Supplementary Material
Phonon-phonon interactions in the polarizarion dependence of Raman scattering

Nimrod Benshalom,1,∗ Maor Asher,1,∗ Rémy Jouclas,2 Roman Korobko,1 Guillaume
Schweicher,2 Jie Liu2, Yves Geerts,2,3 Olle Hellman,4,5,† and Omer Yaffe1,‡

1Department of Chemical and Biological Physics, Weizmann Institute of Science,
Rehovot 76100, Israel.
2Laboratoire de Chimie des Polymères, Universit́e Libre de Bruxelles (ULB),
1050 Brussels, Belgium
3International Solvay Institutes for Physics and Chemistry, 1050 Brussels,
Belgium
4Department of Physics, Chemistry and Biology (IFM), Linköping University,
SE-581 83, Linköping, Sweden
5Department of Molecular Chemistry and Material Science, Weizmann Institute of
Science, Rehovot 76100, Israel

$\ast$ These authors contributed equally.

$\dagger$<EMAIL_ADDRESS>
$\ddagger$<EMAIL_ADDRESS>
##  SI Measurement details

Crystals Growth and characterization: BTBT single-crystals were grown by
leaving a saturated BTBT solution in chloroform in open air at room
temperature until complete chloroform evaporation. We confirmed the crystal
structure and high phase purity of the crystals by performing XRD measurements
(see Sec. SII).

Temperature-Dependent PO Raman: A custom-built Raman system was used to
conduct the Raman measurements. The system included a 785 nm Toptica diode
laser with intensity of around 30 mW on the sample. To control the
polarization of the incident and scattered light for the polarization-
dependent measurements (5∘ steps), rotating half-wave plates and a polarizer-
analyzer combination were used. The system included a 50x objective. Notch
filters are included in the system to allow access to the low-frequency region
($>$10 cm-1) and simultaneous acquisition of the Stokes and anti-Stokes
signal. The system is based on a 1 m long Horiba FHR-1000 dispersive
spectrometer with a 1800 mm-1 grating. The spectral resolution was
approximately 0.15 cm-1. The temperature was set and controlled by a Janis
cryostat ST-500 and a temperature controller by Lakeshore model 335.

##  SII Powder X-ray diffraction measurements of BTBT

Preferred orientation and phase purity of BTBT were analyzed using powder
X-ray diffraction. The experiment was performed at room temperature. The
Compound was finely ground using a mortar and pestle for phase confirmation
measurement. Single crystals were mounted over the sample holder for
preferential orientation analysis. For BTBT the measurements were conducted on
a Panalytical Empyrean diffractometer using Cu–K$\alpha$ radiation
($\lambda=$1.54178 Å). The diffractometer was set up with reflection-
transmission spinner 3.0 configuration, and patterns were collected with
$2\theta$ range between 5.0 and 30.0∘, steps of 0.1∘, time per step of 2.5 s,
and rotation of 1 r/s. A calculated pattern was obtained from known crystal
structure of BTBT [1, 2, 3] using Powder Pattern tool on Mercury software [4,
5]. The crystalline phase of BTBT was confirmed to have the same phase of the
known crystal structures with CSD Refcode PODKEA02 [3] (BTBT) as it can be
observed in Figure S1 (experimental diffraction pattern in red and calculated
diffraction patterns in black). The results show single crystals of BTBT with
highly (100) preferred orientation crystallized along [001] direction.

Figure S1:  X-Ray diffraction patterns calculated from the known crystal
structures (in black), obtained experimentally from powders (in red) and
obtained experimentally from single crystals (in blue) of BTBT.

##  SIII Deconvolution of Raman spectra

To extract the integrated intensity of a spectral feature, we fit the measured
Stokes-shift Raman spectra with the product of the Bose-Einstein distribution
and a multi-damped Lorentz oscillator line shape,

$I_{Raman}(\Omega)=\bigg{(}\frac{1}{e^{\frac{\hbar\Omega}{k_{B}T}}-1}+1\bigg{)}\sum_{i}\frac{c_{i}|\Omega|\Gamma_{i}^{3}}{\Omega^{2}\Gamma_{i}^{2}+(\Omega^{2}-\omega_{i}^{2})^{2}}$
(S1)

Where $\omega_{i}$, $c_{i}$ and $\Gamma_{i}$ are the position, intensity, and
width of each peak, respectively. $\Omega$ is the measured frequency (Raman
shift), $T$ is the temperature, $\hbar$ is the Planck constant and $k_{B}$ is
the Boltzmann constant. The Lorentz in Equation (S1) is a variation of the
Lorentz oscillator, where $c$ is the maximum value of the peak. Given the
fitted Lorentzian parameters, the integrated intensity of the spectral feature
is

$I_{int}=n_{\textrm{BE}}(\omega_{i},T)\frac{2c_{i}\left|\omega_{i}\right|\Gamma_{i}^{2}}{\sqrt{4\omega_{i}^{2}-\Gamma_{i}^{2}}}\left[\frac{\pi}{2}-\tan^{-1}\left(\frac{\Gamma_{i}^{2}-2\omega_{i}^{2}}{\Gamma_{i}\sqrt{4\omega_{i}^{2}-\Gamma_{i}^{2}}}\right)\right],$
(S2)

with $n_{\textrm{BE}}$ standing for an approximated Bose-Einstein constant
pre-factor.

##  SIV The temperature independent polarization-orientation Raman of silicon

Figure S2 presents the PO Raman measurements of a (100) silicon wafer at 10 K
and 300 K with the analysis of the polarization dependence of the integrated
intensity of the prominent Raman peak (the TO phonon at around 520 cm-1). We
performed the measurements and analysis similar to the organic crystals
discussed in the main text using the same optical setup. We present these
results as an example for an inorganic material that shows a temperature
independent PO response, in agreement with the harmonic picture. Furthermore,
according to theory, the intensity of the prominent Raman peak should go down
to zero at the minimum point [6]. In our measurement, the peak intensity drops
to about 1% of its maximum intensity, showing our system suffers very little
leakage.

Figure S2: Temperature dependence of the PO Raman of silicon. (a) and (b)
presents the PO Raman measurement of silicon at 10 K and 300 K, respectively.
The middle and bottom panels present the measurements in parallel and cross
configurations, respectively. The top panel presents the unpolarized spectrum
(sum over all angles). (c) presents the polarization dependence of the
integrated intensity of the prominent Raman peak. The top and bottom panels
present the results for parallel and cross configurations, respectively.

##  SV  Polarization-orientation Raman of chloroform

Figure S3 presents the PO Raman measurement of liquid chloroform at room
temperature. The measurement was performed similarly to the organic crystals
discussed in the main text using the same optical setup. Since there is no
long-range order in a liquid, we expect the measurement to be polarization
independent. This is why measuring the PO response of a liquid is helpful as a
test for the system response to the change in polarization angle. Our results
show that the PO response of the vibrations of chloroform is completely
polarization-independent - proving our systems response is minimal.

Figure S3: Polarization dependent measurement of liquid chloroform in parallel
and cross configurations. The top panel shows a typical spectrum in the
parallel configuration.

##  SVI Temperature dependent low-frequency Raman spectroscopy of BTBT

Figure S4 shows the temperature dependent low-frequency Raman spectroscopy
spectra of BTBT, from 10 K to 290 K at 10 K increments. We see only a gradual
redshift and broadening of the spectrum as temperature increases. No abrupt
changes to the Raman spectrum indicate there is no phase transition in the
measured temperature range.

Figure S4: Temperature dependent low-frequency Raman of BTBT. The spectra were
normalized and shifted up for clarity. The temperature increment is 10 K.

Figure S5 shows the temperature dependence of the peaks position and full-
width-half-maximum (FWHM) of the lattice vibrations, extracted from fitting
the spectra according to Sec. SIII. The trend with temperature of vibrational
frequency and FWHM is primarily linear.

Figure S5: The temperature dependence of the vibrational frequencies (a) and
FWHM (b) of the lattice vibrations of BTBT.

##  SVII Raman polarization-orientation color maps of BTBT

Figure S6 presents the raw data obtained from the PO Raman measurements for
both parallel and cross configurations for a single crystal of BTBT at 10, 80,
150, 200, and 290 K.

Figure S6: Raw PO Raman of BTBT in (a) parallel and (b) cross configurations
at 10 K, 80 K, 150 K, 220 K and 290 K.

##  SVIII Temperature evolution in the Raman polarization-orientation response
of BTBT

Figure S7 presents the temperature evolution of the PO pattern of each
separate peak in the Raman spectrum of BTBT. The results show the PO Raman
response of all BTBT lattice vibrations evolves with temperature.

Figure S7: The temperature evolution of the PO dependence for the low-
frequency peaks of BTBT in (a) parallel and (b) cross configurations. The
intensities were normalized for each mode at each temperature.

##  SIX Fitting the PO response of BTBT

###  SIX.1 Data preparation and factor group analysis for BTBT

Section SVII presents the contour plots of the PO Raman measurement of BTBT at
10, 80, 150 220, and 290 K in parallel and cross configurations. The
measurements were performed as described in Sec. SI. To extract the
polarization dependence of the integrated intensity of each peak we fit the
spectra at each polarization angle to a multi-damped Lorentz oscillator (see
section SIII). The dotted lines in Figures S8 and S9 show the integrated
intensity of each damped Lorentz oscillator with respect to the excitation
polarization angle. Next, for the second-rank Raman tensor formalism we
extract the form of the Raman tensors, as well as the expected number of
lattice modes and their vibrational symmetry, by using factor group analysis
with the relevant space group (monoclinic, $P2_{1}/c$) [7]. For BTBT, factor
group analysis predicts 6 Raman-active lattice libration modes, and indeed our
measured spectra showed exactly six distinguishable Raman peaks.

###  SIX.2 Fitting the PO with a second-rank tensor

The symmetry allowed Raman tensors for $3\text{A}_{g}$ and $3\text{B}_{g}$
are[8]:

$R_{\mathrm{A_{g}}}=\begin{pmatrix}a&0&e\\\ 0&b&0\\\
e&0&c\end{pmatrix},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}R_{\mathrm{B_{g}}}=\begin{pmatrix}0&d&0\\\
d&0&f\\\ 0&f&0\end{pmatrix}.$ (S3)

We perform a global fit to the integrated intensity in both parallel and cross
configurations using the second-rank Raman tensor formalism described by Eq.
$(1)$ in the main text (see Ref. [9] for more details regarding the fitting
process).

Figure S8 shows the results for this fitting procedure at all measured
temperatures. The 10 K fits show that $\omega_{1}$, $\omega_{3}$, and
$\omega_{5}$ are Ag modes, while $\omega_{4}$ and $\omega_{6}$ are Bg modes.
The intensity of $\omega_{2}$ was too weak to extract its polarization
dependence reliably. The fit gets worse for $\omega_{3}$, $\omega_{4}$ and
$\omega_{6}$ as temperature increases, while for $\omega_{1}$ and $\omega_{5}$
the fit is successful in all temperatures.

To further challenge the necessity of a fourth-rank tensor, we attempted
fitting a symmetry-relaxed second-rank tensor:

$\tilde{\mathcal{R}}=\left(\begin{array}[]{ccc}a&d_{1}&e_{1}\\\
d_{2}&be^{i\delta}&f_{1}\\\ e_{2}&f_{2}&c\end{array}\right),$ (S4)

where we introduce a relative phase $\delta$ to account for possible
birefringence effects [10], and allow asymmetric components (effectively
relaxing time-symmetry) [11]. Adapted to our back-scattering geometry,
$\tilde{\mathcal{R}}$ has five independent parameters
$(a,b,d_{1},d_{2},\delta)$. As Fig. S8 demonstrates, this generalized second-
rank tensor form was still unable to fit the observed PO pattern of
$\omega_{4}$.

###  SIX.3 Fitting the PO with a fourth-rank tensor

Applying the symmetry constraints of the relevant space group $(P2_{1}/c)$
along with the intrinsic left and right minor symmetries, and retaining only
the components relevant for our back-scattering geometry, the effective
fourth-rank tensor used for fitting the PO patterns was:

$I_{\mu\nu\xi\rho}=\left(\begin{array}[]{ccc}\left(\begin{array}[]{ccc}a&0&0\\\
0&b&0\\\ 0&0&0\end{array}\right)&\left(\begin{array}[]{ccc}0&d&0\\\ d&0&0\\\
0&0&0\end{array}\right)&\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\\\ \left(\begin{array}[]{ccc}0&d&0\\\ d&0&0\\\
0&0&0\end{array}\right)&\left(\begin{array}[]{ccc}b&0&0\\\ 0&c&0\\\
0&0&0\end{array}\right)&\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\\\ \left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)&\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)&\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right)\end{array}\right)$ (S5)

Figure S9 shows the fit results for the same PO Raman data set by using the
fourth-rank Raman tensor formalism. Here we see that the fit is excellent at
all temperatures. These results are discussed in the main text.

###  SIX.4 Global fit with a two-mode model

The full PO data for spectral features $\omega_{4}$ and $\omega_{5}$ at 10 K
was globally fitted using the two-mode model discussed in the main text, with
birefringence effects incorporated as in the harmonic case of Sec. S9.2,
introducing the additional parameter $\delta$.

In practice, this means _simultaneously_ fitting the spectral range 85-95
$cm^{-1}$ for all 144 spectra (72 polarization angles for each polarization
configuration) to:

$\sigma(\Omega)\propto\sum_{\mu\nu\xi\rho}n_{\mu}n_{\xi}I_{\mu\nu\xi\rho}(\Omega)E_{\nu}E_{\rho},$
(S6)

with

$I_{\mu\nu\xi\rho}(\Omega)=\left[n_{BE}(\Omega,T)+1\right]\left[{\chi^{*}}^{\mu\nu}_{\mathrm{A_{g}}}\chi^{\xi\rho}_{\mathrm{A_{g}}}J_{\mathrm{A_{g}A_{g}}}+{\chi^{*}}^{\mu\nu}_{\mathrm{B_{g}}}\chi^{\xi\rho}_{\mathrm{B_{g}}}J_{\mathrm{B_{g}B_{g}}}\right],$
(S7)

and

$\chi_{\mathrm{A_{g}}}=\begin{pmatrix}a&0&e\\\ 0&be^{i\delta}&0\\\
e&0&c\end{pmatrix},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\chi_{\mathrm{B_{g}}}=\begin{pmatrix}0&de^{i\delta}&0\\\
de^{i\delta}&0&fe^{i\delta}\\\ 0&fe^{i\delta}&0\end{pmatrix}.$ (S8)

Applying our back-scattering geometry to Eq. (S6) gives the following final
fitted expressions for parallel $(\parallel)$ and cross $(\perp)$
configurations:

$\sigma_{\parallel}\propto
J_{\mathrm{AgAg}}a^{2}\cos^{4}(\Delta\theta)+2J_{\mathrm{AgAg}}ab\cos(2\delta)\cos^{2}(\Delta\theta)\sin{{}^{2}}(\Delta\theta)^{2}+\\\
4J_{\mathrm{BgBg}}d^{2}\cos^{2}(\Delta\theta)\sin^{2}(\Delta\theta)+J_{\mathrm{AgAg}}b^{2}\sin^{4}(\Delta\theta),$
(S9) $\sigma_{\perp}\propto
J_{\mathrm{AgAg}}a^{2}\cos^{2}(\Delta\theta)\sin{{}^{2}}(\Delta\theta)^{2}+J_{\mathrm{AgAg}}b^{2}\cos^{2}(\Delta\theta)\sin{{}^{2}}(\Delta\theta)^{2}-\\\
2J_{\mathrm{BgBg}}d^{2}\cos^{2}(\Delta\theta)\sin^{2}(\Delta\theta)-2J_{\mathrm{AgAg}}ab\cos(2\delta)\cos^{2}(\Delta\theta)\sin{{}^{2}}(\Delta\theta)^{2}+\\\
J_{\mathrm{BgBg}}d^{2}\cos^{4}(\Delta\theta)+J_{\mathrm{BgBg}}d^{2}\sin^{4}(\Delta\theta),$
(S10)

where $\Delta\theta=\theta-\theta_{0}$ is determined by the incident
polarization $\theta$, controlled in experiment, and $\theta_{0}$ is a global
angle determined by the crystal orientation. The spectral frequency dependency
is contained within the spectral functions
$J_{\lambda\lambda^{\prime}}(\Omega)$ and the
$\left[n_{BE}(\Omega,T)+1\right]$ factor.

The full derivation of $J_{\lambda\lambda^{\prime}}$ for the two-mode model is
given in the main text, equations $(6),(7),(9)$. This yielded the final
function used in the global fit (using the shorthand $\mathrm{A_{g}}\equiv 1$
and $\mathrm{B_{g}}\equiv 2$):

$J_{11}(\Omega)=\frac{\Gamma_{2}\left(\gamma^{*}\gamma-\Gamma_{1}\Gamma_{2}+\left(\frac{\Omega^{2}-\omega_{1}^{2}}{2\omega_{1}}\right)\left(\frac{\Omega^{2}-\omega_{2}^{2}}{2\omega_{2}}\right)\right)-\left(\frac{\Omega^{2}-\omega_{2}^{2}}{2\omega_{2}}\right)\left(\Gamma_{1}\left(\frac{\Omega^{2}-\omega_{2}^{2}}{2\omega_{2}}\right)+\Gamma_{2}\left(\frac{\Omega^{2}-\omega_{1}^{2}}{2\omega_{1}}\right)\right)}{\left(\gamma^{*}\gamma-\Gamma_{1}\Gamma_{2}+\left(\frac{\Omega^{2}-\omega_{1}^{2}}{2\omega_{1}}\right)\left(\frac{\Omega^{2}-\omega_{2}^{2}}{2\omega_{2}}\right)\right)^{2}+\left(\Gamma_{1}\left(\frac{\Omega^{2}-\omega_{2}^{2}}{2\omega_{2}}\right)+\Gamma_{2}\left(\frac{\Omega^{2}-\omega_{1}^{2}}{2\omega_{1}}\right)\right)^{2}},$
(S11)

with $J_{\mathrm{B_{g}B_{g}}}$ being the same only with swapped irrep indices
$\left(\mathrm{A_{g}}\leftrightarrow\mathrm{B_{g}}\right)$.

The full list of fit parameters is therefore:

  1. 1.

$\omega_{i}$ \- the peak resonance

  2. 2.

$\Gamma_{i}$ \- the peak width

  3. 3.

$\theta_{0}$ \- global incident polarization off-set

The three parameters mentioned above are identical to those used to extract
the integrated intensity in the harmonic analysis. They govern the peak shape,
but not the PO pattern. The parameters that create the PO response in our two-
mode model are:

  1. 4.

$a$ \- susceptibility derivative for $\mathrm{A_{g}}$ normal mode

  2. 5.

$b$ \- susceptibility derivative for $\mathrm{A_{g}}$ normal mode

  3. 6.

$d$ \- susceptibility derivative for $\mathrm{B_{g}}$ normal mode

  4. 7.

$\delta$ \- phase retardance due to birefringence

  5. 8.

$\gamma$ \- off-diagonal self-energy component

which gives a total of five parameters for two spectral peaks, _the same_ as
in the generalized second-rank formalism that failed to account for the PO
response of the _single_ $\omega_{4}$ peak.

Figure S8: Fitting results for the PO Raman response of BTBT at different
temperatures, using a general second-rank Raman tensor. Figure S9: Fitting
results for the PO Raman response of BTBT at different temperatures, showing
the successful fit obtained by using the fourth-rank formalism.

##  SX Raman polarization-orientation measurements of anthracene and pentacene

The full PO Raman spectra of anthracene and pentacene crystals were measured
in a similar fashion to that described in Sec. SI (see full measurement and
irrep assignment details in [9]). The PO Raman spectra at 10 K for both
anthracene and pentacene are presented in Fig. S10.

A procedure similar to that used in BTBT (Sec. SIII) was applied to extract
the integrated intensity of each peak, and fit the PO pattern to both a second
(Sec. SIX.2) and fourth-rank (Sec. SIX.3) formalism (all three crystals share
the same $C_{2h}$ point-group and tensor forms). The results are presented in
figures S11 (second rank), S12 (fourth-rank) for anthracene, and figures S13
(second-rank), S14 (fourth-rank) for pentacene. In both cases the second-rank
formalism fails to capture the PO of all peaks (specifically, $\omega_{2}$,
$\omega_{6}$ for anthracene, and $\omega_{6}$, $\omega_{6}$, $\omega_{7}$ for
pentacene), whereas the fourth-rank formalism consistently yields a perfect
fit.

Figure S10: Temperature dependence of the PO Raman of anthracene (left) and
pentacene (right). Top row shows the unpolarized spectra. Middle and bottom
rows present the PO for parallel and cross configurations, respectively.
Figure S11: Fitting results for the PO Raman response of anthracene at
different temperatures, using a general second-rank Raman tensor. The fit for
peaks $\omega_{2}$, $\omega_{6}$ is unsatisfactory. Figure S12: Fitting
results for the PO Raman response of anthracene at different temperatures,
showing the successful fit obtained by using the fourth-rank formalism. Figure
S13: Fitting results for the PO Raman response of pentacene at different
temperatures, using a general second-rank Raman tensor. The fit for peaks
$\omega_{5}$, $\omega_{6}$, $\omega_{7}$ is unsatisfactory. Figure S14:
Fitting results for the PO Raman response of pentacene at different
temperatures, showing the successful fit obtained by using the fourth-rank
formalism.

##  SXI Theory of inelastic light scattering from crystals at finite
temperatures

###  SXI.1 Vibrational excitations of an anharmonic crystal

In this section we outline the procedure to determine the Raman spectrum of an
anharmonic crystal. We will begin by sketching the procedure to obtain the
phonon-phonon correlation function in the presence of three-phonon
anharmonicity via the equation of motion approach.

If we express the atomic displacements in terms of creation and annihilation
operators we can identify the reciprocal space coefficients needed for the
perturbative expansion, repeated here for clarity and consistency of notation.
Start with the standard creation and annihilation operators for a phonon with
momentum $\bm{q}$ and polarization $s$ (when there is no loss of clarity we
use the compound index $\lambda$ to denote $\bm{q}s$):

$\displaystyle a_{\bm{q}s}=$
$\displaystyle\frac{1}{\sqrt{2N\hbar}}\sum_{i\alpha}\epsilon_{\bm{q}s}^{i\alpha}\left(\sqrt{m_{i}\omega_{\bm{q}s}}u_{i\alpha}-i\frac{p_{i\alpha}}{\sqrt{m_{i}\omega_{\bm{q}s}}}\right)e^{-i\bm{q}\cdot\bm{r}_{i}}$
(S12) $\displaystyle a_{\bm{q}s}^{\dagger}=$
$\displaystyle\frac{1}{\sqrt{2N\hbar}}\sum_{i\alpha}\epsilon_{\bm{q}s}^{i\alpha\dagger}\left(\sqrt{m_{i}\omega_{\bm{q}s}}u_{i\alpha}+i\frac{p_{i\alpha}}{\sqrt{m_{i}\omega_{\bm{q}s}}}\right)e^{i\bm{q}\cdot\bm{r}_{i}}$
(S13) $\displaystyle A_{\bm{q}s}=a_{\bm{q}s}+a_{\bar{\bm{q}}s}^{\dagger}=$
$\displaystyle\frac{2}{\sqrt{2N\hbar}}\sum_{i\alpha}\epsilon_{\bm{q}s}^{i\alpha}\sqrt{m_{i}\omega_{\bm{q}s}}u_{i\alpha}e^{-i\bm{q}\cdot\bm{r}_{i}}$
(S14) $\displaystyle B_{\bm{q}s}=a_{\bm{q}s}-a_{\bar{\bm{q}}s}^{\dagger}=$
$\displaystyle-\frac{2i}{\sqrt{2N\hbar}}\sum_{i\alpha}\epsilon_{\bm{q}s}^{i\alpha}\frac{p_{i\alpha}}{\sqrt{m_{i}\omega_{\bm{q}s}}}e^{-i\bm{q}\cdot\bm{r}_{i}}$
(S15)

Here $\epsilon$ are phonon eigenvectors
($\bm{\epsilon}_{i}\cdot\bm{\epsilon}_{j}=\delta_{ij}$), $m$ atomic masses and
$\omega$ frequencies. The vector $\bm{r}$ is a lattice vector, and $u$ and $p$
are the position and momentum operators. We use the notation
$\bar{\bm{q}}=-\bm{q}$. We introduce scaled eigenvectors (to simplify
notation) via

$\upsilon_{\bm{q}s}^{i\alpha}=\sqrt{\frac{\hbar}{2m_{i}\omega_{\bm{q}s}}}\epsilon_{\bm{q}s}^{i\alpha}$
(S16)

In this notation the matrix elements pertaining to anharmonicity become

$\displaystyle\Phi_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}$
$\displaystyle=\frac{1}{3!}\sum_{ijk\alpha\beta\gamma}\upsilon_{\bm{q}s}^{i\alpha}\upsilon_{\bm{q}^{\prime}s^{\prime}}^{j\beta}\upsilon_{\bm{q}^{\prime\prime}s^{\prime\prime}}^{k\gamma}\Phi_{ijk}^{\alpha\beta\gamma}e^{-i(\bm{q}\cdot\bm{r}_{i}+\bm{q}^{\prime}\cdot\bm{r}_{j}+\bm{q}^{\prime\prime}\cdot\bm{r}_{k})}$
(S17)
$\displaystyle\Phi_{\lambda\lambda^{\prime}\lambda^{\prime\prime}\lambda^{\prime\prime\prime}}$
$\displaystyle=\frac{1}{4!}\sum_{ijkl\alpha\beta\gamma\delta}\upsilon_{\bm{q}s}^{i\alpha}\upsilon_{\bm{q}^{\prime}s^{\prime}}^{j\beta}\upsilon_{\bm{q}^{\prime\prime}s^{\prime\prime}}^{k\gamma}\upsilon_{\bm{q}^{\prime\prime\prime}s^{\prime\prime\prime}}^{l\delta}\Phi_{ijkl}^{\alpha\beta\gamma\delta}e^{-i(\bm{q}\cdot\bm{r}_{i}+\bm{q}^{\prime}\cdot\bm{r}_{j}+\bm{q}^{\prime\prime}\cdot\bm{r}_{k}+\bm{q}^{\prime\prime\prime}\cdot\bm{r}_{l})}$
(S18) so that the anharmonic part of the Hamiltonian becomes $\displaystyle
H_{A}$
$\displaystyle=\Phi_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}A_{\lambda}A_{\lambda^{\prime}}A_{\lambda^{\prime\prime}}+\Phi_{\lambda\lambda^{\prime}\lambda^{\prime\prime}\lambda^{\prime\prime\prime}}A_{\lambda}A_{\lambda^{\prime}}A_{\lambda^{\prime\prime}}A_{\lambda^{\prime\prime\prime}}+\cdots.$
(S19)

In an analogous way we define the coefficients of the expansion of the
polarizability as

$\displaystyle\chi^{\mu\nu}_{\lambda}$
$\displaystyle=\sum_{i\alpha}\upsilon_{\bm{q}s}^{i\alpha}\chi_{i}^{\mu\nu,\alpha}e^{-i\bm{q}\cdot\bm{r}_{i}}\Delta_{\bm{q}}=\sum_{i\alpha}\upsilon_{\bm{\Gamma}s}^{i\alpha}\chi_{i}^{\mu\nu,\alpha}=\chi^{\mu\nu}(s)$
(S20) $\displaystyle\chi^{\mu\nu}_{\lambda\lambda^{\prime}}$
$\displaystyle=\frac{1}{2!}\sum_{ij\alpha\beta}\upsilon_{\bm{q}s}^{i\alpha}\upsilon_{\bm{q}^{\prime}s^{\prime}}^{j\beta}\chi_{ij}^{\mu\nu,\alpha\beta}e^{-i\bm{q}\cdot\bm{r}_{i}}e^{-i\bm{q}^{\prime}\cdot\bm{r}_{j}}\Delta_{\bm{q}\bm{q}^{\prime}}$
(S21)
$\displaystyle=\frac{1}{2!}\sum_{ij\alpha\beta}\upsilon_{\bm{q}s}^{i\alpha}(\upsilon_{\bm{q}s^{\prime}}^{j\beta})^{\dagger}\chi_{ij}^{\mu\nu,\alpha\beta}e^{i\bm{q}\cdot(\bm{r}_{j}-\bm{r}_{i})}$
(S22)
$\displaystyle=\frac{1}{2!}\sum_{ij\alpha\beta}(\upsilon_{\bm{q}s}^{i\alpha})^{\dagger}\upsilon_{\bm{q}s^{\prime}}^{j\beta}\chi_{ij}^{\mu\nu,\alpha\beta}e^{-i\bm{q}\cdot(\bm{r}_{j}-\bm{r}_{i})}=\chi^{\mu\nu}(\bm{q},s,s^{\prime})=\chi^{\mu\nu}(\bar{\bm{q}},s,s^{\prime})^{\dagger}=\chi^{\mu\nu}(\bar{\bm{q}},s^{\prime},s)$
(S23)
$\displaystyle\chi^{\mu\nu}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}$
$\displaystyle=\frac{1}{3!}\sum_{ijk\alpha\beta\gamma}\upsilon_{\bm{q}s}^{i\alpha}\upsilon_{\bm{q}^{\prime}s^{\prime}}^{j\beta}\upsilon_{\bm{q}^{\prime}s^{\prime}}^{k\gamma}\chi_{ijk}^{\mu\nu,\alpha\beta\gamma}e^{-i(\bm{q}\cdot\bm{r}_{i}+\bm{q}^{\prime}\cdot\bm{r}_{j}+\bm{q}^{\prime\prime}\cdot\bm{r}_{k})}$
(S24) Such that $\displaystyle\chi^{\mu\nu}$
$\displaystyle=\chi^{\mu\nu}_{0}+\chi^{\mu\nu}_{\lambda}A_{\lambda}+\chi^{\mu\nu}_{\lambda\lambda^{\prime}}A_{\lambda}A_{\lambda^{\prime}}+\chi^{\mu\nu}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}A_{\lambda}A_{\lambda^{\prime}}A_{\lambda^{\prime\prime}}+\cdots.$
(S25)

###  SXI.2 Raman scattering in terms of the anharmonic Green’s function

If we start from the polarizability-polarizability correlation function and
insert the expansion of the polarizability in terms of phonon coordinates, Eq.
(S25), we get

$\begin{split}\left\langle\chi(t)\chi(0)\right\rangle=&\bm{\chi}_{0}\otimes\bm{\chi}_{0}+\bm{\chi}_{\lambda}\otimes\bm{\chi}_{\lambda^{\prime}}i\overset{>}{G}(A_{\lambda},A_{\lambda^{\prime}})+\\\
+&\bm{\chi}_{\lambda\lambda^{\prime}}\otimes\bm{\chi}_{\lambda^{\prime\prime}\lambda^{\prime\prime\prime}}i\overset{>}{G}(A_{\lambda}A_{\lambda^{\prime}},A_{\lambda^{\prime\prime}}A_{\lambda^{\prime\prime\prime}})+\\\
+&\bm{\chi}_{\lambda}\otimes\bm{\chi}_{\lambda^{\prime}\lambda^{\prime\prime}}i\overset{>}{G}(A_{\lambda},A_{\lambda^{\prime}}A_{\lambda^{\prime\prime}})+\bm{\chi}_{\lambda\lambda^{\prime}}\otimes\bm{\chi}_{\lambda^{\prime\prime}}i\overset{>}{G}(A_{\lambda}A_{\lambda^{\prime}},A_{\lambda^{\prime\prime}})+\\\
+&\bm{\chi}_{\lambda}\otimes\bm{\chi}_{\lambda^{\prime}\lambda^{\prime\prime}\lambda^{\prime\prime\prime}}i\overset{>}{G}(A_{\lambda},A_{\lambda^{\prime}}A_{\lambda^{\prime\prime}}A_{\lambda^{\prime\prime\prime}})+\bm{\chi}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}\otimes\bm{\chi}_{\lambda^{\prime\prime\prime}}i\overset{>}{G}(A_{\lambda}A_{\lambda^{\prime}}A_{\lambda^{\prime\prime}},A_{\lambda^{\prime\prime\prime}}),\end{split}$
(S26)

where
$\overset{>}{G}(A_{\lambda},A_{\lambda^{\prime}}^{\dagger},\tau)\equiv-i\langle
A_{\lambda}(\tau),A_{\lambda^{\prime}}(0)\rangle$, and we have omitted terms
of $A^{4}$ or higher. The physical quantity of interest here is the spectral
function, $J_{\lambda\lambda^{{}^{\prime}}}(\Omega)$, which may also be
written in terms of the retarded Green’s function,
$G^{R}_{\lambda\lambda^{{}^{\prime}}}(\Omega)\equiv-i\theta(\tau)\left\langle\left[A_{\lambda}(\tau),A_{\lambda^{\prime}}^{\dagger}(0)\right]\right\rangle$:

$J_{\lambda\lambda^{{}^{\prime}}}(\Omega)=-\frac{1}{\pi}\text{Im}\left\\{G^{R}_{\lambda\lambda^{{}^{\prime}}}(\Omega)\right\\}=\frac{i}{(n(\Omega,T)+1)}\overset{>}{G}_{\lambda\lambda^{{}^{\prime}}}(\Omega),$
(S27)

with the retarded Green’s function conveniently given in terms of the sum of a
diagonal harmonic part, $g^{0}_{\lambda\lambda^{\prime}}(\Omega)$, and the
self-energy term $\Sigma_{\lambda\lambda^{\prime}}(\Omega)$:

$G^{R}(\Omega)^{-1}=g^{0}(\Omega)^{-1}+\Sigma(\Omega).$ (S28)

Following the work of others [12, 13, 14, 15], we have previously discussed in
detail the solution for the retarded Green’s function of an anharmonic system
$H_{A}$ [16]. In the case of third-order anharmonicity, the self-energy is
given by

$\Sigma_{\lambda\lambda^{\prime}}(\Omega)=-18\sum_{\bm{q}_{1}\bm{q}_{2}s_{1}s_{2}}\Phi^{\lambda
s_{1}s_{2}}_{\bm{q}\bar{\bm{q}}_{1}\bar{\bm{q}}_{2}}\Phi^{\lambda^{\prime}s_{1}s_{2}}_{\bar{\bm{q}}\bm{q}_{1}\bm{q}_{2}}S(s_{1},s_{2},\Omega)$
(S29)

and

$\begin{split}S(s_{a},s_{b},\Omega)=(n_{a}+n_{b}+1)&\left[\frac{1}{(\omega_{a}+\omega_{b}-\Omega)_{p}}-\frac{1}{(\omega_{a}+\omega_{b}+\Omega)_{p}}\right]\\\
+&(n_{a}-n_{b})\left[\frac{1}{(\omega_{b}-\omega_{a}+\Omega)_{p}}-\frac{1}{(\omega_{b}-\omega_{a}-\Omega)_{p}}\right].\end{split}$
(S30)

The constant (zeroth) term in Eq. (S26) does not contribute to the Raman
spectrum. Using the spectral function (S27), the first term becomes

$I^{\textrm{I}}=\bm{\chi}_{\lambda}\otimes\bm{\chi}_{\lambda^{\prime}}\int
i\overset{>}{G}(A_{\lambda}(t),A_{\lambda^{\prime}}(0))e^{-i\Omega
t}dt=\bm{\chi}_{\lambda}\otimes\bm{\chi}_{\lambda^{\prime}}(n(\Omega)+1)J_{\lambda\lambda^{\prime}}(\Omega).$
(S31)

This is sufficient to understand the origin of the polarization dependence
presented in this study. Higher terms in expansion (S26) are usually smaller,
but become significant in Raman inactive structures, where the derivatives in
Eq. (S31) identically vanish. This will happen for any crystal where all
atomic sites constitute a center of inversion. See [16] for a detailed
discussion for the case of rock-salt structure, as well as how higher-order
terms might play a role in finite temperatures.

###  SXI.3 Light scattering in harmonic systems

In harmonic systems all inter-atomic interactions are reduced to terms
quadratic in their displacement from equilibrium in the potential expansion
[17]. These produce ideal, non-interacting normal mode solutions,
$G_{\lambda\lambda^{\prime}}(\Omega)=g^{0}_{\lambda\lambda}(\Omega)\delta_{\lambda\lambda^{\prime}}$
[13]. Using the harmonic spectral function [18], the Stokes component in Eq.
(S31) becomes

$I(\Omega,T)=\sum_{\lambda}\bm{\chi}^{*}_{\lambda}\otimes\bm{\chi}_{\lambda}\frac{\left[n(\Omega,T)+1\right]}{2\omega_{\lambda}}\delta(\Omega-\omega_{\lambda}).$
(S32)

The inadequacy of a purely harmonic treatment in light scattering is already
partially recognized in most Raman studies, at least implicitly. As Eq. (S32)
shows, the frequency dependence of a truly harmonic system should be a delta-
function. In practice, a Lorentzian-like line shape is commonly used, and
explained by the same spectral function $J(\Omega)$, or more explicitly
$\left(\text{Im}\\{\Sigma_{\lambda\lambda}\\}\right)^{-1}$, commonly known as
the phonon lifetime [19]. In addition, thermal expansion is sometimes
incorporated through a quasi-harmonic treatment where the normal modes become
volume dependent [20].

If only diagonal terms are considered (note that the self-energy need not
vanish for this), the generalized fourth-rank expression of Eq. (S6) reduces
back to the temperature _independent_ PO response (Eq. (1) in main text), and
all PO patterns may be explained in terms of a single second-rank tensor
$\mathcal{R}$. This is demonstrated by plugging Eq. (S32) into Eq. (S6). The
incident field, $\bm{E}_{i}$, observed scattering polarization, $\bm{n}$, and
second-rank tensor, $\bm{\chi}_{\lambda}$ are defined as:

$\bm{E}_{i}=\left|\bm{E}_{i}\right|\begin{pmatrix}e_{x}\\\ e_{y}\\\
e_{z}\end{pmatrix};\,\,\,\,\,\,\,\,\,\,\,\bm{n}\equiv\hat{\bm{e}}_{s}=\begin{pmatrix}n_{x}\\\
n_{y}\\\
n_{z}\end{pmatrix};\,\,\,\,\,\,\,\,\,\,\,\mathcal{R}\equiv\bm{\chi}_{\lambda}=\left(\begin{array}[]{ccc}\chi_{xx}&\chi_{xy}&\chi_{xz}\\\
\chi_{yx}&\chi_{yy}&\chi_{yz}\\\
\chi_{zx}&\chi_{zy}&\chi_{zz}\end{array}\right).$ (S33)

The polarization dependence of the scattering cross-section for a single mode
$\lambda$ becomes:

$\displaystyle\sigma\propto$
$\displaystyle\sum_{\mu\nu\xi\rho}n_{\mu}n_{\xi}I_{\mu\nu,\xi\rho}E_{\nu}E_{\rho}$
$\displaystyle=$
$\displaystyle\left[n(\lambda,T)+1\right]\frac{\left|E_{i}\right|^{2}}{2\omega_{\lambda}}\sum_{\mu\nu\xi\rho}n_{\mu}n_{\xi}\left(\bm{\chi}^{*}_{\lambda}\otimes\bm{\chi}_{\lambda}\right)_{\mu\nu\xi\rho}e_{\nu}e_{\rho}$
$\displaystyle\propto$
$\displaystyle\sum_{\mu\nu\xi\rho}n_{\mu}n_{\xi}\chi^{*}_{\nu\mu}\chi_{\rho\xi}e_{\nu}e_{\rho}$
$\displaystyle=$
$\displaystyle\left(\sum_{\nu\mu}\chi^{*}_{\nu\mu}n_{\mu}e_{\nu}\right)\left(\sum_{\rho\xi}\chi_{\rho\xi}n_{\xi}e_{\rho}\right)=\left(\sum_{\nu\mu}\chi_{\nu\mu}n_{\mu}e_{\nu}\right)^{*}\left(\sum_{\rho\xi}\chi_{\rho\xi}n_{\xi}e_{\rho}\right)$
$\displaystyle=$
$\displaystyle\left|\sum_{ij}\chi_{ij}n_{j}e_{i}\right|^{2}=\left|\sum_{i}e_{i}\sum_{j}\chi_{ij}n_{j}\right|^{2}$
$\displaystyle=$
$\displaystyle\left|\hat{\bm{e}}_{i}\cdot\mathcal{R}\cdot\hat{\bm{e}}_{s}\right|^{2},$

which is exactly the original second-rank expression.

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# Neural-network quantum state study of the long-range antiferromagnetic Ising
chain

Jicheol Kim  Department of Physics and Photon Science, Gwangju Institute of
Science and Technology, Gwangju 61005, Korea  Dongkyu Kim  Department of
Physics and Photon Science, Gwangju Institute of Science and Technology,
Gwangju 61005, Korea  Kakao Corporation, Pangyo, Gyeonggi 13529, Korea  Dong-
Hee Kim<EMAIL_ADDRESS>Department of Physics and Photon Science,
Gwangju Institute of Science and Technology, Gwangju 61005, Korea

###### Abstract

We investigate quantum phase transitions in the transverse field Ising chain
with algebraically decaying long-range antiferromagnetic interactions by using
the variational Monte Carlo method with the restricted Boltzmann machine being
employed as a trial wave function ansatz. In the finite-size scaling analysis
with the order parameter and the second Rényi entropy, we find that the
central charge deviates from 1/2 at a small decay exponent
$\alpha_{\mathrm{LR}}$ in contrast to the critical exponents staying very
close to the short-range (SR) Ising values regardless of
$\alpha_{\mathrm{LR}}$ examined, supporting the previously proposed scenario
of conformal invariance breakdown. To identify the threshold of the Ising
universality and the conformal symmetry, we perform two additional tests for
the universal Binder ratio and the conformal field theory (CFT) description of
the correlation function. It turns out that both indicate a noticeable
deviation from the SR Ising class at $\alpha_{\mathrm{LR}}<2$. However, a
closer look at the scaled correlation function for $\alpha_{\mathrm{LR}}\geq
2$ shows a gradual change from the asymptotic line of the CFT verified at
$\alpha_{\mathrm{LR}}=3$, providing a rough estimate of the threshold being in
the range of $2\lesssim\alpha_{\mathrm{LR}}<3$.

##  I Introduction

Artificial neural networks and machine learning have been influencing the
paradigm of physics research with a growing number of applications on various
subjects, including phase transitions and critical phenomena in classical and
quantum many-body systems [1, 2, 3, 4]. In particular, the representation of a
quantum wave function by a neural network [5] provides an alternative
numerical platform combined with the variational Monte Carlo (VMC) method to
find the ground state of a many-body Hamiltonian. The neural-network quantum
state (NQS) has extended its area of applications to the Fermi and Bose
Hubbard models [6, 7], real-time dynamics [5, 8], open quantum systems [9, 10,
11, 12], quantum state tomography [13, 14], frustrated systems [15, 16, 17,
18, 19, 20, 21], and ab initio simulations of molecules [22, 23, 24]. The NQS
ansatz offers the high expressive capacity often measured in terms of
entanglement scaling [25, 26, 27, 28, 29], proposing a complementary tool to
conventional numerical methods for studying quantum criticality.

In this paper, we investigate quantum phase transitions in the transverse
field Ising chain (TFIC) with algebraically decaying long-range (LR)
antiferromagnetic (AF) interactions by employing the NQS ansatz for the VMC
calculations. LR-interacting quantum systems have attracted growing attention,
both theoretical and experimental [30]. The trapped-ion quantum simulation
[31] realized the TFIC Hamiltonian with an LR interaction that maps to the
form of $1/r^{\alpha_{\mathrm{LR}}}$ with a tunable exponent
$\alpha_{\mathrm{LR}}$, providing a controllable experimental platform to
study quantum phase transitions at and out of equilibrium [32, 33, 34]. The
nearest-neighbor-interacting short-range (SR) TFIC is a textbook example of
quantum critical behavior in one dimension (1D) that belongs to the
universality class of the classical two-dimensional (2D) Ising model [35].
However, such quantum-classical correspondence to the universality of critical
phenomena becomes nontrivial in presence of LR interactions. A central
question of how criticality depends on $\alpha_{\mathrm{LR}}$ is still an
active subject of various numerical and analytical studies [30, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62].

We revisit this question in the AF side of the LR interactions for TFIC where
the breakdown of the Ising class in the critical ground state seems to be very
different from what is established in the ferromagnetic (FM) counterpart [47,
48, 57, 49, 50, 51, 56, 58, 52]. Because an exact solution is not available,
constructing the picture of how its criticality deviates from the Ising class
as $\alpha_{\mathrm{LR}}$ decreases relies primarily on the collection of
numerical observations. Despite various numerical studies characterizing the
quantum phase transition in AF-LR-TFIC at equilibrium [53, 54, 57, 55, 56, 59]
and out of equilibrium [56, 60], the picture remains incomplete in some parts,
which requires more numerical evidence for clarification. Using the restricted
Boltzmann machine (RBM) for the NQS ansatz [5], we consider the moments of
staggered magnetization including the order parameter and the Binder ratio,
the two-point correlation function, and the entanglement entropy to examine
the present picture and find clearer signatures of the breakdown of the SR
Ising class and the conformal invariance along the critical line.

We begin with brief reviews of previous results on the characterization of the
criticality. The first study of AF-LR-TFIC [53] using the time-dependent
variational principle (TDVP) found a phase transition for all
$\alpha_{\mathrm{LR}}>0$, where it turned out that the critical exponent of
the correlation function decreases from the SR Ising value for
$\alpha_{\mathrm{LR}}\lesssim 2$. A significant increase in the central charge
from the Ising value $1/2$ was observed at $\alpha_{\mathrm{LR}}\lesssim 1$ in
the TDVP [53] and density matrix renormalization group (DMRG) [54]
calculations, based on which the breakdown of conformal invariance was
proposed [54]. While here we focus on the critical ground state, a violation
of the area law for the entanglement entropy was observed in the offcritical
area [53, 54, 59], and it was shown that the area law of the noncritical
ground state holds for $\alpha_{\mathrm{LR}}>2$ [63].

On the other hand, contrasting evidence was found in the other DMRG
calculations [55, 56], where the estimates of the critical exponents
$\nu\simeq 1$ and $\beta\simeq 1/8$ and the dynamic exponent $z\simeq 1$ were
in agreement with the SR Ising values for all examined $\alpha_{\mathrm{LR}}$
between $0.4$ and $3$. Although, these DMRG estimates of the critical
exponents have not been fully verified in different approaches. Linked cluster
expansion calculations [57] reported $z\nu=1.7(5)$ for
$\alpha_{\mathrm{LR}}=2$ while $z\nu\approx 1$ for $\alpha_{\mathrm{LR}}=9/4$.
Previous quantum Monte Carlo (QMC) calculations with stochastic series
expansion [58] provided lower values of $\nu$ and $\beta$ in its examined
range of $\alpha_{\mathrm{LR}}\geq 2$. However, these TDVP and DMRG results
all together suggest an interesting possibility that some of the exponents can
still be very close to the SR Ising values even for a small
$\alpha_{\mathrm{LR}}$ where the central charge indicates a deviation.

The same scenario was proposed in the study of the Kitaev chain with LR
pairing [61, 62] which becomes equivalent to the Ising chain only in the SR
limit. Along the critical line of a positive chemical potential, the conformal
invariance is broken for $\alpha_{\mathrm{LR}}<2$ while the Ising exponent
$\beta$ is unchanged in the test of a quantity that corresponds to the Ising
order parameter in the SR limit. Although there is no rigorous mapping between
the Kitaev chain and the Ising model at a finite $\alpha_{\mathrm{LR}}$, the
empirical similarity between the scenario of conformal symmetry breakdown in
the Kitaev chain and the previous observations in AF-LR-TFIM motivates us to
revisit the phase transition in this LR Ising system to examine the breakdown
of the Ising universality and the conformal invariance in different numerical
approaches.

Our VMC+RBM calculations investigate this scenario. In the finite-size scaling
(FSS) analysis of the order parameter extracted from the ground-state RBM wave
function for $0.5\leq\alpha_{\mathrm{LR}}\leq 3$, we find that our estimates
of the critical exponents are indeed very close to the SR Ising values for all
the values of $\alpha_{\mathrm{LR}}$ examined, which is in agreement with the
previous DMRG results [55, 56]. On the other hand, we find that the exponent
of the correlation function and the estimate of the central charge exhibit
deviations from the SR Ising values for a small $\alpha_{\mathrm{LR}}$.
Although an increasing numerical difference in the central charge is observed
for a small $\alpha_{\mathrm{LR}}$ between our estimate extracted from the
second Rényi entropy under periodic boundary conditions (PBC) and the previous
measurement with the von Neumann entropy in an open chain [53, 54], both
results agree on the deviation found at $\alpha<1$, supporting the scenario of
conformal invariance breakdown occurring at a sufficiently small
$\alpha_{\mathrm{LR}}$.

To identify the threshold for the breakdown of the SR Ising class beyond the
implications of the critical exponents and the central charge, we additionally
examine the critical Binder ratio [40] and the CFT description of the
universal form of the correlation function [64]. Both tests show a clear
signal of such breakdown at $\alpha_{\mathrm{LR}}<2$, which strengthens the
similar evidence found in the measurement of the critical exponent of the
correlation function. However, above $\alpha_{\mathrm{LR}}=2$, the detailed
view of the scaled correlation function [64] still indicates a gradual
deviation from the asymptotic line predicted by the CFT, raising a possibility
that a precise value of the threshold can be above $\alpha_{\mathrm{LR}}=2$
while it is less than $\alpha_{\mathrm{LR}}=3$ where the description of the
CFT is well verified.

This paper is organized as follows. The AF-LR-TFIC model Hamiltonian and the
numerical details of the VMC+RBM calculations are described in Sec. II. The
main results are given in Sec. III. In the subsections, the FSS analysis for
the estimate of the critical exponents and the extraction of the central
charge from the second Rényi entropy are presented, and then the test of the
critical Binder ratio and the comparison with the CFT prediction of the
correlation function are given to identify the threshold. The conclusions are
given in Sec. IV.

##  II Model and VMC+RBM Calculations

We consider the AF-LR-TFIC Hamiltonian [53] given as

$\hat{H}=\sin\theta\sum_{i<j}J_{ij}\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{j}+\cos\theta\sum_{i}\hat{\sigma}^{z}_{i},$
(1)

where $\theta$ is in the range of $0<\theta<\pi/2$ for the AF coupling, and
the site indices $i$ and $j$ run from $1$ to $L$ in the chain of length $L$.
We impose PBC as the boundary conditions that are necessary for the test of
the CFT description of the correlation function constructed in a cylindrical
space-time geometry. In the implementation of the algebraically decaying LR
interaction under PBC, we choose to write $J_{ij}$ with a range cutoff that
increases with the system size $L$ by adopting the formulation used in the LR-
Kitaev chain [61, 62] as

$J_{ij}=\begin{cases}|i-j|^{-\alpha_{\mathrm{LR}}}&\text{for }|i-j|<L/2,\\\
(L-|i-j|)^{-\alpha_{\mathrm{LR}}}&\text{for }|i-j|>L/2.\end{cases}$ (2)

We choose RBM as an ansatz of a trial wave function for VMC simulations to
find an approximate ground state [5]. A trial state can be written as
$|\Psi\rangle=\sum_{\mathbf{s}}\Psi(\mathbf{s};\mathscr{W})|\mathbf{s}\rangle$
with the visible variables $\mathbf{s}=(s_{1},s_{2},\ldots,s_{L})$ of RBM,
where $s_{i}$ indicates $\sigma^{x}_{i}$ for the $\hat{\sigma}^{x}$-basis
representation of the given Hamiltonian. We impose the translation symmetry
under PBC to reduce the number of variational parameters. Following the
procedures of Ref. [5], after integrating out the hidden layer, one can
express the RBM wave function as

$\Psi(\mathbf{s};\mathscr{W})=e^{a\sum_{j=1}^{L}s_{j}}\prod_{m=1}^{L}\prod_{i=1}^{n_{h}}\cosh\bigg{[}b_{i}+\sum_{j=1}^{L}W_{ij}T_{m}(s_{j})\bigg{]},$
(3)

where the translation operator $T$ is defined as $T_{m}(s_{j})=s_{j+m}$ with
periodicity $s_{j+L}=s_{j}$, and $n_{h}$ is the number of filters given for
the symmetry. On a diagram of RBM, one may illustrate the hidden layer with
$N_{h}=Ln_{h}$ neurons with $L$-fold degeneracy of the neural variables
enforcing the translation invariance. In Eq. (3), there are $(1+n_{h}+Ln_{h})$
RBM parameters of $\mathscr{W}\equiv\\{a,\mathbf{b},\mathbf{W}\\}$ to be
optimized using the VMC method. We adopt complex-valued parameters as
suggested in Ref. [5] for better convergence, although the TFIC Hamiltonian is
stoquastic [65]. We initialize the RBM by setting $a=0$ and assigning Gaussian
random numbers with zero mean and variance of $1/(Ln_{h})$ to $\mathbf{b}$ and
$\mathbf{W}$.

Figure 1: Convergence test of the the RBM wave function in the VMC search for
the ground state. The case with the system of the size $L=64$ for
$\alpha_{\mathrm{LR}}=0.5$ is shown for example. The estimates of (a) energy
density $E_{0}/L$ and (b) relative variance
$(\langle\hat{H}^{2}\rangle-\langle\hat{H}\rangle^{2})/\langle\hat{H}\rangle^{2}$
measured after $2\times 10^{5}$ SR updates are plotted as a function of
$n_{h}$. The insets show the same quantities for a fixed number of filters
$n_{h}=16$ monitored during the SR iterations of the parameter updates. The
data points in the insets represent the averages measured in the logarithmic
bins of iteration numbers. The error bars are measured with ten independent
RBM wave function samples.

In VMC calculations, we optimize the RBM parameters using the stochastic
reconfiguration (SR) method to construct the natural gradient [66, 67, 68].
The SR method can be described as the imaginary-time evolution of a trial
state, providing a new state projected in the space of
$\\{|\Psi\rangle,\partial_{1}|\Psi\rangle,\partial_{2}|\Psi\rangle,\ldots\\}$,
where
$\partial_{i}|\Psi\rangle\equiv\frac{\partial|\Psi\rangle}{\partial\mathscr{W}_{i}}$.
These procedures propose an update of the variational parameter as
$\mathscr{W}_{i}^{\mathrm{new}}=\mathscr{W}_{i}^{\mathrm{old}}+\gamma_{\mathrm{SR}}\delta\mathscr{W}_{i}$,
where $\delta\mathscr{W}_{i}$ is determined by solving the linear equation
$\mathbf{S}\delta\mathscr{W}=-\mathbf{f}$. The essential numerical procedures
are to evaluate the overlap matrix $\mathbf{S}$ and the force vector
$\mathbf{f}$,

$\displaystyle S_{ij}$ $\displaystyle=$
$\displaystyle\left\langle\Delta^{*}_{i}\Delta_{j}\right\rangle_{\mathrm{mc}}-\left\langle\Delta^{*}_{i}\right\rangle_{\mathrm{mc}}\left\langle\Delta_{j}\right\rangle_{\mathrm{mc}},$
(4) $\displaystyle f_{i}$ $\displaystyle=$
$\displaystyle\left\langle\Delta^{*}_{i}E_{\mathrm{loc}}\right\rangle_{\mathrm{mc}}-\left\langle\Delta^{*}_{i}\right\rangle_{\mathrm{mc}}\left\langle
E_{\mathrm{loc}}\right\rangle_{\mathrm{mc}},$ (5)

where the derivative $\Delta_{i}$ and the local energy $E_{\mathrm{loc}}$ are

$\Delta_{i}\equiv\frac{\partial_{i}\Psi(\mathbf{s};\mathscr{W})}{\Psi(\mathbf{s};\mathscr{W})}\quad\text{and}\quad
E_{\mathrm{loc}}\equiv\sum_{\mathbf{s}^{\prime}}\langle\mathbf{s}|\hat{H}|\mathbf{s}^{\prime}\rangle\frac{\Psi(\mathbf{s}^{\prime};\mathscr{W})}{\Psi(\mathbf{s};\mathscr{W})}.$
(6)

The expression $\langle
A\rangle_{\mathrm{mc}}\equiv\sum_{\mathbf{s}}P(\mathbf{s})A(\mathbf{s})$
denotes the Monte Carlo (MC) measurement of $A(\mathbf{s})$ with probability
$P(\mathbf{s})\propto|\Psi(\mathbf{s};\mathscr{W})|^{2}$. We use the conjugate
gradient algorithm with the Jacobi preconditioner to solve the linear equation
without explicitly storing the $S$ matrix following the strategy to reduce
computational costs proposed in Ref. [68]. For numerical stability, we use the
regularization scheme introduced in Ref. [5], where at the $p$-th SR
iteration, $S_{ij}$ is replaced by $S_{ij}(1+\lambda_{p}\delta_{ij})$ with
$\lambda_{p}=\max(\lambda_{0}b^{p},\lambda_{\mathrm{min}})$. We use the
parameters $\lambda_{0}=100$, $b=0.9$, and $\lambda_{\mathrm{min}}=0.01$. The
learning rate $\gamma_{\mathrm{SR}}$ is initially set to $0.1$ and increased
by $0.1$ for every $10000$ SR iterations until it becomes unity.

We monitor the convergence of $|\Psi\rangle$ to the ground state by evaluating
$\langle\hat{H}\rangle$ and the relative variance defined as

$\tilde{\sigma}_{E}\equiv\frac{\langle\hat{H}^{2}\rangle-\langle\hat{H}\rangle^{2}}{\langle\hat{H}\rangle^{2}}.$
(7)

The relative variance $\tilde{\sigma}_{E}$ should be precisely zero when
$|\Psi\rangle$ becomes an exact eigenstate. However, in practice, it does not
decrease below a certain value in VMC simulations. Probable systematic causes
may include the limited expressive power of a finite-size neural network with
a finite $n_{h}$, despite the universal approximation theorem, and the
stochastic fluctuations in MC measurements that can affect the linear solver.

Figure 1 presents an example of the convergence test performed at the critical
point in the system of size $L=64$ for the LR exponent
$\alpha_{\mathrm{LR}}=0.5$. Convergence tends to slow down as
$\alpha_{\mathrm{LR}}$ decreases in this LR-AF system. At the critical point,
it typically takes about an order of $10^{5}$ SR iterations until the energy
and variance become saturated within the scale of their fluctuations over the
iterations. We find that the accuracy level indicated by $\tilde{\sigma}_{E}$
after saturation depends essentially on the number of filters $n_{h}$. In our
VMC calculations for the ground state, we set the convergence criterion as
$\tilde{\sigma}_{E}<10^{-6}$, which, for example, is achieved for $n_{h}>8$ in
Fig. 1. In our tests, $n_{h}=16$ suffices for system sizes up to $L=128$ and
for the values of $\alpha_{\mathrm{LR}}$ that we consider in this study.

##  III Results and Discussions

Using the RBM wave function $\Psi(\mathbf{s})$ obtained in the VMC
optimizations at a given $\theta$, we measure the moments of staggered
magnetization including the AF order parameter, the two-point correlation
function, and the second Rényi entanglement entropy. For a given RBM sample,
the MC averages are calculated with $4\times 10^{8}$ configurations of
$\mathbf{s}$ sampled from the probability distribution
$P(\mathbf{s})\propto|\Psi(\mathbf{s})|^{2}$ using the Metropolis algorithm.
We obtain ten RBM wave function samples from independent VMC calculations. We
find that the standard error of the measurement based on one RBM sample is
typically smaller than the sample-to-sample fluctuations, and thus we estimate
the error bar by the standard deviation of the measurements over the RBM
samples.

In this section, we first present the FSS analysis to estimate the critical
exponents and the central charge for comparison with the previous TDVP and
DMRG results. Then, we proceed to present our additional tests with the
critical Binder ratio and the universal form of the correlation function to
identify the threshold for the breakdown of the SR Ising universality and the
conformal symmetry.

###  III.1 Order parameter and critical exponents

Figure 2: FSS analysis of RBM observables. (a) The estimates of the critical
exponents $\nu$, $\beta$, $\gamma$ are plotted in the range of
$\alpha_{\mathrm{LR}}$ between $0.5$ and $3$. The dotted lines are given for
comparison with the SR Ising values. The FSS collapse tests with the critical
exponents are demonstrated at $\alpha_{\mathrm{LR}}=0.5$ for the data of (b)
the Binder’s cumulant $U_{4}$, (c) the AF order parameter $m_{s}$, and (d) the
susceptibility $\chi_{s}$. The inset of (b) shows the crossing point of
$U_{4}$ locating the critical point $\theta_{c}$.

The emergence of the AF order can be detected by measuring the staggered
magnetization in the input layer of the RBM. In the AF phase, the operator
$\hat{M}_{s}=\sum_{i}(-1)^{i}\hat{\sigma}_{i}^{x}$ in each parity sector of
the $\mathbb{Z}_{2}$ symmetry indicates a finite positive or negative
expectation value. Although our MC sampling does not fix the parity, an
alternative quantity $M_{s}(\mathbf{s})=|\sum_{i}(-1)^{i}s_{i}|$ can
characterize the order-disorder phase transition at the level of the RBM wave
function. We write the order parameter as

$m_{s}=\frac{1}{L}\left\langle M_{s}\right\rangle_{\mathrm{mc}}.$ (8)

Near a critical point $\theta_{c}$, the order parameter measured in a finite
system of size $L$ is expected to behave asymptotically as
$m_{s}(\theta,L)\sim
L^{-\beta/\nu}\mathscr{M}^{(\pm)}_{\mathrm{o}}(|\theta-\theta_{c}|L^{1/\nu})$
with the critical exponents $\beta$ and $\nu$, where
$\mathscr{M}^{(\pm)}_{\mathrm{o}}$ is a size-independent scaling function. The
corresponding susceptibility can also be defined by the fluctuations of
$M_{s}$ as

$\chi_{s}=\left\langle M_{s}^{2}\right\rangle_{\mathrm{mc}}-\left\langle
M_{s}\right\rangle_{\mathrm{mc}}^{2},$ (9)

which is expected to follow the FSS ansatz of $\chi_{s}(\theta,L)\sim
L^{\gamma/\nu}\mathscr{X}_{\mathrm{o}}((\theta-\theta_{c})L^{1/\nu})$
associated with the exponent $\gamma$.

First we determine the critical point $\theta_{c}$ for a given
$\alpha_{\mathrm{LR}}$ by locating a crossing point of the Binder’s fourth-
order cumulant,

$U_{4}=1-\frac{\left\langle
M_{s}^{4}\right\rangle_{\mathrm{mc}}}{3\left\langle
M_{s}^{2}\right\rangle_{\mathrm{mc}}^{2}},$ (10)

between the curves of different $L$’s. The FSS ansatz of the cumulant is given
as
$U_{4}(\theta,L)\sim\mathscr{U}_{\mathrm{o}}((\theta-\theta_{c})L^{1/\nu})$.
Although $\mathscr{U}_{\mathrm{o}}$ becomes independent of $L$ for a large
$L$, a finite-size correction can appear for small $L$’s. The finite-size
correction of the leading order is usually assummed to be in the form of
$\theta^{*}_{L,2L}-\theta_{c}\propto L^{-\tilde{\omega}}$ for a crossing point
$\theta^{*}_{L,2L}$ identified between two adjacent curves of system sizes $L$
and $2L$. We determine $\theta_{c}$ based on this correction-to-scaling ansatz
with the extrapolation to infinite size.

After locating the critical point $\theta_{c}$, we estimate the critical
exponents $\nu$, $\beta$, and $\gamma$ by performing the standard FSS analysis
with the FSS ansatz of $m_{s}$, $\chi_{s}$, and $U_{4}$ in the critical
region. Figure 2 presents an example of the FSS analysis for
$\alpha_{\mathrm{LR}}=0.5$, showing that the data points of different $L$’s
fall well on a common scaling curve with our estimates of the critical
exponents. The numerical estimates of the critical exponents and errors are
measured using the pyfssa package [69, 70]. We tabulate our estimate of
$\theta_{c}$ and the critical exponents in Table 1. Within the error bars, our
estimates of the critical exponents are very close to the SR Ising values for
all the values of $\alpha_{\mathrm{LR}}$ examined as shown in Fig. 2(a), which
is consistent with the previous DMRG results [55, 56].

$\alpha_{\mathrm{LR}}$ | $\theta_{c}$ | $\nu$ | $\beta$ | $\gamma$ | $\eta$ | $c_{\infty}$
---|---|---|---|---|---|---
3.0 | 0.8714(7) | 1.00(4) | 0.128(5) | 1.77(5) | 0.2510(4) | 0.496(5)
2.5 | 0.9041(6) | 1.01(2) | 0.122(3) | 1.76(5) | 0.2491(2) | 0.500(4)
2.0 | 0.9489(7) | 1.00(2) | 0.121(7) | 1.76(7) | 0.2518(7) | 0.502(5)
1.5 | 1.012(1) | 1.00(1) | 0.126(9) | 1.77(4) | 0.2450(24) | 0.508(4)
1.0 | 1.103(1) | 1.01(3) | 0.126(6) | 1.78(7) | 0.2398(35) | 0.491(5)
0.5 | 1.251(1) | 1.01(3) | 0.127(5) | 1.76(4) | 0.2363(15) | 0.454(8)
Table 1: List of the critical points and exponents. Critical exponents $\nu$,
$\beta$, and $\gamma$ are determined in the FSS analysis of the collapse of
the scaling curve. The exponent $\eta$ is measured from the scaling of the
spin-spin correlation function along a fixed $r/L=1/4$ at the critical point
$\theta_{c}$. The central charge $c_{\infty}$ is extracted from the
logarithmic scaling of the second Rényi entropy.

###  III.2 Second Rényi entropy and central charge

The logarithmic system-size scaling of the entanglement entropy at a critical
point in 1D is a useful universal property to measure the central charge of
the CFT that characterizes the phase transition [71, 72, 73]. In the previous
estimate of the central charge using the TDVP [53], DMRG [54], and generalized
Hatree-Fock [59] methods, the von Neumann entanglement entropy was examined
under open boundary conditions (OBC). Instead, we consider the second Rényi
entropy for the measurement using the RBM wave function under PBC. For the
bipartition of a system into subsystems $A$ and $B$, the Rényi entanglement
entropy of an order $n$ for $\rho_{A}$ is written as

$S_{n}(\rho_{A})=\frac{1}{1-n}\ln\mathrm{tr}\rho_{A}^{n},$ (11)

where $\rho_{A}\equiv\mathrm{tr}_{B}\rho$ is the reduced density matrix of $A$
for a pure state $\rho$. The von Neumann entropy is recovered at the limit of
$n=1$. For the universality class fixed by the CFT, the von Neumann and Rényi
entropies at the critical point indicate the same central charge $c$ in the
leading-order FSS behavior. For PBC, the asymptotic scaling behavior of
$S_{n}$ [73] for half-chain bipartition is written as

$S_{n}=\frac{c}{6}\left(1+\frac{1}{n}\right)\ln L+c^{\prime}_{n},$ (12)

where $c^{\prime}_{n}$ is a nonuniversal constant.

Figure 3: Estimate of the central charge. (a) The second Rényi entropy $S_{2}$
of a half chain is plotted at the critical point $\theta_{c}$ as a function of
system size $L$. (b) The effective central charge $c_{\mathrm{eff}}(L)$ is
extrapolated to determine (c) $c_{\infty}$ for the estimate of the central
charge.

The second Rényi entropy $S_{2}$ can be reliably measured in QMC calculations
by using the replica trick [74], which has been successfully applied to the
VMC calculations with the RBM wave function [13]. We consider only $S_{2}$,
but a method was proposed to compute $S_{n}$ of the higher $n$ and to
approximate $S_{1}$ in a different NQS representation [75]. Measuring $S_{2}$
requires two copies of the RBM state, namely $\mathbf{s}^{(1)}$ and
$\mathbf{s}^{(2)}$, sampled from the joint probability distribution
$P(\mathbf{s}^{(1)},\mathbf{s}^{(2)})\propto|\Psi(\mathbf{s}^{(1)})|^{2}|\Psi(\mathbf{s}^{(2)})|^{2}$.
Each copy can be rewritten in a bipartite basis of
$\mathbf{s}\equiv(\mathbf{s}_{A},\mathbf{s}_{B})$, where $\mathbf{s}_{A}$ and
$\mathbf{s}_{B}$ are associated with the subsystems $A$ and $B$. Then, one can
obtain $e^{-S_{2}}$ by measuring the swapping operator on $A$ as

$e^{-S_{2}}=\left\langle\frac{\Psi(\mathbf{s}^{(2)}_{A},\mathbf{s}^{(1)}_{B})}{\Psi(\mathbf{s}^{(1)}_{A},\mathbf{s}^{(1)}_{B})}\frac{\Psi(\mathbf{s}^{(1)}_{A},\mathbf{s}^{(2)}_{B})}{\Psi(\mathbf{s}^{(2)}_{A},\mathbf{s}^{(2)}_{B})}\right\rangle_{\mathrm{mc}}.$
(13)

We extract the central charge from the asymptotic behavior of
$S_{2}(L)=\frac{c}{4}\ln L+c^{\prime}_{2}$. To deal with finite-size
corrections, we measure $S_{2}$ for two system sizes $L$ and $L/2$ to define
the effective central charge,

$c_{\mathrm{eff}}(L)=\frac{4}{\ln 2}[S_{2}(L)-S_{2}(L/2)],$ (14)

which would explicitly reveal finite-size behavior. The central charge is then
formally written as $c=c_{\infty}\equiv\lim_{L\to\infty}c_{\mathrm{eff}}(L)$,
which can be evaluated by extrapolating $c_{\mathrm{eff}}(L)$ to infinite $L$.
Figure 3 describes such extrapolation procedures to evaluate $c_{\infty}$ with
finite-size data of $S_{2}(L)$. We observe that $c_{\mathrm{eff}}(L)$ exhibits
the power-law convergence of $|c_{\mathrm{eff}}(L)-c_{\infty}|\propto 1/L$.
This behavior of $c_{\mathrm{eff}}(L)$ is consistent with the previous
discussion on the finite-size correction of $L^{-1/\nu}$ in the FSS analysis
of the entanglement entropy [76].

Our estimate of $c_{\infty}$ shows good agreement with $c=1/2$ of the SR Ising
class for $\alpha_{\mathrm{LR}}\gtrsim 2$. For $\alpha_{\mathrm{LR}}=1.5$ and
$1$, the values of $c_{\infty}$ are still close to $1/2$ within the deviation
of $0.01$, but the finite-size corrections become systematic and stronger. For
$\alpha_{\mathrm{LR}}=0.5$, the deviation of $c_{\infty}$ from $1/2$ is much
larger than the error bar, implying the breakdown of the conformal symmetry of
the SR Ising class. Our results are obtained from the second Rényi entropy
under PBC, providing an interesting comparison with the previous results based
on the von Neumann entropy under OBC [53, 54, 59]. All studies agree on a
significant deviation from $1/2$ for $\alpha_{\mathrm{LR}}<1$. However, we
observe the tendency of $c_{\infty}$ to decrease below $1/2$, which is in
contrast to the increase of $c$ above $1/2$ previously observed from the von
Neumann entropy under OBC. Although we cannot rule out finite-size influences,
the inconsistent trend of $c$ found in the different measures of entanglement
with different boundary conditions may also be related to the breakdown of the
conformal symmetry.

###  III.3 Critical Binder ratio

Our RBM estimates of the critical exponents and the central charge are
consistent with the combined results of the previous TDVP and DMRG studies,
supporting the scenario that the conformal symmetry is broken at a
sufficiently small $\alpha_{\mathrm{LR}}$ while some of the critical exponents
are very close to the SR Ising values. However, there is still a great
uncertainty in finding the threshold value of $\alpha_{\mathrm{LR}}$ for the
breakdown of the conformal invariance and the universality class of the SR
Ising model. Despite the fact that the deviation in the central charge is
visibly large only at $\alpha_{\mathrm{LR}}=0.5$, the strong finite-size
correction observed at a larger $\alpha_{\mathrm{LR}}$ implies that the
threshold can be much larger than $\alpha_{\mathrm{LR}}=0.5$. Therefore, we
need more reliable indicators that can go beyond the estimates of the critical
exponents and central charge to test the breakdown of the SR Ising class and
the associated conformal invariance.

For such an alternative indicator, we consider the Binder ratio,
$Q\equiv\langle M_{s}^{2}\rangle^{2}_{\mathrm{mc}}/\langle
M_{s}^{4}\rangle_{\mathrm{mc}}$, of the second and fourth moments of the
staggered magnetization. The Binder ratio at a critical point exhibits a
particular value contributing to the universality of the critical behavior,
while the value depends on the boundary conditions and the aspect ratio of the
system (see, for instance, Refs. [77, 78] and references therein). The
critical Binder ratio has been used as a reliable ingredient to identify the
universality class in the classical long-range Ising model [40], which
inspires us to perform the same test of how the Binder ratio depends on
$\alpha_{\mathrm{LR}}$ for the critical RBM wave function of the AF-LR-TFIC.

In the SR limit, at the exact critical point $\theta_{c}=\pi/4$, we obtain the
value of $Q_{\mathrm{SR}}^{*}=0.689(4)$ from the power-law extrapolation of
$Q(L)$ to infinite $L$. This particular value of the ratio has not previously
been known for the AF-TFIC, but it turns out that the corresponding value of
the cumulant $U_{4}^{*}=0.516(3)$ is very similar to the previous MC estimate
of $U_{4}^{*}=0.514(1)$ reported in the classical 2D Ising model subject to
the mixed boundary conditions where the system is periodic in one direction
and open in the other direction [79]. The implicit connection between the
mixed boundary conditions and the cylindrical geometry of our periodic chain
under the imaginary-time evolution at zero temperature may expect the
universal value of the Binder ratio in the SR limit.

Figure 4: Test of the critical Binder ratio. The self-combined ratio
$S_{\mathrm{SR}}^{*}$ and the Binder ratio $Q^{*}$ at the critical point are
plotted as a function of $\alpha_{\mathrm{LR}}$. The data points are
extrapolated to infinite size. The horizontal solid lines indicate the SR
limit.

For a finite $\alpha_{\mathrm{LR}}$, we consider the indicator called the
self-combined Binder ratio proposed in Ref. [40],

$S_{\mathrm{SR}}(L)=\frac{1}{Q_{\mathrm{SR}}^{*}}Q(L)+\frac{1}{Q(L)}Q_{\mathrm{SR}}^{*}-2,$
(15)

which removes the leading-order finite-size correction in $Q(L)$ and thus
exhibits better convergence with increasing $L$ if an accurate value of
$Q_{\mathrm{SR}}^{*}$ is provided. Figure 4 displays the value of
$S_{\mathrm{SR}}^{*}\equiv\lim_{L\to\infty}S_{\mathrm{LR}}(L)$ obtained from
the power-law extrapolation to infinite $L$. It turns out that while
$S^{*}_{\mathrm{SR}}$ is almost zero for $\alpha_{\mathrm{LR}}=3$ and $2.5$,
the deviation of $S^{*}_{\mathrm{SR}}$ appears for
$\alpha_{\mathrm{LR}}\lesssim 2$ and increases as $\alpha_{\mathrm{LR}}$
decreases. The estimate of $Q^{*}\equiv\lim_{L\to\infty}Q(L)$ shows a similar
increase from the value of the SR limit as $\alpha_{\mathrm{LR}}$ decreases,
although it still indicates a slight deviation even for
$\alpha_{\mathrm{LR}}=2.5$ and $3$ where $S^{*}_{\mathrm{SR}}\simeq 0$. This
is consistent with the observation in Ref. [40], verifying that
$S_{\mathrm{LR}}(L)$ converges better at a finite $L$. Our data suggest that
the threshold for the SR Ising universality is possibly around
$\alpha_{\mathrm{LR}}=2$ above which $S_{\mathrm{SR}}^{*}$ is zero within the
error bars.

###  III.4 CFT prediction of the correlation function

Figure 5: Critical exponent of the spin-spin correlation function. The
correlation function $C_{xx}(r)$ at $r=L/4$ is plotted as a function of the
system size $L$. The inset shows the exponent $\eta$ extracted from the data
fitting to $C_{xx}(L/4)\propto L^{-\eta}$. The dotted lines indicating the SR
Ising exponent $\eta=1/4$ are given for comparison.

The other test for the conformal invariance of the SR Ising class concerns the
correlation function to compare its asymptotic scaling behavior with the CFT
description given at the SR limit, following the strategy proposed in Ref.
[64]. The PBC imposed on our RBM wave function is essential for this test. We
consider the spin-spin correlation function,

$C_{xx}(r)=\frac{1}{L-r}\sum_{i=1}^{L-r}\langle\hat{\sigma}^{x}_{i}\hat{\sigma}^{x}_{i+r}\rangle=\frac{1}{L-r}\sum_{i=1}^{L-r}\langle
s_{i}s_{i+r}\rangle_{\mathrm{mc}}\,,$ (16)

where the distance $r$ runs from $1$ to $L/2$ in the periodic chain, and the
average over the sites is taken for better statistics in the MC measurements.

Figure 6: FSS analysis of the spin-spin correlation function. The data
collapse of $L^{2\Delta_{\sigma}}|C_{xx}(r)|$ is examined as a function of
$r/L$ with the exponent $2\Delta_{\sigma}$ being fixed at the SR Ising value
$1/4$. The solid line indicates the CFT-predicted form of $a[\sin(\pi
r/L)]^{-2\Delta_{\sigma}}$ given for comparison with the scaled curve of the
measured correlation function.

The CFT in a cylindrical space-time geometry predicts the asymptotic form of
the two-point correlation function [80, 81]. In the SR limit, the 2D Ising
universality class and its CFT predicts that the correlation function in Eq.
(16) behaves as

$C_{xx}(r)\propto\left(\frac{1}{L\sin(\pi r/L)}\right)^{2\Delta_{\sigma}}$
(17)

with the scaling dimension $\Delta_{\sigma}=1/8$. A partial test of this
prediction includes verification of the scaling dimension that is equivalent
to the exponent $\eta=2\Delta_{\sigma}$ characterizing the algebraic decay of
$C_{xx}(r)\sim r^{-\eta}$ at the critical point. A more comprehensive test is
to directly compare the form of the measured correlation function with the
description of the CFT. This strategy was originally proposed in the projector
QMC simulations [64], which we employ here for the RBM wave function obtained
in the VMC simulations.

First, we measure the critical exponent $\eta$ from the FSS behavior of
$C_{xx}(r)$ along a fixed $r/L=1/4$. We obtain the estimate of $\eta$ from the
linear fit to the ansatz of $C_{xx}(L/4)\propto L^{-\eta}$, which is displayed
as a function of $L$ on the logarithmic scale in Fig. 5. For
$\alpha_{\mathrm{LR}}\geq 2$, the estimate of $\eta$ is consistent with the SR
Ising value $1/4$. However, as $\alpha_{\mathrm{LR}}$ decreases below $2$, it
turns out that the estimate of $\eta$ decreases below $1/4$, implying that the
SR Ising class does not hold below $\alpha_{\mathrm{LR}}=2$. These
observations are consistent with the previous TDVP result [53], where the
threshold value of $\alpha_{\mathrm{LR}}=2.25$ was suggested based on their
estimate of the scaling dimension.

However, it is certainly preferable to find stronger supporting evidence
because the numerical deviation of our estimate of $\eta$ from $1/4$ is rather
small and we cannot rule out finite-size influences in the data fitting. In
fact, as shown in Fig. 5, the data points match well with the lines of
$L^{-1/4}$ within the available system sizes. Additionally, assuming that the
hyperscaling relation still holds, $\eta<1/4$ implies $\gamma>7/4$ and
$\beta<1/8$ if $\nu=1$ is fixed. We cannot argue such small changes of the
exponents in our FSS analysis of data collapse. Within our limited numerical
accuracy, it is thus difficult to precisely determine the breakdown of the
universality class solely on the basis of the critical exponents.

Figure 7: Scaled correlation function for the test of the CFT description.
Equation (18) is examined with ten samples of the RBM wave functions
independently obtained in the VMC simulations at the critical point. The
markers indicate the data of $L=64$ where the fluctuations in the RBM samples
are smaller than the marker size. The gray solid line is the average over the
samples for $L=128$. The shade is filled between the minimum and maximum
magnitudes of the data in the RBM wave function samples for $L=128$.

For the direct test of the CFT-predicted form of the correlation function in
Eq. (17), we perform the FSS analysis with data collapse of
$L^{2\Delta_{\sigma}}|C_{xx}(r)|$ as shown in Fig. 6. We fix the exponent
$2\Delta_{\sigma}$ at the SR Ising value $1/4$ for the test of the SR Ising
class. Despite the fact that the measured values of $\eta$ are not used, we
still observe a good collapse of the data points falling on a common scaling
curve at all $\alpha_{\mathrm{LR}}$ except with slight deviations found at
$\alpha_{\mathrm{LR}}=0.5$ where $\eta\approx 0.236$ is maximally different
from $1/4$. In the graphical comparison between the collapsed data curve and
the CFT prediction of $L^{2\Delta_{\sigma}}|C_{xx}(r)|\propto[\sin(\pi
r/L)]^{-2\Delta_{\sigma}}$, we observe that the data curve starts to deviate
from the CFT prediction at $\alpha_{\mathrm{LR}}\lesssim 2$, and the
deviations increase with decreasing $\alpha_{\mathrm{LR}}$, which is
consistent with the evidence found in the critical Binder ratio.

To compare the measured correlation function with the CFT description in
higher resolution, we examine the scaled correlation function
$C_{\mathrm{sc}}(r/L)$ [64] defined as the ratio of the measured $C_{xx}(r/L)$
and the form predicted in Eq. (18),

$C_{\mathrm{sc}}(r/L)=\left[L\sin\left(\pi\frac{r}{L}\right)\right]^{2\Delta_{\sigma}}\left|C_{xx}(r)\right|,$
(18)

where we also fix $2\Delta_{\sigma}$ at $1/4$. If the CFT of the 2D Ising
universality class holds for this LR system, one can expect an asymptotically
flat tail in this scaled correlation function. Figure 7 displays
$C_{\mathrm{sc}}(r/L)$ for $\alpha_{\mathrm{LR}}$ between $2$ and $3$. At
$\alpha_{\mathrm{LR}}=3$, one can clearly notice the flat line, which
indicates the validity of the CFT description. Although we have not shown
here, our test in the SR limit has shown the same flatness of
$C_{\mathrm{sc}}(r/L)$ as displayed for $\alpha_{\mathrm{LR}}=3$ within the
fluctuations in the examined RBM samples.

It turns out that the shape of $C_{\mathrm{sc}}(r/L)$ becomes more curved as
$\alpha_{\mathrm{LR}}$ decreases. As shown in Fig. 7, the flat tail area in
$C_{\mathrm{sc}}(r/L)$ tends to be harder to identify for a smaller $\alpha$,
such a change in the shape of the correlation function develops gradually as
$\alpha_{\mathrm{LR}}$ decreases below $3$. Within the achievable accuracy of
our present calculations, it is difficult to detect a precise threshold value
of $\alpha_{\mathrm{LR}}$ at which the asymptotic flat line would disappear,
because fluctuations in RBM samples tend to increase with increasing system
size. Our test of $C_{\mathrm{sc}}(r/L)$ suggests that the threshold for the
CFT of the SR Ising class may be higher than $\alpha_{\mathrm{LR}}=2$,
although it still requires much more accurate calculations of the correlation
function or more sensitive indicators to identify a precise threshold.

##  IV Summary and Conclusions

We have studied a quantum phase transition in the AF-LR-TFIC using VMC methods
with the RBM trial wave function ansatz. Based on the FSS analysis to measure
the critical exponents of the order parameter and the estimate of the central
charge from the second Rényi entropy, we have verified the previous TVDP and
DMRG results [53, 54, 55, 56], supporting the scenario [62] that conformal
symmetry is broken at a sufficiently small LR exponent $\alpha_{\mathrm{LR}}$
while some critical exponents are very close to the SR Ising values regardless
of $\alpha_{\mathrm{LR}}$. To identify the threshold for the breakdown of the
SR Ising universality and the conformal symmetry, we have performed two
additional tests that do not rely on the critical exponents and the central
charge. Our first test using the self-combined version of the Binder ratio
[40] finds that the universal Binder ratio holds for
$\alpha_{\mathrm{LR}}\gtrsim 2$ below which the ratio increases significantly
as $\alpha_{\mathrm{LR}}$ decreases. In the test for the CFT description of
the spin-spin correlation function, our FSS analysis finds a qualitative
difference between the measured data curve and the description of the CFT for
$\alpha_{\mathrm{LR}}<2$. The detailed view given by the scaled correlation
function [64] indicates a gradual change that still occurs above
$\alpha_{\mathrm{LR}}=2$, raising a possibility that the threshold may be
larger than $\alpha_{\mathrm{LR}}=2$ while it is less than
$\alpha_{\mathrm{LR}}=3$ where the measured data produce the CFT description.

Our measurement of the critical Binder ratio and the test of the CFT-predicted
form of the correlation function reveal numerical evidence supporting the
breakdown of the Ising universality and the conformal symmetry in the AF-LR-
TFIC. However, our rough estimate of the threshold value of
$\alpha_{\mathrm{LR}}$ for the breakdown also indicates the need for further
intensive studies to precisely determine the threshold with more sensitive
indicators of conformal symmetry, such as Klein bottle entropy [82]. In
addition, the apparent mismatch between the correlation function exponent and
the other critical exponents needs to be further investigated to properly
examine the hyperscaling relations with higher numerical accuracy.

Our VMC+RBM calculations for the FSS analysis of the criticality in this LR
interacting system exemplify the practical applicability of the NQS framework
to the study of quantum phase transitions. While we have only considered a
stoquastic Hamiltonian, the accuracy of the RBM wave function shown in our
analysis of the critical ground state demonstrates its potential as an
alternative or complementary tool to conventional zero-temperature methods.

###### Acknowledgements.

J.K. and D.K. contributed equally to this work. We thank Synge Todo and Hong-
Hao Tu for fruitful discussions in the ASG meeting at the PCS-IBS. This work
was supported by the Basic Science Research Program through the National
Research Foundation of Korea (NRF-2019R1F1A106321) and also by the KIAS
associate member program. Computing resources are provided by the KISTI
supercomputing center (KSC-2021-CRE-0165). We appreciate APCTP and PCS-IBS for
their hospitality during the completion of this work.

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