the axial gauge. The total contribution from $H-O$ chord intersections is

$\frac{1}{d}\sum_{\mu=1}^{d}\left[\frac{\mu-1}{d}\langle\cos\tilde{F}\rangle+\frac{d-\mu}{d}\langle\cos
F\rangle\right]^{\text{\\#$H$-$O$
int.}}=\frac{\int_{\langle\cos{F}\rangle}^{\langle\cos{\tilde{F}}\rangle}\tilde{u}^{\text{\\#$H$-$O$
int.}}d\tilde{u}}{\langle\cos\tilde{F}\rangle-\langle\cos{F}\rangle}+O(d^{-1}).$
(162)

So $O$ behaves like a uniform statistical mixture of all possible
$\tilde{D}_{\mu}$.

##  Appendix C Algebra and motion on the hypercube

We would like to take advantage of the simplicity of the hypercube geometry
and try to gain some elementary intuition of particle motions on this Fock-
space graph. To that end let us compute the equation of motion for the
position operator. It is useful to first list the alegbraic relations among
$D_{\mu}$, $V_{\mu}$ and $X_{\mu}$ operators:

$\begin{split}&D_{\mu}D_{\nu}=(\cos F_{\mu\nu})D_{\nu}D_{\mu}-(i\sin
F_{\mu\nu})V_{\nu}V_{\mu},\\\ &V_{\mu}V_{\nu}=(\cos
F_{\mu\nu})V_{\nu}V_{\mu}-(i\sin F_{\mu\nu})D_{\nu}D_{\mu},\\\
&D_{\mu}V_{\nu}=(\cos F_{\mu\nu})V_{\nu}D_{\mu}+(i\sin
F_{\mu\nu})D_{\nu}V_{\mu},\\\ &V_{\mu}D_{\nu}=(\cos
F_{\mu\nu})D_{\nu}V_{\mu}+(i\sin F_{\mu\nu})V_{\nu}D_{\mu},\\\
&[D_{\mu},X_{\nu}]=[V_{\mu},X_{\nu}]=[X_{\mu},X_{\nu}]=0,\end{split}$ (163)

for $\mu\neq\nu$ and that

$\begin{split}&V_{\mu}D_{\mu}=-D_{\mu}V_{\mu}=i\sigma_{\mu}^{3}=2iX_{\mu},\\\
&V_{\mu}X_{\mu}=-X_{\mu}V_{\mu}=-iD_{\mu}/2,\\\
&D_{\mu}X_{\mu}=-X_{\mu}D_{\mu}=iV_{\mu}/2,\\\
&D_{\mu}^{2}=V_{\mu}^{2}=(2X_{\mu})^{2}=\mathbb{1}.\end{split}$ (164)

Among other things this shows $\\{D_{\mu}\\}$, $\\{V_{\mu}\\}$ and
$\\{X_{\mu}\\}$ form an algebra under multiplication and addition. We can also
augment this alegbra by including $\tilde{D}_{\mu}$ and $\tilde{V}_{\mu}$, and
use

$\displaystyle
T_{\mu}^{\pm}\tilde{T}_{\nu}^{\pm}=\tilde{T}_{\nu}^{\pm}T_{\mu}^{\pm}e^{i\frac{F_{\mu\nu}+\tilde{F}_{\mu\nu}}{2}},\
T_{\mu}^{\pm}\tilde{T}_{\nu}^{\mp}=\tilde{T}_{\nu}^{\mp}T_{\mu}^{\pm}e^{-i\frac{F_{\mu\nu}+\tilde{F}_{\mu\nu}}{2}}.$
(165)

The end result is essentially to replace all the $F_{\mu\nu}$ in equation
(163) by an average $(F_{\mu\nu}+\tilde{F}_{\mu\nu})/2$, for example

$\begin{split}&D_{\mu}\tilde{D}_{\nu}=\left(\cos\frac{F_{\mu\nu}+\tilde{F}_{\mu\nu}}{2}\right)\tilde{D}_{\nu}D_{\mu}-\left(i\sin\frac{F_{\mu\nu}+\tilde{F}_{\mu\nu}}{2}\right)\tilde{V}_{\nu}V_{\mu},\\\
&D_{\mu}\tilde{V}_{\nu}=\left(\cos\frac{F_{\mu\nu}+\tilde{F}_{\mu\nu}}{2}\right)\tilde{V}_{\nu}D_{\mu}+\left(i\sin\frac{F_{\mu\nu}+\tilde{F}_{\mu\nu}}{2}\right)\tilde{D}_{\nu}V_{\mu},\\\
\end{split}$ (166)

As we have seen in equation (21) that $\dot{X}_{\mu}=V_{\mu}$, now let us
derive a “Newtonian” (second derivative in time) equation in the following
way:

$\begin{split}&\dot{X}_{\mu}D_{\mu}=V_{\mu}D_{\mu}=2iX_{\mu},\\\
\implies&\ddot{X}_{\mu}D_{\mu}+\dot{X}_{\mu}\dot{D}_{\mu}=2i\dot{X}_{\mu}=2iV_{\mu},\\\
\implies&\ddot{X}_{\mu}D_{\mu}^{2}+\dot{X}_{\mu}\dot{D}_{\mu}D_{\mu}=2iV_{\mu}D_{\mu}=-4X_{\mu},\\\
\implies&\ddot{X}_{\mu}+\dot{X}_{\mu}\dot{D}_{\mu}D_{\mu}+4X_{\mu}=0,\end{split}$
(167)

where we have used identities presented in equations (163) and (164). The
harmonic term $4X_{\mu}$ is solely due to the finiteness of the hypercube and
is present even if the magnetic field is turned off. The second term in the
last line is nonzero if and only if the magnetic field is nonzero, so it must
play the role of Lorentz force. To see this more clearly, we write the second
term as

$\begin{split}\dot{X}_{\mu}\dot{D}_{\mu}D_{\mu}&=iV_{\mu}[H,D_{\mu}]D_{\mu}=i\sum_{\nu\neq\mu}V_{\mu}[D_{\nu},D_{\mu}]D_{\mu}\\\
&=i\sum_{\nu\neq\mu}V_{\mu}D_{\nu}-i\sum_{\nu\neq\mu}V_{\mu}D_{\mu}D_{\nu}D_{\mu}\\\
&=i\sum_{\nu\neq\mu}(1-\cos F_{\mu\nu})V_{\mu}D_{\nu}-\sum_{\nu\neq\mu}\sin
F_{\mu\nu}D_{\mu}V_{\mu}\\\ &=i\sum_{\nu\neq\mu}(1-\cos
F_{\mu\nu})V_{\mu}D_{\nu}-D_{\mu}\sum_{\nu\neq\mu}\sin
F_{\mu\nu}\dot{X}_{\nu},\end{split}$ (168)

where we have used multiple identities in equations (163) and (164). So far
all the equations above are exact. To see the “continuum limit” of the
motion,888See the discussion below equation (49). we take a small $F_{\mu\nu}$
and Taylor expand equation (167) to the first power of $F_{\mu\nu}$ with fixed
$d$, and this gives

$\dot{X}_{\mu}\dot{D}_{\mu}D_{\mu}=-\sigma^{1}_{\mu}\sum_{\nu}F_{\mu\nu}\dot{X}_{\nu}+O(F_{\mu\nu}^{2}),$
(169)

where we used $D_{\mu}(t)=\sigma^{1}_{\mu}+O(F_{\mu\nu})$. Hence in the
continuum limit the Newtonian equation becomes

$\ddot{X}_{\mu}-\sigma^{1}_{\mu}\sum_{\nu}F_{\mu\nu}\dot{X}_{\nu}+4X_{\mu}=0.$
(170)

The second term is almost what one expects from a classical Lorentz force
except here we have an extra $\sigma_{\mu}^{1}$ coefficient. The
$\sigma_{\mu}^{1}$ coefficient seems to suggest there is no truly classical
Lorentz force on the hypercube.999The conventional continuum limit would have
a shrinking lattice spacing $a$ with some length scale $L=na$ fixed, we can
repeat the same derivation of the equation of motion and the upshot is the
replacement of $\sigma^{1}$ by $a\Sigma^{1}+1$ where $\Sigma^{1}$ is the
hopping on the large lattice and $1$ comes from the fact that $D_{\mu}$ needs
a constant counter-term to become a well-defined continuum derivative. As
$a\to 0$, we get the conventional Lorentz force. In fact without the
$\sigma_{\mu}^{1}$ coefficient we will be in trouble: the motion becomes
exactly the same as a harmonic oscillator in a uniform magnetic field, which
is integrable, contradicting the chaotic physics of the Parisi model we have
known so far. A useful comparison is $d=2$ lattice with a large linear size,
which is a Landau problem for Bloch electrons whose spectrum is known to be
complicated but integrable WIEGMANN1994495 . Note that the Lorentz force term
is a sum over $d$ terms, so it could dominate over the harmonic potential even
when each $F_{\mu\nu}$ is very small. This is consistent with our earlier
result that the system is fast-scrambling when $F_{\mu\nu}$ is sent to zero
after the large $d$ limit is taken. Moreover, according to a numerical
analysis of the Parisi model jia_unpublished , it indeed looks like any
nonzero small magnetic field would lead to random matrix statistics and hence
quantum chaos. It should be said that if we want to take the large $d$ limit,
the truncation to linear order in (169) is not valid unless $\left\langle
F^{2}\right\rangle\ll 1/d^{2}$ because $O(F^{2})$ term involves summing
$d^{2}$ terms. Nevertheless equation (170) could be useful, for example it
gives us a hint on what kind of scaling a localizing potential must have so
that our model can have a localized phase. For example, we can add to the
Parisi Hamiltonian a term

$-\sum_{\mu}\mathcal{E}_{\mu}\frac{\sigma^{3}_{\mu}}{2}=-\sum_{\mu}\mathcal{E}_{\mu}X_{\mu}$
(171)

where $\mathcal{E}_{\mu}$ can be interpreted as the components of an electric
field acting on the charged particle, and hence has the effect of localizing
the particle on hypercube vertices. This electric term would change the
equation of motion (170) to

$\ddot{X}_{\mu}-\sigma^{1}_{\mu}\sum_{\nu}F_{\mu\nu}\dot{X}_{\nu}+4X_{\mu}-\sigma^{1}_{\mu}\mathcal{E}_{\mu}=0.$
(172)

Since the Lorentz force term sums over $d$ random numbers, we estimate its
magnitude to be roughly $\sqrt{d\left\langle F_{\mu\nu}^{2}\right\rangle}$.
Therefore $\mathcal{E}_{\mu}$ must at least scale as $\sqrt{d\left\langle
F_{\mu\nu}^{2}\right\rangle}$ for it to be a real competition with the Lorentz
force term. This is naturally satisfied in a scaling regime that has an
extensive free energy (see section 7.4):

$\left\langle
F_{\mu\nu}^{2}\right\rangle\sim\frac{1}{{d}},\quad\left\langle\mathcal{E}_{\mu}^{2}\right\rangle\sim
1.$ (173)

However this scaling is much less understood.

##  Appendix D Subleading moments at $q=1$ and $q=0$

The subleading moments involve two pieces of contributions, one is from the
$d(d-1)\cdots$ factor of the leading lattice paths, which (for $M_{2p}$) is
equal to

$-\frac{1}{d}\binom{p}{2}M^{\rm QH}_{2p},$ (174)

where $M^{\rm QH}_{2p}$ denotes the $q$-Hermite moments; the other is from the
subleading lattice paths (with repeatedly traversed lattice directions), whose
values can be computed by equation (90) (the same computation was called the
“merge and delete” procedure of intersection graphs in Jia_2020 ). At $q=1$
and $q=0$ subleading lattice path contributions can be figured out for general
$p$:

$\displaystyle\frac{1}{d}\frac{1}{3}\binom{p}{2}(2p-1)!!\quad$ for $q=1$,
(175) $\displaystyle\frac{1}{d}\binom{2p}{p-2}\frac{p}{2}\quad$ for $q=0$.
(176)

Hence the total subleading corrections at $q=1$ and $q=0$ are101010We use the
fact that $M^{\rm QH}_{2p}(q=1)=(2p-1)!!$ and $M^{\rm
QH}_{2p}(q=0)=\frac{1}{p+1}\binom{2p}{p}$.

$\displaystyle\delta M_{2p}=$
$\displaystyle-\frac{1}{d}\frac{2}{3}\binom{p}{2}(2p-1)!!\quad\text{for
$q=1$,}$ (177) $\displaystyle\delta M_{2p}=$
$\displaystyle-\frac{1}{d}\binom{2p}{p-2}\quad\text{for $q=0$.}$ (178)

From which we can work out the subleading corrections to spectral densities

$\displaystyle\delta\rho=$
$\displaystyle\frac{1}{d}\frac{d^{4}}{dx^{4}}\left[-\frac{1}{12}\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}\right]\quad\text{for
$q=1$,}$ (179) $\displaystyle\delta\rho=$
$\displaystyle\frac{1}{d}\frac{d^{2}}{dx^{2}}\left[\frac{1}{120\pi}(4-x^{2})^{\frac{3}{2}}(2-3x^{2})\right]\quad\text{for
$q=0$.}$ (180)

For $q=0$ the subleading density diverges rapidly near the spectral edges
$x=\pm 2$, this density should be understood in the distributional sense:

$\int f(x)\delta\rho(x)dx=\frac{1}{d}\int
f^{\prime\prime}(x)\left[\frac{1}{120\pi}(4-x^{2})^{\frac{3}{2}}(2-3x^{2})\right]dx.$
(181)

##  Appendix E Degenerate perturbation for the first excited states

The general form of equation (125) is

$\begin{split}\langle m,n|\sqrt{d}H|0\rangle=&\sum_{\mu\neq
m,n}\left[\sin\left(\frac{F_{\mu m}}{4}\right)\sin\left(\frac{F_{\mu
n}}{4}\right)\prod_{\nu\neq\mu,m,n}\cos\left(\frac{F_{\mu\nu}}{4}\right)\right]\\\
&\quad+i\sin\left(\frac{F_{mn}}{4}\right)\left[\prod_{\nu\neq
m,n}\cos\left(\frac{F_{m\nu}}{4}\right)-\prod_{\nu\neq
m,n}\cos\left(\frac{F_{n\nu}}{4}\right)\right].\end{split}$ (182)

Matrix elements of the form $\langle m|\sqrt{d}H|n\rangle$ are also simple to
compute. Using the notations of equation (118), we get the matrix elements of
$H$ using the wavefunctions of the first excited states ($d$ states) of
$H_{0}$:

$\begin{split}\langle
m|\sqrt{d}H|m\rangle=-(d-2)\cos^{d-1}\left(\frac{\phi}{4}\right),\end{split}$
(183)

and the off-diagonal elements ($m\neq n$) are

$\langle
m|\sqrt{d}H|n\rangle=\cos^{d-3}\left(\frac{\phi}{4}\right)\sin^{2}\left(\frac{\phi}{4}\right)\sum_{\mu\neq
m,n}S_{\mu m}S_{\mu
n}-2i\cos^{d-2}\left(\frac{\phi}{4}\right)\sin\left(\frac{\phi}{4}\right)S_{mn}$
(184)

for the Parisi’s disorder distribution. In full generality, we have

$\begin{split}\langle m|\sqrt{d}H|m\rangle&=-\sum_{\mu\neq
m}\prod_{\nu\neq\mu}\cos\left(\frac{F_{\mu\nu}}{4}\right)+\prod_{\nu\neq
m}\cos\left(\frac{F_{m\nu}}{4}\right)\\\ \langle
m|\sqrt{d}H|n\rangle&=\sum_{\mu\neq m,n}\left[\sin\left(\frac{F_{\mu
m}}{4}\right)\sin\left(\frac{F_{\mu n}}{4}\right)\prod_{\nu\neq
m,n,\mu}\cos\left(\frac{F_{\mu\nu}}{4}\right)\right]\\\
&\quad\quad-i\sin\left(\frac{F_{mn}}{4}\right)\left[\prod_{\nu\neq
m,n}\cos\left(\frac{F_{m\nu}}{4}\right)+\prod_{\nu\neq
m,n}\cos\left(\frac{F_{n\nu}}{4}\right)\right],\quad m\neq n,\end{split}$
(185)

We can then numerically diagonalize the $d$ by $d$ matrix above and its lowest
energy will approximate the first excited state energy of $H$. However we did
not need this in our main text.

##  Appendix F State mixing in Madelcena-Qi model

In this appendix we demonstrate that the unperturbed ground state only mixes
with a small fraction of the excited states in the MQ model. The MQ
Hamiltonian is of the form

$H_{\rm MQ}=H_{\rm SYK,L}+H_{\rm
SYK,R}+i\mu\sum_{k=1}^{d}\psi_{L}^{k}\psi_{R}^{k}.$ (186)

The $\psi_{L}^{k}$ and $\psi_{R}^{k}$ operators are two sets of Majoranas
defined on the left and right spatial sites with the anticommutation relation

$\\{\psi_{a}^{k},\psi_{b}^{l}\\}=2\delta_{ab}\delta^{kl},$ (187)

where $a,b=L,R$ and $k,l=1,\ldots,d$. $H_{\rm SYK,L}$ is the standard $p$-body
SYK Hamiltonian built from the $\psi_{L}$, and $H_{\rm SYK,R}$ is built from
$\psi_{R}$ but with a time reversal. As stated in section 7, the coupling term
is identical to the zero-flux Parisi Hamiltonian, so to parallel the
discussion in the main text, we treat $H_{\rm SYK,L}+H_{\rm SYK,R}$ as a
perturbation and study the mixing between the ground state and the excited
states of the coupling term under this perturbation. Following garcia2019b ,
we define ladder operators to generate the eigenstates of the coupling term:

$c^{\pm}_{k}=\frac{1}{\sqrt{2}}\left(\psi_{R}^{k}\pm i\psi_{L}^{k}\right),$
(188)

then

$i\sum_{k=1}^{d}\psi_{L}^{k}\psi_{R}^{k}=\sum_{k=1}^{d}c^{+}_{k}c^{-}_{k}-d$
(189)

and the commutator

$\left[i\sum_{k=1}^{d}\psi_{L}^{k}\psi_{R}^{k},\ c_{l}^{\pm}\right]=\pm
2c_{l}^{\pm}.$ (190)

Hence we can label the eigenstates in the identical way as we did for the
zero-flux Parisi Hamiltonian in equation (118), now with

$|m_{1},\ldots,m_{k}\rangle:=c^{+}_{m_{1}}\ldots c^{+}_{m_{k}}|0\rangle,$
(191)

where the ground state $|0\rangle$ satisfies

$c_{k}^{-}|0\rangle=0$ (192)

for all $k=1,\ldots,d$. Now consider the mixing

$\begin{split}&\langle m_{1},\ldots,m_{l}|H_{L}|0\rangle\\\ =&\langle
0|c^{-}_{m_{1}}\ldots c^{-}_{m_{l}}H_{L}|0\rangle\\\
=&\sum_{i_{1}<\ldots<i_{p}}J_{i_{1}\ldots i_{p}}\langle 0|c^{-}_{m_{1}}\ldots
c^{-}_{m_{l}}\psi_{L}^{i_{1}}\ldots\psi_{L}^{i_{p}}|0\rangle.\end{split}$
(193)

If $l>p$, there is at least one annihilation operator that anticommutes with
every fermion in any given set of
$\\{\psi_{L}^{i_{1}},\ldots,\psi_{L}^{i_{p}}\\}$. Then, we can just move this
annilhilation operator all the way to the right and annihilate $|0\rangle$.
This means the ground state only mixes with excited states up to
$|m_{1},\ldots,m_{p}\rangle$, which only includes a vanishingly small fraction
of all states as $d\to\infty$.

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# Giant effective $g$-factor due to spin bifurcations in polariton condensates

A. Bochin  School of Physics and Engineering, ITMO University, Kronverksky Pr.
49, bldg. A, St. Petersburg, 197101, Russia  I. Chestnov  School of Physics
and Engineering, ITMO University, Kronverksky Pr. 49, bldg. A, St. Petersburg,
197101, Russia  A. Nalitov  Moscow Institute of Physics and Technology,
Institutskiy per., 9, Dolgoprudnyi, Moscow Region, Russia 141701

###### Abstract

We predict giant susceptibility of spin-bifurcating polariton condensates to
externally applied permanent magnetic field. In the presence of spin-
anisotropic polariton-polariton interactions, the condensate spontaneously
takes an elliptically polarised state, whose perturbation dynamics can be
interpreted in terms of the presence of strong effective magnetic field
significantly surpassing the external one. Surprisingly, this behaviour of the
addressed strongly out-of-equilibrium system in the vicinity of a critical
point exhibits intriguing analogy with the second-order phase transition. The
predicted field-enhancement effect can be utilized for creation of
topologically nontrivial states of Bogoliubov’s excitations existing on top of
the polariton condensate.

Magneto-optic phenomena lifting degeneracy of photonic modes in the presence
of external magnetic field are remarkable manifestations of light-matter
coupling. As an example, Zeeman effect for electron excitations in dielectrics
or semiconductors causes energy splitting of otherwise degenerate circularly
polarised photonic modes, which, in turn, results in magneto-optic Faraday
effect or Kerr rotation [1]. The possibility of local time-reversal symmetry
breaking for electromagnetic waves due to magneto-optic coupling is crucial
for engineering topological photonic states and suppressing backscattering in
signal transmission [2, 3].

Although the magneto-optic effect is weak in the optical frequency domain, it
can be significantly enhanced in the strong coupling regime, where mixed
light-matter quasi-particles, such as exciton-polaritons, emerge. In
particular, topological exciton-polariton states were demonstrated in optical
cavity lattices subject to strong magnetic fields [4]. In addition,
incorporating ferromagnetic materials in strongly coupled optical cavities was
proposed recently to reach giant values of the effective $g$-factor, which
quantifies the magneto-optic coupling strength [5].

The full potential of strongly coupled systems supporting exciton polaritons
is revealed in the nonlinear regime, where the macroscopic coherent states
associated with bosonic condensates are formed. In particular, spin-
anisotropic polariton-polariton interaction results in Larmor precession of
the condensate pseudospin in a self-induced effective magnetic field even in
the absence of an externally applied one [7].

In the thermodynamic equilibrium limit, the externally applied field and the
effective self-induced one are opposite and even exactly compensate each other
below a certain threshold, giving rise to the spin Meissner effect [8]. In
contrast, nonequilibrium condensates can develop spin polarisation and thus
produce effective magnetic fields spontaneously due to the spin bifurcation
mechanism [9]. Similarly, coherently driven nonequilibrium polariton states
also exhibit spontaneous spin polarisation and spin multistability [10]. Note
that the interplay between the equilibrium effect of locked polarisation
direction in the spin Meissner regime and a spontaneous choice of spin
polarisation typical to driven-dissipative condensates is still under
discussion [11, 12].

Artificial gauge [13] and Zeeman [14] fields emerging in spatially structured
optical systems can replace the external magnetic field giving rise to
topological photonic states. Similarly, the effective field in spontaneously
spin-polarised condensate lattices was shown to result in topologically
nontrivial excitation spectra and unidirectional edge states [15]. However,
the underlying spin bifurcation mechanism requires a delicate balance and
degeneracy of the spin states, rendering this system extremely susceptible to
external symmetry breaking factors. This inspires investigation of the
possibility to exploit this sensitivity to control the band topological
invariants in polariton condensate lattices.

In this work, we show that the effective field of a single spin-bifurcating
condensate is not only aligned with the external Zeeman field, but can also
significantly surpass it in magnitude. Magnification of the weak Zeeman field
with the strong and aligned effective field can be considered as a giant
enhancement of the effective $g$-factor. We demonstrate that the proposed
enhancement is most pronounced in the vicinity of the critical bifurcation
point and identify the range of the optimal parameters.

The system of interest represents a condensate of exciton-polaritons whose
coherent state is governed by the driven-dissipative Gross-Pitaevskii equation
for a spinor wave function
$\mathbf{\Psi}=\left(\Psi_{+},\Psi_{-}\right)^{\intercal}$:

$\displaystyle i\dot{\Psi}_{\pm}=$ $\displaystyle-$
$\displaystyle\frac{i}{2}\left(i\Delta+W-\Gamma-\eta N_{\rm
pol}\right)\Psi_{\pm}-\frac{1}{2}\left(\varepsilon+i\gamma\right)\Psi_{\mp}$
(1) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(\alpha_{1}\left|\Psi_{\pm}\right|^{2}+\alpha_{2}\left|\Psi_{\mp}\right|^{2}\right)\Psi_{\pm},$

where $\Psi_{+(-)}$ stands for spin-up (down) projections of the polariton
state on the structure growth axis corresponding to the right (left) circular
polarisation of the emitted photons. Here $\Gamma$ is the polariton decay
rate, $W$ is the rate of stimulated scattering from the incoherent particle
reservoir, $\eta$ is the strength of the gain saturation proportional to the
condensate occupation $N_{\rm pol}$, while $\varepsilon$ and $\gamma$ are
respectively the energy splitting and the loss-rate difference of the two
linearly polarised polariton states. Finally, $\alpha_{1(2)}$ is the
interaction constant for polaritons with the same (opposite) spin.

Following Ref. [9], we treat the system using the classical pseudospin vector
$\mathbf{S}=1/2\mathbf{\Psi}^{{\dagger}}\bm{\sigma}\mathbf{\Psi}$ defined via
Pauli vector $\bm{\sigma}$ with the magnitude $S$ proportional to the
condensate occupation and the direction characterising its spin state. Note
that the pseudospin $\mathbf{S}$ is directly connected to the Stokes
parameters of the laser emission as $S=S_{0}$, $S_{x}=S_{1}$, $S_{y}=S_{2}$,
$S_{z}=S_{3}$. The pseudospin dynamics is described with the following system
of equations:

$\displaystyle\dot{S}_{x}$ $\displaystyle=$ $\displaystyle(W-\eta
S-\Gamma)S_{x}-\gamma S-(\alpha S_{z}+\Delta)S_{y},$ (2a)
$\displaystyle\dot{S}_{y}$ $\displaystyle=$ $\displaystyle(W-\eta
S-\Gamma)S_{y}+\varepsilon S_{z}+(\alpha S_{z}+\Delta)S_{x},$ (2b)
$\displaystyle\dot{S}_{z}$ $\displaystyle=$ $\displaystyle(W-\eta
S-\Gamma)S_{z}-\varepsilon S_{y}.$ (2c)

In analogy with the classical spin in magnetic field, the pseudospin vector
$\mathbf{S}$ is precessing about the $z$-axis due to the mixed effect of
Zeeman splitting $\Delta$ from the external out-of-plane magnetic field and a
self-induced effective field $\alpha S_{z}$ [6, 7]. The latter is governed by
the interaction constant $\alpha{=\alpha_{1}-\alpha_{2}}>0$, which accounts
for the strong spin anisotropy of polariton-polariton interactions.

In what follows, we use the dimensionless form of system (2):

$\displaystyle\dot{s}_{x}$ $\displaystyle=$
$\displaystyle(p-s)s_{x}-gs-(as_{z}+\delta)s_{y},$ (3a)
$\displaystyle\dot{s}_{y}$ $\displaystyle=$
$\displaystyle(p-s)s_{y}+s_{z}+(as_{z}+\delta)s_{x},$ (3b)
$\displaystyle\dot{s}_{z}$ $\displaystyle=$ $\displaystyle(p-s)s_{z}-s_{y},$
(3c)

with the effective splitting field $\delta=\Delta/\varepsilon$, the
dimensionless time $\tau=t\varepsilon$ and a couple of dimensionless
parameters $a=\alpha/\eta$, $g=\gamma/\varepsilon$. We also define the
effective pump strength $p=(W-\Gamma)/\varepsilon$ and the normalized Stokes
vector $\mathbf{s}=\eta\mathbf{S}/\varepsilon$.

In the absence of an external magnetic field, $\delta=0$, a couple of linearly
polarised ($s_{z}=s_{y}=0$) trivial solutions of system (3) are characterised
by the magnitudes $s_{\pm}(p)=p\pm g$. These states are pinned to the
$x$-axis, $s_{x}=\mp s_{\pm}^{(0)}$. At $g>0$ (the upper linearly polarised
mode dissipates stronger than the lower one), the $s_{-}$-state is dynamically
unstable against weak perturbations according to the regular Lyapunov
stability analysis. The more populated state $s_{+}$ is stable below the
critical pumping $p_{c}=(1-ag+g^{2})/a$, characterised by the condensate
population $s_{c}=(g^{2}+1)/a$.

At $p=p_{c}$, the $s_{+}$-state is destabilized via a pitchfork bifurcation,
which has a supercritical character if $a>g-g^{-1}$ and a subcritical one
otherwise (see the phase diagram in Fig. 1(a)). This gives rise to a couple of
nontrivial elliptically polarised ($s_{z}\neq 0$) states with

$\displaystyle s_{x}=(p-s)s/g,\>\ \ \ s_{y}=(p-s)s_{z},$ (4a) $\displaystyle
s_{z}=\pm{s\over g}\sqrt{g^{2}-(p-s)^{2}\over 1+(p-s)^{2}},$ (4b)

and the condensate population $s$ given by the positive root of the quadratic
equation

$\left({a\over g}-1\right)s^{2}-\left({a\over g}-2\right)ps-(p^{2}+1)=0.$ (5)

Note that at $g<0$, the symmetry-breaking bifurcation does not alter stability
properties of $s_{\pm}$-states. In this regime, the more populated state
$s_{-}$ always remains the only stable linearly polarised solution. In what
follows, we focus on the case $g>0$, corresponding to experimental conditions
[9].

Figure 1: Fig. 1. (a) The phase diagram of the spontaneous symmetry breaking
pitchfork bifurcation at $\delta=0$. Pseudospin projection $s_{z}$ (left axis)
and field enhancement parameter $\chi$ (right axis) in the supercritical
regime at $g=0.1$ and $a=2.28$ [9]. The solid lines correspond to stable
states while dashing shows dynamically unstable solutions. (c) The same as on
panel (b) but in the subcritical regime at $g=3$ and $a=1$. The dotted blue
curve corresponds to the field-enhancement $\chi$ of the upper stable
elliptically polarised state which appears in the fold bifurcation.

To describe the impact of the external magnetic field, we start with the limit
of perturbatively weak field $\delta$. In the first approximation, stationary
solutions acquire a correction
$\mathbf{s^{(1)}}=\left(s_{x}^{(1)},s_{y}^{(1)},s_{z}^{(1)}\right)^{\intercal}$
linear in $\delta$:

$\mathbf{s}^{(1)}=\delta\times\left[J\left(\mathbf{s}^{(0)}\right)\right]^{-1}\left(s_{y}^{(0)},-s_{x}^{(0)},0\right)^{\intercal},$
(6)

where $J(\mathbf{\mathbf{s}^{(0)}})$ is the Jacobi matrix of the system (3)
calculated at the unperturbed fixed point solution $\mathbf{s}^{(0)}$.

In the case of the linearly polarised fixed point $s_{+}$ with
$s_{x}^{(0)}=-(p+g)$ which is stable below the critical pumping $p_{c}$, the
correction (6) provides a non-zero interaction-driven field
$as_{z}^{(1)}=\delta s^{(0)}\left/\left(p_{c}-p\right)\right.$. The condensate
response can thus be characterized by the field enhancement parameter
$\chi=as_{z}^{(1)}/\delta$ and the effective $g$-factor

$g_{\rm eff}=1+\chi={s_{c}\over p_{c}-p}$ (7)

responsible for the condensate pseudospin precession frequency $\delta g_{\rm
eff}$ according to equations (3).

Since $g_{\rm eff}>0$ at $p<p_{c}$, the induced field is aligned with the
external one. Note that expression (7) diverges at the critical point
similarly to the susceptibility behaviour in Landau theory of second order
phase transitions, and linearly scales with the condensate population, as
shown in Fig. 1(b). Moreover, as $p$ approaches $p_{c}$ from above,
$p\rightarrow p_{c}^{+}$, the field enhancement parameter $\chi$, which is now
governed by the response of the elliptical states (4), has a similar
asymptotic behavior. However, it is important to note that $g_{\rm eff}>0$
only for the state whose built-in field is aligned with the external one,
$s_{z}^{(0)}/\delta>0$ (the upper elliptical state in Fig. 1(b)). For the
anti-aligned state, a real Zeeman splitting reduces the total effective field.
That is why we consider the aligned states only. In the supercritical case
their response diverges as $s_{c}/(2|p-p_{c}|)$ near $p=p_{c}$. However, in
the subcritical regime, dynamically unstable states emerging from the
pitchfork exhibit a negative response, $as_{z}^{(1)}/\delta<0$, reducing the
effective field magnitude, – see the red dashed lines in Fig. 1(c).

Figures 1(b) and 1(c) show that close to the critical pumping, polariton
pseudospin perceives up to hundredfold increase in the Zeeman field magnitude.
One of the striking manifestations of this strong field enhancement effect can
be observed in the topological properties of Bogoliubov excitations in
polariton lattices [15, 16]. Since emergence of nontrivial topological phases
typically requires time-reversal symmetry breaking, the strong magnetic field
is needed – up to 5 T in Ref. [4]. However, the elementary excitations
emerging in the condensate are also subject to the effective field, created by
the condensate pseudospin polarisation and potentially significantly exceeding
the external one.

In order to estimate the efficiency of the proposed field-enhancement
principle for manipulating topology of bogolons, we proceed with the analysis
of the dynamics of single condensate elementary excitations. In particular, we
focus on the precession of weakly perturbed stationary pseudospin in the
presence of finite magnetic field $\delta$.

Dynamics of the perturbation $\bm{\mathfrak{s}}$ of the fixed point state
$\mathbf{s}$ is governed by the linearized system (3), i.e. its Jacobi matrix,
$\dot{\bm{\mathfrak{s}}}=J(\mathbf{s})\bm{\mathfrak{s}}$. Therefore, its
naturally expected that the excitation experiences the same effective field as
the pseudospin state, $g_{\rm eff}\delta=as_{z}^{(0)}+\delta$. However, due to
the intrinsic non-Hermiticity of $J$, the evolution of the weak perturbation
strongly differs from the corresponding pseudospin dynamics.

The $\bm{\mathfrak{s}}(t)$-evolution can be given in terms of eigenvalues
$\lambda_{i}$ and eigenvectors $\mathbf{v}_{i}$ of $J(\mathbf{s})$. Since the
Jacobian of (3) is purely real, it has either three real or a single real and
two complex conjugate eigenvalues. We are interested in the latter case where
the perturbation dynamics reads:

$\displaystyle\bm{\sigma}(t)=C_{1}e^{-\gamma_{p}t}\left[\mathbf{u}_{1}\cos(\Omega
t+\varphi)-\right.$ $\displaystyle\left.\mathbf{u}_{2}\sin(\Omega
t+\varphi)\right]+C_{2}\mathbf{v}_{3}e^{-\Gamma_{p}t}.$ (8)

Here $\mathbf{v}_{1}=\mathbf{u}_{1}+i\mathbf{u}_{2}$ is the eigenvector
corresponding to either of complex conjugate eigenvalues
$\lambda_{1,2}=-\gamma_{p}\pm i\Omega$, $\mathbf{v}_{3}$ is the eigenvector
for purely real $\lambda_{3}=-\Gamma_{p}$. Coefficients $C_{1,2}$ and
$\varphi$ are real.

Figure 2: Fig. 2. (a) A schematic evolution of the Bogoliubov’s elementary
excitation $\bm{\mathfrak{s}}(t)$ shown with the blue line in the pseudospin
space. The end-point to which the spiral curls up corresponds to the
dynamically stable stationary state $\mathbf{s}$. The pale orange plane
contains vectors $\mathbf{u}_{1,2}$ while the vector $\bm{\Omega}$ is its
normal. (b) Comparison between definitions of the effective magnetic field
acting on the Bogoliubov’s elementary excitation no top of the stable
pseudospin states at $\delta=0.1$, $a=2.28$ and $g=0.1$. The $p$-dependencies
of the precession frequency $\Omega$ and the effective magnetic field
${\Omega}_{z}$ illustrated in the panel (a). The vertical gray line indicates
position of the fold bifurcation, – cf. with Fig. 1(c).

The long-term evolution (Giant effective $g$-factor due to spin bifurcations
in polariton condensates) of a weak excitation shown in Fig. 2(a) represents
the dumped rotation with the frequency $\Omega$ in the plane determined by
$\mathbf{u}_{1,2}$. The normal
$\mathbf{n}\perp(\mathbf{u}_{1},\mathbf{u}_{2})$ assigns direction to the
effective magnetic field $\mathbf{\Omega}=\mathbf{n}\Omega$ whose
$z$-component $\Omega_{z}$ is responsible for time-reversal symmetry breaking
for Bogoliubov’s excitations. The axis of pure relaxation $\mathbf{v}_{3}$ is
misaligned with $\mathbf{\Omega}$ in contrast to the problem of a classical
spin in magnetic field.

In general, $\mathbf{\Omega}$ differs from the effective field $g_{\rm
eff}\delta$ acting on a pseudospin itself. However, in the limit of strong
pumping, the excitation dynamics is dominated by the interaction-driven field
$as_{z}^{(0)}$ that implies $\Omega_{z}\approx g_{\rm eff}\delta$. In
contrast, in the vicinity of the bifurcation point, where the giant
enhancement of the external field occurs, this simple asymptotics fails.

The values of $\Omega_{z}$ and $g_{\rm eff}\delta$ near the fold (saddle-node)
bifurcation are compared in Fig. 2(b). Despite significant difference in the
involved dynamics, the magnitudes of the effective field for a pseudospin
$\mathbf{s}$ and for Bogoliubov’s excitations $\bm{\mathfrak{s}}$ follow the
same trend. In addition, the existing mismatch quickly vanishes with the
increase of the real field amplitude $\delta$.

In the case of coupled condensate lattices, the topological gap is expected to
open in the Bogoliubov’s excitation spectrum in the presence of symmetry
breaking and spin-anisotropic interactions [15]. Moreover, in the effective
field approximation, the uniform spin polarisation is equivalent to a
conservative polariton Zeeman field and reduces to Hermitian spin-splitting
diagonal terms in the Bogoliubov matrix. These terms responsible for
topological gap opening are proportional to both the uniform spin polarisation
and the strength of spin-anisotropic interaction, rendering the corresponding
effective Zeeman field identical to the self-induced field $as_{z}$ in Eqs.
(3).

In order to find the field-enhancement strength beyond perturbative approach,
we numerically search for stable stationary solutions of system (3). The
resulting $g$-factor $g_{\rm
eff}=\left.\left(as_{z}+\delta\right)\right/\delta$ is shown in Fig. 3(a) on
the $(\delta,p)$ parameter plane. Even away from the region of fast
divergence, polariton condensate is able to amplify external magnetic field up
to an order of magnitude.

The best results can be obtained at the weak magnetic field where $g_{\rm
eff}$ diverges according to the perturbation theory. This regime, however, has
a significant drawback: at strong pumping which favours large $g_{\rm eff}$,
polariton condensate exhibits bistability with two stable pseudospin
configurations with opposite directions of the interaction-driven effective
field. Under non-resonant excitation, the building-up of internal
magnetization in the spin-bifurcation event occurs spontaneously with randomly
selected direction. In particular, in the field-free case $\delta\rightarrow
0$, the condensate excited above the critical pumping $p_{c}$ occupies either
of two elliptically polarised states (4) with equal probabilities [9].

Figure 3: Fig. 3. (a) Effective $g$-factor as a function of the pump $p$ and
the external magnetic field strength $\delta$. It is assumed that the
condensate occupies a stable pseudospin state with $as_{z}/\delta>0$. (b) The
probability $\mathcal{P}$ of excitation of the state whose interaction-driven
field is co-directed with the external one. A bistable regime with
$\mathcal{P}<1$ occurs in the region on the right bottom corner of the
($p$,$\delta$) parameter plane. The dash-dotted line indicates position of the
fold bifurcation which gives birth to the stable state with anti-aligned
effective field. The descending ladder of probability cuts are shown with gray
lines which are duplicated in panel (a). Each data point was obtained by a
direct numerical solution of (3) with random initial conditions and averaging
over 200 realizations.

However, in the broken symmetry case of finite $\delta$, the balance between
spin-up and spin-down states shifts towards the state with the aligned built-
in magnetic field corresponding to positive $g_{\rm eff}$. In the
configuration shown in Fig. 1(c), it is the upper state with $s_{z}>0$.

The probability $\mathcal{P}$ of occupation of the aligned state is shown in
Fig. 3(b) in the parameter space spanned by $p$ and $\delta$. Within the
domain of $\mathcal{P}=1$, the condensate supports a single stable
configuration. Far away from the low-$\delta$ limit, this regime provides
quite small $g_{\rm eff}$ about few units, – see Fig. 3(a). Above the critical
pumping (dash-dotted line) corresponding to the fold bifurcation shown in Fig.
1(c), the probability is continuously declining and converges to
$\mathcal{P}=0.5$ as $\delta\rightarrow 0$ or $p\rightarrow\infty$. Therefore,
the optimal regime, where strong field enhancement is combined with the high
predictability of the condensate magnetization direction, is reached in a
close vicinity to the bifurcation conditions.

In conclusion, we summarize the obtained results. Intrinsically non-
equilibrium bosonic condensate of exciton polaritons can spontaneously develop
strong effective magnetic fields due to the spin bifurcation phenomenon [9].
Since the very first studies [6, 7] this field was associated with the so-
called self-induced Larmor precession of the Stokes vector of the light
emitted by the condensate. Here we demonstrated that this phenomenon is
controllable with an external permanent magnetic field and has a pronounced
manifestation at the level of Bogoliubov’s excitations [15]. The
nonequilibrium condensate exhibits a very sensitive response to the applied
magnetic field providing its strong enhancement. The enhancement strength can
be characterised by the effective $g$-factor which exceeds one hundred and
strongly depends on the pump intensity and the external field magnitude.

The probabilistic character of the steady condensate spin polarisation and
thus the direction of the effective field reduces the available parameter
range down to the region near the fold bifurcation of the condensate
pseudospin. Outside of this range, in particular, where spin multistability
takes place, polariton spin fluctuations are expected to destabilize the
condensate at sufficiently high temperatures [17, 18].

In addition, we notice a peculiar connection with the Landau theory of second-
order phase transitions. The magnetic-field susceptibility of the condensate
diverges near the critical point where a circular polarisation $s_{z}\neq 0$
appears spontaneously. For the symmetry-broken states, the divergence at
$p=p_{c}$ is twice slower.

The obtained results pave the way to further investigations of the topological
properties of Bogoliubov’s excitations in the lattices of driven-dissipative
polariton condensates.

Acknowledgements. The work of I.C. (analysis of the non-Hermitian dynamics,
text writing) is supported by the Russian Science Foundation Grant No.
22-72-00061. The work of A.N. (analytical calculations, work supervision, text
writing) is supported by he Russian Science Foundation under Grant No.
22-12-00144.

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