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# Supplementary Information for quadrupolar magnetic excitations in an
isotropic spin-1 antiferromagnet

A. Nag<EMAIL_ADDRESS>Diamond Light Source, Harwell Campus,
Didcot OX11 0DE, United Kingdom  A. Nocera<EMAIL_ADDRESS>Stewart
Blusson Quantum Matter Institute, University of British Columbia, Vancouver,
British Columbia, V6T 1Z4 Canada  Department of Physics Astronomy, University
of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1  S. Agrestini
M. Garcia-Fernandez  A. C. Walters  Diamond Light Source, Harwell Campus,
Didcot OX11 0DE, United Kingdom  Sang-Wook Cheong  Rutgers Center for Emergent
Materials, Rutgers University, Piscataway, NJ, USA.  S. Johnston
<EMAIL_ADDRESS>Department of Physics and Astronomy, The University of
Tennessee, Knoxville, Tennessee 37966, USA  Ke-Jin Zhou
<EMAIL_ADDRESS>Diamond Light Source, Harwell Campus, Didcot OX11
0DE, United Kingdom

## Supplementary Note 1: Experimental details

The single crystal of Y2BaNiO5 was pre-aligned using Laue diffraction so that
its chain direction lies in the RIXS scattering plane (Fig. S1a). The momentum
transfer resolution is less than 0.011 ($2\pi/c$) (see Fig. S1b). Negative and
positive values of $q_{\parallel}$ represent x-ray grazing-incident and
grazing-exit geometries, respectively. The strongest specular elastic
reflection was seen close to $q_{\parallel}=-0.03$, as shown in Fig. 2(a) of
main text. This position originates from a combination of the high quality
mirror-like cleaved surface of the crystal and the arrangement of optical
elements used for collecting the scattered x-rays (see Fig. S1c, d).

Supplementary Figure S1: a RIXS geometry for probing spin excitations in
Y2BaNiO5. Throughout the paper, projection of momentum transfer
($q_{\parallel}$) in units of $2\pi/c$ along the chain direction is presented.
b The momentum transfer resolution for our experimental setup as a function of
$q_{\parallel}$. c A comparison of the RIXS spectra recorded at $\pm
q_{\parallel}$. The agreement between the two RIXS spectra confirm that our
assigned $q_{\parallel}=0.00$ corresponds to the true specular reflection
condition. d A schematic representation of the optical arrangement for
scattered X-rays from sample at I21 RIXS beamline, Diamond Light Source. The
$k_{f}$ and true momentum transfer values are determined by the scattered beam
collected by a plane parabolic mirror $\sim 4^{\circ}$ higher than the direct
scattered beam direction. The incidence angle on the sample for specular
reflection condition for the direct beam is, therefore, $\sim 2^{\circ}$ lower
than that for the true specular reflection condition at $q_{\parallel}=0.00$.
Usually this direct beam is weak and does not reach the detector. However, due
to the high quality mirror like cleaved surface of the crystal, a strong
specular elastic reflection corresponding to this direct beam is seen close to
$q_{\parallel}=-0.03$ in Fig. 2(a) of the main text.

## Supplementary Note 2: RIXS data fitting

Fig. S2 shows the extracted values of coefficients $C_{0}$, $C_{1}$ and
$C_{2}$ of the three different types of excitations considered for fitting the
RIXS data as described in Methods of main text. Fig. S3 shows the fit profiles
of the different components. Fig. S4 shows the fitting of the $\Delta
S_{\mathrm{tot}}=1$ triplet excitations at $q_{\parallel}=0.47$.

Supplementary Figure S2: a, b, c Coefficients $C_{1}$, $C_{2}$, and $C_{0}$
accounting for changes in the RIXS scattering cross section with varying x-ray
incidence for $\Delta S_{1}$, $\Delta S_{2}$, and $\Delta S_{0}$ excitations,
respectively. The coefficient values are determined by fitting RIXS spectra
with dynamical spin susceptibilites obtained from DMRG using Eq. (5) given in
Methods. Error bars are least square fit errors. Also shown by the continuous
lines are the integrated dynamical spin structure factors
$A_{S}(q_{\parallel})=\int S(q_{\parallel},\omega)d\omega$ from DMRG for the
$\Delta S_{1}$, $\Delta S_{2}$, and $\Delta S_{0}$ excitations. d Total RIXS
intensities for $\Delta S_{1}$, $\Delta S_{2}$, and $\Delta S_{0}$
excitations.  Supplementary Figure S3: Ni $L$-edge RIXS line spectra from
Y2BaNiO5 at 11 K, with fits derived from the spin susceptibilities obtained
from DMRG using Eq. (5) given in Methods. Corresponding $q_{\parallel}$ value
is written in each panel.  Supplementary Figure S4: a-f Temperature dependent
RIXS spectra at $q_{\parallel}=0.47$. Shaded regions are fits to $\Delta
S_{\mathrm{tot}}=1$ triplet excitations using resolution-convoluted damped
harmonic oscillator functions weighted by the Bose factor. The extracted
amplitudes have been used in Fig. 4d of main text. The dashed lines indicate
contributions from the $\Delta S_{\mathrm{tot}}=2$ and multi-magnon
excitations.

## Supplementary Note 3: Influence of temperature on the boundaries of
two-$\Delta S_{1}$ excitations.

In a simple picture, if one considers a continuum from pairs of non-
interacting $\Delta S_{1}$ triplets due to single spin-flips at multiple
sites, then bandwidth reduction of each would manifest as the overall raising
and lowering of the lower and upper boundaries of the continuum, respectively.
In Fig. S5a and b, we show the change of such continuum boundaries upon
changing the Haldane gap energy $\Delta_{H}$ using the semi-quantitative
Haldane dispersion relation given in the caption of Fig. 2 of main text for
two $\Delta S_{1}$ triplets. In Fig. S5c and d, we show the change of the
continuum boundaries upon changing the velocity $v$ of the $\Delta S_{1}$
triplets and in Fig. S5e and f, the combined effect of $\Delta_{H}$ and $v$.
As temperature is raised, due to the increase (decrease) of the $\Delta_{H}$
($v$), the energy of the lower (upper) boundary of the continuum at
$q_{\parallel}=0$ increases (decreases). Similar effect is seen on the two
components of the $\Delta S_{2}$ excitations in Fig. 4 of main text.

Supplementary Figure S5: a, The influence of Haldane gap $\Delta_{H}$ on the
boundaries of two-$\Delta S_{1}$ excitations. The arrow indicates the
hardening of the lower boundary with increasing $\Delta_{H}$. b, The variation
in the energies of the upper and lower boundaries at $q_{\parallel}=0$ with
increasing $\Delta_{H}$ extracted from the curves in panel a. c The influence
of the velocity $v$ of the $\Delta S_{1}$ excitations on the boundaries of
two-$\Delta S_{1}$ excitations. The arrow indicates the softening of the upper
boundary with decreasing $v$. d, The variation in the energies of the upper
and lower boundaries at $q_{\parallel}=0$ with decreasing $v$ extracted from
the curves in panel c. e The influence of $\Delta_{H}$ and $v$ on the
boundaries of the two $\Delta S_{1}$ excitations. The arrow indicates the
hardening (softening) of the lower (upper) boundary with increasing
(decreasing) $\Delta_{H}$ ($v$). f, The variation in the energies of the upper
and lower boundaries at $q_{\parallel}=0$ with increasing (decreasing)
$\Delta_{H}$ ($v$), extracted from the corresponding curves in panel e.

## Supplementary Note 4: Real-space representations for the dynamical
structure factors

Here, we provide the real-space representation of the dynamical structure
factors studied in the main text. For a finite system, the dynamical structure
factor is given by

$S_{\alpha}(q_{\parallel},\omega)=\frac{1}{L}\sum^{L}_{j=1}e^{\mathrm{i}q_{\parallel}(j-c)}\Big{\\{}-\frac{1}{\pi}{\rm
Im}\Big{[}\langle\psi|S^{\alpha}_{j}\frac{1}{\omega-H+\mathrm{i}\eta}S^{\alpha}_{c}|\psi\rangle\Big{]}\Big{\\}},$
(S1)

where the operator $S^{\alpha}_{j}$ is applied at the center $j=c$ of the long
chains with open boundaries[1, 2, 3], and $|\psi\rangle$ represents the ground
state of the system.

Using the Liouville[4] and the purification approaches[5, 6, 7], the dynamical
structure factor in real space and a finite temperature $T=1/\beta$ is

$S_{\alpha}(q_{\parallel},\omega)=\frac{1}{L}\sum^{L}_{j=1}e^{\mathrm{i}q_{\parallel}(j-c)}\Big{\\{}-\frac{1}{\pi}{\rm
Im}\Big{[}\langle\psi(\beta)|S^{\alpha}_{j}\frac{1}{\omega-{\cal
L}+\mathrm{i}\eta}S^{\alpha}_{c}|\psi(\beta)\rangle\Big{]}\Big{\\}},$ (S2)

where ${\cal L}=H\otimes I+I\otimes(-H)$ is the Liouville operator and
$|\psi(\beta)\rangle$ is the purified thermal state given by
$|\psi(\beta)\rangle={e^{-\beta(H\otimes I)/2}}|\psi_{\infty}\rangle$. Here,
the Hamiltonian acts only on the physical sites of the system and
$|\psi_{\infty}\rangle$ is the maximally entangled infinite temperature state
(more details are provided in the Supplementary Note 4).

## Supplementary Note 5: Magnon-like states unveiled by the dynamical spin
correlations in Haldane spin chains

In this section we analyze the excitations encoded in the three dynamical
correlation functions introduced in the main text in Eq. (1) in terms of
single triplet excitations or _magnon_ states in the Haldane chain. Following
Eq. (5) in Ref. [8], a single magnon state in an infinite chain with momentum
$q_{\parallel}$ and spin quantum number $S^{z}=\alpha$ has the form

$M_{q_{\parallel},\alpha}|\psi\rangle=|q_{\parallel},\alpha\rangle=\sum_{j}e^{\mathrm{i}q_{\parallel}j}c^{{\dagger}}_{j}(q_{\parallel},\alpha)|\psi\rangle,$
(S3)

where the operator $c^{{\dagger}}_{j}(q_{\parallel},\alpha)$ includes the
$S^{+}_{j}$ operator and products of multiple spin operators in the vicinity
of site $j$. In other words, single magnons are not constructed by applying
only local spin operators
$S^{1}_{q_{\parallel}}=\frac{1}{\sqrt{L}}\sum_{j}e^{\mathrm{i}q_{\parallel}j}S^{+}_{j}$.
When $S^{1}_{q_{\parallel}}$ is applied to the ground state, the resulting
state can be expressed at low energy by a multi-magnon expansion

$S^{1}_{q_{\parallel}}|\psi\rangle=\Big{(}\sum_{q_{\parallel},\alpha}t^{1}(q_{\parallel},\alpha)M_{q_{\parallel},\alpha}+\sum_{q^{1}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma}s^{1}(q^{(1)}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma)M_{q^{(1)}_{\parallel},\beta}M_{q^{(2)}_{\parallel},\gamma}+...\Big{)}|\psi\rangle,$
(S4)

where two-magnon and higher-order multi-magnon states naturally appear. Note
that $t^{1}(q_{\parallel},\alpha)$ and
$s^{1}(q^{(1)}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma)$ encapsulate the
projection of the $S^{1}_{q_{\parallel}}|\psi\rangle$ state on the single- and
two-magnon wave functions, respectively. The expression above tells us that
the dynamical correlation function $S_{1}(q_{\parallel},\omega)$ encodes
information about single magnon excitations, but also two-magnon and three-
magnon excitations as well [9, 10] (with a reduced spectral weight). By
symmetry arguments, it is easy to prove that $S^{0}_{q_{\parallel}}$ and
$S^{2}_{q_{\parallel}}$ operators can only induce states with single or
multiple pairs of magnon excitations

$\displaystyle S^{0}_{q_{\parallel}}|\psi\rangle$
$\displaystyle=\Big{(}t^{0}(q_{\parallel})+\sum_{q^{1}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma}s^{0}(q^{(1)}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma)M_{q^{(1)}_{\parallel},\beta}M_{q^{(2)}_{\parallel},\gamma}+...\Big{)}|\psi\rangle$
(S5) $\displaystyle S^{2}_{q_{\parallel}}|\psi\rangle$
$\displaystyle=\Big{(}\sum_{q^{1}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma}s^{2}(q^{(1)}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma)M_{q^{(1)}_{\parallel},\beta}M_{q^{(2)}_{\parallel},\gamma}+...\Big{)}|\psi\rangle.$
(S6)

Here, $s^{0(2)}(q^{(1)}_{\parallel},\beta,q^{(2)}_{\parallel},\gamma)$
describes the projection of the $S^{0(2)}_{q_{\parallel}}|\psi\rangle$ states
on the two-magnon wave function.

Note that the dynamical correlator $S_{0}(q_{\parallel},\omega)$ defined in
the main text naturally contains an elastic peak since the
$S^{0}_{q_{\parallel}}$ operator is equal to the Hamiltonian operator at
$q_{\parallel}=0$ and $t^{0}(q_{\parallel}=0)=E_{\rm gs}$.

## Supplementary Note 6: Bound state of $\Delta S_{2}$ excitations

In this section, we discuss the physical origin of the sharp peak observed in
the $\Delta S_{2}$ spectrum close to $q_{\parallel}=0$ with an energy of
$\simeq 136$ meV (see Figs. 2e and 2f of the main text). In low-dimensional
systems, a van Hove singularity in magnetic density of states can give rise to
sharp peaks at momentum values where the magnetic dispersion has stationary
points. The sharp peaks observed in our study could therefore originate either
from a van Hove singularity in the triplet density of states, or from the
formation of a bound state of the triplet pairs _above_ the upper boundary of
the continuum formed (equivalent to two-magnons).

Supplementary Figure S6: a, $S_{2}(q_{\parallel},\omega)$ from DMRG. b,
$\rho_{2}(q_{\parallel},\omega)$ obtained from joint $\Delta S_{1}$ density of
states. Both panels are over plotted with the lower and upper boundaries of
the continuum from pairs of non-interacting triplets (equivalent to the two-
magnon continuum). c, $S_{2}(q_{\parallel}=0,\omega)$,
$\rho_{2}(q_{\parallel}=0,\omega)$ and $S_{1}(q_{\parallel}=0.25,2\omega)$.
The top $x$-axis is the energy scale for $S_{1}(q_{\parallel}=0.25,\omega)$.
d, One magnon dispersion relationship fitted from DMRG and experimental RIXS
data. We expand it around $q_{\parallel}\simeq\pm 0.25$ obtaining free
quasiparticles with _negative_ mass. Here negative mass just means that the
triplet energy cannot be bigger than $\omega_{S_{1}}(q_{\parallel}\simeq
0.25)=64.5$ meV.

Adopting the dispersion of $\Delta S_{1}$ triplet excitations [11, 12, 13, 14]
obtained from our DMRG calculations
$\omega^{2}_{S_{1}}(q_{\parallel})=\Delta^{2}_{\mathrm{H}}+v^{2}\mathrm{sin}^{2}q_{\parallel}+\alpha^{2}\mathrm{cos}^{2}\frac{q_{\parallel}}{2}$,
stationary points are located at $q_{\parallel}=0$ and $\simeq 0.25$
($2\pi/c$) (see Fig. S6d). At $q_{\parallel}\simeq 0.25$, where the $\Delta
S_{1}$ excitation has the highest energy and thus can contribute to the
highest triplet pair energy, the $\Delta S_{1}$ density of states has a
square-root singularity $\sim(|\omega-\omega_{S_{1}}(q_{\parallel}\simeq
0.25)|)^{-1/2}$. Ignoring matrix elements effects, the triplet pair density of
states can be assumed to be proportional to the _joint_ $\Delta S_{1}$ density
of states
$\rho_{2}(q_{\parallel},\omega)=\sum_{k_{\parallel}}\delta(\omega-\omega_{S_{1}}(k_{\parallel})-\omega_{S_{1}}(q_{\parallel}-k_{\parallel}))$,
which is shown in Fig. S6b. Fig. S6c shows the DMRG calculated
$S_{2}(q_{\parallel}=0,\omega)$ with the sharp peak at an energy of $\sim 136$
meV. Compared to it, $\rho_{2}(q_{\parallel}=0,\omega)$ has an asymmetric peak
corresponding to the van Hove singularity at $\sim 125$ eV (lower by $\sim 11$
meV). Evidently, the sharp component of the $\Delta S_{2}$ excitation does not
arise from the singularities in the magnetic density of states if the triplets
are non-interacting and therefore have negligible bandwidth renormalisation.

In Fig. S6c, $S_{1}(q_{\parallel}=0.25,2\omega)$ is plotted by doubling the
energy scale, to show that the maximum energy of a pair of non-interacting
triplets can be $~{}129$ meV ($\sim 64.5\times 2$, the true energy scale for
$S_{1}(q_{\parallel}=0.25,\omega)$ is given on top of the panel). The sharp
peak obtained in the $\Delta S_{2}$ excitations is therefore $\sim 7$ meV
higher than the non-interacting scenario at $q_{\parallel}=0$. Also, shown in
Fig. S6a are the lower and upper boundaries of the continuum from pairs of
non-interacting triplets (equivalent to the two-magnon continuum). There, the
sharp peaks for the $\Delta S_{2}$ excitation appear above the upper boundary
of the continuum throughout the probed region of reciprocal space. The small
positive value therefore suggests a weak _repulsive_ interaction between the
triplets formed after a quadrupolar $\Delta S_{2}$ excitation [8], and the
sharpness of the peaks suggests that a bound state of the triplet pairs forms.

We here provide a semi-quantitative analysis to show that at sufficiently high
energies, a pair of triplet excitations form a bound state above the two-
triplet continuum in presence of weak repulsion between them. The $\Delta
S_{1}$ dispersion can be expanded around $q_{\parallel}\simeq\pm 0.25$, which
are the likely momentum states contributing to the formation of the highest
energy triplet pair state at $q_{\parallel}\simeq 0$. From Fig. S6d, we obtain

$E(q_{\parallel}\simeq\pm 0.25+\delta
q_{\parallel})\simeq\omega_{S_{1}}(q_{\parallel}=\pm
0.25)+\frac{\hslash^{2}(\delta q_{\parallel})^{2}}{2m^{*}}.$ (S7)

A Schrödinger equation can thus be set up for these two quasiparticles
resulting from independent $\Delta S_{1}$ excitations:

$\Big{[}-\frac{\hslash^{2}}{2m^{*}}\frac{\partial^{2}}{\partial
r_{1}^{2}}-\frac{\hslash^{2}}{2m^{*}}\frac{\partial^{2}}{\partial
r_{2}^{2}}+V(r_{1},r_{2})\Big{]}\Psi(r_{1},r_{2})=(E-2\omega_{S_{1}}(q_{\parallel}\simeq
0.25))\Psi(r_{1},r_{2}).$ (S8)

Since we are interested in the triplet pair state at $q_{\parallel}\simeq 0$,
we can neglect the center of mass motion ($\delta q^{\rm
cm}_{\parallel}=\delta q_{\parallel}=0$ and
$E=\tilde{E}+\frac{\hslash^{2}(\delta
q_{\parallel})^{2}}{2m_{cm}^{*}}=\tilde{E}$) and consider just the dynamics in
the relative coordinate $r=r_{1}-r_{2}$:

$\Big{[}-\frac{\hslash^{2}}{2m^{*}}\frac{\partial^{2}}{\partial
r^{2}}+V(r)\Big{]}\Psi(r)=(\tilde{E}-2\omega_{S_{1}}(q_{\parallel}\simeq
0.25))\Psi(r).$ (S9)

Assuming a repulsive $\delta$-like potential among the $\Delta S_{tot}=2$
excitations [8], $V(r)=\lambda\delta(r)$, with $\lambda>0$ and noting that the
effective mass to be negative $m^{*}=-|m^{*}|$ we get

$\Big{[}-\frac{\hslash^{2}}{2|m^{*}|}\frac{\partial^{2}}{\partial
r^{2}}-\lambda\delta(r)\Big{]}\Psi(r)=E^{\prime}\Psi(r),$ (S10)

where we have multiplied the equation by $-1$ and then redefined
$E^{\prime}=-\tilde{E}+2\omega_{S_{1}}(q_{\parallel}\simeq 0.25)$. Elementary
quantum mechanics then predicts the existence of a bound state with energy

$E^{\prime}_{B}=-\frac{|m^{*}|\lambda^{2}}{2\hslash^{2}}.$ (S11)

Eq. (S11) implies the existence of a bound state above the continuum, since
for $E^{\prime}<0$ we have $\tilde{E}>2\omega_{S_{1}}(q_{\parallel}\simeq
0.25)$, and $\tilde{E}=2\omega_{S_{1}}(q_{\parallel}\simeq
0.25)+|E^{\prime}_{B}|$. From this expression, we can _a posteriori_ estimate
the strength of the quasiparticle interaction $\lambda$ (which has units of
energy times length, so we divide by the lattice unit $c$) assuming
$|E^{\prime}_{B}|\simeq 7$ meV:

$\lambda/c=\sqrt{4\frac{\hslash^{2}}{2|m^{*}|}|E^{\prime}_{B}|}\simeq
39~{}\text{meV}\simeq 1.6J.$ (S12)

This is a rather weak interaction if we compare it with the $\Delta S_{1}$
bandwidth $W\simeq 2.3J$ and therefore validates our previously estimated
energy scales of the triplet pair density of states and continuum boundaries.

To further confirm this estimate, we simulated a Bose Hubbard model on a 1D
lattice of a finite length with open boundary conditions with an on-site
attractive interaction $U_{b}=-|U_{b}|$ with $N_{b}=2$ bosons

$H=-t_{b}\sum_{i}(b^{{\dagger}}_{i}b^{\phantom{\dagger}}_{i+1}+{\rm
h.c.})+\frac{U_{b}}{2}\sum_{i}n^{b}_{i}(n^{b}_{i}-1).$ (S13)

We have assumed $t_{b}=0.575J$ such that the bandwidth is $W_{b}=4t_{b}=2.3J$.
We find that on a long chain of $L=64$ sites, the system shows a bound state
with energy $E_{B}=E(U_{b}=-1.675J)-E(U_{b}=0)\simeq-0.28J$ $\sim-6.7$ meV,
consistent with our estimation above using an attractive $\delta$-like
potential between the quasiparticles.

## Supplementary Note 7: Computational details to reproduce the DMRG results

Here we provide instructions on how to reproduce the DMRG results used in the
main text. The results reported in this work were obtained with DMRG++
versions 6.01 and PsimagLite versions 3.01. The DMRG++ computer program [15]
can be obtained with:

    git clone https://github.com/g1257/dmrgpp.git
    git clone https://github.com/g1257/PsimagLite.git

The main dependencies of the code are BOOST and HDF5 libraries. To compile the
program:

    cd PsimagLite/lib; perl configure.pl; make
    cd ../../dmrgpp/src; perl configure.pl; make

The DMRG++ documentation can be found at `
https://g1257.github.io/dmrgPlusPlus/manual.html` or can be obtained by doing
`cd dmrgpp/doc; make manual.pdf`. In the description of the DMRG++ inputs
below, we follow very closely the description in the supplemental material of
Ref. [16], where similar calculations were performed. Other inputs and scripts
can be made available upon request.

a) Obtaining zero-temperature spectra

The $T=0$ results can be reproduced as follows. We first run `./dmrg -f
inputGS.ain -p 12` to obtain the ground state wave-function and ground state
energy with 12 digit precision using the `-p 12` option. The `inputGS.ain` has
the form

    ##Ainur1.0
    TotalNumberOfSites=100;
    NumberOfTerms=2;

    ### 1/2(S^+S^- + S^-S^+)
    gt0:DegreesOfFreedom=1;
    gt0:GeometryKind="chain";
    gt0:GeometryOptions="ConstantValues";
    gt0:dir0:Connectors=[1.0];

    ### S^zS^z part
    gt1:DegreesOfFreedom=1;
    gt1:GeometryKind="chain";
    gt1:GeometryOptions="ConstantValues";
    gt1:dir0:Connectors=[1.0];

    Model="Heisenberg";
    HeisenbergTwiceS=2;
    SolverOptions="twositedmrg";
    InfiniteLoopKeptStates=100;
    FiniteLoops=[[49, 1000, 0],
    [-98, 1000, 0],
    [98, 1000, 0],
    [-98, 1000, 0],
    [98, 1000, 0]];

    # Keep a maximum of 1000 states, but allow SVD truncation with
    # tolerance 1e-10 and minimum states equal to 100
    TruncationTolerance="1e-10,100";
    # Symmetry sector for ground state S^z_tot=0
    TargetSzPlusConst=100

Here we showed the input for S=1 (note `HeisenbergTwiceS=2`). The parameter
TargetSzPlusConst should be equal $Sz+L$, where $Sz$ is the targeted $Sz$
sector and $L$ is the system size. The next step is to calculate dynamics for
the $S^{0}(q_{\parallel},\omega)$, $S^{1}(q_{\parallel},\omega)$ and
$S^{2}(q_{\parallel},\omega)$ spectral functions using the saved ground state
as an input. For simplicity, we discuss the $S^{1}(q_{\parallel},\omega)$
first. It is convenient to do the dynamics run in a subdirectory `S1`, so `cp
inputGS.ain S1/inputS1.ado` and add/modify the following lines in
`inputS1.ado`

    SolverOptions="twositedmrg,restart,minimizeDisk,CorrectionVectorTargeting";
    # The finite loops now start from the final loop of the gs calculation
    FiniteLoops=[[-98, 2000, 2],
    [98, 2000, 2],
    [-98, 2000, 2],
    [98, 2000, 2]];
    TruncationTolerance="1e-7,100";

    # RestartFilename is the name of the GS .hd5 file (extension is not needed)
    RestartFilename="../inputGS";

    # The weight of the g.s. in the density matrix
    GsWeight=0.1;
    # Legacy, set to 0
    CorrectionA=0;
    # Fermion spectra has sign changes in denominator.
    # For boson operators (as in here) set it to 0
    DynamicDmrgType=0;
    # The site(s) where to apply the operator below. Here it is the center site.
    TSPSites=[49];
    # The delay in loop units before applying the operator. Set to 1
    TSPLoops=[1];
    # If more than one operator is to be applied, how they should be combined.
    # Irrelevant if only one operator is applied, as is the case here.
    TSPProductOrSum="sum";
    # How the operator to be applied will be specified
    string TSPOp0:TSPOperator=expression;
    # The operator expression
    string TSPOp0:OperatorExpression="splus";
    # How is the freq. given in the denominator (Matsubara is the other option)
    CorrectionVectorFreqType="Real";
    # This is a dollarized input, so the
    # omega will change from input to input.
    CorrectionVectorOmega=$omega;
    # The broadening for the spectrum in omega + i*eta
    CorrectionVectorEta=0.25;
    # The algorithm
    CorrectionVectorAlgorithm="Krylov";
    #The labels below are ONLY read by manyOmegas.pl script
    # How many inputs files to create
    #OmegaTotal=600
    # Which one is the first omega value
    #OmegaBegin=-2.0
    # Which is the "step" in omega
    #OmegaStep=0.025
    # Because the script will also be creating the batches,
    # indicate what to measure in the batches
    #Observable=sminus

Then all individual inputs (one per $\omega$ in the correction vector
approach) can be generated and submitted using the `manyOmegas.pl` script
which can be found in the `dmrgpp/src/script` folder:

    perl manyOmegas.pl inputS1.ado BatchTemplate.pbs <test/submit>.

It is recommended to run with `test` first to verify correctness, before
running with `submit`. Depending on the machine and scheduler, the
`BatchTemplate` can be e.g. a PBS or SLURM script. The key is that it contains
a line `./dmrg -f $$input "<X0|$$obs|P2>" -p 12` which allows `manyOmegas.pl`
to fill in the appropriate input for each generated job batch. After all
outputs have been generated,

    perl procOmegas.pl -f inputS1.ado -p
    perl pgfplot.pl

can be used to process and plot the results (these scripts are also given in
`dmrgpp/src/script` folder).

For the calculation of the quadrupolar spectral function
$S^{2}(q_{\parallel},\omega)$ we would have to substitute in the file above
the lines

    OperatorExpression="splus*splus";
    #Observable=sminus*sminus

These lines will allow to apply the operator $(S^{+}_{c})^{2}$ at the center
of the chain and to measure its hermitian conjugate $(S^{-}_{j})^{2}$ on all
others sites when we run with `./dmrg -f $$input "<X0|$$obs|P2>"`.

For the calculations of the $S^{0}(q_{\parallel},\omega)$ we would need to
perform four separate calculations:

$\displaystyle S^{0}_{\rm part-1}(q_{\parallel},\omega)$
$\displaystyle=\frac{1}{L}\sum^{L}_{j=1}e^{iq_{\parallel}(j-c)}\Big{\\{}-\frac{1}{\pi}{\rm
Im}\Big{[}\langle\psi|S^{z}_{j}S^{z}_{j+1}\frac{1}{\omega-H+i\eta}S^{z}_{c}S^{z}_{c+1}|\psi\rangle\Big{]}\Big{\\}}$
(S14) $\displaystyle S^{0}_{\rm part-2}(q_{\parallel},\omega)$
$\displaystyle=\frac{1}{L}\sum^{L}_{j=1}e^{iq_{\parallel}(j-c)}\Big{\\{}-\frac{1}{\pi}{\rm
Im}\Big{[}\langle\psi|(S^{+}_{j}S^{-}_{j+1}+S^{-}_{j}S^{+}_{j+1})\frac{1}{\omega-H+i\eta}S^{z}_{c}S^{z}_{c+1}|\psi\rangle\Big{]}\Big{\\}}$
(S15) $\displaystyle S^{0}_{\rm part-3}(q_{\parallel},\omega)$
$\displaystyle=\frac{1}{L}\sum^{L}_{j=1}e^{iq_{\parallel}(j-c)}\Big{\\{}-\frac{1}{2\pi}{\rm
Im}\Big{[}\langle\psi|(S^{+}_{j}S^{-}_{j+1}+S^{-}_{j}S^{+}_{j+1})\frac{1}{\omega-H+i\eta}S^{+}_{c}S^{-}_{c+1}|\psi\rangle\Big{]}\Big{\\}}$
(S16) $\displaystyle S^{0}_{\rm part-4}(q_{\parallel},\omega)$
$\displaystyle=\frac{1}{L}\sum^{L}_{j=1}e^{iq_{\parallel}(j-c)}\Big{\\{}-\frac{1}{2\pi}{\rm
Im}\Big{[}\langle\psi|(S^{+}_{j}S^{-}_{j+1}+S^{-}_{j}S^{+}_{j+1})\frac{1}{\omega-H+i\eta}S^{-}_{c}S^{+}_{c+1}|\psi\rangle\Big{]}\Big{\\}}$
(S17)

We should then combine the results as $S^{0}(q_{\parallel},\omega)=S^{0}_{\rm
part-1}(q_{\parallel},\omega)+S^{0}_{\rm
part-2}(q_{\parallel},\omega)+\frac{1}{2}[S^{0}_{\rm
part-3}(q_{\parallel},\omega)+S^{0}_{\rm part-4}(q_{\parallel},\omega)]$. For
simplicity, we here only provide instructions for the calculation of the
spectral function $S^{0}_{\rm part-1}(q_{\parallel},\omega)$. This needs the
following modifications in the `inputS0part1.ado`

    # minimizeDisk option should be removed
    SolverOptions="twositedmrg,restart,CorrectionVectorTargeting";
    # The finite loops need to have save option 3 at the last loop.
    # This is to prepare observation of two-point correlations
    FiniteLoops=[[-98, 2000, 2],
    [98, 2000, 2],
    [-98, 2000, 2],
    [98, 2000, 3]];
    string TruncationTolerance="1e-7,100";

    # The sites where to apply the operator S^{z}_{c}S^{z}_{c+1}.
    # Here center and center+1 sites
    TSPSites=[49,50];
    # The delay in loop units before applying the operator. Set to 0
    TSPLoops=[0,0];
    TSPProductOrSum="product";
    # How the operator to be applied will be specified
    string TSPOp0:TSPOperator=expression;
    string TSPOp0:OperatorExpression="sz";
    string TSPOp1:TSPOperator=expression;
    string TSPOp1:OperatorExpression="sz";

In this case, the `BatchTemplate` needs to contain the lines running `dmrg`
and the `observe` executables

        ./dmrg -f $$input  -p 12
        ./observe -f $$input -p 12 "<gs|sz;sz|P2>"

Straightforward modifications of the script `manyOmegas.pl` need to performed
to gather the resulting real space correlations and their Fourier Transform.
Input files for the calculation of $S^{0}_{\rm part-2}(q_{\parallel},\omega)$,
$S^{0}_{\rm part-3}(q_{\parallel},\omega)$ and $S^{0}_{\rm
part-4}(q_{\parallel},\omega)$ and associated scripts for processing the
numerical results can be provided upon request.

b) Obtaining finite-temperature spectra

As noted in the Methods section in the main text, finite temperature $T>0$
calculations proceed in three steps (see also Refs. [5, 4]): We first prepare
the system in a $T\rightarrow\infty$ (inverse temperature $\beta=1/T=0$) state
using a fictitious “entangler” Hamiltonian $H_{E}$ acting in an enlarged
Hilbert space. In the purification approach, the geometry of the system can be
described as a spin ladder, with physical sites on one leg (with even sites
0,2,4,…) and ancilla sites on the other (with odd sites 1,3,5,…).

The entangler Hamiltonian is chosen such that its ground state
$|\psi_{\infty}\rangle$ corresponds to the $T\rightarrow\infty$ state of the
physical system when the ancilla degrees of freedom are traced out, $\rho={\rm
Tr}\left[|\psi_{\infty}\rangle\langle\psi_{\infty}|\right]_{A}$. We use a
conventional entangler Hamiltonian $H_{E}$ conserving only the total
magnetization of the enlarged physical+ancilla system.

We find the ground state of $H_{E}$ by running `./dmrg -f Entangler.ain -p
12`, where `Entangler.ain` for L=32 sites (32 physical and 32 ancillas) is
given by

    ##Ainur1.0
    TotalNumberOfSites=64;
    NumberOfTerms=2;
    gt0:DegreesOfFreedom=1;
    gt0:GeometryKind="ladder";
    gt0:LadderLeg=2;
    gt0:GeometryOptions="ConstantValues";
    gt0:dir0:Connectors=[0.0];
    gt0:dir1:Connectors=[-1.0];

    gt1:DegreesOfFreedom=1;
    gt1:GeometryKind="chain";
    gt1:GeometryOptions="ConstantValues";
    gt1:dir0:Connectors=[0];

    Model="Heisenberg";
    integer HeisenbergTwiceS=2;
    SolverOptions="twositedmrg,MatrixVectorOnTheFly";
    InfiniteLoopKeptStates=100;
    FiniteLoops=[[ 31, 1000, 0],
    [-62, 1000, 0],
    [ 62, 1000, 0],
    [-62, 1000, 0]];
    # Keep a maximum of 1000 states, but allow
    # density matrix truncation with tolerance and minimum states as below
    TruncationTolerance="1e-8,100";
    TargetSzPlusConst=64

Second, the system is cooled through evolving in imaginary time with the
physical Hamiltonian $H$ acting only on physical sites $e^{-\beta(H\otimes
I)/2}$, where $I$ is the identity operator in the ancilla space. To compute
the thermal state of the system $|\psi(\beta)\rangle={e^{-\beta(H\otimes
I)/2}}|\psi_{\infty}\rangle$, we used the Krylov algorithm for time evolution
with an imaginary time step $\Delta\beta=0.025/J$. The time evolution is done
with an evolution operator $exp[-\beta^{\prime}H/2]$ where we define
$\beta^{\prime}=\beta/2$ in units of $1/J$.

Technically, we compute the imaginary time evolution with `./dmrg -f
EvolutionT.ain -p 12`, where `EvolutionT.ain` reads (we report only the main
differences with respect to the previous input)

    gt0:GeometryKind="ladder";
    gt0:LadderLeg=2;
    gt0:GeometryOptions="none";
    gt0:dir0:Connectors=[1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, ...];
    gt0:dir1:Connectors=[0.0, 0.0, 0.0, 0.0, ...];

    gt1:GeometryKind="ladder";
    gt1:LadderLeg=2;
    gt1:GeometryOptions="none";
    gt1:dir0:Connectors=[1.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0, 0.0, ...];
    gt1:dir1:Connectors=[0.0, 0.0, 0.0, 0.0, ...];

    string PrintHamiltonianAverage="s==c";
    string RecoverySave="@M=100,@keep,1==1";
    SolverOptions="twositedmrg,restart,TargetingAncilla";

    # Notice the save option 3 at the last loop corresponding to the
    # physical temperature of interest (in this specific case 250K)
    # This allows to measure static correlation functions
    FiniteLoops=[[ 62, 2000, 2],[-62, 2000, 2],[ 62, 2000, 2],[-62, 2000, 2],
    [ 62, 2000, 2],[-62, 2000, 2],[ 62, 2000, 2],[-62, 2000, 2],
    [ 62, 2000, 2],[-62, 2000, 2],[ 62, 2000, 2],[-62, 2000, 2],
    [ 62, 2000, 2],[-62, 2000, 2],[ 62, 2000, 2],[-62, 2000, 2],
    [ 62, 2000, 2],[-62, 2000, 2],[ 62, 2000, 2],[-62, 2000, 2],
    [ 62, 2000, 2],[-62, 2000, 2],[ 62, 2000, 2],[-62, 2000, 3]];
    RestartFilename="Entangler";
    TSPTau=0.025;
    TSPTimeSteps=5;
    TSPAdvanceEach=62;
    TSPAlgorithm="Krylov";
    TSPSites=[1];
    TSPLoops=[0];
    TSPProductOrSum="sum";
    GsWeight=0.1;
    string TSPOp0:TSPOperator=expression;
    string TSPOp0:OperatorExpression="identity";

Above, the `gt1:dir0` connectors give the interactions couplings along the leg
direction of the two-leg ladder geometry, so the alternating pattern `[1.0,
0.0, 1.0, 0.0,...]` just means that we evolve in imaginary time the physical
sites rather than the ancillas. The imaginary time $\beta^{\prime}$ can be
obtained with
`h5dump -d /Def/FinalPsi/TimeSerializer/Time <hd5>`, where `<hd5>` should be
replaced with the name of the `hd5` file of interest. The targeted
$\beta^{\prime}$ value is given by $\beta^{\prime}(J,T)=1/(2JT)$, where J and
T are given in Kelvin. Finally, the arguments to `RecoverySave` mean that we
keep a maximum of 100 `hd5` outputs, and output one in every loop (when the
condition `1==1` holds). We also observe dipolar as well as quadrupolar
correlation functions at finite temperature to extract their characteristic
correlation lengths. This is done by doing

        ./observe -f EvolutionT.ain -p 12 "<time|sz;sz|time>"
        ./observe -f EvolutionT.ain -p 12 "<time|splus*splus;sminus*sminus|time>"

Finally, the dynamical spin structure factor is calculated using the operator
${\cal L}=H\otimes I+I\otimes(-H)$ with the real space expression in Eq. S2.
For the DMRG++ inputs, the dynamics calculation proceeds similarly to the
$T=0$ case, but with number of sites and precision as in the preceding $T>0$
step. We do, however, need to additionally add/modify the following lines

    gt0:dir0:Connectors=[1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0, ...];
    gt0:dir1:Connectors=[0.0, 0.0, 0.0, 0.0, ...];
    gt1:dir0:Connectors=[1.0, -1.0, 1.0, -1.0, 1.0, -1.0, 1.0, -1.0, ...];
    gt1:dir1:Connectors=[0.0, 0.0, 0.0, 0.0, ...];
    RestartFilename="Recovery23EvolutionT.hd5";
    SolverOptions="CorrectionVectorTargeting,restart,twositedmrg,minimizeDisk,fixLegacyBugs";
    integer RestartSourceTvForPsi=0;
    vector RestartMappingTvs=[-1, -1, -1, -1];
    integer RestartMapStages=0;
    # This is the structure of FiniteLoops for dynamics
    FiniteLoops=[[-98, 2000, 2],[98, 2000, 2],[-98, 2000, 2],[98, 2000, 2]];
    string TruncationTolerance="1e-7,100";

Above, the restart filename should be chosen to match the `hd5` file of
interest. All rung couplings are zero. As before, we have abbreviated the
arrays of coupling constants.

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electrostatic potential landscape changes, thus shifting the edges in real
space. This shift affects the transparency of the barriers between the edges.

Therefore, in order to compare the single-source and double-source current
correlation functions, a rescaling of the double-source transmission is
required (see below for detailed steps). In addition, following Fig. S3a,
transmission probabilities $W_{A}$, $W_{B}$ greatly deviate from the non-
interacting chiral Luttinger liquid prediction (i.e., $W_{A,B}\propto
I_{+}^{(2\nu-2)/(2\nu-1)}$, denoted by the gray dashed line of the inset of
Fig. S3a), which implies the energy and/or voltage dependence of the bare
transmission probabilities $T_{A}$ and $T_{B}$ in the Hamiltonian. In
particular, the voltage dependence can be again related to the global
electrostatics of the setup. In this work, this dependence is included by
working with differential noises.

Below, we list the steps of the data processing that lead to Fig. 4 of the
main text. More specifically, Fig. 4 contains four sets of data: the
“theoretical data” [obtained from Eq. (3) of the main text], and the rescaling
of the experimental data.

Theoretical data:

To obtain the “theoretical data” following Eq. (3) of the main text, one needs
to know the tunneling parameters ($T_{A}$, $T_{B}$ and $T_{C}$), and the
ultraviolet cutoff $\tau_{0}$, from the experimental data. Here, we take the
following steps.

(i) The effective transmission probabilities $W_{A}(V_{sA},V_{sB}=0)$ and
$W_{B}(V_{sA}=0,V_{sB})$ for the single-source cases, defined as

$W_{A}(V_{sA},0)=\frac{\partial I_{A0}}{\partial V_{sA}}\frac{h}{\nu
e^{2}},\quad\ W_{B}(0,V_{sB})=\frac{\partial I_{B0}}{\partial
V_{sB}}\frac{h}{\nu e^{2}},$ (S59)

are experimentally obtained by applying a small AC voltage on top of a DC
background voltage (shown in Fig. S3b).

(ii) Next, we find the values of $W_{C}(V_{sA},V_{sB}=0)$ and
$W_{C}(V_{sA}=0,V_{sB})$ for the single-source scenario. These two quantities,
defined as

$W_{C}(V_{sA},0)=\frac{\partial I_{\text{T}}}{\partial I_{A0}}\nu,\quad\
W_{C}(0,V_{sB})=-\frac{\partial I_{\text{T}}}{\partial I_{B0}}\nu,$ (S60)

are also experimentally obtained by applying a weak AC bias on top of a
relatively large DC background.

(iii) We obtain the value of the ultraviolet cutoff, with Eq. (S21), or more
specifically,

$\displaystyle\tau_{0}$ $\displaystyle=\frac{\hbar}{\nu
V_{sA}}\frac{I_{\text{T}}(V_{sA},0)}{S_{\text{T}}(V_{sA},0)}\frac{2-5\nu+2\nu^{2}}{2[\nu+W_{A}(V_{sA},0)\sin(2\pi\nu)]}$
(S61) $\displaystyle=\frac{\hbar}{\nu
V_{sB}}\frac{I_{\text{T}}(0,V_{sB})}{S_{\text{T}}(0,V_{sB})}\frac{2-5\nu+2\nu^{2}}{2[\nu+W_{B}(0,V_{sB})\sin(2\pi\nu)]}.$

Afterwards, we obtain the values $T_{A}$, $T_{B}$ and $T_{C}$, using

$\displaystyle W_{A}(V_{sA},0)$
$\displaystyle=T_{A}(V_{sA},0)\frac{\sin(2\pi\nu)\Gamma(1-2\nu)}{2\pi^{2}}\left(\frac{\tau_{0}\nu
eV_{sA}}{\hbar}\right)^{2\nu-2},$ (S62) $\displaystyle W_{B}(0,V_{sB})$
$\displaystyle=T_{B}(0,V_{sB})\frac{\sin(2\pi\nu)\Gamma(1-2\nu)}{2\pi^{2}}\left(\frac{\tau_{0}\nu
eV_{sB}}{\hbar}\right)^{2\nu-2},$ $\displaystyle W_{C}(V_{sA},0)$
$\displaystyle\\!=\\!T_{A}(V_{sA},0)\ T_{C}[I_{+}(V_{sA})]\
\frac{2\nu^{2}}{\pi^{2}}\frac{(1\\!-\\!\cos
2\pi\nu)\left[\nu^{2}\frac{e}{\hbar}V_{sA}\\!+\\!\frac{I_{A0}(V_{sA})}{e}\sin
2\pi\nu\right]}{(2\\!-\\!3\nu\\!+\\!2\nu^{2})(2\\!-\\!4\nu\\!+\\!2\nu^{2})(2\\!-\\!5\nu\\!+\\!2\nu^{2})}\tau_{0},$
$\displaystyle W_{C}(0,V_{sB})$ $\displaystyle\\!=\\!T_{B}(0,V_{sB})\
T_{C}[I_{+}(V_{sB})]\ \frac{2\nu^{2}}{\pi^{2}}\frac{(1\\!-\\!\cos
2\pi\nu)\left[\nu^{2}\frac{e}{\hbar}V_{sB}\\!+\\!\frac{I_{B0}(V_{sB})}{e}\sin
2\pi\nu\right]}{(2\\!-\\!3\nu\\!+\\!2\nu^{2})(2\\!-\\!4\nu\\!+\\!2\nu^{2})(2\\!-\\!5\nu\\!+\\!2\nu^{2})}\tau_{0}.$

Notice that in Eq. (S62), all transmission coefficients are expressed as
functions of the corresponding voltages. The coefficients $T_{A}$, $T_{B}$ and
$T_{C}$, which would be constants in a conventional Luttinger liquid theory,
depend on voltage through the energy and/or voltage dependence of bare
transmission probabilities. This leads to the deviation of the voltage
dependencies of currents and noises from the expressions predicted by the
Luttinger liquid theory, where voltage enters in the form of an anomalous
scaling. With the above procedure, we have thus included the extra (non-
Luttinger) voltage dependence of coefficients that appear in the Hamiltonian,
(i.e., $\zeta_{A}$, $\zeta_{B}$ and $\zeta_{C}$).

(iv) To obtain entanglement pointer [given by Eq. (4) of the main text], we
also need to know the value of the product $T_{A}T_{B}T_{C}$, with the
equality

$T_{A}T_{B}T_{C}\\!=\\!\sqrt{W_{A}W_{B}W_{C}^{2}}\frac{\pi^{4}}{\nu^{3}}\left(\frac{\tau_{0}\nu
eV}{\hbar}\right)^{1-2\nu}\frac{(2\\!-\\!3\nu\\!+\\!2\nu^{2})(2\\!-\\!4\nu\\!+\\!2\nu^{2})(2\\!-\\!5\nu\\!+\\!2\nu^{2})}{\sin(2\pi\nu)[1-\cos(2\pi\nu)]\Gamma(1-2\nu)},$
(S63)

where explicit and implicit dependencies on $V_{sA}=V_{sB}=V$ have been
omitted for simplicity. The factor $W_{C}^{2}$ under the square root is the
product of two single-source contributions: $W_{C}^{2}=W_{C}(V,0)W_{C}(0,V)$.
The dependence of $T_{A}T_{B}T_{C}$ on $I_{+}$ is presented by Fig. S3c. To
avoid influence from strong fluctuation, we neglect points that greatly
deviate from the fitting $\sim I_{+}^{4/3}$ (red line of Fig. S3c). This
dependence of $T_{A}$, $T_{B}$ and $T_{C}$, as analyzed at the beginning of
this section, leads to the deviation of the $I_{+}(V)$ dependence predicted by
the Luttinger liquid theory (where $I_{+}\propto V^{2\nu-1}$). The product
$T_{A}T_{B}T_{C}$ obtained this way from the measured data has a meaning of
“would-be” product of “virtual” Luttinger-liquid transmission probabilities
that would yield the same current at given voltages in the experiment.

(v) We are now ready to calculate the differential noise, which is defined as
$s_{2}(I_{+})\equiv\partial_{I_{+}}S_{2}(I_{+})$ (shown in Fig. S3d). The
integral of $s_{2}(I_{+})$ yields the blue points in Fig. 3c of the main text.

Figure S3: Data used to obtain Fig. 3c of the main text. a, Transmissions
$W_{A}$ and $W_{B}$ at diluters, for single-source and double-source
situations. The $x$-axis refers to the non-equilibrium current through the
corresponding diluter. The single-source and double-source data agrees
amazingly with each other. For each value of the non-equilibrium current,
$W_{A}$ and $W_{B}$ have a small deviation, indicating different barrier
amplitudes at upper and lower diluters. b, Effective transmission at the
central QPC, for single-source settings. c, The value $T_{A}T_{B}T_{C}$
obtained from Eq. (S63). When calculating differential noise, we would like to
omit contributions from the points that greatly deviate from the fit $\sim
I_{+}^{4/3}$. d, The differential noise obtained from Eq. (3) of the main
text. Its integral over $I_{+}$ leads to blue points of Fig. 3c of the main
text. e, The linear fit (the gray dashed line) of the transmission probability
$W_{C}$ at the central QPC. f, The differential noise, obtained after the
rescaling of the double-source noise. It is rather noisy, in comparison to
that of d. Its integral over $I_{+}$ leads to red points of Fig. 3c of the
main text.

Rescaling of the experimental data:

In this part, we will use current as the argument of $W_{C}$ for clarity of
the description of the procedure. We consider the same values of total current
$I_{+}$ in the single-source (where $I_{+}=I_{A0}$ or $I_{+}=I_{B0}$) and
double-source (where $I_{A0}=I_{B0}=I_{+}/2$) scenarios.

(i) For a given value of $I_{+}$, read out the data of single-source effective
transmission probabilities $W_{C}(I_{+},0)$ and $W_{C}(0,I_{+})$, following
the definition of Eq. (S60). We also obtain the double-source effective
transmission amplitudes $W_{C}(I_{+}/2,I_{+}/2)$, defined as

$W_{C}(I_{+}/2,I_{+}/2)\equiv\nu\frac{\partial
I_{\text{T}}(I_{+},I_{-})}{\partial I_{-}}\Bigg{|}_{I_{-}=0}.$ (S64)

Once again, it is experimentally obtained by applying a weak AC voltage on top
of a DC background. However, different from single-source scenarios, the
measurement of $W_{C}(I_{+}/2,I_{+}/2)$ requires the application of equal AC
bias in both sources. For double-source situations, the total non-equilibrium
current is larger than that of single-source ones. We thus have to take a
linear fit (shown in Fig. S3e), to rescale $W_{C}(I_{+}/2,I_{+}/2)$ for the
entire relevant range of $I_{+}$.

(ii) For each value of $I_{+}$, we obtain the differential cross correlation
for double-source correlations, i.e.,
$s_{AB,\text{double}}(I_{+})\equiv\partial_{I_{+}}S_{AB}(I_{+}/2,I_{+}/2)$. We
then rescale the double source differential noise, following

$s_{AB,\text{modified}}(I_{+})\equiv
s_{AB,\text{double}}(I_{+})\frac{W_{C}(I_{+},0)+W_{C}(0,I_{+})}{2W_{C}(I_{+}/2,I_{+}/2)}.$
(S65)

We further use $s_{2,\text{modified}}(I_{+})\equiv
s_{AB,\text{modified}}(I_{+})-s_{AB,\text{single}}(I_{+},0)-s_{AB,\text{single}}(0,I_{+})$,
where the latter two are the single-source differential noises.

(iii) Finally, we integrate $s_{2,\text{modified}}$ over the relevant current
range, to obtain the modified double-source noise

$S_{AB,\text{modified}}(I_{+})\equiv\int_{0}^{I_{+}}dIs_{2,\text{modified}}(I).$
(S66)

The plot of the function $s_{2,\text{modified}}(I_{+})$ is presented in Fig.
S3f.

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