# The free energy of twisting spins in Mn3Sn

Xiaokang Li1,∗, Shan Jiang3,1, Qingkai Meng1, Huakun Zuo1, Zengwei Zhu1,∗,
Leon Balents2,4 and Kamran Behnia3 (1) Wuhan National High Magnetic Field
Center and School of Physics, Huazhong University of Science and Technology,
Wuhan 430074, China
(2)Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, California 93106-4030, USA
(3) Laboratoire de Physique et d’Étude des Matériaux
(ESPCI - CNRS - Sorbonne Université), PSL Research University, 75005 Paris,
France
(4) Canadian Institute for Advanced Research, Toronto, Ontario, Canada

###### Abstract

The magnetic free energy is usually quadratic in magnetic field and depends on
the mutual orientation of the magnetic field and the crystalline axes. Tiny in
magnitude, this magnetocrystalline anisotropy energy (MAE) is nevertheless
indispensable for the existence of permanent magnets. Here, we show that in
Mn3Sn, a non-collinear antiferromagnet attracting much attention following the
discovery of its large anomalous Hall effect, the free energy of spins has
superquadratic components, which drive the MAE. We experimentally demonstrate
that the thermodynamic free energy includes terms odd in magnetic field
($\mathcal{O}(H^{3})+\mathcal{O}(H^{5})$) and generating sixfold and twelve-
fold angular oscillations in the torque response. We show that they are
quantitatively explained by theory, which can be used to quantify relevant
energy scales (Heisenberg, Dzyaloshinskii-Moriya, Zeeman and single-ion
anisotropy) of the system. Based on the theory, we conclude that, in contrast
with common magnets, what drives the MAE in Mn3Sn is the field-induced
deformation of the spin texture.

Aligned spins located on two adjacent atoms are affected by the anisotropic
electrostatic forces connecting their orbital angular momenta van Vleck
(1937). This magnetocrystalline anisotropy energy (MAE), a consequence of the
spin-orbit coupling, is remarkably small ($\sim 60\mu$ eV/atom in Co and $\sim
1\mu$ eV/atom in Fe and Ni). Since it is the outcome of the competition
between energies many orders of magnitude larger, it is hard to calculate from
first principles Daalderop  _et al._ (1990); Halilov  _et al._ (1998).

Mn3Sn, a noncollinear antiferromagnet with an inverse triangle spin structure
located on a breathing kagome lattice Tomiyoshi and Yamaguchi (1982) has
attracted much attention following the observation of a large anomalous Hall
effect(AHE) Nakatsuji  _et al._ (2015) with a sizeable net Berry curvature
near the Fermi level Yang  _et al._ (2017). The discovery was followed by the
observation of various counterparts of AHE, including the anomalous Nernst
Ikhlas  _et al._ (2017); Li  _et al._ (2017) and the anomalous thermal Hall
effects Li  _et al._ (2017); Sugii  _et al._ (2019); Xu  _et al._ (2020), as
well as the anomalous magneto-optical Kerr effect Higo  _et al._ (2018); Balk
_et al._ (2019). These are room-temperature effects requiring a small magnetic
field. Therefore, Mn3Sn is potentially attractive in the field of
antiferromagnetic spintronics Šmejkal  _et al._ (2018); Baltz  _et al._
(2018); Kimata  _et al._ (2019); Tsai  _et al._ (2020) or as a Nernst
thermopile Ikhlas  _et al._ (2017); Li  _et al._ (2021). The peculiar spin
texture of Mn3Sn has been subject of several studies Cable  _et al._ (1993);
Liu and Balents (2017); Suzuki  _et al._ (2017); Park  _et al._ (2018);
Zelenskiy  _et al._ (2021); Miwa  _et al._ (2021). The magnetic Hamiltonian
includes Heisenberg and Dzyloshinskii-Moriya spin-spin interaction terms
dominating by far the small single-ion anisotropy term Liu and Balents (2017).
A study of torque magnetometry Duan  _et al._ (2015) quantified the latter.
Previous experiments have documented that magnetic domain walls are chiral Li
_et al._ (2018) and host a topological Hall effect associated with a finite
skyrmionic number Li  _et al._ (2017).

Here, we present a study of magnetization and angle-dependent magnetic torque
in Mn3Sn single crystals and find an additional term in the magnetic free
energy with an intriguingly delicate anisotropy. Magnetization, on top of its
zero-field (spontaneous) and field-linear components, has a previously
undetected term, which is quadratic in magnetic field. This implies that the
free energy has a component proportional to the cube of the magnetic field
Treves (1962). We show that, in contrast to ordinary ferromagnets, this
second-order magnetization is the driver of the magneto-crystalline
anisotropy. This conclusion is based on our torque magnetometry data, which
uncovers a field-dependent MAE. The angular oscillations of the torque signal
in addition to its two-fold, $K_{2}$, and sixfold, $K_{6}$, components
detected previously Duan  _et al._ (2015), have an additional twelve-fold,
$K_{12}$, component. The field dependence of the two intrinsic shape
independent terms ($K_{6}\propto H^{3}$ and $K_{12}\propto H^{5}$ ) establish
that they are caused by the second-order magnetization. Both the six-fold and
twelve-fold anisotropies are quantitatively reproduced by a theoretical
calculation based on the Hamiltonian of ref. Liu and Balents (2017). Fits to
the experimental quantifies the Zeeman energy, the Heisenberg (J) and the
Dzyaloshinskii-Moriya (DM) interaction term, and finally the single-ion-
anisotropy energy, the cost of misalignment between spin lattice and crystal
axes. The analysis demonstrates that the unusual features are driven by field-
induced twisting of the spins, which result from the frustrated competition of
the above four energy scales, particular to the inverse triangular spin
structure.

Figure 1: A quadratic term in the magnetization of Mn3Sn: (a) The spin texture
of Mn3Sn. Mn magnetic moments are locating on an AB-stacked breathing kagome
lattice. (b-c) In-plane, $M_{ab}$, and the out-of-plane, $M_{c}$,
magnetization in Mn3Sn at 300 K, as a function of magnetic field swept from 14
T to -14 T. (d-e) The residual component of $M_{ab}$ after subtracting its
dominant linear term $M_{1}(\propto O(H))$. It is plotted with the linear (d)
and quadratic (e) $x$-coordinate respectively. There is an additional
component, $M_{2}(\propto O(H^{2}))$. (f-g) The residual component of $M_{c}$.
The sign of its second-order magnetization is opposite to $M_{ab}$. Figure 2:
Anisotropy of the quadratic magnetization : (a) Experimental configuration for
the angular magnetization. (b-c) Comparison of $M_{ab}$ and it’s residual
after subtracting the linear background, with four different angles, 0∘, 30∘,
60∘ and 90∘ respectively. (d) The angle dependence of the $M_{0}$, $M_{1}$,
and $M_{2}$. Only the latter show angular oscillations with a periodicity of
60 degrees.

Centimeter-size Mn3Sn single crystals were grown by the vertical Bridgman
technique Li  _et al._ (2018). As shown in Fig. 1a, Mn atoms form a breathing
kagome lattice. In this inverse triangle spin lattice Tomiyoshi and Yamaguchi
(1982); Nakatsuji  _et al._ (2015), large magnetic moments of the Mn atoms
cancel each other. As seen in Fig. 1b-c, a weak, and mainly in-plane,
ferromagnetism persists, which may be caused by a slight deviation of spin
orientations from a perfect triangular alignment Tomiyoshi and Yamaguchi
(1982); Nakatsuji  _et al._ (2015). A collection of six Mn atoms located in
two adjacent layers constitutes the simplest unit of the spin texture and has
been treated as a magnetic octupole spin cluster Suzuki  _et al._ (2017).

Our first new finding is shown in Fig. 1d-g. After subtracting a linear
background ($M_{1}(\propto\mathcal{O}(B)$), we find that in addition to the
spontaneous magnetization, M0, there is an additional field-dependent term
(Fig. 1d) in magnetization. As seen in Fig. 1e, this term is quadratic in
magnetic field. Such a term is also present in the out-of-plane magnetization
too, albeit with a smaller amplitude and an opposite sign (See Fig. 1f-g). We
conclude that the magnetization consists of three terms:

$M_{total}=M_{0}+M_{1}(\propto\mathcal{O}(H))+M_{2}(\propto\mathcal{O}(H^{2})).$
(1)

The first term is the zero-field spontaneous magnetization associated with the
weak residual ferromagnetism. The second term ($M_{1}$) is the dominant linear
magnetization resolved in previous studies of magnetization Nakatsuji  _et
al._ (2015). The third term, $M_{2}$, not detected in previous studies is
quadratic in field. It represents a second-order correction to the
magnetization response Treves (1962); Gorodetsky and Treves (1964); Kharchenko
_et al._ (1995); Liang  _et al._ (2017). Since the magnetization is the
partial derivative of the magnetic free energy with respect to the magnetic
field ($M=\partial F_{M}/\partial H$), a finite $M_{2}$ implies an additional
term for the magnetic free energy, which can be written as:

$F_{M}=\sum_{i}M_{0,i}H_{i}+\frac{1}{2}\sum_{i,j}\chi_{i,j}H_{i}H_{j}+\frac{1}{3}\sum_{i,j,k}C_{i,j,k}H_{i}H_{j}H_{k}.$
(2)

Here, $C_{i,j,k}$ is a $3\times 3\times 3$ tensor, which represents the second
odd term in the field dependence of the free energy.

The angular variation of these three terms was investigated by performing the
angular dependent magnetization measurements, using the set-up shown in Fig.
2. Rotation was achieved by changing the sharp angle of a quartz wedge held
between the sample and the sample-holder. Fig. 2b and c show magnetization and
its residual after subtracting the linear background as a function of field
for four different angles in $xy$-plane.

The angle dependence of in-plane $M_{0}$, $M_{1}$, and $M_{2}$ are plotted in
Fig. 2d. Angular variation is undetectable for $M_{0}$ and $M_{1}$, but as
large as 37 percent for $M_{2}$ and displays a clear six-fold symmetry, with
minima and maxima separated by 30 degrees. We conclude that the third term of
free energy $F_{3}\propto O(H^{3})$ has a sixfold anisotropic component. In
other words, $M_{2}=M_{2,0}+M_{2,6}cos(6\phi)$. At 13 T,
$M_{2,0}=4.4~{}m\mu_{B}$/f.u., and $M_{2,6}=0.8~{}m\mu_{B}$/f.u. As we will
see below, the magnitude of the latter term is confirmed with a higher
precision by measurements of the magnetic torque, $\tau$, which quantifies the
angular derivative of the magnetic free energy($\tau=\partial
F_{M}/\partial\theta$).

Figure 3: Magnetic torque measurements: (a) The home-made experimental setup
and its photograph (top) with a capacitive torque magnetometer, rotating the
field in the $xy$-plane of the Mn3Sn sample. (b) The angle dependent torque
responses for magnetic fields up to 6 T. (c) The angle dependent torque
responses for magnetic fields larger than 6 T. As the field increases, the
shape of oscillations evolve, indicating the emergence of an additional
component. $H\parallel x$ corresponds to 0∘. Figure 4: Components of the
torque response: (a) Fit to the 13 T data with an expression, which has only a
two-fold and a six-fold term ($\tau=K_{2}\cdot
sin(2\cdot(\phi+\phi_{2}))+K_{6}\cdot sin(6\cdot(\phi+\phi_{6}))$). The
mismatch is obvious. (b) The residual torque component after subtracting the
data and the previous fit. It shows a clear twelve-fold symmetry. (c) A fit
which includes an additional twelve-fold symmetry component ($K_{12}\cdot
sin(12\cdot(\phi+\phi_{12})$)). (d) Fitting parameters $K_{2}(H)$, $K_{6}(H)$
and $K_{12}(H)$ as a function of magnetic field. $K_{2}(H)$ follows $H^{3/2}$
and is much smaller than $K_{6}(H)$. $K_{6}(H)$ follows $H^{3}$. $K_{12}(H)$
follows $H^{5}$ at the beginning. (e) A comparison of $K_{2}(H)$, $K_{6}(H)$
and $K_{12}(H)$ in three different samples with different cross sections.
Here, the sample with the rectangular cross section has a larger aspect ratio
($l_{x}/l_{y}$ = 5.6), compared to the samples with square and triangle cross
sections ($l_{x}/l_{y}\simeq 1$). The data reported by Duan et al. are also
shown. Figure 5: Magneto-crystalline anisotropy driven by field-induced twist
of non-aligned spins: (a) In absence of magnetic field, the spin lattice is
triangular. The magnetic field distorts the spin triangle (in white), which is
no more isomorphic to the lattice triangle (in gray). The deformation angles
$\eta_{i}$ quantify the distortion. (b) Interaction between one spin and its
immediate neighbors favor clockwise and anticlockwise twists. The Zeeman
effect favors alignment of all spins with magnetic field. Single-ion
anisotropy causes in-equivalency between the two perpendicular orientations of
the spin triangle with respect to the lattice triangle. (c) When the magnetic
field rotates, there is a lag angle $\psi$ between the magnetic field and the
total magnetization . (d) The experimental $K_{6}$ and $K_{12}$ (symbols)
compared to theoretical expectation (solid line) using $\mu=3\mu_{B}$,
$J=20.1meV$, $D/J=0.18$, $K/J=0.0065$. (e) The deformation angles $\eta_{i}$
and lag angle $\psi$ at 3 T, 8 T and 13 T predicted by theory SM (2021). The
three tilt angles follow the same pattern with an angular delay of
$60^{\circ}$. With increasing magnetic field, a secondary oscillation with a
periodicity of $30^{\circ}$ emerges on top of a $60^{\circ}$ periodicity.

Fig. 3a shows a sketch and a photograph of the torque set-up. The applied
magnetic field rotates in the ab-plane of the Mn3Sn sample, and the magnetic
torque along the c-axis is detected by the variation in the capacitance
between two metal plates. Fig. 3b shows the angular dependence of the torque
signal below 6 T resolving a clear six-fold symmetry. However, as the magnetic
field increases further, a deviation becomes visible. As seen in Fig. 3c, when
the magnetic field increases from 6 T to 13 T, the shape of each oscillation
evolves. Instead of being sine-like, it becomes sawtooth-like. When the field
is along a high symmetry axis ($x,-x,y,-y$), torque vanishes and the free
energy is at its extremum, as one expects. Remarkably, however, there is a
widening difference between the slopes along the two high-symmetry
orientations. This is caused by the emergence of an additional angle-dependent
term.

In addition to the two-fold and six-fold terms detected by Duan et al.Duan
_et al._ (2015), our high-field data has an additional term with twelve-fold
symmetry. This can be seen in Fig. 4a, which shows that the difference between
the experimental data and the sum of the two first terms ($K_{2}\cdot
sin(2\cdot(\theta+\phi_{2}))+K_{6}\cdot sin(6\cdot(\theta+\phi_{6}))$). The
residual has a clear twelve-fold anisotropy ($K_{12}\cdot
sin(12\cdot(\theta+\phi_{12}))$). The evolution of the fitting parameters,
$K_{2}$, $K_{6}$, and $K_{12}$ with magnetic field is shown in Fig. 4d and e.
$K_{2}(H)$ follows $H^{3/2}$ and $K_{6}(H)$ follows $H^{3}$ and remains the
largest in the whole field range. Finally, $K_{12}(H)$ shows a steep increase,
which is close to $H^{5}$ at low fields.

The strong field dependence of the torque components of Mn3Sn is in sharp
contrast with what is seen in ferromagnetsGraham (1958), as confirmed by our
own data on bcc Fe (See the supplementSM (2021)). Additionally, it implies
that a single magnetic field measurement (5 T in the case of ref. Duan  _et
al._ (2015), see Fig.4e) is not sufficient to extract the magnotcrystalline
anisotropy. Let us now consider the information brought by these results on
the components of magnetic free energy.

The two-fold term, $K_{2}$, evolves much slower than $H^{3}$ and strongly
depends on the sample aspect ratio (See Fig. 4e). The first feature indicates
that it is not caused by the second-order magnetization and the second feature
implies a role played by sample geometry. $K_{2}$ can be caused by a finite
angle between the applied field and magnetization, which can, in principle,
occur in any sample with finite size.

The six-fold term, $K_{6}$, shows a field dependence clearly linking it to the
field-cubic free energy ($F_{3}\propto\mathcal{O}(H^{3})$), confirming what
was deduced from measurements of angle-dependent magnetization. Both sets of
data lead to the conclusion that there is a component of free energy with
sixfold angular symmetry and cubic field dependence ($F^{(3,6)}\propto
H^{3}cos6\theta$). A consistency check can be done by comparing the magnitude
of $K_{6}$, the angle derivative and $M_{2,6}$, the field derivative. At 13 T,
$K_{6}$ is 4700- 5600 $Jm^{-3}$, implying $F^{(3,6)}\approx 0.34\pm 4~{}\mu
eV/f.u.$ and $M_{2,6}$ is $0.8~{}m\mu_{B}$ per f.u. corresponding to
$F^{(3,6)}\approx 0.20~{}\mu eV/f.u.$. The list of all components of the free
energy identified by our experiments are given in Table 1.

Component | Gives rise to | Amplitude at 13 T ($\mu$ eV/Mn) | Accounted by theory? | Comments
---|---|---|---|---
$F^{(1,ab)}\propto H$ | M0 | 1.7 | Yes | residual FM
$F^{(1,c)}\propto H$ | M0 | 0.11 | Canting? | residual FM
$F^{(2,ab)}\propto H^{2}$ | M1 | 21.4 | Yes | paramagnetism
$F^{(2,c)}\propto H^{2}$ | M1 | 27.9 | Yes | paramagnetism
$F^{(3,0)}\propto H^{3}$ | M2,0 | 0.37 | No | Field-induced correction to FM?
$F^{(3,6)}\propto H^{3}cos6\theta$ | M${}_{2,6}/K_{6}$ | 0.067/0.115 | Yes | Field-induced tilt of spins
$F^{(5,12)}\propto H^{5}cos12\theta$ | $K_{12}$ | 0.018 | Yes | secondary correction to the tilt
$F^{(1.5,2)}\propto H^{3/2}cos2\theta$ | $K_{2}$ | 0.028 | No | Boundary-related
Table 1: Components of the magnetic free energy in Mn3Sn identified by
measurements of magnetization ($M_{0}$, $M_{1}$ and $M_{2}$) and torque
($K_{2}$, $K_{6}$ and $K_{12}$).

Let us now show that theory provides a satisfactory account of the existence
and the amplitude of $K_{6}$ term as well as the emergence and rapid growth of
the secondary $K_{12}$ term with increasing magnetic field.

Following Liu and Balents Liu and Balents (2017), the energy per magnetic unit
cell (six spins) consists of the sum of four termsSM (2021): These are
Heisenberg: $4J\sum_{i}\bm{S}_{i}\cdot\bm{S}_{i+1}$; Dzyaloshinskii-Moriya
(DM): $4D\sum_{i}\bm{\hat{z}}\cdot\bm{S}_{i}\times\bm{S}_{i+1}$; Single-ion-
anisotropy (SIA) : $-2K\sum_{i}(\bm{S}_{i}\cdot\bm{\hat{e}}_{i})^{2}$; and
Zeeman: $-2\mu\sum_{i}\bm{H}\cdot\bm{S}_{i}$. For $D>0$ and in absence of SIA
and Zeeman terms, the ground state is an anti-chiral state with in-plane
spins. A finite magnetic field will distort the spin triangles. The distortion
angles, $\eta_{i}$ (See Fig.5 ) are small, because in our window of
investigation ($H<14$T), one has $K\ll J$ and $\mu H\ll J$. Introducing a
small parameter $r\ll 1$, with $K/J,~{}\mu H/J$ of $O(r)$, one can expand
$\eta_{1,2}$ in a formal series in $r$ :

$\eta_{i}=\sum_{n=1}^{\infty}\eta_{i,n}r^{n},$ (3)

By minimizing the energy terms over $\eta_{i,n}$, the small distortions of the
triangle at each order can be quantified. This leads to an expression for the
free energy per unit cell. The first term is linear in magnetic field (See the
supplementSM (2021)):

$F^{(1,ab)}=\frac{K\mu H}{J+\sqrt{3}D}$ (4)

The quadratic term SM (2021) has slightly different expressions for in-plane
and out-of-plane orientations of magnetic field is:

$\begin{split}F^{(2,ab)}=\frac{(\mu H)^{2}}{2J}(1-\frac{\sqrt{3}D}{J})\\\
F^{(2,c)}=\frac{(\mu H)^{2}}{2J}(1-\frac{D}{\sqrt{3}J})\end{split}$ (5)

Therefore, one expects the quadratic free energy to be larger for the out-of-
plane orientation of the magnetic field, in agreement with what is seen
experimentally (See Table 1). For in-plane configuration, the first correction
to the quadratic term has a $cos6\theta$ angle dependence. Its amplitude is
equal to:

$F^{(3,6)}=\frac{(K+\mu H)^{2}((3J+7\sqrt{3}D)K+4\sqrt{3}D\mu
H)}{36(J+\sqrt{3}D)^{3}}$ (6)

The field dependence of this term is close to $H^{3}$. As one can see in Fig.
5d, this expression provides an excellent account of the field and angular
dependence of the experimentally observed $K_{6}$. The next term has a
$sin^{2}(6\theta)$ angle dependence whose amplitude is equal to :

$\begin{split}F^{(5,12)}=\frac{(K+\mu H)^{2}}{72(J+\sqrt{3}D)^{5}\mu
HK}((3J+7\sqrt{3}D)K^{2}+\\\ 2(J+4\sqrt{3}D)\mu HK+2\sqrt{3}D(\mu
H)^{2})^{2}\end{split}$ (7)

The field dependence of this term is close to $H^{5}$ and it accounts for the
emergence of $K_{12}$ in the torque data and its field dependence. (See Fig.
5d).

The agreement between theory and experiment allows us to extract the energy
scales of the system. Taking the magnetic moment of each Mn atom to be
$\mu=3~{}\mu_{B}$, as reported by neutron diffraction studies Brown  _et al._
(1990); Tomiyoshi (1982), we extracted $J$, $D$, and $K$. The results are
summarized in table 2. It shows what is yielded by fitting the torque data
with equations 6 and 7. Alternatively, one can use the magnetization data and
equations 4 and 5, the results are given in the second row of table 2. As seen
in the table 2, $J=J_{1}+J_{2}=20.1$ meV, which is a third larger than what is
yielded by magnetization. The most plausible source of this difference is the
existence of a finite orbital contribution to magnetization Sandratskii and
Kübler (1996) and assuming a 30 percent orbital contribution to M1 would lead
to consistency. We note that our result is fairly close to the what has been
reported by a study of magnon dispersion by inelastic neutron scattering (18
meV)Park  _et al._ (2018). Finally, our study pins down the values for $K$ and
$D$ and demonstrates the capacity of torque magnetometry to quantify the
magnitude of Dzyaloshinskii-Moriya interaction in a frustrated magnet.

Parameter | J (meV) | D/J | K/J
---|---|---|---
Torque | 20.1(8) | 0.18(2) | 0.0065(5)
Magnetization | 13.8 | 0.18 | 0.0058
Table 2: The energy scales of Mn3Sn extracted from the torque and
magnetization data using the theoretical expressions of the free energy.

In summary, we resolved different components of the magnetic free energy in
Mn3Sn. In addition to the dominant term, which is even in magnetic field, it
includes odd terms with superquadratic field dependence and presenting sixfold
and twelvefold angular oscillations. We showed that theory invoking a field-
induced twist in the orientation of in-plane spins can successfully explain
the presence of these terms and their amplitude can be used to extract all
energy scales of the system. The model also makes experimental predictions of
specific twists which may be tested in the future in neutron scattering and
anomalous Hall effect measurements. It is remarkable that the simple localized
spin description used here works in such quantitative detail for this highly
itinerant magnet.

This work was supported by the National Science Foundation of China (Grant
No.51861135104 and No.11574097) and The National Key Research and Development
Program of China (Grant No.2016YFA0401704). K. B was supported by the Agence
Nationale de la Recherche (ANR-18-CE92-0020-01; ANR-19-CE30-0014-04). S. J.
acknowledges a PhD scholarship by the China Scholarship Council (CSC). X. L.
acknowledges the China National Postdoctoral Program for Innovative Talents
(Grant No.BX20200143) and the China Postdoctoral Science Foundation (Grant
No.2020M682386). L.B. was supported by the NSF CMMT program under Grant No.
DMR-2116515.

*<EMAIL_ADDRESS>
*<EMAIL_ADDRESS>

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Supplemental Material for “The free energy of twisting spins in Mn3Sn”

##  S1 Samples and Methods

Mn3Sn single crystals used in this work were cut to desired dimensions by a
wire saw from a centimeter-size mother sample grown by the vertical Bridgman
method Li  _et al._ (2018). A large-size sample (#1) was used for the
magnetization measurements and three samples (#2, #3, #4) with different
cross-sections were used for the magnetic torque measurements (See the table
S1 for details). We also studied two bcc Fe samples, cut from the same
commercial single crystal, orientated along (110). One sample was cut to a
thin-disk shape, with dimensions of $1.3~{}mm$ (diameter) $\times$ $0.12~{}mm$
(thickness). Another sample was cut to a cuboid shape, with dimensions of
$0.39~{}mm\times 0.36~{}mm\times 0.6~{}mm$.

All magnetization measurements were performed in a physical property
measurement system (Quantum design PPMS-16) with a vibrating sample
magnetometer (VSM) option. All magnetic torque measurements were performed in
a TeslatronPT (Oxford Instruments) with a homemade rotated probe equipped with
the capacitive torque magnetometer. A capacitance bridge (AH-2550A) was used
to measure the variation of the capacitance between two metal plates
proportional to change in the magnetic torque of the sample. Measurement of
the rotation angle was calibrated by a Hall chip.

| #1 | #2 | #3 | #4
---|---|---|---|---
Cross-section | Square | Triangle | Square | Rectangular
$l_{x}/l_{y}$ |  $\sim$1 | 1.15 | 1.2 | 5.6
Mass (mg) | 30.0 | 3.12 | 4.4 | 1.15
Technique | Magnetization | Torque | Torque | Torque
$K_{2}$ at 5T ($kJ/m^{3}$) | $\backslash$ | 0.10 | 0.11 | 0.34
Table S1: Details of Mn3Sn samples used in this work. Figure S1:
Reproducibility of data: The magnetic torque responses in different Mn3Sn
samples, with triangle, square, rectangular cross-section respectively (a).
Temperature dependent magnetic torque responses in Mn3Sn samples (# triangle),
varying from 300 K to 100 K (b), and 50K (c). Figure S2: Magnetization and the
magnetic torque in bcc Fe single crystal (thin disk sample): (a-b)
Magnetization. Saturating at 0.2 T (a). No observable quadratic term (b). The
demagnetizing factor is approach to 0, in the plane direction of the thin
disk. (c-g) The magnetic torque. The photograph of the measurements, the thin
disk sample is fixed with a quartz holder(c). The angular torque responses at
different field, varying from 0.13 T to 0.35 T (d), 0.5 T to 3 T (e). The fit
of the data at 1 T, with the formula ($\tau=K_{2}\cdot
sin(2\cdot(\theta+\phi_{2}))+K_{4}\cdot sin(4\cdot(\theta+\phi_{4}))$) (f).
Field dependence of $K_{2}$ and $K_{4}$ (g). Figure S3: Magnetization and the
magnetic torque in bcc Fe single crystal (cuboid sample). (a-b) Magnetization.
Saturating at 0.5 T (a). No observable quadratic term (b). (c-e) The magnetic
torque. The angular torque responses at different field, varying from 0.2 T to
0.5 T (c), 1T to 4 T (d). Field dependence of $K_{2}$ and $K_{4}$ (e).

##  S2 Reproducibility of data

As seen in Fig. S1a, we repeated the torque measurements in three Mn3Sn
samples with different cross-section. All the samples show the same six-fold
oscillation accompanied by a deviation at high field, suggesting the $K_{6}$
and $K_{12}$ are the intrinsic responses in Mn3Sn. While in the sample #4
(with a rectangular cross-section) the two-fold oscillation was more
prominent. This indicates that the origin of $K_{2}$ is linked to the boundary
geometry. As argued previously Li  _et al._ (2018, 2021), there is a lag angle
between the applied magnetic field and magnetization in Mn3Sn and this lag
angle increases with increasing aspect ratio of the sample Li  _et al._ (2018,
2021). This provides an explanation for the larger $K_{2}$ in the sample with
the largest aspect ratio (Fig. 4e and see the table S1 for details).

In the case of the sample with a triangular cross section, we repeated the
torque measurements at different temperatures. The data is presented in Fig.
S1b-c. When we compare the high field torque data at different temperature, we
find the $K_{6}$ and $K_{12}$ are robust from 300 K down to 100 K, as seen in
Fig. S1b. But the signal collapsed suddenly at 50 K, nearing the transition
temperature of the spin glass state Nakatsuji  _et al._ (2015), as seen in
Fig. S1c. Our data suggests that the angle-dependent magnetic free energy has
little temperature dependence and implies that this is a property of the
triangular spin texture of the AF state which is destroyed at low temperature.

##  S3 Magnetization and magnetic torque in bcc iron

In order to compare with Mn3Sn, we measured the magnetization of a disk-shape
bcc Fe single crystal at 250 K, with the field along (110) plane. As seen in
Fig. S2a-b. Unlike Mn3Sn, the dominant component in bcc Fe is the spontaneous
magnetization, which saturates at 0.2 T, as seen in Fig. S2a. Focusing on the
signal at high field region and having subtracted a linear background no
quadratic magnetization is detectable (Fig. S2b).

We also measured the magnetic torque in the same crystal at 250 K, with the
field rotating in (110) plane. Fig. S2c shows the photograph of the
measurements. Fig. S2d-e show the angular torque responses in presence of
different magnetic fields. A clear four-fold oscillations combined with the
two-fold component can be observed. Fig. S2f shows the fit of the data at 1 T.
From which we extracted the amplitude of two-fold ($K_{2}$) and four-fold
($K_{4}$) components, and plotted them as a function of magnetic field in Fig.
S2g. It can be seen that the dominant $K_{4}(H)$ begin to saturate above 0.2
T, same to the literature Graham (1958), but the $K_{2}(H)$ still maintained a
slow growth.

Comparison of the torque angular oscillations in bcc Fe and in Mn3Sn reveals a
fundamental difference. In Fe, the toque (that is the angular dependence of
the magnetic free energy) does not change with magnetic field (at least above
the coercive field). In Mn3Sn, torque increases strongly with magnetic field
($\propto H^{3}$). In the first case, the angular dependence of magnetic free
energy is caused by single-ion-anisotropy and is field independent. In the
second case, the larger the magnetic field, the larger the field-induced twist
of the spins and the larger the energy cost of the rotation.

We also repeated the magnetization and magnetic torque of bcc Fe single
crystal in another cuboid sample. As seen in Fig. S3, the magnetization
responses have the higher saturation field ($H_{s}$), about 0.5 T, and the in-
plane curves are obviously separated, suggesting a larger anisotropy
originated from the shape anisotropy. In contrast to the case in Fig. S2, the
torque responses in cuboid sample show a obvious two-fold oscillation, and the
four-fold component only emerges above 1 T, as seen in Fig. S3. Moreover, the
$K_{2}(H)$ is about ten times larger, imply the shape anisotropy can have a
remarkable contribution to the magnetic anisotropy in bcc Fe. However, it
doesn’t appear in Mn3Sn case, as seen in Fig. S1.

##  S4 Comparison with previous studies on bcc iron

Our data is compatible with what was previously reported in the case of bcc
Fe. However, there is a difference of notations. The parameters Ki used in
this work are distinct from those traditionally used in the case of cubic
ferromagnets. In the latter case, the free energy is written as

$E=K_{0}+K_{1}^{*}(\alpha_{1}^{2}\alpha_{2}^{2}+\alpha_{2}^{2}\alpha_{3}^{2}+\alpha_{3}^{2}\alpha_{1}^{2})+K_{2}^{*}(\alpha_{1}^{2}\alpha_{2}^{2}\alpha_{3}^{2})+...$
(S1)

Here $\alpha_{i}$ are direction cosines of the spontaneous magnetization with
respect to the cubic axes Bozorth (1936). How, when ($hkl$) = (110), the
torque can be written as
$\tau=K_{1}^{*}(2sin2\theta+3sin4\theta)/8+K_{2}^{*}(sin2\theta+4sin4\theta-3sin6\theta)/64$.
Therefore, there is a simple relationship between K${}_{i}^{*}$ and Ki:
$K_{1}^{*}=\frac{8}{5}(4K_{2}-K_{4})$ and
$K_{2}^{*}=\frac{64}{5}(2K_{4}-3K_{2})$.

As seen in Fig. S2f, K4 saturates to about 17 kJ/m3 (1 kJ/m3 = 10$\cdot$ 103
ergs/cm3). On the other hand the saturated value for K2 can be estimates to be
11 kJ/m3. This would yield a K${}_{1}^{*}$ and K${}_{2}^{*}$ of 43.2 and 12.8
kJ/m3. As an see in the table S2, these values are close to what was reported
previously Bozorth (1936); Graham (1958).

Component | Expression | This work value (kJ/m3) | Previously reported (kJ/m3)
---|---|---|---
$\tau$ (This work) |  $K_{2}\cdot sin2\theta+K_{4}\cdot sin4\theta$ Bozorth (1936) | $\backslash$ | $\backslash$
$\tau$ (Literature) | $K_{1}^{*}\cdot(2sin2\theta+3sin4\theta)/8+K_{2}^{*}\cdot(sin2\theta+4sin4\theta-3sin6\theta)/64$ | $\backslash$ | $\backslash$
K2 | $K_{1}^{*}/4+K_{2}^{*}/64$ | 11 (estimated) |  $\sim$ 12 at 2 TGraham (1958)
K4 | $3K_{1}^{*}/8+K_{2}^{*}/16$ | 17 |  $\sim$ 16 at 2 T Graham (1958)
K${}_{1}^{*}$ | $\frac{8}{5}(4K_{2}-K_{4})$ | 43.2 | 40-52.5 Bozorth (1936) or 48 $\pm$ 1 Graham (1958)
K${}_{2}^{*}$ | $\frac{64}{5}(2K_{4}-3K_{2})$ | 12.8 | -17-29 Bozorth (1936) or 0 $\pm$ 5 Graham (1958)
Table S2: Torque’s expressions and values for bcc Fe in (110) plane.

##  S5 Theory of energy from microscopic spin model

Here we extend the small anisotropy expansion of the earlier paper of Liu and
Balents to higher order in the presence of an applied field. We consider the
bulk energy of a uniform state. The basic unit of the structure is a set of 6
spins, forming a triangle in one layer and its counterpart in the neighboring
layer. The spins in the second layer are assumed to be equal to their
counterparts in the first layer. Then the energy depends upon three so far
unspecified spins $\bm{S}_{i}$ with $i=1,2,3$. Then we write the energy per
unit cell – i.e. per 6 spins as

$E_{u.c.}=\sum_{i}\left[4J\bm{S}_{i}\cdot\bm{S}_{i+1}-2K(\bm{S}_{i}\cdot\bm{\hat{e}}_{i})^{2}+4D\bm{\hat{z}}\cdot\bm{S}_{i}\times\bm{S}_{i+1}-2\mu\bm{H}\cdot\bm{S}_{i}\right],$
(S2)

where we use the convention $\bm{S}_{i+3}=\bm{S}_{i}$. The factors account for
the two copies of the triangles etc. We take the spins to be unit vectors so
factors of spin length and magnetic moments should be absorbed into the above
parameters. The parameter $J=J_{1}+J_{2}$.

We assume $K\ll D\ll J$ and $\mu H\ll J$, and $D>0$, and obtain an expansion
of the ground state energy. For $D>0$, the ground state for $K=\mu H=0$ is an
anti-chiral state with spins in the plane.

###  S5.1 Expansion to fourth order for in-plane fields

The most important case is for an in-plane field. So take

$\bm{H}=H(\cos\theta,\sin\theta,0).$ (S3)

We can further assume in this case that the spins remain in the plane even for
non-zero $K$ and $H$, hence

$\bm{S}_{i}=(\cos\phi_{i},\sin\phi_{i},0).$ (S4)

We then write the angles as

$\phi_{1}=\phi+\eta_{1},\qquad\phi_{2}=\phi-\frac{2\pi}{3}+\eta_{2},\qquad\phi_{3}=\phi-\frac{4\pi}{3}-\eta_{1}-\eta_{2}.$
(S5)

Here $\phi$ gives the order parameter angle, and $\eta_{1},\eta_{2}$ are small
distortions of the triangle.

To be systematic, we introduce a small parameter $r\ll 1$, and let
$K\rightarrow Kr$ and $H\rightarrow Hr$, and then expand $\eta_{1,2}$ in a
formal series in $r$ and minimize the energy order by order in $r$. This is
effectively an expansion in $K/J$ and $\mu H/J$, which may safely be
considered small parameters. There is a priori no need to assume $D\ll J$ as a
second small parameter but since in reality it is small, it is sometimes
convenient to simplify very cumbersome algebraic expressions, and we will
occasionally use it.

This procedure can be carried out at fixed $\phi$.

So assuming this condition, we can systematically write

$\eta_{i}=\sum_{n=1}^{\infty}\eta_{i,n}r^{n},$ (S6)

and expand the full energy order by order in $r$. We successively minimize
terms beginning at $O(r^{2})$ in the energy over $\eta_{i,n}$ which appear in
these expressions. This determines the small distortions of the triangle at
each order and results in a fully determined expansion of the energy:

$E_{u.c.}=\sum_{n=0}^{\infty}E_{u.c.}^{(n)},$ (S7)

where

$\displaystyle E_{u.c.}^{(0)}=$ $\displaystyle-6J-6\sqrt{3}D,$ (S8)
$\displaystyle E_{u.c.}^{(1)}=$ $\displaystyle-3K,$ (S9) $\displaystyle
E_{u.c.}^{(2)}=$ $\displaystyle-\frac{(\mu H)^{2}+K^{2}+2\mu
HK\cos(\theta+\phi)}{2\left(\sqrt{3}D+J\right)},$ (S10) $\displaystyle
E_{u.c.}^{(3)}=$
$\displaystyle-\frac{1}{36(J+\sqrt{3}D)^{3}}\Big{[}(3J+7\sqrt{3}D)K^{3}\cos(6\phi)+6(J+3\sqrt{3}D)\mu
HK^{2}\cos(5\phi-\theta)$ $\displaystyle+3(J+5\sqrt{3}D)(\mu
H)^{2}K\cos(4\phi-2\theta)+4\sqrt{3}D(\mu H)^{3}\cos(3\phi-3\theta)\Big{]}.$
(S11)

These expressions are a bit complicated and rather ugly at higher orders.
Higher order terms are negligible for the effects we discuss here.

The energy should now be minimized over the order parameter angle $\phi$. One
can see that the third and second order terms have different angular
dependence, hence there is a competition between the two in determining this
angle.

To proceed, we will assume that the second order term is dominant, being lower
order. This is true unless $H$ becomes very small, because the second order
term’s angular dependence vanishes for $H=0$. Comparing the coefficient of
$\cos(\phi+\theta)$ from the second order term and the coefficient of
$\cos(6\phi)$ from the third order term, we see this implies the condition

$\mu H\gg K^{2}/J.$ (S12)

When this is true, (and recall we assumed $\mu H,K\ll J$) the second order
term is parametrically larger, so $\phi$ must be close (but not equal!) to the
minimum of this term. Hence we can write $\phi=-\theta+\psi$, and we expect
$\psi\ll 1$. Therefore we expand $E_{u.c.}^{(2)}$ to second order in $\psi$
(it is quadratic around its minimum) and $E_{u.c.}^{(3)}$ to first order in
$\psi$, and minimize over $\psi$. This leads to the final expression for the
energy

$\displaystyle E_{u.c.}=-6J-6\sqrt{3}D-3K$ $\displaystyle-\frac{(\mu
H+K)^{2}}{2(J+\sqrt{3}D)}\Big{[}1+\frac{(3J+7\sqrt{3}D)K+4\sqrt{3}D\mu
H}{18(J+\sqrt{3}D)^{2}}\cos(6\theta)$
$\displaystyle+\frac{\left((3J+7\sqrt{3}D)K^{2}+2(J+4\sqrt{3}D)\mu
HK+2\sqrt{3}D(\mu H)^{2}\right)^{2}}{36(J+\sqrt{3}D)^{4}\mu
HK}\sin^{2}(6\theta)\Big{]}.$ (S13)

We see that by including the small shift in the angle ($\psi$), we generated a
12-fold harmonic (contained in the $\sin^{2}(6\theta)$ factor. If we take the
high field limit, $\mu H\gg K$ (this is a stronger condition than Eq. (S12))
the expression simplifies further to

$\displaystyle E_{u.c.}=-6J-6\sqrt{3}D-3K-\frac{(\mu
H)^{2}}{2(J+\sqrt{3}D)}\Big{[}$ $\displaystyle 1+\frac{2\sqrt{3}D\mu
H}{9(J+\sqrt{3}D)^{2}}\cos(6\theta)+\frac{D^{2}(\mu
H)^{3}}{3(J+\sqrt{3}D)^{4}K}\sin^{2}(6\theta)\Big{]}.$ (S14)

In solving for the energy, we obtain the angles $\eta_{1},\eta_{2}$ and
$\psi$:

$\displaystyle\eta_{1}=$ $\displaystyle\frac{\mu
H+K}{3(J+\sqrt{3}D)}\Big{[}\sin 2\theta-\frac{\mu H+2K}{6\mu
HK(J+\sqrt{3}D)^{2}}\left(2\sqrt{3}D(\mu H)^{2}+2(J+4\sqrt{3}D)\mu
HK+(3J+7\sqrt{3}D)K^{2}\right)\cos 2\theta\sin 6\theta\Big{]},$
$\displaystyle\eta_{2}=$ $\displaystyle\frac{\mu
H+K}{3(J+\sqrt{3}D)}\Big{[}\cos(2\theta+\frac{\pi}{6})$
$\displaystyle+\frac{\mu H+2K}{6BK(J+\sqrt{3}D)^{2}}\left(2\sqrt{3}D(\mu
H)^{2}+2(J+4\sqrt{3}D)\mu
HK+(3J+7\sqrt{3}D)K^{2}\right)\cos(2\theta-\frac{\pi}{3})\sin 6\theta\Big{]},$
$\displaystyle\psi=$ $\displaystyle\frac{B+K}{6\mu
HK(J+\sqrt{3}D)^{2}}\left(2\sqrt{3}D(\mu H)^{2}+2(J+4\sqrt{3}D)\mu
HK+(3J+7\sqrt{3}D)K^{2}\right)\sin 6\theta.$ (S15)

Despite the complexity, here are a few features:

  * •

A $H^{2}$ term appears first at order $n=2$, and has no angular dependence.
Because terms successively decrease with increasing $n$, this is the dominant
contribution to the $H^{2}$ term. It represents term in the magnetization
linear in field, i.e. just a susceptibility. The $D$ in the denominator is
subdominant, and so the susceptibility is just order $1/J$.

  * •

The $H^{3}$ term appears first at order $n=3$. Examining this expression we
see that this term however is linear in $D$ and vanishes for $D=0$. The third
order contribution has a pure $\cos 6\theta$ dependence.

  * •

The $\cos 12\theta$ harmonic is formally of fourth order (e.g. in Eq. (S14)
its coefficient is order $H^{5}/K$). One can check that no additional 12-fold
contribution arises if we include the $E_{u.c.}^{(4)}$ correction.

Many of these features may match experimental observations. A few notable
differences from theory are:

  * •

There are no 2-fold harmonics. They can appear only from effects that break
the six-fold structural symmetry of the solid, e.g. finite size shape effects,
or strain due to magnetostriction in the ordered state.

  * •

The energy is analytic in $H$. This means that the fractional power appearing
experimentally in the two-fold anisotropy must arise from additional physics
not included here, which is hardly surprising since the two-fold anisotropy
itself is not present in this model.

  * •

The $H^{3}$ term has a pure $\cos(6\theta)$ dependence on the angle, i.e.
there is no angle-independent part. A small angle-independent part does arise
at higher orders in the expansion but it is negligible. This means that the
angle-independent $H^{3}$ term seen in experiment must arise from effects
beyond the simple model studied here.

###  S5.2 Arbitrary spin angle

It is too cumbersome to carry out the full expansion to 6th order for applied
fields in arbitrary directions, but we can do so for the first few orders. In
this case, we can repeat similar manipulations but not assuming in-plane
spins. Suppressing the details, we obtain the following energy to second order
in $H,K$:

$\displaystyle E_{u.c.}=$
$\displaystyle-6J-6\sqrt{3}D-3K-\frac{1}{2J}\left(1-\frac{\sqrt{3}D}{J}\right)(K+\mu
H_{xy})^{2}$ $\displaystyle-\frac{1}{2J}\left(1-\frac{D}{\sqrt{3}J}\right)(\mu
H_{z})^{2}+O(r^{3}).$ (S16)

From this we can see that the linear in field magnetization is almost
isotropic, i.e. the fractional difference in differential susceptibility in
the two directions is order $D/J$.


# Sofic Lie Algebras

Cameron Cinel Department of Mathematics, University of California, San Diego,
La Jolla, CA 92093-0112, USA Email<EMAIL_ADDRESS>
(March 3, 2022)

###### Abstract

We introduce and study soficity for Lie algebras, modelled after linear
soficity in associative algebras. We introduce equivalent definitions of
soficity, one involving metric ultraproducts and the other involving almost
representations. We prove that any Lie algebra of subexponential growth is
sofic. We also prove that a Lie algebra over a field of characteristic 0 is
sofic if and only if its universal enveloping algebra is linearly sofic.
Finally, we give explicit families of almost representations for the Witt and
Virasoro algebras.

Key words and phrases: linearly sofic algebra, universal enveloping algebra,
metric ultraproduct, subexponential growth

Mathematics Subject Classification: 17B35, 03C20, 17B68

##  1 Introduction

Sofic groups were first introduced by Misha Gromov [Gro99] in relation to the
Gottschalk surjunctiviy surjecture conjecture (1976). The were later named
sofic by Weiss [Wei00]. We recall the definitions of sofic groups via
ultraproducts. For more details of ultraproducts in relation to soficity, see
§2 of [Pes08].

For $n\in\mathbb{N}$, we denote the symmetric group on $n$ letters by $S_{n}$.
We consider the normalized Hamming distance

$d_{n}(\sigma,\tau)=\frac{1}{n}|\\{i\mid\sigma(i)\neq\tau(i)\\}|.$

######  Definition 1.1.

A group $G$ is sofic if there exists and ultrafilter $\omega$ on $\mathbb{N}$
and a sequence of natural numbers $(n_{k})$ such that $G$ embeds into the
metric ultraproduct $\prod_{k\to\omega}(S_{n_{k}},d_{n_{k}})$.

In [ES05], this definition was reformulated without the notion of
ultraproducts. It was shown that a group is sofic if and only if it can be
locally embedded in some symmetric group almost homomorphically, up to some
arbitrarily small error. Similar ultraproducts can be constructed from any
family of groups equipped with bi-invariant metrics, such as unitary groups
over $\mathbb{C}$ (where subgroups are called hyperlinear).

In particular, for any field $F$, the group $GL_{n}(F)$ can be endowed the
normalized rank metric

$\rho_{n}(A,B)=\frac{1}{n}\text{rank}(A-B).$

Groups embedable into ultraproducts of $GL_{n}(F)$ with this metric are called
linearly sofic. In [AP17] the notion of linear soficity was extended to
associative algebras by considering metric ultraproducts of $M_{n}(F)$. In
particular, it was shown that a group $G$ is linearly sofic if and only if its
group algebra $\mathbb{C}[G]$ is linearly sofic.

The goal of this paper to do analogous work for Lie algebras. Our first
section deals with the ultraproduct construction and definition of soficity
for Lie algebras as well as basic properties. We then provide an equivalent
characterization for soficity using almost representations for our Lie
algebra.

We also provide examples of sofic Lie algebras using families of almost
representations. In particular we prove one of our primary results

######  Theorem 4.2.

Any Lie algebra of subexponential growth is sofic.

We also provide explicit almost representations to show soficity for the Witt
and Virasoro algebras. This is analogous to soficity in groups and associative
algebras, where amenability, and therefore subexponential growth, implies
soficity.

Finally, we show that soficity for Lie algebras and associative algebras is
compatible.

######  Theorem 5.2.

Let $L$ be a Lie algebra over a field of characteristic 0. Then $L$ is sofic
if and only if its universal enveloping algebra $U(L)$ is sofic.

This result should be compared to other work on the growth of Lie algebras,
notably the theorem in [Smi76] showing that Lie algebras of subexponential
growth have universal enveloping algebras of subexponential growth.

##  2 Ultraproducts

We define our notion of soficity for Lie algebras using the ultra product
construction, similar to that of sofic groups and associative algebras. For
$n\in\mathbb{N}$, we call the function

$\displaystyle\rho_{n}:M_{n}(F)$ $\displaystyle\to[0,1]$ $\displaystyle a$
$\displaystyle\mapsto\frac{1}{n}\text{rank}(a)$

the normalized rank function. Let $\omega$ be a non-principal ultrafilter on
$\mathbb{N}$ and $n_{k}$ be a sequence of natural numbers such that
$\lim_{k\to\infty}n_{k}=\infty$. We can extend the $\rho_{n_{k}}$’s to a
function

$\displaystyle\rho_{n_{k}}:M_{n_{k}}(F)$ $\displaystyle\to[0,1]$
$\displaystyle(a_{k})$
$\displaystyle\mapsto\lim\limits_{k\to\omega}\rho_{n_{k}}(a_{k}).$

In [AP17], the pre-image $\rho_{\omega}^{-1}(0)$ was shown to be an ideal.
Thus we can construct the ultra-product of the matrix rings, denoted

$\prod_{k\to\omega}M_{n_{k}}(F)/\ker\rho_{\omega}:=\prod_{k=1}^{\infty}M_{n_{k}}(F)/\rho_{\omega}^{-1}(0).$

As an associative algebra, this ultra-product has a natural Lie algebra
structure via the commuatator bracket, which we denote

$\prod_{k\to\omega}\mathfrak{gl}_{n_{k}}(F)/\ker\rho_{\omega}:=\left(\prod_{k\to\omega}M_{n_{k}}(F)/\ker\rho_{\omega}\right)^{(-)}.$

######  Definition 2.1.

We denote the metric ultraproduct of the Lie algebras
$\mathfrak{gl}_{n_{k}}(F)$ as
$\prod_{k\to\omega}\mathfrak{gl}_{n_{k}}(F)/\ker\rho_{\omega}$, called a
universal sofic Lie algebra.

For any subalgebra $L\subset\mathfrak{gl}_{n}(F)$, we can restrict the
normalized rank map and hence create different metric ultraproducts.

######  Lemma 2.1.

For any field $F$,

$\prod_{k\to\omega}\mathfrak{gl}_{n_{k}}(F)/\ker\rho_{\omega}\cong\prod_{k\to\omega}\mathfrak{sl}_{n_{k}}(F)/\ker\rho_{\omega}.$

###### Proof.

Clearly, we can embed
$\prod_{k\to\omega}\mathfrak{sl}_{n_{k}}(F)/\ker\rho_{\omega}\hookrightarrow\prod_{k\to\omega}\mathfrak{gl}_{n_{k}}(F)/\ker\rho_{\omega}$.
Moreover, for any sequence
$(A_{k})\in\prod_{k\to\omega}\mathfrak{gl}_{n_{k}}(F)/\ker\rho_{\omega}$, we
can consider the second sequence $(A_{k}-E_{11}\text{tr}(A))$. Then

$\rho_{\omega}((A_{k})-(A_{k}-E_{11}\text{tr}(A)))=\lim_{k\to\omega}\frac{1}{n_{k}}=0$

so $(A_{k})=(A_{k}-E_{11}\text{tr}(A))$ in the ultra-product. Thus the natural
embedding is a surjection. ∎

We now define the map object of this paper.

######  Definition 2.2.

A Lie algebra $L$ over a field $F$ is called linearly sofic if there exists a
Lie algebra embedding of $L$ into some metric ultraproduct of some
$\mathfrak{gl}_{n_{k}}(F)$’s.

######  Proposition 2.2.

For an algebraically closed field $F$, the associative algebra
$\prod_{k\to\omega}M_{n_{k}}(F)/\ker\rho_{\omega}$ is simple.

###### Proof.

If $\omega$ is a principal ultrafilter, this is trivially true. So we assume
that $\omega$ is free.

Denote $R:=\prod_{k\to\omega}M_{n_{k}}(F)/\ker\rho_{\omega}$. Suppose $0\neq
x\in R$. Let $\delta:=\rho_{\omega}(x)$. Since $\delta>0$, there exists
$n\in\mathbb{N}$ such that $\frac{1}{n}<\delta$. Thus, if $(B_{k})\in\prod
M_{n_{k}}(F)$ is a representative for $x$, there exists some $S\in\omega$ such
that $\rho_{n_{k}}(B_{k})\geq\frac{1}{n}$ for every $k\in S$.

Let $J_{k}$ be the Jordan normal form of $B_{k}$. Then $J_{k}$ row equivalent
to

$A^{(m)}_{k}:=\sum_{i=1}^{\lfloor\frac{n_{k}}{n}\rfloor}E_{\lfloor\frac{mn_{k}}{n}\rfloor+i,\lfloor\frac{mn_{k}}{n}\rfloor+i}$

where $0\leq m\leq n-1$. Thus, there exists $P_{k},Q_{k}\in M_{n_{k}}(F)$ such
that

$B_{k}=P_{k}A^{(m)}_{k}Q_{k}.$

Define $C_{k}\in M_{n_{k}}(F)$ via

$C_{k}=\begin{cases}\sum_{m=0}^{n-1}A^{(m)}_{k},&k\in S\\\ 0,&k\in
S^{C}\end{cases}.$

Then if $y=[C_{k}]\in R$, we have that $y\in(x)$, the ideal generated by $x$.
Moreover, we notice that for $k\in S$,

$\text{rank}(C_{k}-I_{n_{k}})\leq n-1.$

Let $\varepsilon>0$. Since $\omega$ is a free ultrafilter, $S$ must be
infinite. Therefore, there exists an infinite subset $S^{\prime}\subset S$
such that $\rho_{n_{k}}(C_{k}-I_{n_{k}})<\varepsilon$. Notice that
$S^{\prime}\in\omega$ since otherwise, $(S^{\prime})^{C}\cap S\in\omega$
making $\omega$ a principal ultrafilter. Therefore
$\rho_{\omega}(y-[I_{n_{k}}])=0$ and $(x)=R$. ∎

######  Corollary 2.2.1.

For an algebraically closed field $F$, the non-trivial ideals of
$\prod_{k\to\omega}\mathfrak{gl}_{n_{k}}(F)/\ker\rho_{\omega}$ are contained
in its center.

###### Proof.

This follows from Theorem 2 of [Her61]. ∎

##  3 Almost representations

Just as with sofic groups and linearly sofic associative algebras, linearly
sofic Lie algebras can be defined via families of maps that are Lie algebra
homomorphisms up to a small error. We follow a similar approach to section
11.2 of [AP17].

######  Definition 3.1.

Let $L$ be a Lie algebra, $W\subset L$ a finite dimensional subspace, $V$ a
finite dimensional vector space, and $\varepsilon>0$. A linear map
$\varphi:W\to\mathfrak{gl}(V)$ is called and $\varepsilon$-almost
representation of $W$ if there exists a subspace $V_{\varepsilon}\subset V$
such that:

  1. 1.

for all $x,y\in W$ such that $[x,y]\in W$,

$\varphi([x,y])|_{V_{\varepsilon}}=(\varphi(x)\varphi(y)-\varphi(y)\varphi(x))|_{V_{\varepsilon}};$

  2. 2.

$\dim V$-$\dim V_{\varepsilon}\leq\varepsilon\cdot\dim V$.

Clearly a morphism from a Lie algebra to a universal sofic Lie algebra gives
rise to a family of almost representations of the Lie algebra. Similarly, a
family of almost representations on a covering of a Lie algebra gives a
morphism to a universal sofic Lie algebra. However, not all families of almost
representations will correspond to embeddings. In particular, some elements of
our Lie algebra will have to be mapped to arbitrarily small rank
transformations via the family of almost representations. This gives rise to
to the following subspace of our Lie algebra.

######  Definition 3.2.

For a Lie algebra $L$, the sofic radical of $L$, denote $SR(L)$, is defined as
follows: $p\in SR(L)$ if for every $\delta>0$, there exists a finite
dimensional $W\subset L$ containing $p$ and $n_{\delta}>0$ such that if
$0<\varepsilon<n_{\delta}$ and $\varphi_{W,\varepsilon}:W\to\mathfrak{gl}(V)$
is an $\varepsilon$-almost representation, then

$\dim\text{Im }\varphi_{W,\varepsilon}(p)<\delta\cdot\dim V.$

The sofic radical is essentially the collection of ”bad” elements of our Lie
algebra when it comes to trying to make embeddings of it into a universal
sofic Lie algebra. This view is summarized by the following lemma and
corollary.

######  Lemma 3.1.

For a Lie algebra $L$, $p\in SR(L)$ if and only if for every Lie algebra
homomorphism

$\Theta:L\to\prod_{\omega}\mathfrak{gl}_{n_{k}}(F)/\rho_{\omega}$

we have that $\Theta(p)=0$.

###### Proof.

Suppose $p\in SR(L)$ and let
$\Theta:L\to\prod_{\omega}\mathfrak{gl}_{n_{k}}(F)/\rho_{\omega}$ be a Lie
algebra homomorphism with lifts $\theta_{k}:L\to\mathfrak{gl}_{n_{k}}(F)$. Fix
$\delta>0$ and choose $n_{\delta}$ and finite dimensional $W\subset L$ from
the definition of the sofic radical. Choose $0<\varepsilon<n_{\delta}$. Then
there exists $S\in\omega$ such that $\theta_{k}|_{W}$ is an
$\varepsilon$-almost representation for $W$ for every $k\in S$. Therefore
$\dim\text{Im }\theta_{k}(p)<\delta n_{k}$. In other words,
$\rho_{n_{k}}(\theta_{k}(p))<\delta$ for every $k\in H$. Therefore
$\rho_{\omega}(\Theta(p))<\delta$. Since $\delta$ was arbitrary, we get that
$\Theta(p)=0$.

Now suppose that $p\in L\setminus SR(L)$. Then there exists $\delta>0$ such
that for any finite dimensional $W\subset L$ containing $p$ and $n>0$, there
exists $0<\varepsilon<n$ and $\varepsilon$-almost representation
$\varphi:L\to\mathfrak{gl}(V)$ such that $\dim\text{Im
}\varphi(p)\geq\delta\cdot\dim V$.

Let $W_{k}$ be an increasing sequence of finite dimensional subspaces of $L$
containing $p$ that cover $L$. Then there exists a sequence of
$\varepsilon_{k}$ that converges to 0 and $\varepsilon_{k}$-almost
representations $\theta_{k}:W_{k}\to\mathfrak{gl}(V_{k})$ such that
$\dim\text{Im }\theta_{k}(p)\geq\delta\cdot\dim V_{k}$. Define a map
$\hat{\Theta}:L\to\prod\mathfrak{gl}(V_{k})$ by
$\hat{\Theta}(x)=(\hat{\theta}_{k}(x))$ where

$\hat{\theta}_{k}(x)=\begin{cases}\theta_{k}(x),&x\in W_{k}\\\ 0,&x\in
L\setminus W_{k}\end{cases}.$

Choose a non-principal ultrafilter $\omega$ of $\mathbb{N}$ and let
$\Theta:L\to\prod_{\omega}\mathfrak{gl}(V_{k})/\rho_{\omega}$ be the
composition of $\hat{\Theta}$ with the quotient map. Then since
$\varepsilon_{k}\to 0$, we have that $\Theta$ is a Lie algebra homomorphism
and $\rho_{\omega}(\Theta(p))\geq\delta>0$. ∎

From this lemma, we get the following corollary

######  Corollary 3.1.1.

For a Lie algebra $L$, $SR(L)$ is an ideal. Moreover, $SR(L/SR(L))=(0)$.

###### Proof.

The lemma show that

$SR(L)=\bigcap_{\Theta\in S}\ker\Theta$

where

$S=\left\\{\Theta:L\to\prod_{\omega}\mathfrak{gl}_{n_{k}}(F)/\rho_{\omega}\mid\Theta\text{
is a Lie algebra homomorphism}\right\\}$

so it is clear that $SR(L)$ is an ideal.

Now suppose $0\neq p\in L/SR(L)$ and let $q\in L$ be a pre-image of $p$. Then
$q\notin SR(L)$ so there exists a Lie algebra homomorphism
$\Theta:L\to\prod_{\omega}\mathfrak{gl}_{n_{k}}(F)/\rho_{\omega}$ such that
$\Theta(q)\neq 0$. Since $SR(L)\subset\ker\Theta$, we can get a map
$\hat{\Theta}$ by composing $\Theta$ with the quotient $L\to L/SR(L)$. Then we
have that $\hat{\Theta}(p)=\Theta(q)\neq 0$ so $p\notin SR(L/SR(L))$. ∎

We can now use the sofic radical to characterize sofic Lie algebras.

######  Theorem 3.2.

A Lie algebra $L$ is sofic if and only if $SR(L)=(0)$.

###### Proof.

The forward implication is trivial so we only show the reverse implication.

Let $L$ be a Lie algebra such that $SR(L)=0$. Then for every $p\in
L\setminus(0)$, there exists a Lie algebra homomorphism
$\Theta_{p}:L\to\prod_{\omega}\mathfrak{gl}_{n_{k,p}}(F)/\rho_{\omega}$ such
that $\Theta_{p}(p)\neq 0$.

Let $\\{x_{i}\\}_{i\in\mathbb{N}}\subset L$ be a basis for $L$ as a vector
space. We shall construct maps
$\Psi_{m}:L\to\prod_{\omega}\mathfrak{gl}_{n_{k,m}}/\rho_{\omega}$ such that
$\ker\Psi_{m}\cap\text{span}\\{x_{1},\dots,x_{m}\\}=(0)$ for every
$m\in\mathbb{N}$.

Let $\Psi_{1}=\Theta_{x_{1}}$. Now suppose for $m\in\mathbb{N}_{\geq 2}$, we
have a map $\Psi_{m-1}$ as above. Then

$\dim(\Psi_{m-1}\cap\text{span}\\{x_{1},\dots,x_{m}\\})\leq 1.$

If the dimension is 0, let $\Psi_{m}=\Psi_{m-1}$. Otherwise, choose a non-zero
element $y_{m}$ in the intersection. In this case, we define
$\Psi_{m}=\Psi_{m-1}\oplus\Theta_{y_{m}}$. Then if
$z\in\ker\Psi_{m}\cap\text{span}\\{x_{1},\dots,x_{m}\\}$, we have that
$z\in\ker\Psi_{m-1}\cap\text{span}\\{x_{1},\dots,x_{m}\\}$. Thus $z=\alpha
y_{m}$ for some $\alpha\in F$. However, $z\in\ker\Theta_{y_{m}}$ so $z=0$.

Now we construct a map
$\Phi:L\to\prod_{\omega}\mathfrak{gl}_{n_{k}}(F)/\rho_{\omega}$ such that
$\ker\Phi\subset\bigcap\ker\Psi_{m}$. Let $n_{k}=n_{k,1}\cdots n_{k,k}$.
Tensor each component of $\Psi_{m}$ with an appropriately sized identity
matrix gives us maps
$\hat{\Psi}_{m}:L\to\prod_{\omega}\mathfrak{gl}_{\hat{n}_{k,m}}(F)/\rho_{\omega}$
where $\hat{n}_{k,m}=n_{k}$ if $m\leq k$ and $n_{k,m}$ otherwise. Let
$\hat{\psi}_{k,m}:L\to\mathfrak{gl}_{n_{k}}(F)$ be a lift of $\hat{\Psi}_{m}$
for $m\leq k$.

Define a map $\varphi_{k}:A\to\mathfrak{gl}_{2^{k}n_{k}}$ by

$\varphi_{k}=(\hat{\psi}_{k,1}\otimes\text{Id}_{2^{k-1}}\oplus(\hat{\psi}_{k,2}\otimes\text{Id}_{2^{k-2}})\oplus\cdots\oplus(\hat{\psi}_{k,k}\otimes\text{Id}_{1})\oplus\text{Id}_{n_{k}}.$

Let $\Phi=\prod_{\omega}\varphi_{k}/\rho_{\omega}$.

A direct calculation gives us that for any $x\in L$

$\displaystyle\rho_{\omega}(\Phi(x))$
$\displaystyle=\lim_{k\to\omega}\frac{\text{rank}(\varphi_{k}(x))}{2^{k}n_{k}^{k}}$
$\displaystyle=\lim_{k\to\omega}\frac{1}{2^{k}n_{k}^{k}}\sum_{i=1}^{k}\text{rank}((\hat{\psi}_{k,1}\otimes\text{Id}_{2^{k-1}})(x))$
$\displaystyle=\lim_{k\to\omega}\frac{1}{n_{k}^{k}}\sum_{i=1}^{k}\frac{\text{rank}(\hat{\psi}_{k,1}(x))}{2^{i}}$
$\displaystyle=\lim_{k\to\omega}\sum_{i=1}^{k}\frac{\text{rank}(\psi_{k,1}(x))}{2^{i}n_{k,i}}$
$\displaystyle=\sum_{i=1}^{k}\frac{1}{2^{i}}\rho_{\omega}(\Psi_{i}(x))$

where $\psi_{k,m}:L\to\mathfrak{gl}_{n_{k,m}}$ is a lift of $\Psi_{m}$.

Therefore $\ker\Phi\subset\bigcap\ker\Psi_{m}=(0)$, so we have that $\Phi$ is
injective. ∎

##  4 Examples of Sofic Lie Algebras

By using the sofic radical, we can determine if particular Lie algebras is
sofic.

######  Proposition 4.1.

Any abelian Lie algebra is sofic.

###### Proof.

Suppose $L$ is an abelian Lie algebra over a field $F$ and suppose that $0\neq
p\in L$. Then for any finite dimensional subspace $p\in W\subset L$ and
$\varepsilon>0$, there exists an $\varepsilon$-almost representation
$\varphi:W\to F\cong\mathfrak{gl}_{1}(F)$ such that $\varphi(p)=1$ and
$\varphi(W\setminus Fp)=0$. Thus $\dim\text{Im}\varphi(p)=1=\dim F$ so
$p\notin SR(L)$. ∎

For groups, those of subexponential growth are amenable and therefore sofic.
We show the same result holds true for Lie alebras.

######  Theorem 4.2.

Any Lie algebra of subexponential growth is sofic.

###### Proof.

Let $L$ be a Lie algebra of subeponential growth generated by the set
$X\subset L$. Let $V_{n}$ denote the words in $L$ of length at most $n$. We
inductively create a basis for $L$ as follows. Let $\\{x_{1},\dots,x_{\dim
V_{1}}\\}$ be a basis for $V_{1}$. For $n\geq 2$, let $\\{x_{\dim
V_{n-1}+1},\dots,x_{\dim V_{n}}\\}$ be the preimage in $L$ of a basis for
$V_{n}/V_{n-1}$. We also define $W_{n}$ to be the subspace of $U(L)$ spanned
by words of length at most $n$ and $\gamma(n)=\dim W_{n}$.

For $m>n$, we define a linear map $\varphi_{n,m}:V_{n}\to\mathfrak{gl}(W_{m})$
where

$\varphi_{n,m}(x_{i})(x_{j_{1}}\cdots
x_{j_{k}})=\begin{cases}x_{i}x_{j_{1}}\cdots x_{j_{k}},&x_{i}x_{j_{1}}\cdots
x_{j_{k}}\in W_{m}\\\ 0,&\text{otherwise}\end{cases}.$

We notice on that for $v\in W_{m-n}$ and $x,y\in V_{n}$,

$[\varphi_{n,m,k}(x),\varphi_{n,m,k}(y)](v)=\varphi_{n,m,k}([x,y])(v).$

Let us show that for a fixed $n$ that

$\lim_{m\to\infty}\frac{\gamma(m-n)}{\gamma(m)}=1.$

Since $L$ is of subexponential growth, we have that $\gamma$ is a function of
subexponential growth from [Smi76]. To see this suppose that
$f:\mathbb{R}\to\mathbb{R}$ is a function of subexponential growth and $d>0$.
We have that if

$\lim_{x\to\infty}\frac{f(x)}{f(x+d)}=\frac{1}{L}<1$

Then there exists $x_{0}\in\mathbb{R}$ such that

$f(x_{0}+nd)\geq L^{n}f(x_{0}).$

Therefore $f(x)\geq L^{\frac{x-x_{0}}{d-1}}f(x_{0})$ and is therefore not of
subexponential growth.

Thus for any $n\in\mathbb{N}$ and $\varepsilon>0$, we can find $\varphi_{n,m}$
such that

$\frac{\dim W_{m}-\dim W_{n-m}}{\dim W_{m}}<1-(1-\varepsilon)=\varepsilon.$

Now suppose $p\in L\setminus\\{0\\}$, $V\subset L$ is a finite dimensional
subspace containing $p$, and $\varepsilon>0$. Then there exists
$n\in\mathbb{N}$ such that $V\subset V_{n}$. Thus for a sufficiently large
$m\in\mathbb{N}$, we have an $\varepsilon$-almost representation
$\psi:=\varphi_{n,m}|_{V}$ for $V$. On $W_{m-n}$, we have that $\psi(p)$ works
like left multiplication by $p$, considered as an element of $U(L)$. Thus
$\dim\text{Im}(\psi(p))\geq\dim W_{m-n}$. By our choice of $m$ to make $\psi$
an $\varepsilon$-almost representation, we have that

$\dim W_{m-n}>(1-\varepsilon)\dim W_{m}.$

Hence the normalized rank of $\psi(p)$ is greater than $1-\varepsilon$.

Thus for any non-zero $p\in L$, any finite dimensional subspace $p\in V\subset
L$, and $\varepsilon>0$, we can find an $\varepsilon$-almost representation
$\psi$ for $V$ such that the normalized rank of $\psi(p)$ is at least 1/2.
Specifically, we can choose $\psi$ to be a $\min\\{\varepsilon,1/2\\}$-almost
representation. Therefore $p\notin SR(L)$ so $SR(L)=(0)$ and $L$ is sofic. ∎

We now provide explicit families of almost representations for two particular
Lie algebras. Though both of the examples are of subexponential growth, their
families of almost homomorphisms are not based on their subspaces of words of
particular length.

######  Example 4.3.

The Witt algebra $L_{W}$, which is the Lie algebra of derivations for
$\mathbb{C}[t,t^{-1}]$, is sofic.

We can cover $L_{W}$ by the finitely dimensional subspaces

$V_{n}=\sum_{i=-n}^{n}\mathbb{C}x_{i}$

where $x_{i}=-t^{n+1}\frac{d}{dt}$. We also consider the finite dimensional
vector spaces

$W_{n}=\sum_{i=-n}^{n}\mathbb{C}t^{i}\subset\mathbb{C}[t,t^{-1}].$

For $m\geq n$, define a linear map
$\varphi_{n,m}:V_{n}\to\mathfrak{gl}(W_{m})$ via

$\varphi_{n,m}(x_{i})(t^{j})=\begin{cases}-jt^{i+j},&j\leq m-i\\\
0,&\text{otherwise}\end{cases}$

and

$\varphi_{n,m}(x_{-i})(t^{j})=\begin{cases}-jt^{j-i},&j\geq i-m\\\
0,&\text{otherwise}\end{cases}$

for $i\geq 0$. For any $n\in\mathbb{N}$ and $\varepsilon>0$, $m$ can be made
sufficiently large so as to make $\varphi_{n,m}$ and $\varepsilon$-almost
homomorphism.

######  Example 4.4.

The Virasoro algebra $L_{V}$, which is the unique central extension of
$L_{W}$, is sofic.

To construct the family of almost representations, we need to use the Verma
modules of $L_{V}$. For more information on the Verma modules, see [WY86].

We have a basis of $L_{V}$ given by the $x_{i}$’s from $L_{W}$ and a central
element $c$. Define the subspaces $\mathfrak{h}=\mathbb{C}x_{0}+\mathbb{C}c$
and

$\mathfrak{n}_{+}=\sum_{k=1}^{\infty}\mathbb{C}x_{k}.$

Given $\lambda\in\mathfrak{h}^{*}$, we define the Verma module to be the space

$M(\lambda)=U(L_{V})\otimes_{U(\mathfrak{h}\oplus\mathfrak{n}_{+})}\mathbb{C}$

where the action of $\mathfrak{h}\oplus\mathfrak{n}_{+}$ on $\mathbb{C}$ is
given by

$(h+x)\cdot\alpha=\lambda(h)\alpha.$

For $m,d\in\mathbb{N}$, we consider the subspaces

$M(\lambda)_{m,d}=\text{span}\\{x_{-i_{1}}\cdots x_{-i_{k}}\otimes 1\mid 0\leq
k\leq r,0\leq i_{k}\leq\cdots\leq i_{1}\leq m\\}.$

We then define linear maps
$\varphi_{n,m,d}:V_{n}+\mathbb{C}c\to\mathfrak{gl}(M(\lambda)_{m,d})$ via
$\varphi_{n,m,d}(c)=\text{id}_{M(\lambda)_{m,d}}$ and

$\varphi_{n,m,d}(x_{r})(x_{-i_{1}}\cdots x_{-i_{k}})=\begin{cases}0,&k=d\text{
or }r\geq i_{1}-m\\\ x_{k}x_{-i_{1}}\cdots
x_{-i_{k}},&\text{otherwise}\end{cases}.$

Just as in the case with $L_{W}$, for a fixed $n,d\in\mathbb{N}$ and
$\varepsilon>0$, we can choose $m$ large enough to make $\varphi_{n,m,d}$ an
$\varepsilon$-almost representation.

##  5 Soficity of Universal Enveloping Algebras

In [AP17], it was shown that a group is linearly sofic if and only if its
corresponding group algebra is itself linearly sofic. For Lie algebras, we
have a similar construction with universal enveloping algebras. This section
is spent on showing equivalence of soficity for Lie algebras and universal
enveloping algebras.

We first need the following result from [AP17].

######  Proposition 5.1.

Let $A$ be a unital algebra and $\omega\subset\mathscr{P}(\mathbb{N})$ an
ultrafilter. If we have a set $\\{\Theta^{i}\\}_{i\in I}$ of unital algebra
morphisms from $A$ to $\prod_{\omega}M_{n_{k,i}}(F)/\rho_{\omega}$, then there
exists a unital algebra homomorphism
$\psi:A\to\prod_{\omega}M_{m}(F)/\rho_{\omega}$ such that

$\ker\psi\subset\bigcap_{i\in I}\ker\Theta^{i}.$

We can now tackle our main result:

######  Theorem 5.2.

Let $L$ be a Lie algebra over a field $F$ of characteristic 0. Then $L$ is
sofic if and only if $U(L)$ is sofic.

###### Proof.

Since $L$ embeds in $U(L)$ the reverse direction is trivial. Thus we only
prove the forward direction.

Since $L$ is sofic, there exists a Lie algebra homomorphism

$\Theta:L\to\prod_{\omega}\mathfrak{gl}_{n_{k}}(F)/\rho_{\omega}=:A.$

Let $\Theta_{1}=\Theta$ and

$\Theta_{j}=\underbrace{1\otimes\cdots\otimes 1}_{j-1\text{
times}}\otimes\Theta$

for $j\geq 2$. Define a new Lie algebra homomorphism
$\Theta^{i}:L\to\bigotimes_{j=1}^{i}A$ by $\Theta^{0}=0$, $\Theta^{1}=\Theta$,
and $\Theta^{i}=\Theta^{i-1}\otimes 1+1\otimes\Theta_{i-1}$. Since finite
tensor product of matrix rings are again matrix rings, we use the proposition
to get a unital algebra homomorphism

$\psi:U(L)\to\prod_{\omega}{M_{m}(F)}/\rho_{\omega}$

such that

$\ker\psi\subset\bigcap_{i=0}^{\infty}\widetilde{\Theta}^{i}.$

We claim that $\psi$ is injective.

Let $\\{x_{j}\\}_{j\in J}\subset L$ be a basis. First suppose that
$x_{j_{1}}\cdots x_{j_{d}}\in\ker\psi$ is a monomial for $d\geq 2$. Then by
construction

$\widetilde{\Theta}^{i}(x_{j_{1}}\cdots x_{j_{d}})=0$

for all $i\geq 1$. Notice that $\Theta^{d}(x_{j_{1}}\cdots x_{j_{d}})$ will
consist of a linear sum of pure tensors of $1$ and products of the
$\Theta(x_{j_{i}})$’s. In fact, we have two sets of pure tensors: one set
consisting of those with at least one 1 and at least one degree 2 or greater
monomial and the other set consisting of pure tensors with only linear
monomials and no 1’s, i.e. pure tensors of the form
$x_{j_{\sigma(1)}}\otimes\cdots\otimes x_{j_{\sigma(d)}}$ for some $\sigma\in
S_{d}$. In fact we get exactly one such pure tensor for every $\sigma\in
S_{d}$. Call the first set the lower order tensors and the second set the high
order tensors.

Now notice that $\widetilde{\Theta}^{d-1}(x_{j_{1}}\cdots x_{j_{d}})$ is a sum
of low order tensors of $\widetilde{\Theta}^{d}(x_{j_{1}}\cdots x_{j_{d}})$
with one of the 1’s removed. Thus, viewing $0$ as $0\otimes
0\otimes\cdots\otimes 0$ ($d-1$ times), we can tensor the equation

$\widetilde{\Theta}^{d-1}(x_{j_{1}}\cdots x_{j_{d}})=0$

by 1 in $d$ spots and adding up all the new equations shows that the low order
tensors of $\widetilde{\Theta}^{d}(x_{j_{1}}\cdots x_{j_{d}})$ sum to 0.
Therefore, we get that

$\sum_{\sigma\in
S_{d}}\Theta(x_{j_{\sigma(1)}})\otimes\cdots\otimes\Theta(x_{j_{\sigma(d)}})=0.$

Now suppose that $i>d$. Then we can see that in the sum of pure tensors in
$\widetilde{\Theta}^{i}(x_{j_{1}}\cdots x_{j_{d}})$, every pure tensor has at
least one 1 in it. Thus repeating the same process of the tensor the equation
$\widetilde{\Theta}^{i-1}(x_{j_{1}}\cdots x_{j_{d}})=0$ with $1$ in $i$ spots
and summing up, we get back to the equation
$\widetilde{\Theta}^{i}(x_{j_{1}}\cdots x_{j_{d}})=0$.

Now we are ready to prove injectivity of $\psi$. Suppose that
$a\in\ker\psi\setminus\\{0\\}$. Put an order on the indexing set $J$ of the
basis for $L$. Then by the Poincaré-Birkoff-Witt Theorem, we can uniquely
write $a$ as

$a=\sum\limits_{i=1}^{n}\alpha_{i}x_{j^{(i)}_{1}}^{e^{(i)}_{1}}\cdots
x_{j^{(i)}_{d_{i}}}^{e^{(i)}_{d_{i}}}+\alpha_{0}$

where $\alpha_{i}\in F$, $\alpha_{i}\neq 0$ for $i\geq 1$, $e^{(i)}_{j}\geq 1$
for every $i,j$, $d_{i}\geq 1$, and $j^{(i)}_{k}<j^{(i)}_{k+1}$. By applying
$\widetilde{\Theta}^{0}$, we get rid of every term but $\alpha_{0}$, so we see
that $\alpha_{0}=0$. Let

$D_{i}=\sum\limits_{k=1}^{d_{i}}e^{(i)}_{k}$

be the degree of the $i$-th monomial and let $D=\max\\{D_{i}\\}_{1\leq i\leq
n}$. Since we know that $\Theta$ is injective on $L$, we get a contradiction
if $D=1$. Thus assume that $D\geq 2$.

Repeating the same process we did on monomials, take the equation
$\widetilde{\Theta}^{D-1}(a)=0$, tensor it with 1 in the $D$ possible spots
and add up all the equations. Then we subtract that sum from the equation
$\widetilde{\Theta}^{D}(a)=0$. For every $i$ such that $D_{i}<D$, the terms
coming from the $i$-th monomial will vanish in the new equation. Thus we are
on left with the monomials of degree exactly $D$. The equation we are left
with is a linear combination of pure tensors of the form
$\Theta(x_{j_{1}})\otimes\cdots\otimes\Theta(x_{j_{D}})$. Since our monomials
are ordered, the only way to have such pure tensors are equal is if they come
from the same original monomial. Therefore we have a non-trivial linear
combination. However, $\\{\Theta(x_{j})\\}_{j\in J}\subset A$ is linearly
independent since $\Theta$ is injective, so all the weights in the linear
combination must be 0. Since $F$ is characteristic, we get that $\alpha_{i}=0$
for every $i$ such that $D_{i}=D$. This is a contradiction so no such $a$
exists and $\ker\psi=(0)$. ∎

######  Remark 5.2.1.

This equivalence of soficity is particularly useful when combined with
Proposition 11.6 from [AP17] stating amenable algebras without zero divisors
are sofic. A particular application is a simplification of Theorem 4.2 for Lie
algebras over fields of characteristic 0. Indeed, if $L$ is such a Lie
algebra, by Theorem 7 of [Smi76], $U(L)$ is also of subexponential growth.
From [Ele03], we see that $U(L)$ is amenable algebra with no zero divisors.
Therefore $U(L)$, and thus $L$, is sofic.

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