# Digital quantum simulation of dynamical topological invariants on near-term
quantum computers

Huai-Chun Chang  Department of Mathematical Sciences, National Chengchi
University, Taipei 11605, Taiwan  Hsiu-Chuan Hsu<EMAIL_ADDRESS>Graduate
Institute of Applied Physics, National Chengchi University, Taipei 11605,
Taiwan  Department of Computer Science, National Chengchi University, Taipei
11605, Taiwan

###### Abstract

Programmable quantum processors are suitable platforms for simulating quantum
systems, of which topological phases are of particular interest. We simulate
the quench dynamics of a one-dimensional system on IBM Q devices. The
topological properties of the dynamics are described by the dynamical
topological invariants, the dynamical winding number and the time-dependent
Berry phase, which are simulated with the quantum circuit model. The results
show that despite the noise present in the current quantum computers, the
dynamical topological invariants are robust. Moreover, to investigate the
influence of open quantum system, we analytically solve the master equation in
Lindblad form and show that the dynamical winding number and the change in
Berry phase are not affected by the dissipation. This study sheds light on the
robustness of topological phases on the noisy intermediate-scale quantum
computers.

##  I Introduction

The most appropriate tool for simulating quantum systems is the quantum
computer because the fundamental of nature is quantum mechanics, as proposed
by Feynmann in 1982 Feynman (1982). Since then, the scientific and engineering
communities have been pursuing the realization of the device and exploring the
applications. It has been proved that simulations of quantum systems is
plausible due to the universality of the gate-based quantum comuting
processors Lloyd (1996); Tacchino  _et al._ (2020). Recently, the quantum
technologies have experienced a great breakthrough and improved the
fabrication of the quantum processors Arute  _et al._ (2019); Jurcevic  _et
al._ (2021). However, the current quantum computers are at the stage of Noisy
Intermediate-Scale Quantum (NISQ) era Preskill (2018). The quality and number
of qubits are still limited. The capabilities and applications of NISQ
computers have been explored in many subjects. For example, several studies
have demonstrated the quantum simulation of non-equillibirum quantum systems
Smith  _et al._ (2019); Babukhin  _et al._ (2020); Fauseweh and Zhu (2021),
many-body states Smith  _et al._ (2019); Rahmani  _et al._ (2020); Kirmani
_et al._ (2021), open quantum systems Head-Marsden and Mazziotti (2019); Del
Re  _et al._ (2020); García-Pérez  _et al._ (2020); Head-Marsden  _et al._
(2021); Kamakari  _et al._ (2021), dynamical quantum phase transitions Guo
_et al._ (2019) and topological phases of matter Murta  _et al._ (2020); Ji
_et al._ (2020); Mei  _et al._ (2020); Chen  _et al._ (2021); Xiao  _et al._
(2021).

Moreover, phase factor is one of the most important aspect that distinguishes
quantum mechanics from classical mechanics Dirac (1972); Yang (2013). In
classical numerical simulations of quantum systems, one could simulate the
wave functions. However, experiments can only obtain the squared modulus of
the wave function without the information of phase. For a more realistic
simulation of quantum systems, quantum computer is a suitable platform for
controlling the quantum states and allowing the measurement of the phase with
a careful design of the quantum circuits Murta  _et al._ (2020); Cian  _et
al._ (2021); Xiao  _et al._ (2021).

The study of the phase factor is the center of the topological matters in
modern condensed matter physics. The Berry phase, the phase difference of the
wave function when the parameters of the system change in a closed path Berry
(1984), has been found to present in many materials. In solids, the Berry
phase is the phase difference of the Bloch states as momentum changes in a
closed path in the Brillouin zone Thouless  _et al._ (1982); Kohmoto (1985).
The Berry phase has direct consequence on the electronic properties of
materials Xiao  _et al._ (2010), such as anomalous Hall effect Jungwirth  _et
al._ (2002), topological insulators Hasan and Kane (2010) and Weyl semimetals
Yan and Felser (2017).

The study of topology has extended to non-equilibrium states. Recently, the
topological phases of quench dynamics have been explored Yang  _et al._
(2018); Gong and Ueda (2018); Chang (2018); Sun  _et al._ (2018); Hu and Zhao
(2020). Quench dynamics consider the time evolution of a quantum state, which
is the ground state of the initial Hamiltonian $H_{i}$, under a sudden change
of the Hamiltonian to $H_{f}$. As the quantum state evolves unitarily with the
quench Hamiltonian $H_{f}$, the topology of the equilibrium quench Hamiltonian
can be detected from the dyanmics, such as the dynamical winding numbers Zhu
_et al._ (2020) and band-inversion surfaces Zhang  _et al._ (2019a, b) .
Moreover, by treating time as an additional dimension, the dynamical
topological invariants can be defined on a momentum-time torus in quenching
one-dimensional systems. For example, previous studies have proposed time-
dependent Berry phase Hsu  _et al._ (2021), dynamical Chern number Yang  _et
al._ (2018); Chang (2018); Hsu  _et al._ (2021) and the Skyrmion texture in
the momentum-time domain Wang  _et al._ (2019); Guo  _et al._ (2019).

Several studies have studied topological phases on NISQ computers. Viyuela et
al. Viyuela  _et al._ (2018) simulated topological phases interacting with
thermal baths and observed topological Uhlmann phases. Murta et al. Murta  _et
al._ (2020) proposed a Berry phase estimation algorithm that removes the
dynamical phase while preserves the geometric phase. Mei et al. Mei  _et al._
(2020) showed quantum simulation of topological insulators and the boundary
modes. Xiao et al. Xiao  _et al._ (2021) demonstrated the robustness of the
topological invariants against noise by computing Chern number on quantum
computers. Quantum simulation on NISQ devices opens a new direction for
investigating the topological properties of the quantum states.

In this work, we simulate quench dynamics on the cloud quantum computer
provieded by IBM Q and found the topological robustness of the topology in
quench dynamics on NISQ devices. We compare the results of the simulations of
quantum circuits on qasm_simulator provided by the Qiskit API Abraham and et
al. (2019) and IBM Q devices. The dynamical winding number and Berry phase
were computed and shown to reflect the topological properties. Moreover, we
consider the dynamics in the open quantum system described by the master
equation and show that the topological invariants are robust against
dissipation. The analytical results agree with that of quantum simulation. The
rest of the paper is organized as follows. In sec. II, the quench protocol and
the quantum circuit model for computing topological properties are introduced.
In sec. III, the simulation results and analysis are presented. In sec. IV,
the solution and discussion of the master equation for the dynamics in the
open quantum system are given. Finally, the conclusion is given in sec. V

##  II The Quench dynamics

In this study, we consider the quench dynamics in one-dimension. The initial
states are the ground state of $H_{i}=-\sigma_{z}$. Thus, the pseudospins are
polarized along the computational basis. The initial quantum state is suddenly
quenched by the single-particle Su-Schrieffer-Heeger (SSH) Hamiltonian Su  _et
al._ (1979) $\mathcal{H}_{0}(k)$

$\displaystyle\mathcal{H}_{0}(k)$ $\displaystyle=$ $\displaystyle
h_{x}(k)\sigma_{x}+h_{y}(k)\sigma_{y},$ (1)

where

$\displaystyle h_{x}(k)$ $\displaystyle=$ $\displaystyle g_{f}-\cos(k)$
$\displaystyle h_{y}(k)$ $\displaystyle=$ $\displaystyle\sin(k)$ (2)

$\sigma_{x,y}$ are Pauli matrices and act on the sublattices $a,b$. The
lattice constant is taken to be one. The eigen-energy is $E_{\pm}=\pm
E_{k}=\pm\sqrt{h_{x}^{2}+h_{y}^{2}},=\pm\sqrt{\sin^{2}(k)+(g_{f}-\cos(k))^{2}}$.
The topology of the Hamiltonian is characterized by winding number
$\mathit{w}$. The topological phase is in the regime $|g_{f}|<1$ with
$\mathit{w}=1$. While the trivial phase is in the regime $|g_{f}|>1$ with
$\mathit{w}=0$. $g_{f}=1$ is the phase transition point. The winding number is
the number of times the pseudospin winds about the origin on the $x-y$ plane
as the parameter $k$ changes from $0$ to $2\pi$ Vanderbilt (2018).

Two topological invariants defined to characterize the dynamics are studied.
The first is the dynamical winding number. At $t>0$, the pseudospin starts to
precess about $(h_{x}(k),h_{y}(k))$. For a fixed time $t$, the trajectory of
the pseduospin projected on the $x-y$ plane as $k$ varies from $0$ to $2\pi$
reflects the topology of the quench Hamiltonian. For the topological quench
Hamiltonian, this trajectory makes full revolutions about the origin. In
contrast, for the trivial quench Hamiltonian, the trajectory does not encircle
the origin. The dynamics of the pseudospin is characterized by the dynamical
winding number

$\displaystyle
w_{dyn}=\frac{1}{2\pi}\int_{0}^{2\pi}dk\frac{\partial\eta_{yx}}{\partial k},$
(3)

where

$\displaystyle\eta_{yx}={\rm
Im}\log[{\langle\sigma_{x}\rangle}+i{\langle{\sigma_{y}\rangle}}].$ (4)

In the noiseless situations, for the quench protocol considered here,
$\eta_{yx}$ equals to ${\rm Im}\log[(h_{y}-ih_{x})(\sin(2E_{k}t))]$ with the
Planck constant $\hbar$ is set to one. The details can be found in Appendix A.
Because the principal value of the complex log lies in $(-\pi,\pi]$, when the
psuedospin makes one full revolution on the $x-y$ plane, $\eta_{yx}$ shows a
discontinuous jump between $-\pi$ and $\pi$. For the topological dynamics with
$w_{dyn}=1$, $\eta_{yx}$ shows a discontinuous $2\pi$ jump. In contrast, for
trivial dyanmics, $\eta_{yx}$ is a smooth function.

The dynamical winding number can be regarded as the detection of bulk topology
of the quench Hamiltonian. Nevertheless, one can treat time as an additional
dimension and define a topological invariant on the momentum-time space. This
topological invariant is the dynamical Chern number, which is defined on the
momentum-time torus which $k\in[0,2\pi],t\in[0,\pi/(2E_{k})]$. Since the
topology is robust against smooth deformation, it is equivalent to rescaling
$E_{k}=1$, for which the torus becomes $k\in[0,2\pi],t\in[0,\pi/2]$ Yang  _et
al._ (2018); Chang (2018); Guo  _et al._ (2019). The dynamical Chern number
can be obtained by integrating the time derivative of the Berry phase Gresch
_et al._ (2017); Kuno (2019); Hsu  _et al._ (2021)

$\displaystyle
C_{dyn}=\frac{1}{2\pi}\int^{\pi/2}_{0}dt\frac{\partial\gamma(t)}{\partial t},$
(5)

where $\gamma(t)$ is the time-dependent Berry phase given by the overlap
matrix between neighboring k-points Vanderbilt (2018); Kuno (2019)

$\displaystyle\gamma(t)$ $\displaystyle=$ $\displaystyle{\rm
Im}\log[\prod_{k}M_{k,k+\delta k}],$ $\displaystyle M_{k,k+\delta k}(t)$
$\displaystyle=$ $\displaystyle\langle\psi(k)|\psi(k+\delta k)\rangle.$ (6)

$|\psi(k)\rangle$ is the quantum state of the pseudospin at time $t$ with
$E_{k}=1$. The Berry phase is the total accumulated phase as the parameter $k$
changes from $0$ to $2\pi$. Analytically, the Berry phase is shown to be
$\gamma(t)=-2\pi\mathit{w}\sin^{2}t$ (details are given in Appendix B),
proportional to the winding number of $\mathcal{H}_{0}$ Chang (2018). The
change in Berry phase from $t=0$ to $t=\pi/2$ is $2\pi\mathit{w}$. Due to the
interval of the principal value of the complex log, the Berry phase
$\gamma(t)$ shows a discontinuous jump between $-\pi$ and $\pi$ for
topological dynamics, while exhibits a smooth function for trivial dynamics.
Thus, we compute the Berry phase and determine the topology by the Berry phase
flow (the change of Berry phase in a period taking into account the $2\pi$
jump).

The dynamical winding number and Berry phase are computed on the quantum
computers and quantum circuit simulators on a classical device. First, we
compute the in-plane pseudospins to obtain the dynamical winding number. The
quantum circuit for this purpose is shown in Fig. 1. We map the time evolution
operator to the U-gate provided by qiskit API. The details can be found in
Appendix A. Because the measurement is in the computational basis, i.e. along
$z$-axis, a rotation on the quantum state is performed before the measurement
of $\langle\sigma_{x,y}\rangle$. Second, we compute the Berry phase flow. We
added an ancilla bit and apply the control unitary bewteen the ancilla and
target bit to compute the phase difference between state at $k$ and $k+\delta
k$ Xiao  _et al._ (2021). To aqcuire the real(imaginary) part of the phase, a
$(-)\pi/2$ rotation about $y(x)$-axis is performed before the measurement, as
shown in Fig.2. It is equivalent to the measurement of the expectation values
of $\sigma_{x,y}$, respectively. From the measurement outcome of the circuit,
the Berry phase is obtained by

$\displaystyle\gamma(t)={\rm
Im}\log[\prod_{k}{\langle\sigma_{x}\rangle_{k}}+i{\langle{\sigma_{y}\rangle_{k}}}],$
(7)

where $\langle\sigma_{x,y}\rangle_{k}=\langle\psi(k)|\psi(k+\delta k)\rangle$
computed with the quantum circuit. In our simulation for Berry phase, $E_{k}$
is normalized to $1$ for the same reason as mentioned in the previous
paragraph.

Figure 1:  The circuit model for measuring the time evolution of the
pseudospinors. The rotation angles for the unitary gates are
$\theta=2E_{k}t_{i}\Delta t$, where $\Delta t=\frac{\pi}{T},\ T=40$ and $\
t_{i}\in\\{1,2,3,\cdots,20\\}$, $\phi=-{\rm
Im}(\log(h_{x}+ih_{y}))-\frac{\pi}{2},\lambda=-\phi$. $R_{y}$ is the rotation
gate about $y$-axis, $S^{\dagger}$ is the Hermitian conjugate of the phase
gate and $H$ is the Hadamard gate. The three qubits are used to calculate
$\langle\sigma_{x,y,z}\rangle$, respectively.  Figure 2: The quantum circuit
for calculating the phase between two adjacent momenta. $a_{0}$ is the ancilla
bit for measuring the phase. $q_{0}$ is the target bit of which the quantum
state represents the quench dynamics. $H$ gate is the Hadamard gate, $R$ gate
refers to $+(-)\pi/2$ rotation about $y(x)$-axis when measuring the
real(imaginary) part of the phase. For the unitary gates,
$\theta=2t,\phi_{k}=-{\rm Im}(\log(h_{x}(k)+ih_{y}(k)))-\frac{\pi}{2}$ and
$\lambda_{k}=-\phi_{k}$.

##  III Results of the Digital quantum simulations

In this section, we present the results of quantum simulation and discuss the
quench dynamics of the pseudospins. To investigate whether the Berry phase
flow is robust on NISQ computers, we choose the quench Hamiltoians to be near
the topological phase transition. In all the experiments, the number of
samplings is taken to be 8192, the maximum shots on IBM Q machines.

###  III.1 Dynamical winding number

Since the winding number corresponds to that the in-plane pseudospins winds
around the origin by multiples of $2\pi$, we monitor the dynamical azimuthal
angles swept by the pseudospins. Fig. 3(a,b) shows $\eta_{yx}$ simulated on
ibmq_lima. In (a), the quench Hamiltonian is topological with $g_{f}=0.8$. A
discontinuous jump can be observed near $k\approx 1.8\pi$, indicating the full
revolution on the $x-y$ plane. In Fig. 3(b), the quench Hamiltonian is trivial
with $g_{f}=1.2$. There is no $2\pi$ discontinuous jump for the trivial case.

From the dynamical azimuthal angles, we obtain the dynamical winding number.
In principle, the dynamical winding number is obtained with Eq. (3) which
requires dense mesh in digital simulations. Because of the limited
availabitliy of the NISQ devices, we only use sparse mesh in quantum
simulation. Thus, we extract the difference of the jump of $\eta_{yx}$ at the
discontinuity in unit of $2\pi$ as the dynamical winding number. The results
are presented in Fig. 3 (c). The results do not show significant deviation
between the qasm_simulator and ibmq_lima. The winding number is not exactly
quantized to unity since the $k$ mesh is not dense enough. We performed
convergence test on the qasm_simulator. Fig. 3(d) shows that as the $k$ mesh
density increases, the dynamical winding number becomes closer to
quantization. Even though the dynamical winding number is not well quantized
due to sparse mesh, the dynamical azimuthal angles show the a $\sim 2\pi$ jump
as a signature of topological dynamics.

Figure 3: (a)The winding angle at $t=\pi/40$ and $t=8\pi/40$ obtaind with
ibmq_lima for quench Hamiltonian $g_{f}=0.8$. (b) The same as (a) but for
$g_{f}=1.2$. (c) The winding number extracted from the discontinuity of
$\eta_{yx}$ simulated on different devices. (d) The convergence test for the
dynamical winding number on qasm simulators. The curves are for different $k$
mesh as shown in the legend.

###  III.2 Berry phase

Figure 4: Berry phase for topological quench dynamics. (a, b) The Berry phase
obatained on ibmq_lima on for $g_{f}=0.8,1.2$, respectively. (c, d) The Berry
phase obtained on ibmq_toronto for $g_{f}=0.8,1.2$, respectively. qasm-err
denotes the results for simulating the unitary errors on the qasm_simulator.

We execute the quantum circuit (Fig. 2) to compute the time-dependent Berry
phase. The circuits are performed on noiseless qasm_simulators and IBM Q
machines. The results for $g_{f}=0.8$ and $g_{f}=1.2$ are presented in Fig. 4.
For qasm_simulators, as indicated by the red curve with circles, the $2\pi$
jump is exactly at $t=\pi/4$ for topological dynamics. We present the results
on two IBM Q devices, ibmq_lima and ibmq_toronto. For each device, we present
results of three experiments taken on the same day. Different runs of data
show the same behavior. For ibmq_lima, the results for $g_{f}=0.8$, as
presented in Fig. 4(a), show a clear discontinuity near $t=\pi/4$, giving rise
to a nearly $2\pi$ Berry phase flow from $t=0$ to $t=\pi/2$. The first
experiment, as labeled by ’lima0’ in the figure, shows random spikes. The
similar spikes are observed for the trivial case, as shown in Fig. 4(b). These
sudden spikes do not contribute to Berry phase flow in the period of time
$[0,\pi/2]$ because the change of the phases on each side of the spike
cancels. Thus, the results in 4(b) indicate trivial dynamics.

For ibmq_toronto, the results for $g_{f}=0.8$ show a clear discontinuity, as
shown in Fig. 4(c). The position of dicontinuity shifts by $\sim\pi/8$ to the
right of $\pi/4$. The shift of the discontinuity is attributed to the unitary
error, which is discussed in detail in the next paragraph. There is a
background Berry phase $\pi/2$, which is not shown on qasm_simulator. The
background phase does not change the Berry phase flow. The results for
$g_{f}=1.2$ are shown in Fig. 4(d). The Berry phase is near $\pi/2$ at all
instants. There is no Berry phase flow, indicating trivial dynamics. For both
parameters, there is a $\sim\pi/2$ background Berry phase on ibmq_toronto, but
not on ibmq_lima.

We simulate the error on qasm_simulator by adding a global phase $\delta$ to
the target bit in the control-unitary gate (Fig. 2). This error gives rise to
a background Berry phase, but only shifts the position of the $2\pi$ jump by
$\sim\pi/16$. To take into account the shift observed in Fig. 4(c) , an error
$\delta_{t}$ is added to the polar angles in the unitary and controlled-
unitary gates. It is found that with $\delta=0.03\pi$ and
$\delta_{t}=-0.12\pi$, the resultant Berry phase captures the trend on
ibmq_toronto, as shown by the black dashed line in Fig. 4 (c,d). Therefore, we
conclude that the background Berry phase is a result of the accumulated global
phase due to the systematic unitary gate errors. The shift of the $2\pi$ jump
is a consequence of both errors in global phase and polar angles.

Comparing to the results obtained from ibmq_lima, ibmq_toronto shows less
fluctuations and no random spikes. Among the quantities for device
characetization, including average control-not errors, readout errors,
depolarizing time ($T_{1}$) and decoherence time ($T_{2}$), the largest
difference between the two devices is the time constants. The average
depolarizing and decoherence time for ibmq_toronto are $T_{1}=101\mu
s,T_{2}=122\mu s$, while for ibmq_lima are $T_{1}=66\mu s,T_{2}=99\mu s$. The
longer $T_{1},T_{2}$ constants preserve the qubit states and allows more
precise measurement outcome. Therefore, the cleaner result on ibmq_toronto is
attributed to the longer time constants.

##  IV The master equation

In experiment settings, the qubits are coupled to environments that lead to
dissipation. In order to gain insight on how dissipation affects the quench
dynamics, we analytically solve the master equation that describes the open
quantum system. After quench, the qubit dynamics evolves according to
$\mathcal{H}_{0}+H_{int}$, where $\mathcal{H}_{0}$ is given in Eq. (1) and
$H_{int}$ describes the interaction between the qubit and radiation fields.
With dipole approximation and rotating wave approximation, the interaction is
written as $H_{int}=\vec{d}(e^{i\Omega t}\sigma_{+}+e^{-i\Omega
t}\sigma_{-})$, where $\vec{d}$ is the tansition matrix element of the dipole
operator, $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$ denotes transition
between two levels and $\Omega$ is the angular frequency that is the same as
the energy gap of the quench Hamiltonian. With weak coupling limit and Born-
Markovian approximations, the evolution of the density matrix is given by the
Lindblad equation Carollo  _et al._ (2003); Heinz-Peter and Petruccione (2007)

$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle-i[\mathcal{H}_{0}(k),\rho]$ (8) $\displaystyle+$
$\displaystyle\gamma_{0}(N+1)(\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\left\\{\sigma_{+}\sigma_{-},\rho\right\\})$
$\displaystyle+$
$\displaystyle\gamma_{0}N(\sigma_{+}\rho\sigma_{-}-\frac{1}{2}\left\\{\sigma_{-}\sigma_{+},\rho\right\\}),$

where $\rho$ is the density matrix, $N$ is the Bose-Einstein distribution,
$\gamma_{0}$ is the spontaneous emission rate. The Planck constnt $\hbar$ is
set to one. $\gamma_{0}\sigma_{-}$ describes the spontaneous emission,
$\gamma_{0}N\sigma_{-(+)}$ describes the thermally induced emission
(absorption). The details of the derivation are shown in Appendix C. After
solving for $\rho$ at low temperature, where $\gamma_{0}\approx\gamma$, the
psuedospins are given by

$\displaystyle\langle\sigma_{x}(t)\rangle$ $\displaystyle=$ $\displaystyle
h_{y}F^{\prime}(\gamma,k,t)$ $\displaystyle\langle\sigma_{y}(t)\rangle$
$\displaystyle=$ $\displaystyle-h_{x}F^{\prime}(\gamma,k,t),$ (9)

where

$\displaystyle F^{\prime}(\gamma,k,t)=4e^{-\frac{3\gamma
t}{4}}\frac{(8E_{k}^{2}+5\gamma^{2})\sin(\frac{\omega
t}{4})+\omega\gamma\cos(\frac{\omega
t}{4})}{\omega(4E_{k}^{2}+\gamma^{2})}-\frac{4\gamma}{4E_{k}^{2}+\gamma^{2}}$
(10)

and $\omega=\sqrt{64E_{k}^{2}-\gamma^{2}}$. The expression of $F^{\prime}$ is
complicated, but only its sign affects the measurement of $\eta_{yx}$ [Eq.
(4)]. As shown in Fig. 5, at a damping rate $\gamma=4E_{k}$, which is twice
the energy gap of the quench Hamiltonian, $\eta_{yx}$ shows a total $2\pi$
difference when $k$ changes from $0$ to $2\pi$.

Figure 5: $\eta_{yx}$ as a function of $k$ at $t=0.2\pi$ given by Eq. (9). The
overdamping case $\gamma=4E_{k}$ is shown by red solid line, compared to the
dissipationless case with $\gamma=0$ shown by the black solid line.

To see how dissipation affects the pseudospins’ dynamics on the Bloch sphere,
we look into the z-component

$\displaystyle\langle\sigma_{z}\rangle=e^{-\frac{3\gamma
t}{4}}\frac{2{\omega}\left(2E_{k}^{2}+\gamma^{2}\right)\cos\left(\frac{{\omega}t}{4}\right)-2\gamma\left(\gamma^{2}-14E_{k}^{2}\right)\sin\left(\frac{{\omega}t}{4}\right)}{{\omega}\left(4E_{k}^{2}+\gamma^{2}\right)}-\frac{\gamma^{2}}{4E_{k}^{2}+\gamma^{2}}.$
(11)

The geometric phase is a consequence of the dynamics of the pseudospins Wang
_et al._ (2019). Thus, we visualize the pseudospin texture as a function of
time and momentum given by Eq. (9) and 11. As explained in Sec. II, $E_{k}$ is
normalized to $1$ in the calculation. Fig. 6 (a) shows the texture for
$g_{f}=0.8$ in the dissipationless limit when $\gamma$ is set to $0$. At each
moment, the $x,y$ components of the pseudospinors winds clockwisely as $k$
varies from $0$ to $2\pi$. While the dissipative case with $\gamma=0.5E_{k}$
is shown in Fig. 6 (b). The $x,y$ components wind clockwisely. The major
difference with the dissipationless limit is that the $z$-components decays
faster. It shows that the pseudospinor wraps the Bloch sphere for a shorter
period of time, as denoted by the black dotted line on the figure. Thus, the
$2\pi$ jump in Berry phase can be observed in the presence of dissipation at a
shorter time.

Figure 6: The pseudospin texture in the momentum-time space. (a) Without
dissipation, $\gamma=0$. (b) $\gamma=0.5E_{k}$. The color denotes the
magnitude of the $z$ components. The arrows represent the $x,y$ direction of
the pseudospinors. The black dotted line denotes the boundary of time for the
momentum-time torus.

##  V Conclusion

In this study, we performed quantum simulations of quench dynamics on IBM Q
devices. The dynamical winding number and the Berry phase flow were simulated
with quantum circuits and computed on ibmq_lima and ibmq_toronto. Despite the
unitary errors and randomness observed in NISQ processors, the quantum
simulation of quench dynamics shows robust signatures for identifying
topological and trivial dynamics. Moreover, the influence of open quantum
system on the topology is addressed in this paper. To study the dissipation,
we analytically solve the master equation in the Lindblad form. We found that
with dissipation, the dyanmical winding number and Berry phase flow are
robust. The analytic solution provides insight on the topological quench
dynamics in open quantum systems.

## Acknowledgments

We would like to thank Po-Yao Chang, Liang-Yan Hsu, Jhih-Shih You and Geng-
Ming Hu for valuable discussions. We acknowledge the IBM Q service via NTU-IBM
Q hub. This work is supported by the Ministry of Science and Technology (MOST)
in Taiwan with Grant No. 108-2112-M-004-002-MY2.

##  Appendix A Expectation values of pseudospins

Given the quench Hamiltonian, the unitary operator for time evolution is

$\displaystyle U(t)=\cos(E_{k}t)-i\sin(E_{k}t)\hat{h}\cdot\bm{\sigma},$ (12)

where $\hat{h}=\hat{h}_{x}\hat{x}+\hat{h}_{y}\hat{y}$ with
$\hat{h}_{x,y}=h_{x,y}/\sqrt{h_{x}^{2}+h_{y}^{2}}$. For simulation on IBM Q
simulators and machines, we utilize the U-gate Abraham and et al. (2019)

$\displaystyle
U(\theta,\phi,\lambda)=\begin{pmatrix}\cos(\theta/2)&-e^{i\lambda}\sin(\theta/2)\\\
e^{i\phi}\sin(\theta/2)&e^{i(\phi+\lambda)}\cos(\theta/2)\end{pmatrix}$ (13)

with $\theta=2E_{k}t,\phi={\rm Im}\log(h_{x}+ih_{y})-\pi/2,\lambda=-\phi$. It
can be shown that with this choice of parameter, the U-gate is exactly the
time evolution operator.

For an initial state

$\displaystyle|\psi_{k}(t=0)\rangle=\begin{pmatrix}a\\\ b\end{pmatrix}$ (14)

the time evolution due to the Hailtonian [(Eq. 1)] is given by

$\displaystyle|\psi_{k}(t)\rangle=\begin{pmatrix}a\cos(E_{k}t)-ib\sin(E_{k}t)(\hat{h}_{x}-i\hat{h}_{y})\\\
b\cos(E_{k}t)-ia\sin(E_{k}t)(\hat{h}_{x}+i\hat{h}_{y})\end{pmatrix}.$ (15)

For the initial states consiedered in the main text $(1\ \ 0)^{T}$ , the
expectation values of spinors are

$\displaystyle\langle\sigma_{x}(t)\rangle$ $\displaystyle=$
$\displaystyle\hat{h}_{y}\sin(2E_{k}t)$ (16)
$\displaystyle\langle\sigma_{y}(t)\rangle$ $\displaystyle=$
$\displaystyle-\hat{h}_{x}\sin(2E_{k}t)$ (17)
$\displaystyle\langle\sigma_{z}(t)\rangle$ $\displaystyle=$
$\displaystyle\cos(2E_{k}t).$ (18)

Thus, the azimuthal angle swept by the in-plane components is given by ${\rm
Im}\log[\langle\sigma_{x}\rangle+i\langle\sigma_{y}\rangle]=\log[(h_{y}-ih_{x})\sin(2E_{k}t)]$.

##  Appendix B The analytical expression for the Berry phase

The Berry phase can be calculated analytically by

$\displaystyle\gamma(t)$ $\displaystyle=$ $\displaystyle-{\rm
Im}\int_{0}^{2\pi}dk\langle\psi_{k}(t)|\partial_{k}\psi_{k}(t)\rangle.$ (19)

It is the continuum version of Eq. (7) in the main text. Using Eq. (15) with
$(a\ \ b)=(1\ \ 0)$ and $E_{k}=1$, one obtains

$\displaystyle\gamma(t)$ $\displaystyle=$ $\displaystyle-{\rm
Im}\int_{0}^{2\pi}dk\langle\psi_{k}(t)|\frac{d\phi}{dk}\partial_{\phi}\psi_{k}(t)\rangle$
$\displaystyle=$ $\displaystyle-{\rm
Im}\int_{0}^{2\pi}dk\frac{d\phi}{dk}\sin^{2}t$ $\displaystyle=$
$\displaystyle-2\pi\mathit{w}\sin^{2}t,$ (21)

where $\mathit{w}$ is the winding number of the quench Hamiltonian. Thus, the
dynamical Chern number is
$C_{dyn}=-\mathit{w}\sin^{2}t\Bigr{|}^{\pi/2}_{0}=-\mathit{w}$.

##  Appendix C Solving the master equation

Consider the qubit under the influence of the electromagnetic field, such
process can be described by the quantum master equation Carollo  _et al._
(2003); Heinz-Peter and Petruccione (2007)

$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle-i[\mathcal{H}_{0}(k),\rho]$ $\displaystyle+$
$\displaystyle\gamma_{0}(N+1)(\sigma_{-}\rho\sigma_{+}-\frac{1}{2}\left\\{\sigma_{+}\sigma_{-},\rho\right\\})$
$\displaystyle+$
$\displaystyle\gamma_{0}N(\sigma_{+}\rho\sigma_{-}-\frac{1}{2}\left\\{\sigma_{-}\sigma_{+},\rho\right\\})$

where the symbols are explained below Eq. (8) in the main text.

The density matrix can be expanded as $\rho=\frac{1}{2}(I+\vec{a}\cdot\sigma)$
and the master equation [Eq. (8)] can be written as the following form Zhang
_et al._ (2021)

$\displaystyle\frac{d}{dt}\begin{pmatrix}a_{x}\\\ a_{y}\\\
a_{z}\end{pmatrix}=\mathcal{L}(k)\begin{pmatrix}a_{x}\\\ a_{y}\\\
a_{z}\end{pmatrix}+\begin{pmatrix}0\\\ 0\\\ -\gamma_{0}\end{pmatrix},$ (22)

where

$\displaystyle\mathcal{L}(k)=\begin{pmatrix}-\gamma/2&0&2h_{y}(k)\\\
0&-\gamma/2&-2h_{x}(k)\\\ -2h_{y}(k)&2h_{x}(k)&-\gamma\end{pmatrix}.$ (23)

To solve Eq. (23), we follow Ref. Heinz-Peter and Petruccione (2007); Zhang
_et al._ (2021); Noh and Jhe (2010). First, we find the stationary solution
where the time derivative of the density matrix is zero. The stationary
solution is

$\displaystyle a_{x}^{0}=\frac{-4h_{y}\gamma_{0}}{4E_{k}^{2}+\gamma^{2}}$ (24)
$\displaystyle a_{y}^{0}=\frac{4h_{x}\gamma_{0}}{4E_{k}^{2}+\gamma^{2}}$ (25)
$\displaystyle a_{z}^{0}=\frac{-\gamma_{0}\gamma}{4E_{k}^{2}+\gamma^{2}}.$
(26)

Second, we define $\vec{a}^{c}=\vec{a}-\vec{a}^{0}$ such that $\vec{a}^{c}$
satisfies the homogeneous differential equation

$\displaystyle\frac{d\vec{a}^{c}}{dt}=\mathcal{L}(k)\vec{a}^{c}$ (27)

The left and right eigenvalues of $\mathcal{L}(k)$ are
$\lambda_{0}=\frac{-\gamma}{2}$,
$\lambda_{1,2}=\frac{-3\gamma}{4}\mp\frac{i\omega}{4}$, where
$\omega=\sqrt{64E_{k}^{2}-\gamma^{2}}$. Thus, the solution to $\vec{a}^{c}$ is
$\vec{a}^{c}=\sum_{\ell}S_{\ell}|R\ell\rangle e^{\lambda_{\ell}t}$ with
$S_{\ell}=\langle L\ell|{a}^{c}(t=0)\rangle$, where $\langle L\ell|$ and
$|R\ell\rangle$ are the left and right eigenvector of the $\ell$th eigenvalue,
respectively, $|{a}^{c}(t=0)\rangle$ denotes the initial column vector
$\vec{a}^{c}$. The left and right eigenvectors satisfy $\langle
L\ell|R\ell^{\prime}\rangle=\delta_{\ell,\ell^{\prime}}$. For the quench
protocol considered in the main text, the initial state is given by
$\vec{a}^{c}(t=0)=(0,0,1)^{T}-\vec{a}^{0}$. At low temperature, one can
approximate $\gamma_{0}=\gamma$ which gives a simple form of the solution

$\displaystyle a_{x}^{c}(t)$ $\displaystyle=$ $\displaystyle
h_{y}F(\gamma,k,t)$ (28) $\displaystyle a_{y}^{c}(t)$ $\displaystyle=$
$\displaystyle-h_{x}F(\gamma,k,t)$ (29) $\displaystyle a_{z}^{c}(t)$
$\displaystyle=$ $\displaystyle e^{-\frac{3\gamma
t}{4}}\frac{2{\omega}\left(2E_{k}^{2}+\gamma^{2}\right)\cos\left(\frac{{\omega}t}{4}\right)-2\gamma\left(\gamma^{2}-14E_{k}^{2}\right)\sin\left(\frac{{\omega}t}{4}\right)}{{\omega}\left(4E_{k}^{2}+\gamma^{2}\right)},$
(30) $\displaystyle\text{where }F(\gamma,k,t)$ $\displaystyle=$ $\displaystyle
4e^{-\frac{3\gamma t}{4}}\frac{(8E_{k}^{2}+5\gamma^{2})\sin(\frac{\omega
t}{4})+\omega\gamma\cos(\frac{\omega t}{4})}{\omega(4E_{k}^{2}+\gamma^{2})}$
(31)

Thus, the pseudospinors are given by
$\langle\sigma_{i}\rangle(t)=Tr\frac{1}{2}\left[\rho\sigma_{i}\right]=a_{i}^{c}(t)+a_{i}^{0}$.
It can be shown that in the limit $\gamma=0$,
$F(\gamma,k,t)=\frac{\sin(2E_{k}t)}{E_{k}}$ and $a_{z}(t)=\cos(2E_{k}t)$,
which agree with the results in Appendix A. Moreover, at $t=0$,
$F(\gamma,k,t)=\frac{4\gamma}{4E_{k}^{2}+\gamma^{2}}$ and the solution
recovers the initial condition $\vec{a}(t=0)=(0,0,1)^{T}$

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# Noise crosscorrelations can induce instabilities in coupled driven models

Sudip Mukherjee<EMAIL_ADDRESS><EMAIL_ADDRESS>Barasat Government
College, 10, KNC Road, Gupta Colony, Barasat, Kolkata 700124, West Bengal,
India

###### Abstract

We study the effects of noise cross-correlations on the steady states of
driven, nonequilibrium systems, which are described by two stochastically
driven dynamical variables, in one dimension. We use a well-known
stochastically driven coupled model with two dynamical variables, where one of
the variables is autonomous being independent of the other, whereas the second
one depends explicitly on the former. Introducing cross-correlations of the
two noises in the two dynamical equations, we show that depending upon the
details of the nonlinear coupling between the dynamical fields, such cross-
correlations can induce instabilities in the models, that are otherwise stable
in the absence of any cross-correlations. We argue that this is reminiscent of
the roughening transition found in the Kardar-Parisi-Zhang equation in
dimensions greater than two. Phenomenological implications of our results are
discussed.

##  I Introduction

Time-dependent statistical descriptions of condensed matter systems are often
made in terms of continuum, Langevin equations of the relevant dynamical
variables driven by noises chaikin . The noises represent inherent microscopic
stochasticity of the dynamics. In equilibrium systems, such stochasticity
arises from thermal fluctuations. Conditions of thermal equilibrium, known as
the fluctuation-dissipation theorem (FDT) chaikin , ensures that the damping
in the system is proportional to the noise variance, which is assumed to be
Gaussian-distributed with a zero mean. The proportionality constant in fact
gives the temperature. In nonequilibrium systems, there is no FDT, and hence,
the damping and the noise variance are independent of each other. As a result,
nonequilibrium steady states (NESS) are far more diverse and complex than
their equilibrium counterparts.

Physical descriptions of many natural driven systems involve coupled dynamics
of several degrees of freedom. Prominent examples include driven symmetric
mixture of a miscible binary fluid binfluid  and magnetohydrodynamics mhd .
Stochastically driven binary fluid equations of the velocity and concentration
gradient binfluid1  and magnetohydrodynamics (MHD) equations of the velocity
and magnetic fields abmhd  have been used to study turbulence in these
systems. In equilibrium systems, conditions of thermal equilibrium, e.g., in
the form an FDT, ensures that the noise statistics have no role to play in
determining the thermodynamic properties of the system. For instance,
relaxational dynamics both without and with a conservation law for the order
parameter halpin  refer to the same equal-time, thermal equilibrium
properties. In contrast to equilibrium systems, in the absence of any FDT,
varying the noise statistics can result into distinctly different NESS in
driven systems. For instance, the Kardar-Parisi-Zhang (KPZ) equation driven by
white noises kpz ; stanley , and its conserved counterpart (the CKPZ equation)
driven by conserved noises ckpz  have very different universal properties.

Introduction of noise cross-correlations in a stochastically driven coupled
model necessarily changes the noise distribution, Whether or not this can lead
to a new NESS, is a question of basic importance in nonequilibrium statistical
mechanics. In fact, there are examples where non-zero cross-correlations of
the two noises in the two dynamical equations are found to affect the scaling
properties of the NESS. Presence of such noise cross-correlations in driven
systems cannot be ruled out on the basis of any symmetry arguments or physical
principles. Simpler reduced models have been proposed and further studied to
explore the role of noise cross-correlations. For instance, noise cross-
correlations in a nonconserved relaxational model for the complex scalar field
turns out to be generally a relevant perturbation on the equilibrium states of
the model near a critical point niladri-noise . Subsequently, by using a
coupled Burgers model originally proposed in abfrey1 , Refs. abfrey1 ; abfrey2
have been shown that noise cross-correlations can lead to continuously varying
scaling exponents in the NESS. Nonetheless, general understanding of the
effects of noise cross-correlations on the NESS of driven models still remains
largely at an early stage.

In this work, we revisit the issue of the effects of noise crosscorrelations
on the NESS of coupled driven models. Since we are interested in studying a
question of principle, it suffices to work with simple models where explicit
calculations can be performed relatively easily, but still nontrivial results
can be obtained. To that end, we use a one-dimensional (1D) model ertas ;
barabasi ; jkb-mode-coup , where one of the fields $v(x,t)$ in autonomous, and
satisfies the well-known 1D Burgers equation fns . The second dynamical field
$b$ is dynamically influenced by $v$, and follows an equation similar to the
well-known passive scalar equation with a compressible velocity field. We
investigated the universal spatio-temporal scaling properties. Using a
specific structure of the noise crosscorrelations, we show that depending upon
the specific model at hand, either it introduces instabilities to the NESS
characterised by well-known scaling exponents, or is irrelevant (in the
renormalisation group or RG sense), leaving the long wavelength scaling
properties unaffected. In the former case, the ultimate steady state that the
instability by the noise cross-correlations lead to cannot be determined from
the low order perturbation theory employed here. The remainder of this article
is organised as follows. In Section II, we have introduced the models we have
used, along with the form of the noise cross-correlations. Then in Section
IV.2, we have discussed the details of the dynamic RG analysis on the model.
In Section V, we summarise and conclude. Some of the intermediate technical
details including the one-loop Feynman diagrams are given in Appendix for
interested readers.

##  II Model

Noise crosscorrelations can generically exist in any multi-variable system
described by stochastically driven dynamical equations for the dynamical
variables. Since we are trying to answer questions of principles, we use a
simple, purpose-built 1D model that suffices for our purposes. The model
consists of two dynamical variables $v(x,t)$ and $b(x,t)$. Out of these two,
we assume $v$ to be autonomous, i.e., independent of $b$, and follows the
well-known 1D Burgers equation fns . This is given by

$\frac{\partial v}{\partial
t}=\nu\partial_{xx}v+\frac{\lambda_{1}}{2}\partial_{x}v^{2}+f_{v}.$ (II.1)

Here, $\nu>0$ is a diffusivity, and $\lambda_{1}$ a non-linear coupling
constant, and can be of any sign. Further, $f_{v}$ is a conserved noises.
Hereafter, we refer to $v$ as the “Burgers velocity field” fns .

The second field $b$ is assumed to be advected by the Burgers velocity $v$
passively, i.e., it has no effect on the dynamics of $v$. The dynamics of $b$
follows jkb-mode-coup

$\frac{\partial b}{\partial
t}=\mu\partial_{xx}b+\lambda_{2}\partial_{x}(vb)+f_{b}.$ (II.2)

Here, $\mu>0$ is a diffusivity, and $\lambda_{2}$ a non-linear coupling
constant, and can be of any sign. Further, $f_{b}$ is a conserved noise.
Following the nomenclature of Ref. abfrey1 , we call $b$ as the “Burgers
magnetic field”. Equations (II.1) and (II.2) in fact can be obtained from the
coupled Burgers equations without the feedback term in the equation of $v$;
see Refs. diamond ; abfrey1 . These can also be written in terms of
nonconserved height or displacement fields; see, e.g., Refs. ertas ; lahiri .

The noises $f_{v}$ and $f_{b}$ have zero mean, and are Gaussian-distributed.
Their joint probability distribution is characterised by the three variances
(written in the Fourier space and functions of frequency $\omega$ and
wavevector $k$)

$\displaystyle\langle f_{v}(k,\omega)f_{v}(-k,-\omega)\rangle$
$\displaystyle=$ $\displaystyle 2D_{v}k^{2},$ (II.3) $\displaystyle\langle
f_{b}(k,\omega)f_{b}(-k,-\omega)\rangle$ $\displaystyle=$ $\displaystyle
2D_{b}k^{2},$ (II.4) $\displaystyle\langle
f_{v}(k,\omega)f_{b}(-k,-\omega)\rangle$ $\displaystyle=$ $\displaystyle
2iD_{\times}k|k|,$ (II.5)

where $D_{v},\,D_{b}>0$ necessarily, but the sign of $D_{\times}$ is
arbitrary. Reality of the noises means the noise variance matrix constructed
from (II.3)-(II.5) must have real, non-negative eigenvalues, which in turn
implies $D_{v}\,D_{b}\geq D_{\times}^{2}$. Equation (II.5) gives the noise
cross-correlation here, which is purely imaginary and odd in $k$, i.e.,
$D_{\times}(k)=-D_{\times}(-k)$. The structure of (II.5) is dictated by the
symmetry properties of (II.1) and (II.2). In line with abfrey1 , we assume $v$
to be a pseudo-scalar field and $b$ to be a scalar field (i.e., a vector and a
pseudo-vector at dimensions $d>1$). This means Eqs. (II.1) and (II.2) are
invariant under $x\rightarrow-x,\,v\rightarrow-v,\,b\rightarrow b$. These
symmetries hold true even if there is a “mean $b$”, i.e., $\langle
b\rangle\neq 0$ abfrey1 . On the other hand, if $\langle b\rangle=0$, the
model has a higher symmetry: Equations (II.1) and (II.2) are also invariant
under $x\rightarrow-x,\,v\rightarrow-v,\,b\rightarrow-b$. While our subsequent
calculations specialise for $\langle b\rangle=0$, we continue to impose
invariance under (II.1) and (II.2) are invariant under
$x\rightarrow-x,\,v\rightarrow-v,\,b\rightarrow b$ only. This symmetry implies
the cross-correlation function $\langle v(x,t)b(0,0)\rangle$ is an odd
function of $x$. This in turn means, as in Ref. abfrey1 , $\langle
v(k,t)b(-k,0)\rangle$ is purely imaginary and odd in $k$. Since $\langle
v(k,t)b(-k,0)\rangle$ proportional to the noise cross-correlations, and the
noises in (II.1) and (II.2) must follow the symmetries of the corresponding
dynamical variables, (II.5) follows directly foot1 .

##  III Galilean invariance

Model equations (II.1) and (II.2) are invariant under a pseudo-Galilean
transformation $x\rightarrow x+v_{0}t,\,v\rightarrow v+v_{0},\,t\rightarrow
t,\,\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial
t}+\lambda_{1}v_{0}\partial_{x}$, when $\lambda_{1}=\lambda_{2}$. When
$\lambda_{1}\neq\lambda_{2}$, there is no Galilean invariance. However, as our
calculations below show (see also Refs. ertas ; abfrey1 ; abfrey2 ; abfrey3
for related discussions) even if $\lambda_{1}\neq\lambda_{2}$, i.e., even when
they are unequal microscopically, Galilean invariance is recovered and appears
as an emergent symmetry in the long wavelength limit, i.e.,
$\lambda_{1}=\lambda_{2}$ in the renormalised theory, so long as
$\lambda_{1}\lambda_{2}>0$ holds. In contrast, if $\lambda_{1}\lambda_{2}<0$,
Galilean invariance is not restored even in the long wavelength limit. Hence,
Galilean invariance is genuinely broken even in the renormalised theory in
this case. These are the two distinct cases, which we discuss separately
below. We further consider a third case, in which we set $\lambda_{1}=0$. Thus
in this case, $v$ satisfies the linear 1D diffusion equation. The
corresponding equation in terms of a height field is the well-known Edward-
Wilkinson equation stanley , forced by a nonconserved noise.

##  IV Scaling

We are interested in calculating the scaling exponents which characterise the
time-dependent correlation functions of $v$ and $b$:

$\displaystyle C_{v}(r,t)\equiv\langle v({r},t)v(0,0)\rangle$ $\displaystyle=$
$\displaystyle|r|^{2\chi_{v}}f_{v}(|r|^{z_{v}}/t),$ (IV.1) $\displaystyle
C_{b}(r,t)\equiv\langle b({r},t)b(0,0)\rangle$ $\displaystyle=$
$\displaystyle|r|^{2\chi_{b}}f_{b}(|r|^{z_{b}}/t).$ (IV.2)

or their Fourier transformed versions

$\displaystyle C_{v}(k,\omega)\equiv\langle|v({k},\omega)|^{2}\rangle$
$\displaystyle=$ $\displaystyle
k^{2\tilde{\chi}_{v}}\tilde{f}_{v}(k^{z_{v}}/\omega),$ (IV.3) $\displaystyle
C_{b}(k,\omega)\equiv\langle|b({k},\omega)|^{2}\rangle$ $\displaystyle=$
$\displaystyle k^{2\tilde{\chi}_{b}}\tilde{f}_{b}(k^{z_{b}}/\omega),$ (IV.4)

in the long wavelength limit. Here $\chi_{v}$ and $z_{v}$ are the roughness
and dynamic exponents of $v(x,t)$; $\chi_{b}$ and $z_{b}$ are respectively the
corresponding roughness and dynamic exponents of $b(x,t)$; $\tilde{\chi}_{v}$
($\tilde{\chi}_{b}$) can be connected to $\chi_{v}$ ($\tilde{\chi}_{b}$) by
Fourier transform, giving

$\tilde{\chi}_{v}=1+\chi_{v}+z_{v},\;\;\tilde{\chi}_{b}=1+\chi_{b}+z_{b}.$
(IV.5)

Further, $f_{v,b}(|r|^{z}/t)$ and $\tilde{f}_{v,b}(k^{z}/\omega)$ are
dimensionless scaling functions of their respective arguments. Notice that we
have allowed for two different dynamic exponents. If $z_{v}=z_{b}$, then one
gets strong dynamic scaling, else, if $z_{v}\neq z_{b}$, weak dynamic scaling
ensues dibyendu .

###  IV.1 Linear theory

The linear limit of the model equations is obtained by setting all nonlinear
term to zero, i.e., by setting $\lambda_{1}=0=\lambda_{2}$. In this limit, all
the two point correlations can be calculated exactly. We have

$\displaystyle\langle|v(k,\omega)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{2D_{v}k^{2}}{\omega^{2}+\nu^{2}k^{4}},$ (IV.6)
$\displaystyle\langle|b(k,\omega)|^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{2D_{b}k^{2}}{\omega^{2}+\mu^{2}k^{4}},$ (IV.7)
$\displaystyle\langle v(k,\omega)b(-k,-\omega)\rangle$ $\displaystyle=$
$\displaystyle\frac{2ik|k|D_{\times}}{(-i\omega+\nu k^{2})(i\omega+\mu
k^{2})}.$ (IV.8)

These give the exact exponent values $\chi_{v}=\chi_{b}=-1/2$, which may be
obtained by inverse Fourier transforming the above correlators, and
$z_{v}=z_{b}=2$, corresponding to strong dynamic scaling in the linear theory.
If noise crosscorrelations vanish, the linearised equations actually admit an
FDT. In fact, if $D_{\times}=0$, $v$ and $b$ fully decouple at the linear
level, and by using FDT one can identify two “temperatures” $T_{v}=D_{v}/\nu$
and $T_{b}=D_{b}/\mu$ in the linear theory, which are in general unequal. But
a non-zero noise cross-correlation breaks FDT even at the linear level, making
it impossible to identify any temperature-like quantity. Lastly, it is
straightforward to show by using that (IV.6)-(IV.8) that the ratios of the
equal-time correlators $\langle v(x,0)v(0,0\rangle,\,\langle
b(x,0)b(0,0)\rangle,\,\langle v(x,0)b(x,0)\rangle$ are all just numbers.

###  IV.2 Nonlinear effects

The presence of the nonlinear terms no longer allows enumeration of the exact
scaling exponents for (II.1) and (II.2), unlike in the linear theory. Thus
perturbative treatments are necessary. Naïve perturbation theory produces
diverging corrections to the model parameters. These divergences may be
systematically handled within the framework of dynamic RG halpin .

While the dynamic RG procedure is already well-documented in the literature
halpin , we give below a brief outline of this method for the convenience of
the readers. It is useful to first cast the model equations (II.1) and (II.2)
into a dynamic generating functional by introducing dynamic conjugate fields
$\tilde{v}(x,t)$ and $\tilde{b}(x,t)$; see Ref. janssen , see also Appendix A
and Appendix B for some intermediate details. The dynamic generating
functional is then averaged over the Gaussian distribution of the noises
$f_{v}$ and $f_{b}$ with variances (II.3), (II.4) and (II.5). The momentum
shell dynamic RG procedure consists of integrating over the short wavelength
Fourier modes of $v(x,t),\,b(x,t),\,\tilde{v}(x,t)$ and $\tilde{b}(x,t)$ in
the generating functional. This is then followed by rescaling of lengths and
time. In particular, we follow the standard approach of initially restricting
wavevectors to lie in a Brillouin zone: $|q|<\Lambda$, where $\Lambda$ is an
ultra-violet cutoff of order the inverse of the lattice spacing $a$, although
its precise value is unimportant so far as the scaling in the long wavelength
limit is considered. The fields $v(x,t),\,b(x,t)$ and their dynamic conjugates
$\tilde{v}(x,t),\,\tilde{b}(x,t)$ are then split into two parts, a high and
low wave vector parts
$v(x,t)=v^{>}(x,t)+v^{<}(x,t),\,b(x,t)=b^{>}(x,t)+b^{<}(x,t)$ and
$\tilde{v}(x,t)=\tilde{v}^{>}(x,t)+\tilde{v}^{<}(x,t),\,\tilde{b}(x,t)=\tilde{b}^{>}(x,t)+\tilde{b}^{<}(x,t)$,
where $(v,b)^{>}(x,t)$ and $(\tilde{v},\tilde{b})^{>}(x,t)$ are non-zero in
the high wavevector range $\Lambda/b<k<\Lambda,\,b>1$, whereas
$(v,b)^{<}(x,t)$ and $(\tilde{v},\,\tilde{b})^{<}(x,t)$ are non-zero in the
low wavevector range $k<\Lambda/b$. Next, $v^{>}(x,t),\,b^{>}(x,t)$ and
$\tilde{v}^{>}(x,t),\,\tilde{b}^{>}(x,t)$ are to be integrated out in the
dynamic generating functional. Of course, this integration cannot be done
exactly, but is done perturbatively in the anharmonic couplings in Appendix A.
This perturbation theory is usually represented by Feynman diagrams, with the
order of perturbation theory given by the number of loops in the diagrams that
we calculate; see Appendix LABEL:resc. Next to this perturbative step, we
rescale length by $x=x^{\prime}\exp(l)$, in order to restore the UV cutoff
back to $\Lambda$. We further rescale time by $t=t^{\prime}\exp(l\,z)$,
whether or not $z$ is the actual dynamic exponent will be found as we go
along. This is then followed by rescaling of $v^{<}(x,t),\,b^{<}(x,t)$ and
$\tilde{v}^{<}(x,t),\,\tilde{b}^{<}(x,t)$, the long wave length parts of
$v(x,t),\,b(x,t)$ and $\tilde{v}(x,t),\,\tilde{b}(x,t)$; see Appendix
LABEL:resc. We discuss (i) $\lambda_{1}\lambda_{2}>0$ (Case I), (ii)
$\lambda_{1}\lambda_{2}<0$ (Case II), and (iii) $\lambda_{1}=0\,\lambda_{2}>0$
(Case III) separately below. Before we discuss the RG results in details, we
note that the dynamics of $v$, independent of $b$, follows the well-known 1D
Burgers equation fns , for the scaling exponents $\chi_{u}$ and $z_{u}$ are
known exactly, thanks to the Galilean invariance and FDT fns ; stanley . This
gives $\chi_{v}=-1/2,\,z_{v}=3/2$, which are exact. The corresponding scaling
exponents of $b$, however, cannot be obtained exactly, necessitating
perturbative approaches. The relevant one-loop Feynman diagrams for the model
parameters and the noise strengths are given in Appendix B.1. Independent of
the sign of $\lambda_{1},\lambda_{2}$, the critical dimension of the model is
two. Since we are interested in 1D, less than the critical dimension, we use a
fixed dimension RG scheme, same as that used in the RG calculations on the 1D
KPZ equation stanley .

###  IV.3 Case I: Renormalisation group analysis

The one-loop Feynman diagrams upon evaluation produces the discrete recursion
relations. These are given in Appendix B.2. This procedure is followed by
rescaling of space, time and the fields together with
${\mathcal{L}}=e^{dl}\approx 1+dl$ for small $dl$ (here $\mathcal{L}$ is a
running scale factor, not to be confused with the system size, which we
formally take to be infinity), which ultimately give the following RG
recursion relations:

$\displaystyle\frac{d\nu}{dl}$ $\displaystyle=$
$\displaystyle\nu\left[z-2+\frac{g}{4}\right],$ (IV.9)
$\displaystyle\frac{d\mu}{dl}$ $\displaystyle=$
$\displaystyle\mu\left[z-2+\frac{g\psi^{2}}{2(1+P)P}+\frac{g(1-P)\psi^{2}}{(1+P)^{2}P}\right],$
(IV.10) $\displaystyle\frac{dD_{v}}{dl}$ $\displaystyle=$ $\displaystyle
D_{v}\left[z-2\chi_{v}-3+\frac{g}{4}\right],$ (IV.11)
$\displaystyle\frac{dD_{b}}{dl}$ $\displaystyle=$ $\displaystyle
D_{b}\left[z-2\chi_{b}-3+\frac{g\psi^{2}}{P(1+P)}-\frac{4\Gamma\alpha
g\psi^{2}}{(1+P)^{3}}\right],$ $\displaystyle\frac{d\lambda_{1}}{dl}$
$\displaystyle=$ $\displaystyle\lambda_{1}\left[\chi_{v}+z-1\right],$ (IV.13)
$\displaystyle\frac{d\lambda_{2}}{dl}$ $\displaystyle=$
$\displaystyle\lambda_{2}\bigg{[}\chi_{v}+z-1-\frac{\psi^{2}g}{(1+P)^{2}}+\frac{\psi
g(3+P)}{2(1+P)^{2}}$ (IV.14) $\displaystyle-$
$\displaystyle\frac{g\psi}{2(1+P)}\bigg{]},$
$\displaystyle\frac{dD_{\times}}{dl}$ $\displaystyle=$ $\displaystyle
D_{\times}\left[z-\chi_{v}-\chi_{b}-3\right].$ (IV.15)

Here, $g\equiv\frac{\lambda_{1}^{2}D_{v}}{\nu^{3}}$ is the dimensionless
coupling constant, $P\equiv\frac{\mu}{\nu}$ is the dimensionless magnetic
Prandtl number, $\Gamma\equiv D_{\times}^{2}/D_{v}^{2},\,\alpha\equiv
D_{v}/D_{b}$. All these are non-negative by construction. For reasons of
notational convenience, we define $\Phi\equiv\alpha\Gamma$. In addition,
$\psi\equiv\lambda_{2}/\lambda_{1}$, which in the present case in positive.
Notice that there are no relevant one-loop corrections to $D_{\times}$. This
is because the vertices in Eqs. (II.1) and (II.2) are ${\cal O}(k)$, and hence
cannot generate any relevant corrections to the noise crosscorrelation (II.5),
which scales as $k|k|$ abfrey2 . This reason for the lack of renormalisation
of $D_{\times}$ actually holds to all orders in the perturbation theory,
making $D_{\times}$ unrenormalised to any order in the perturbation theory.

Flow equations (IV.9)-(IV.15) can be used to calculate the flow equations for
$g,P,\Gamma,\alpha$. We find

$\displaystyle\frac{dg}{dl}=g\left[1-\frac{g}{2}\right],$ (IV.16)
$\displaystyle\frac{dP}{dl}=Pg\left[\frac{\psi^{2}}{2P(1+P)}+\frac{\psi^{2}(1-P)}{P(1+P)^{2}}-\frac{1}{4}\right],$
(IV.17) $\displaystyle\frac{d\psi}{dl}=\psi
g\bigg{[}-\frac{\psi^{2}}{(1+P)^{2}}+\frac{\psi}{(1+P)^{2}}\bigg{]},$ (IV.18)
$\displaystyle\frac{d\Phi}{dl}=\Phi
g\left[-\frac{1}{4}-\frac{\psi^{2}}{1+P}+\frac{4\Phi\psi^{2}}{(1+P)^{3}}\right].$
(IV.19)

At the RG fixed point, $dg/dl=0=dP/dl=d\psi/dl=d\Phi/dl$. This gives
$g^{*}=2,\,\psi^{*}=1$ and $P^{*}=1$ as the stable RG fixed point; here and
below a superscript ∗ refers to the RG fixed point value of any quantity.
Notice that $\psi^{*}=1$ is obtained from (IV.18) for any $P$. Notice that
$\psi=0=g$ is also a fixed point of (IV.16) and (IV.18); it is however
globally unstable. Unsurprisingly, $g^{*}$ is same as that for the 1D Burgers
equation fns ; stanley , since $g$ depends only on the parameters of (II.1),
which is autonomous. This in turn gives $\chi_{v}=-1/2,\,z_{v}=3/2$, which are
exact, due to the Galilean invariance and FDT of the 1D Burgers equation fns ;
stanley ; frey-2-loop . Further, $P^{*}=1$ means at the RG fixed point
$\nu^{*}=\mu^{*}$, i.e., the two renormalised diffusivities are equal, even if
they were unequal microscopically. This further implies that the fields $v$
and $b$ have the same dynamic exponent: $z_{v}=z_{b}=z$. This is an example of
strong dynamic scaling dibyendu . Furthermore, at the RG fixed point
$\psi^{*}=1$ implies that $\lambda_{1}^{*}=\lambda_{2}^{*}$ at the RG fixed
point, such that the fixed point is Galilean invariant, even if it were not so
microscopically (but with $\lambda_{1}\lambda_{2}>0$ microscopically). Thus,
the Galilean invariance is an emergent symmetry, even though it is absent
microscopically, a statement that holds even in the presence of noise
crosscorrelations just as it does in its absence ertas .

The RG flow of $\Phi$ requires careful attention. We see that the fixed point
condition $d\Phi/dl=0$ produces the following fixed points, at each of which
the spatial scaling of $b$ given by $\chi_{b}$ is analysed:

(i) $\Phi^{*}=0$, a stable fixed point. At this RG fixed point, the noise
crosscorrelations effectively vanish, and the long wavelength scaling
properties of the system are identical to that without it: $\chi_{b}=-1/4$.

(ii) There is a second fixed point given by the condition

$-\frac{1}{4}-\frac{\psi^{2}}{1+P}+\frac{4\Phi\psi^{2}}{(1+P)^{3}}=0,$ (IV.20)

giving $\Phi^{*}=3/2\equiv\Phi_{c1}$ obtained by using $\psi^{*}=1=P^{*}$,
which is linearly unstable. This implies if $\Phi(\ell=0)<\Phi_{c1}$, the
system flows to the fixed point $\Phi^{*}=0$, meaning a steady state that is
statistically identical to a state having no crosscorrelations at the
microscopic level in the long wavelength limit, and hence the scaling
exponents have the values identical to their values without crosscorrelations.
At this unstable fixed point $\chi_{b}=-7/8$, different from its value when
$\Phi^{*}=0$, i.e., without crosscorrelations. However,
$\Phi(\ell=0)>\Phi_{c1}$, then the system reaches a steady state not known
from perturbation theories. Thus in this case, noise crosscorrelations remain
relevant in the RG sense.

(iii) Given that $\Phi^{*}=\Phi_{c1}$ is a linearly unstable fixed point, such
that if the “initial value” $\Phi(l=0)>\Phi_{c1}$, $\Phi(l)$ flows away as $l$
grows, there should be another “strong coupling” fixed point presumably
stable, but cannot be accessed in this one-loop perturbation theory. This
indicates an instability of the zero crosscorrelation state, induced by noise
crosscorrelations. It is instructive to find out how $\Phi(l)$ diverges for
sufficiently large $l$. As $\Phi(l)$ grows, (IV.19) reduces to

$\frac{d\Phi}{dl}\approx\frac{4g\Phi^{2}\psi^{2}}{(1+P)^{3}}={\Phi^{2}}.$
(IV.21)

Solving this, we find

$\Phi(l)=\frac{\Phi(\ell=0)}{\ell\Phi(l=0)-1}.$ (IV.22)

This shows that $\Phi(\ell)$ diverges as $\ell\rightarrow l_{c}\equiv
1/\Phi(\ell=0)$ from below. In other words, $\phi(\ell)$ diverges as the
system size reaches a non-universal threshold $a_{0}\exp(l_{c})$, where
$a_{0}$ is a small-scale cutoff. This happens so long as the “initial” or
microscopic value $\Phi(l=0)>\Phi_{c1}$. What is the nature of the steady
state in this case? We note from (LABEL:flowDb) that as $\Phi(\ell)$ grows,
$\chi_{b}$ decreases continuously. This is of course unphysical. We cannot in
fact follow the flow of $\Phi(\ell)$ all the way to $\ell\rightarrow l_{c}$,
as the perturbation theory breaks down long before that.

Combining the fixed points of $\psi$ and $\Phi$, we thus find the following
fixed points in the $\psi-\phi$ plane for $g=2$:

(i) The origin (0,0). This has quite an interesting stability property - it is
marginally unstable along the $\psi$-direction, but stable along the
$\Phi$-direction! Naturally, the flow along the $\Phi$-axis is towards the
origin.

(ii) A globally stable fixed point $(1,0)$. At this fixed point, the long
wavelength scaling properties are identical to those with zero noise
crosscorrelations.

(iii) A fixed point $(1,3/2)$ that is stable along the $\psi$-direction, but
unstable along the $\Phi$-direction.

(iv) A putative globally stable “strong coupling” fixed point, which cannot be
captured in our perturbation theory.

The presence of the several fixed points suggests the existence of one or more
separatrix, which separates different regions of the phase space having
distinct behaviours. Since the origin is stable along the $\Phi$-axis, but
unstable along the $\psi$-axis, there should be a separatrix originating from
the origin delineating these behaviours. Linearising (IV.18) and (IV.19) about
(0,0), and defining $\Gamma_{1}=\Phi/\psi$ as the slope of the separatrix near
the origin, we set $d\Gamma_{1}/dl=0$ for the separatrix. This gives
$\Gamma_{1}=0$ near the origin, i.e., the $\psi$-axis. In the same way, we
linearise about the unstable fixed point $(1,3/2)$, and define
$\tilde{\Gamma}_{1}=\delta\Phi/\delta\psi$, where $\delta\Phi$ and
$\delta\psi$ are (small) fluctuations of $\Phi$ and $\psi$ from their fixed
point values. For a separatrix that passes through $(1,3/2)$, we set
$d\tilde{\Gamma}/dl=0$ giving

$\delta\Phi=-\frac{3}{4}\delta\psi,$ (IV.23)

giving the separatrix near the fixed point (1,3/2).

A schematic RG flow diagram of the model in the $\psi-\Phi$ plane is shown in
Fig. 1.

Figure 1: Schematic RG flow diagram in the $\psi-\Phi$-plane for Case I.
Arrows show the RG flow directions. The filled square represents the unstable
fixed point, and the filled circle on the $\psi$-axis is a stable fixed point.
The red broken line is the separatrix (IV.23).

At this point, it is instructive to draw a formal analogy of this instability
with the roughening transition in the KPZ equation at $d>2$, a transition
between the smooth phase and the perturbatively inaccessible rough phase. This
can be accessed, e.g., by increasing the noise strength. Similarly in the
present problem, by increasing $\Phi$ one can observe a transition from a
perturbatively accessible phase having no effects of the noise
crosscorrelations to a perturbatively inaccessible phase, where noise
crosscorrelations should be relevant in the RG sense, via an unstable fixed
point that is reminiscent of a critical point.

While one-loop perturbative RG cannot predict the nature of the steady states
near the putative strong coupling fixed point, we note that in the special
limit with $\lambda_{1}=\lambda_{2}$ and $\nu=\mu$ “initially” (i.e.,
microscopically), these conditions remain satisfied under mode eliminations.
It is therefore reasonable to expect that even at the strong coupling fixed
point, these should hold in the long wavelength limit at least when these are
satisfied microscopically. This in turn means $z_{v}=z_{b}=3/2$ (strong
dynamic scaling) at the strong coupling fixed point. Roughness exponent
$\chi_{b}$ however cannot be estimated in this way. Nonetheless, given that
$\chi_{b}\neq\chi_{v}$ at the unstable fixed point $\Phi^{*}=\Phi_{c1}$, we
are tempted to speculate that $\chi_{b}\neq\chi_{v}$ at the strong coupling
fixed point as well. Furthermore, since $\Phi$ should be non-zero, at this
strong coupling fixed point, we expect at this fixed point
$\chi_{b}<\chi_{b}(\Phi=0)$. In addition, the topology of the RG flow lines
suggest that at the perturbatively inaccessible strong coupling fixed point
$\Phi^{*}>\Phi_{c1}$, the fixed point value of $\phi$ at the unstable fixed
point. If this is the case, then we must have the hierarchy $\chi_{b}({\rm
strong\,coupling})<\chi_{b}(\Phi=\Phi_{c1})<\chi_{b}(\Phi=0)$. This runs in
contrast to the KPZ equation at $d>2$, where the strong coupling phase is also
the “rough phase”, being rougher than both the smooth phase and at the
roughening transition. Physical implication of what this means is not
immediately clear to us. Numerical studies of the equations of motion or mode
coupling approaches should help in this regard.

The different scaling exponents obtained in Case I are presented in a tabular
form in Table 1 below.

Case I fixed points and scaling exponents ($g=2$)
---
$\psi=0,\,\Phi=0$ |  $\psi=1,\,\Phi=0$ |  $\psi=1,\,\Phi=3/2$ |  $\psi={\rm notknown,}\,\Phi\rightarrow{\rm large}$
Linearly unstable along $\psi$, stable along $\Phi$ |  Linearly fully stable |  Linearly stable along $\psi$, unstable along $\Phi$ |  Strong coupling, presumably fully stable
Linear theory for $b$, $\chi_{b}=-1/2$ |  $\chi_{b}=-1/4,z_{b}=3/2$ |  $\chi_{b}=-1,z_{b}=3/2$ |  $z_{b}=3/2$ if $P=1$ microscopically, $\chi_{b}$ not known
Table 1: Fixed points in the $\psi-\Phi$ plane and the associated scaling
exponents (with $g=2$) in Case I (see text).

Although we cannot obviously follow the RG flow of $\Phi(l)$, starting from
$\Phi(l=0)>\Phi_{c1}$, all the way to infinity as $\Phi(l)$ appears to diverge
as $l\rightarrow l_{c}$ from below, it is possible to speculate about the
nature of the phases in this region of the parameter space. For this, we are
guided by the fact that $\psi=1$ and $P=1$ is maintained by the perturbation
theory, and hence at the strong coupling fixed point also. Given that the form
of the noise crosscorrelations, as given in (II.5), does not break the
Galilean invariance by itself, this persuades us to speculate that
$\psi^{*}=1$ (corresponding to Galilean invariance) and $P^{*}=1$ are stable
at the strong coupling fixed point. All that the noise crosscorrelations can
do is to generate a non-zero fixed value of $\Phi$, leading to
$\chi_{b}\neq\chi_{v}$, something that already happens at the unstable fixed
point. This suggests the existence of another stable fixed point located at
$\psi=1$ and $\Phi>\Phi_{c1}$. This is an Occam’s razor-style argument which
allows us to draw the simplest RG flow lines that are one hand physically
intuitive and also consistent with the perturbatively obtained flow lines; see
Fig. 2.

Figure 2: Conjectured global RG flows constructed by using Occam’s razor style
arguments diagram in the $\psi-\Phi$-plane for Case I. Arrows show the RG flow
directions. The filled square represents the unstable fixed point, and the
filled circle in a stable fixed point. The red square is the speculated,
presumably globally stable fixed point. The red, broken line is the
separatrix, drawn schematically as an extension of (IV.23), which we do not
expect to meet any of the axes at any finite distance from the origin.

###  IV.4 Case II: Renormalisation group analysis

We now consider the case with $\psi<0$, or $\lambda_{1}\lambda_{2}<0$. As in
Case I, there are no relevant corrections to $D_{\times}$. To proceed further,
we assume $\lambda_{1}>0$ without any loss of generality. Then
$\lambda_{1}\lambda_{2}<0$ implies $\lambda_{2}<0$. Writing $\psi=-|\psi|$,
flow equation (IV.18) takes the form

$\displaystyle\frac{d|\psi|}{dl}$ $\displaystyle=$
$\displaystyle|\psi|g\bigg{[}-\frac{\psi^{2}}{(1+P)^{2}}-\frac{|\psi|}{(1+P)^{2}}\bigg{]}.$
(IV.24)

Thus, $\psi^{*}=0$ is the only RG fixed point, which is stable. Further,
$g^{*}=2$ at the RG fixed point, which is stable as before. With $\psi^{*}=0$,
Eq. (II.2) effectively decouples from Eq. (II.1); as a result, fluctuation
corrections to $\mu$ vanish, which means $z_{b}=2$. However, the fluctuation
corrections to $\nu$ remains unaffected, giving $z_{v}=3/2$ as before.
Therefore $z_{v}>z_{b}$, giving $P^{*}\rightarrow 0$ at the RG fixed point.
This gives weak dynamic scaling ertas ; barabasi ; jkb-mode-coup . These in
turn gives

$\frac{d\Phi}{dl}=-\frac{\Phi}{2}<0,$ (IV.25)

in the long wavelength limit. This means $\Phi$ flows to zero rapidly near the
RG fixed point in the thermodynamic limit. Thus, the effects of the noise
cross-corrections are irrelevant in the RG sense when
$\lambda_{1}\lambda_{2}<0$: even if noise crosscorrelations are present
microscopically, the long wavelength scaling properties of the steady states
are same as those without it. A schematically drawn RG flow diagram in the
$|\psi|-\Phi$-plane is shown in Fig. 3. Unsurprisingly, the origin (0,0) is
the only stable fixed point in the flow diagram. The flows along both the
$\psi$ and $\Phi$-directions are towards the origin.

Figure 3: Schematic RG flow diagram in the $\psi-\Phi$-plane for Case II.
Arrows show the RG flow directions. The filled circle represents the globally
stable fixed point. Here, noise crosscorrelations are generally irrelevant
(see text).

###  IV.5 Case III: Renormalisation group analysis

We now consider the case with $\lambda_{1}=0$. Thus, not only $v$ is
autonomous, it follows a linear equation, and its scaling exponents are of
course exactly known: we have $\chi_{v}=-1/2,\,z=2$. In this case, the sign of
$\lambda_{2}$ has no significance. We first define a new dimensionless
coupling constant $\tilde{g}=\lambda_{2}^{2}D_{v}/\nu^{3}$, which plays the
role of $g$ here. The RG flow equations now read

$\displaystyle\frac{d\nu}{dl}$ $\displaystyle=$
$\displaystyle\left[z-2\right],$ (IV.26) $\displaystyle\frac{d\mu}{dl}$
$\displaystyle=$
$\displaystyle\mu\left[z-2+\frac{\tilde{g}}{2(1+P)P}+\frac{\tilde{g}(1-P)}{(1+P)^{2}P}\right],$
(IV.27) $\displaystyle\frac{dD_{v}}{dl}$ $\displaystyle=$ $\displaystyle
D_{u}\left[z-2\chi_{v}-3\right],$ (IV.28) $\displaystyle\frac{dD_{b}}{dl}$
$\displaystyle=$ $\displaystyle
D_{b}\left[z-2\chi_{b}-3+\frac{\tilde{g}}{P(1+P)}-\frac{4\Phi\tilde{g}}{(1+P)^{3}}\right],$
(IV.29) $\displaystyle\frac{d\lambda_{2}}{dl}$ $\displaystyle=$
$\displaystyle\lambda_{2}\left[\chi_{v}+z-1-\frac{\tilde{g}}{(1+P)^{2}}\right].$
(IV.30)

Since $\lambda_{1}=0$, there are no diagrammatic corrections to $D_{\times}$.
Hence, the flow of $D_{\times}$ follows the same equation (IV.15). We have
noted above that $z_{v}=2$ gives the dynamic exponent of $v$. Does $z_{b}=2$
for $b$ as well? If it is so, then $P^{*}$ must be finite at the RG fixed
point. The flow of $P$ can be calculated by using (IV.26) and (IV.27) as given
above. We find

$\frac{dP}{dl}=P\tilde{g}\left[\frac{1}{2P(1+P)}+\frac{1-P}{P(1+P)^{2}}\right].$
(IV.31)

This has a stable fixed point at $P^{*}=3$, meaning $\mu^{*}=3\nu^{*}$. This
further means that like $v$, even $b$ has a dynamic exponent $z_{b}=2$.
Furthermore, the flow equation for $\tilde{g}$ reads

$\frac{d\tilde{g}}{dl}=\tilde{g}\left[1-\frac{2\tilde{g}}{(1+P)^{2}}\right].$
(IV.32)

Therefore, at the RG fixed point $\tilde{g}^{*}=8$, using $P^{*}=3$. Then,
proceeding as for Case I, we find

$\frac{d\Phi}{dl}=\Phi\tilde{g}\left[-\frac{1}{P(1+P)}+\frac{4\Phi}{(1+P)^{3}}\right].$
(IV.33)

Flow equation (IV.33) gives the following fixed points, which interestingly
are qualitatively similar to Case I:

(i) We have $\Phi^{*}=0$, a stable fixed point. At this RG fixed point, noise
crosscorrelations are irrelevant in the RG sense, and the long wavelength
scaling properties of the model is statistically identical to its zero noise
crosscorrelation version ertas ; barabasi ; jkb-mode-coup . In particular,
$\chi_{b}=-1/6$ and $z_{b}=2$.

(ii) Then there is a linearly unstable fixed point
$\Phi^{*}=4/3\equiv\Phi_{c2}$. Thus, if the initial value
$\Phi(l=0)<4/3\equiv\Phi_{c2}$, $\Phi(l)$ flows to zero, rendering noise
crosscorrelations irrelevant in the RG sense. At this unstable fixed point,
$\chi_{b}(\Phi_{c2})=-1/2<\chi_{b}(\Phi=0)$. On the other hand, if
$\Phi(l=0)>\Phi_{c2}$, $\Phi(l)$ grows indefinitely as $l$ grows. Again as in
Case I, this indicates an instability, induced by noise crosscorrelations. In
fact, proceeding as in Case I, we can show that $\Phi(l)$ diverges as
$l\rightarrow\tilde{l}_{c}\equiv 2/\Phi(l=0)$, a non-universal value.

Similar to Case I, a separatrix can be obtained that passes through $(8,4/3)$
in the $\tilde{g}-\psi$ plane. Following the procedure outlined in Case I, we
find

$\frac{\delta\Phi}{\delta\tilde{g}}=0$ (IV.34)

as the equation for separatrix near the fixed point $(8,4/3)$, where
$\delta\tilde{g}$ and $\delta\psi$ are small deviations of $\tilde{g}$ and
$\psi$ from their fixed point values.

(iii) Again as in Case I, given that if $\Phi(l=0)>\Phi_{c2}$, $\Phi(l)$
grows, another (presumably stable) fixed point $\Phi^{*}>4/3$ should exist,
whose actually value cannot be obtained in the present one-loop perturbation
theory. Based on our one-loop theory, no inference can be drawn about the
scaling properties of this “strong coupling” state. However, given that if one
has $P=3$ microscopically, it remains so under mode eliminations, we still
expect $z_{v}=z_{b}=2$. By using arguments similar to Case I, we expect
$\chi_{b}<\chi_{b}(\Phi=0)$ at this fixed point. By using arguments similar to
Case I above, we again expect an analogous hierarchy
$\chi_{b}(\Phi>\Phi_{c2})<\chi_{b}(\Phi=\Phi_{c2})<\chi_{b}(\Phi=0)$.
Numerical approaches should provide qualitative results and additional
physical insight about the strong coupling phase.

The different scaling exponents obtained in Case III are presented in a
tabular form in Table 2 below.

Case III fixed points and scaling exponents ($\tilde{g}=8$)
---
$\Phi^{*}=0$ |  $\Phi^{*}=4/3$ |  $\Phi\rightarrow\infty$
Linearly stable |  Linearly unstable |  Strong coupling
$\chi_{b}=-1/6,z_{b}=3/2$ |  $\chi_{b}=-1/2,z_{b}=3/2$ |  $z_{b}=3/2$ if $P=3$ microscopically, $\chi_{b}$ not known
Table 2: Fixed points and the associated scaling exponents (with
$\tilde{g}=8$) in Case III (see text)

A schematically drawn RG flow diagram in the $\tilde{g}-\Phi$-plane is shown
in Fig. 4. There is a stable fixed point at $\tilde{g}=8,\,\Phi=0$ and an
unstable fixed point at $\tilde{g}=8,\,\Phi=4$. Further, it is clear from
(IV.33) that the $\Phi$-axis (i.e., $\tilde{g}=0$) is a marginal direction.
Naturally, the origin, another fixed point, is unstable along the $\tilde{g}$
direction, but marginal along the $\Phi$-direction.

Figure 4: Schematic RG flow diagram in the $\tilde{g}-\Phi$-plane for Case I.
Arrows show the RG flow directions. The filled square represents the unstable
fixed point, and the filled circle represents a stable fixed point. The red
broken line is the separatrix (IV.34).

Notice that in Case III, similar to Case I, the divergence of $\Phi(l)$ for a
sufficiently large initial value $\Phi(l=0)$ is reminiscent of the divergence
of the dimensionless coupling constant in the higher dimensional ($d>2$) KPZ
equation in the long wavelength limit, resulting into a perturbatively
inaccessible strong coupling rough phase. In an analogy to the rough phase in
the $d>2$ KPZ equation, we are tempted to speculate the perturbatively
inaccessible steady states with a large but presumably finite $\Phi$ as a type
of strong coupling phase.

Similar to Case I, one may use Occam’s razor type arguments draw the global RG
flow lines, which match with the flow diagram in (4). Due to the obvious
qualitative similarity between Case I and Case III, such a global RG flow
diagram in the $\tilde{g}-\Phi$ plane in Case III should have the same
topology as the corresponding diagram for Case I in its $\psi-\Phi$ plane as
shown in Fig. 2.

##  V Summary and outlook

In this work, we have studied the effects of noise crosscorrelations on the
steady states of a 1D coupled driven model. Specifically, one of the dynamical
variables $v$ follows the well-known Burgers equation, and evolves
autonomously, being independent of the second dynamical field $b$. The second
dynamical field $b$ is passively advected by the “Burgers velocity” $v$, and
follows an equation that closely resembles the well-known passive scalar model
passive-scal . We have analysed the long wavelength properties of this model
in the presence of finite noise crosscorrelations, whose effects depends upon
the precise nature of the model. We consider three different cases, delineated
by the nonlinear coupling constants. For instance in Case I with Galilean
invariance appearing as an emergent symmetry, where the advective coupling
constant $\lambda_{2}$ in the $b$-equation has the same sign as the advective
nonlinearity $\lambda_{1}$ in the Burgers equation for $v$, a sufficiently
strong noise crosscorrelation amplitude above a finite threshold can
destabilise the system, whereas its microscopic values weaker than the
threshold render noise crosscorrelations irrelevant in the RG sense. In the
latter case, the model is identical to the one without noise crosscorrelations
in the long wavelength limit. In contrast, in Case II, where $\lambda_{2}$ and
$\lambda_{1}$ have opposite signs, noise crosscorrelations are generically
irrelevant in the RG sense. We have also considered yet another case denoted
Case III here, where $\lambda_{1}=0$, making $v$ follow the linear diffusion
equation. In this case, similar to Case I, noise crosscorrelations with
amplitudes greater than a threshold lead to instabilities, whereas when below
the threshold, it is irrelevant in the RG sense. In the unstable cases, the
eventual steady states cannot be obtained from our calculations. Numerical
simulations of equivalent lattice models, or direct numerical solutions of the
model equations can verify our perturbative results, and also shed light on
the instabilities and the resulting unknown steady states. The existence of
unstable fixed points in Case I and Case III, and the associated
perturbatively inaccessible putative strong coupling phases have strong
resemblance with the well-known roughening transition in the KPZ equation at
$d>2$. Quite interestingly, the scaling exponents of $b$ at the unstable fixed
point in both Case I and Case III are less than their values at zero noise
crosscorrelations. This suggests that the field $b$ actually fluctuates less
at the unstable critical point. The RG analysis fails to capture the strong
coupling phases in Case I and Case III. Mode coupling methods may be useful in
extracting the scaling exponents in these strong coupling phases jkb-mode-coup
. It may however be noted that in Case III there are no parameter regimes
where mode coupling theories may be straightforwardly applied to the strong
coupling regime. This is because in Case III vertex corrections play a
significant role for any nonzero $\lambda_{2}$. This is in contrast to Case I,
where if the model is Galilean invariant (i.e., $\lambda_{1}=\lambda_{2}$)
microscopically, there are no relevant vertex corrections. On the whole, we
thus see that the precise effects of the noise crosscorrelations depend quite
sensitively on the details of the models under consideration.

In this work, we have confined ourselves to studying only conserved noises.
Similar calculations as here can be performed with noise variances than scale
differently with $k$, e.g., long-range noises. We speculate similar effects
will be visible even in that case.

We have assumed the noise crosscorrelations to be imaginary and odd in Fourier
wavevector $k$. What would happen if we chose a different structure for the
noise crosscorrelations, e.g., real and even in $k$? Straightforward
perturbation theory generates a term of the form $\partial_{xx}v$ in the
fluctuation-corrected (II.2). This is not surprising, since a real and even
noise crosscorrelation implies that $v$ and $b$ have the same properties under
parity inversions; see, e.g., discussions in Ref. abfrey1 . This then no
longer rules out a diffusive $v$ term in (II.2) abfrey1 . Given that $v$ is
autonomous, such a term in (II.2) effectively acts like another additive noise
for $b$, whose statistics is given by the statistics of $v$. It would be
interesting to study how noise crosscorrelations in that case affects the
known scaling and stability of the NESS without it.

We have used a simple, purpose-built, minimal coupled 1D model to study the
role of noise crosscorrelations. However, the question of the effects of noise
crosscorrelations should be important in hosts of natural systems, going much
beyond such simple driven 1D models. RG studies, for instance, on active XY
models astik1 ; astik2 , passive scalar models sudip-multi , can give
interesting insights on the precise role of noise crosscorrelations on the
NESS of more complex models. We hope our work here will provide impetus on
further studies on this topic.

Acknowledgement:- S.M. thanks the SERB, DST (India) for partial financial
support through the TARE scheme [file no.: TAR/2021/000170] (2022).

##  Appendix A Action functional

We give the actine functional ${\cal S}$ corresponding to Eqs. (II.1) and
(II.2). It is defined via the generating functional $\cal Z$ given by

${\cal Z}=\int{\cal D}v{\cal D}b{\cal D}\tilde{v}{\cal D}\tilde{b}\exp(-{\cal
S}),$ (A.1)

where $\cal S$ is given by

$\displaystyle{\cal S}$ $\displaystyle=$
$\displaystyle-\int\frac{dk}{2\pi}\frac{d\omega}{2\pi}\left[D_{v}k^{2}|\tilde{v}(k,\omega)|^{2}+D_{b}k^{2}|\tilde{b}(k,\omega)|^{2}+iD_{\times}k|k|\tilde{v}(k,\omega)\tilde{b}(-k,-\omega)\right]$
(A.2) $\displaystyle-$ $\displaystyle\int
dxdt\left[\tilde{v}\left(\partial_{t}v-\frac{\lambda_{1}}{2}\partial_{x}v^{2}-\nu\partial_{xx}v\right)+\tilde{b}\bigg{(}\partial_{t}b-\lambda_{2}\partial_{x}(vb)-\mu\partial_{xx}b\bigg{)}\right].$

##  Appendix B Details of the RG analysis

In this Section, we discuss some details of the momentum shell RG procedure
applied on our model.

###  B.1 Feynman diagrams

In this Section, we give the one-loop Feynman diagrams for the model
parameters in (II.2). The corresponding one-loop diagrams are standard; see,
e.g., Refs. stanley .

Figure 5: One-loop Feynman diagram that corrects $\mu$.

Figure 6: One-loop Feynman diagrams that correct $D_{b}$. The bottom diagram
carries the contribution from the noise crosscorrelations (see text).

Figure 7: One-loop Feynman diagrams which correct $\lambda_{2}$.

###  B.2 Discrete recursion relations

$\displaystyle D_{v}^{<}$ $\displaystyle=$ $\displaystyle
D_{v}+\frac{\lambda_{1}^{2}D_{v}^{2}}{4\nu^{3}}\frac{\delta l}{\Lambda\pi},$
(B.1) $\displaystyle D_{b}^{<}$ $\displaystyle=$ $\displaystyle
D_{b}+\bigg{[}\frac{\lambda_{2}^{2}D_{v}D_{b}}{\nu\mu(\nu+\mu)}-\frac{4\lambda_{2}^{2}D_{\times}^{2}}{(\nu+\mu)^{3}}\bigg{]}\frac{\delta
l}{\Lambda\pi},$ (B.2) $\displaystyle D_{\times}^{<}$ $\displaystyle=$
$\displaystyle D_{\times},$ (B.3) $\displaystyle\nu^{<}$ $\displaystyle=$
$\displaystyle\nu+\frac{\lambda_{1}^{2}D_{v}}{4\nu^{2}}\frac{\delta
l}{\Lambda\pi},$ (B.4) $\displaystyle\mu^{<}$ $\displaystyle=$
$\displaystyle\mu+\left[\frac{\lambda_{2}^{2}D_{v}}{2\nu(\nu+\mu)}+\frac{\lambda_{2}^{2}D_{v}(\nu-\mu)}{\nu(\nu+\mu)^{2}}\right]\frac{\delta
l}{\Lambda\pi},$ (B.5) $\displaystyle\lambda_{1}^{<}$ $\displaystyle=$
$\displaystyle\lambda_{1},$ (B.6) $\displaystyle\lambda_{2}^{<}$
$\displaystyle=$
$\displaystyle\lambda_{2}-\bigg{[}\frac{\lambda_{2}^{3}D_{v}}{\nu(\nu+\mu)^{2}}-\frac{\lambda_{2}^{2}\lambda_{1}(3\nu+\mu)D_{v}}{2\nu^{2}(\nu+\mu)^{2}}$
(B.7) $\displaystyle+$
$\displaystyle\frac{\lambda_{2}^{2}\lambda_{1}D_{v}}{2\nu^{2}(\nu+\mu)}\bigg{]}\frac{\delta
l}{\Lambda\pi}.$

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