitons and the speed of light is

$\Delta\upsilon^{\prime}=c-\upsilon_{massless}={{3\beta\omega_{g}^{2}}\mathord{\left/{\vphantom{{3\beta\omega_{g}^{2}}c}}\right.\kern-1.2pt}c}={{3\beta_{0}\omega_{g}^{2}}\mathord{\left/{\vphantom{{3\beta_{0}\omega_{g}^{2}}{M_{p}^{2}c^{3}}}}\right.\kern-1.2pt}{M_{p}^{2}c^{3}}}.$
(21)

From the equation above, it is clear that the GUP slows down the speed of
gravitons. In our previous work [20], we assumed that $\beta_{0}\sim 1$.
However, if this assumption is made, this will lead to a non-zero but
virtually unmeasurable effect of the GUP. Conversely, without this assumption,
one can set an improved upper bound on the GUP parameter by comparing
$\Delta\upsilon^{\prime}$ with Eq. (13), which is

${{3\beta_{0}\omega_{g}^{2}}\mathord{\left/{\vphantom{{3\beta_{0}\omega_{g}^{2}}{M_{p}^{2}c^{3}}}}\right.\kern-1.2pt}{M_{p}^{2}c^{3}}}<5.6\times
10^{-12}{\rm{m/s}}.$ (22)

Since the signal of gravitational wave event GW150914 is peaked at
$150{\rm{Hz}}$, here we set $\omega_{g}\approx 6.024\times 10^{-13}{\rm{eV}}$.
Therefore, the upper bound on the GUP parameter $\beta_{0}$ turns out to be

$\beta_{0}<2.3\times 10^{60}.$ (23)

By comparing our upper bound on $\beta_{0}$ with the results in Table. 1, it
is easy to see that our result is weaker than those set by the tunneling
current in a scanning tunneling microscope, the position measurement, the
hydrogen lamb shift, the 87Rb cold-atom-recoil experiment and the Landau
levels, whereas it is more stringent than those derived in a previous work
[34]. However, it is believe that the accuracy of $\beta_{0}$ will get
increased by several orders of magnitude with more accurate measurements in
the further.

Table 1:  Current experimental bounds on the GUP parameter $\beta_{0}$ Measurement/Experiment | $\beta_{0}$ | Refs. |
---|---|---|---
Electron tunneling | $10^{21}$ | [28, 29] |
Gravitational bar detectors | $10^{33}$ | [30] |
Electroweak measurement | $10^{34}$ | [28, 29] |
Lamb shift | $10^{36}$ | [28, 29] |
87Rb cold-atom-recoil experiment | $10^{39}$ | [35] |
Landau levels | $10^{50}$ | [28, 29] |
Perihelion precession |  |  |
(Solar system data) | $10^{69}$ | [34] |
Perihelion precession |  |  |
(Pulsar PRS B 1913+16 data) | $10^{71}$ | [34] |
Modified mass-temperature relation | $10^{78}$ | [34] |
Light deflection | $10^{78}$ | [34] |

##  4 Bounds on the GUP parameters $\alpha_{0}$

In this section we apply the formalism given above to the GUP II case, so the
effect of the GUP-induced term can be measurable. Now, let us define the
operators for the position $x_{i}$ and momentum $p_{i}$ in GUP II as follows

$\displaystyle x_{i}=x_{0i};\quad p_{i}=p_{0i}\left({1-\alpha
p+2\alpha^{2}p^{2}}\right),$ (24)

where $p^{2}=g_{ij}p^{0i}p^{0j}$, thus $p=\sqrt{g_{ij}p^{0i}p^{0j}}$, $x_{0i}$
and $p_{0i}$ satisfying the canonical commutation relations
$\left[{x_{0i},p_{0j}}\right]=i\hbar\delta_{ij}$. The $p_{0i}$ is the the
momentum at low energies (having the standard representation in position space
$p_{0i}={{-i\hbar d}\mathord{\left/{\vphantom{{-i\hbar
d}{dx_{i}}}}\right.\kern-1.2pt}{dx_{i}}}$), and the $p_{i}$ as that at lower
energies. Similar to the GUP I case, the modified dispersion relation is
derived as [46]

$\omega^{2}=m^{2}c^{4}+p^{2}c^{2}\left({1-\alpha p}\right)^{2}.$ (25)

Using Eq. (8) and the massless dispersion relation $\omega^{2}=p^{2}c^{2}$,
and by ignoring the higher order term of $\mathcal{O}\left(\alpha\right)$, the
speed of massless gravitons is

$\tilde{\upsilon}_{massless}=c\left({1-{{2\alpha\omega_{g}}\mathord{\left/{\vphantom{{2\alpha\omega_{g}}c}}\right.\kern-1.2pt}c}}\right),$
(26)

where $\omega_{g}$ is the graviton energy. Therefore, difference between the
modified speed of gravitons and the speed of photon is given by

${\Delta\upsilon}^{\prime\prime}=c-\tilde{\upsilon}_{massless}={2\alpha\omega_{g}}<5.6\times
10^{-12}{\rm{m/s}}.$ (27)

From the inequality stated above, it is easy to see that the GUP corrected
speed of gravitons is smaller than the speed of light. In order to obtained
the upper bound on the GUP parameter, we need to compare the
${\Delta\upsilon}^{\prime\prime}$ with Eq. (13), the result reads

$\alpha<4.5.$ (28)

In Eq. (28), we still set $\omega_{g}=6.024\times 10^{-13}{\rm{eV}}$ at a
frequency of $\nu=150{\rm{Hz}}$. Next, according to Eq. (2), the upper bound
on the GUP parameter $\alpha_{0}$ is given by

$\alpha_{0}<1.8\times 10^{20}.$ (29)

It should be stressed that the GUP parameter $\alpha_{0}$ is attached to a
term linear in $p$, whereas the GUP parameter $\beta_{0}$ is attached to a
term quadratic in $p$. Therefore, we should compare
$\left({\alpha_{0}}\right)^{2}$ with $\left({\beta_{0}}\right)$. It is obvious
that the upper bound on $\left({\alpha_{0}}\right)^{2}$ is 20 orders better
than that on $\beta_{0}$, which indicates that the gravitational wave
measurements are more suitable for studying effects of the GUP II proposal.

Table. 2 shows the bounds on $\alpha_{0}$ set by high-energy physics and other
experiments.

Table 2:  Current experimental bounds on GUP parameter $\alpha_{0}$ Measurement/Experiment | $\alpha_{0}$ | Refs. |
---|---|---|---
Anomalous magnetic moment of the muon | $10^{8}$ | [15] |
Lamb shift | $10^{10}$ | [31] |
Electron tunneling | $10^{11}$ | [31] |
87Rb cold-atom-recoil experiment | $10^{14}$ | [35] |
Electroweak measurement | $10^{17}$ | [31] |
Superconductivity | $10^{17}$ | [15] |
Landau levels | $10^{23}$ | [31] |

Now, compared with the results in Table. 2, one can find that the improved
upper bound in Eq. (29) is weaker than that set by the anomalous magnetic
moment of the muon, electron tunneling, hydrogen Lamb shift, 87Rb cold-atom-
recoil experiment, position measurement and superconductivity experiment,
whereas it is more stringent than that set Landau levels. In addition, it
seems that our result is exceeding the stringent bound set by electroweak
scale which is around $10^{17}$. We think the reason for this is that the
spins of the black holes could not be measured accurately, which leads to the
energy of gravitons ($6.024\times 10^{-13}{\rm{eV}}$) is far weaker than the
energy of electroweak measurements ($240{\rm{GeV}}$). With more accurate
measurements in the further, the upper bound on $\alpha_{0}$ will reduce
significantly with time [47, 48]. Therefore, our result is compatible with the
bound set by electroweak scale. In particular, the LIGO Scientific and Virgo
Collaborations recently reported the second gravitational wave event,
GW151226, with the peak amplitude of the gravitational wave train
$\nu=450{\rm{Hz}}$ [49]. Using the same method, we find that the energy of
gravitons is $\omega^{\prime}_{g}=h\nu=1.8612\times 10^{-12}{\rm{eV}}$ and the
difference between the gravitons and speed of light becomes
$\Delta\upsilon^{\prime}=6.2\times 10^{-13}$, which leading to the improved
upper bound $\alpha_{0}^{\prime}<6.7\times 10^{18}$. Obviously,
$\alpha_{0}^{\prime}$ is more stringent than $\alpha_{0}$.

##  5 Conclusion

In previous study, it was generally assumed that the GUP parameters are of the
order of unity. However, this assumption would lead to the effect of the GUP
too small to be measured. By contrast, if this assumption is not made, one can
constrain the GUP parameters based on previous experiments. In this letter, we
considered two proposals regarding the GUP and used the data from the
gravitational wave event GW150914 to calculate the modified speed of
gravitons. Finally, we obtained improved upper bounds on the GUP parameters
$\beta_{0}$ and $\alpha_{0}$. The results showed that the modified speed of
gravitons is related to the effect of the GUP. When the GUP parameters
approach zero, the modifications reduce to the original cases. Furthermore, we
obtained $10^{60}$-level upper bound on $\beta_{0}$ and $10^{20}$-level upper
bound on $\alpha_{0}$. Comparing the two kinds of GUP parameters, it is easy
to see that the gravitational wave measurements are more suitable for studying
effects of the GUP II proposal. Furthermore, it was shown that the upper
bounds on GUP parameters $\beta_{0}$ and $\alpha_{0}$ are weaker than that set
by electroweak measurement but not incompatible with it. With more accurate
measurements in the further, the upper bounds on $\beta_{0}$ and $\alpha_{0}$
will reduce significantly with time. Finally, it is interesting to note that
many studies have predicted that pulsars (spinning neutron stars) are
promising candidates for producing the gravitational wave signals [50, 51,
52]. If these theories are proved by the gravitational wave detectors, one
could obtain many new results in the future.

Acknowledgements

The authors thank Klaus Doerbecker, Fabio Scardigli, Roberto Casadio, Karissa
Liu, and the anonymous referees for helpful suggestions and enlightening
comments, which helped to improve the quality of this paper. This work was
supported by the Natural Science Foundation of China (Grant No. 11573022).

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