Quantum Contact Process: Critical Dynamics

• Quantum Contact Process is a quantum many-body model with active and inactive states, exhibiting absorbing-state transitions via coherent spin flips and spontaneous decay.
• It integrates coherent spin dynamics with dissipative decay to produce continuous, discontinuous, and metastable phase transitions that deviate from classical directed percolation.
• The model benchmarks advanced numerical methods and inspires topological variants, offering practical insights into tensor-network simulations and non-Hermitian dynamics.

Searching arXiv for the cited quantum contact process papers to ground the article in published work.
The quantum contact process (QCP) is a quantum many-body generalization of the classical contact process in which local active and inactive degrees of freedom are retained, but branching and coagulation are implemented through coherent spin dynamics while spontaneous decay remains dissipative. In its standard formulation, the QCP is an open spin-(\tfrac12) lattice system with an exact absorbing state, and it has been studied as a model of absorbing-state criticality, as a benchmark for tensor-network simulation of dissipative dynamics, and, in a distinct no-jump formulation, as a non-Hermitian many-body problem with exceptional-point singularities [1902.04515], [2209.10718]. Subsequent work has also examined quantum-to-classical crossover, metastability and bistability in steady states, and a coherent topological variant realized through Rydberg facilitation on a one-dimensional lattice [2004.02672], [2510.16836], [2604.03184].

1. Microscopic definition and absorbing-state structure

In the standard one-dimensional QCP, each lattice site is a two-level system representing an active/occupied state and an inactive/empty state. The local basis is written either as ({|\bullet\rangle,|\circ\rangle}) or as ({|\uparrow\rangle,|\downarrow\rangle}), with ( |\bullet\rangle ) or (|\uparrow\rangle) denoting the active state and ( |\circ\rangle ) or (|\downarrow\rangle) the inactive state [1902.04515], [2510.16836]. The dynamics is Markovian and Lindbladian:
[
\partial_t \rho = -i[H,\rho] + \sum_\mu \left(L_\mu \rho L_\mu^\dagger - \frac12 {L_\mu^\dagger L_\mu,\rho}\right).
]

The coherent part of the dynamics is neighbor-assisted spin flipping. In the notation used in the one-dimensional open-system studies,
[
H=\Omega\sum_{j=1}^{L-1}\left(\hat{\sigma}{j}^{x}\hat{n}{j+1}+\hat{n}j\hat{\sigma}{j+1}^{x}\right),
\qquad
\hat n_j=\hat\sigma_j^+\hat\sigma_j^-,
]
or equivalently
[
H=\Omega \sum_{k=1}^{L-1}\left( \sigma_1^{(k)}n^{(k+1)}+ n^{(k)}\sigma_1^{(k+1)}\right),
]
with (\sigma_1) the local flip operator [1902.04515], [2510.16836]. These terms realize coherent branching and coherent coagulation: a spin can flip only in the presence of neighboring activity.

The dissipative part is local spontaneous decay,
[
L_j=\sqrt{\Gamma}\,\hat\sigma_j^-,
\qquad
\mathcal D[\rho] = \Gamma\sum_{j=1}^{L} \left( \hat\sigma_j^-\rho \hat\sigma_j^+ -\frac12{\hat n_j,\rho} \right),
]
or, with (\gamma) instead of (\Gamma),
[
\mathcal{D}[\rho]=\gamma\sum_{k=1}^L\left(\sigma_-^{(k)}\rho\sigma_+^{(k)}-\frac{1}{2}\left{n^{(k)},\rho\right}\right).
]
The competition between (\Omega) and (\Gamma) or (\gamma) is the central control parameter of the model [1902.04515], [2510.16836].

The fully inactive product state,
[
|0\rangle=\bigotimes_{k=1}^L|\circ\rangle,
\qquad
\rho_0=|0\rangle\langle 0|,
]
or equivalently (|\downarrow\downarrow\cdots\downarrow\rangle), is an exact absorbing state. It is absorbing because every coherent facilitation term contains a number operator (n) and therefore annihilates the empty lattice, while the decay jumps also vanish on that state [1902.04515]. This absorbing-state structure is not an incidental detail: it is the defining nonequilibrium constraint of the QCP and the source of both its critical phenomenology and its numerical difficulty.

A broader Lindblad version introduces incoherent branching and coagulation in addition to coherent facilitation,
[
\hat L_\ell^{(b)} = \sqrt{\kappa}\,(\hat n_{\ell-1}+\hat n_{\ell+1})\hat\sigma_\ell^+,
\qquad
\hat L_\ell^{(c)} = \sqrt{\kappa}\,(\hat n_{\ell-1}+\hat n_{\ell+1})\hat\sigma_\ell^-,
]
with coherent Hamiltonian
[
H_S = \omega \sum_{\ell=1}^{N} (\hat n_{\ell-1}+\hat n_{\ell+1})\hat\sigma_\ell^x.
]
Here (\omega) controls coherent quantum branching/coagulation, (\kappa) controls incoherent classical branching/coagulation, and (\gamma) is set to (1) [2004.02672].

2. Order parameters, observables, and scaling diagnostics

The basic distinction is between an absorbing phase, in which activity dies out and the system approaches the empty configuration, and an active phase, in which the excitation density remains nonzero. In steady-state language, the principal order parameter is the averaged density
[
\bar n_{\rm ss}=\frac1L\sum_{j=1}^L \langle \hat n_j\rangle_{\rm ss},
]
so that (\bar n_{\rm ss}=0) defines the absorbing phase and (\bar n_{\rm ss}\neq 0) an active phase [2510.16836]. In real-time studies, the corresponding time-dependent observable is
[
n(t)=\frac{1}{L}\sum_k {\rm Tr}\left(\rho(t)n^{(k)}\right),
]
with critical decay analyzed through
[
n(t) \approx t^{-\delta}\, f\left(\left(\Omega-\Omega_c\right) t^{1/\nu_\parallel}\right).
]
At criticality this reduces to (n(t)\sim t^{-\delta}) [1902.04515].

For spreading from a single seed, the standard observables are the survival probability (P_{\rm sur}(t)), the total activity (N_a(t)), and the seed density (n_{\rm seed}(t)). In the notation of the seed-based critical-dynamics study,
[
P_{\rm sur}(t)=1-\mathrm{Tr}[\rho(t)\rho_a],\qquad
N_a(t)=\sum_k n(t,k),\qquad
n_{\rm seed}(t)=n(t,k=x_{\rm seed}),
]
with critical scaling forms
[
P_{\rm sur}(t)\sim t^{-\delta},\qquad
N_a(t)\sim t^{\Theta},\qquad
n_{\rm seed}(t)\sim t^{\Theta-1/z}.
]
These are the natural QCP analogues of classical spreading observables in absorbing-state criticality [1907.02433].

Static criticality is commonly probed through
[
n_{\rm qs}\approx |\Omega-\Omega_c|^\beta,
\qquad
\xi_\perp \approx |\Omega-\Omega_c|^{-\nu_\perp},
]
with (\xi_\perp) extracted from the connected density-density correlator
[
C(r)=\langle n^{(r)}n^{(0)}\rangle-\langle n^{(0)}\rangle^2 \sim e^{-r/\xi_\perp}.
]
An effective exponent
[
\delta_{\rm eff}(t)=-\frac{1}{\log b}\log\frac{n(tb)}{n(t)}
]
is used to diagnose whether time evolution is approaching algebraic decay [1902.04515].

A representative summary of formulations discussed in the literature is:

Formulation	Generator	Principal focus
Open-system QCP	Lindblad master equation with coherent facilitation and local decay	Absorbing-state transition [1902.04515]
Steady-state 1D QCP	Lindblad dynamics analyzed by MF, CMF, and Liouvillian spectra	Bistability, metastability, Liouvillian gap [2510.16836]
No-jump QCP	Effective non-Hermitian Hamiltonian from postselected no-jump trajectories	Exceptional-point-induced continuous transition [2209.10718]
Quantum-to-classical crossover QCP	Lindblad dynamics with coherent and incoherent branching/coagulation	Initial-condition-dependent crossover to DP [2004.02672]
Topological QCP/QXP	Coherent facilitated Hamiltonian on a topological lattice	Domain-space SSH/AAH dynamics [2604.03184]

3. One-dimensional phase transition: continuous, discontinuous, and protocol-dependent descriptions

A central issue in the QCP literature is the nature of the one-dimensional absorbing-state transition. Real-time tensor-network simulations in the thermodynamic limit found strong evidence for a continuous transition at approximately
[
\Omega_c \approx 6\gamma,
]
with estimated exponents
[
\delta \approx 0.36,\qquad \nu_\parallel \approx 1,\qquad \beta \approx 0.39,\qquad \nu_\perp \approx 0.5,
]
and a summarized approximate set
[
\delta=0.36,\quad \beta=0.36,\quad \nu_\parallel=1.00,\quad \nu_\perp=0.5.
]
These values were reported to differ markedly from classical (1d) directed percolation, for which the same study quotes
[
\delta=0.16,\quad \beta=0.28,\quad \nu_\parallel=1.73,\quad \nu_\perp=1.10.
]
The conclusion of that work was that the one-dimensional QCP shows strong evidence for a continuous absorbing-state transition with non-DP critical behavior [1902.04515].

A later tensor-network and quantum-trajectories analysis of seed dynamics reached a related but numerically distinct conclusion. Using quantum jump Monte Carlo and TEBD, it reported
[
\delta = 0.26 \pm 0.04,\qquad \Theta = 0.26 \pm 0.05,\qquad z = 1.61 \pm 0.16,
]
and argued that the exponent (\delta) is incompatible with both (1d) DP and (2d) DP values, thereby strengthening the claim that the QCP is not in the directed-percolation universality class [1907.02433].

By contrast, a steady-state analysis of the one-dimensional QCP using single-site mean field, cluster mean field, and Liouvillian spectra concluded that the transition is effectively discontinuous and bistable in the thermodynamic limit. In that work, the mean-field active branches appear for
[
\Omega/\Gamma > \frac{1}{\sqrt2},
]
with onset at
[
S_1:\quad \Omega=\Gamma/\sqrt2,\qquad n_{\rm ss}=0.25.
]
Going beyond single-site mean field, finite-size extrapolation of the Liouvillian gap gave
[
\Omega_c/\Gamma \approx 5.83,
]
and cluster mean-field calculations up to cluster size (L=13) were reported to converge toward the same scale. The resulting picture is one of coexistence between an always-stable absorbing state and a stable active branch, accompanied by an unstable active branch [2510.16836].

These conclusions are not directly equivalent, because they interrogate different observables and dynamical regimes. The real-time tensor-network studies infer criticality from long-time decay and quasi-stationary behavior of trajectories started in active states or from single-seed spreading [1902.04515], [1907.02433]. The steady-state bistability study emphasizes Liouvillian gap closing, coexistence of fixed points, and metastable plateaus close to the transition [2510.16836]. Taken together, this suggests that the one-dimensional QCP is unusually sensitive to how the thermodynamic transition is operationally accessed.

4. Numerical methods, absorbing-state bias, and metastability

The QCP is also a benchmark problem for numerical methods because the absorbing state is an exact steady state for all parameters. This creates a failure mode for algorithms that target the stationary state directly: they can be biased toward the simple weakly entangled absorbing product state even when an active stationary state should exist in the thermodynamic limit [1902.04515].

For that reason, one prominent strategy is to study real-time evolution rather than solve the steady-state problem directly. The thermodynamic-limit simulations of the one-dimensional QCP used TEBD and iTEBD in Liouville space, with time step
[
0.01 \le dt \le 0.1,
]
bond dimensions
[
\chi \le 1300,
]
and reachable times
[
t \approx 50
]
in units of (\gamma^{-1}) [1902.04515]. Near criticality, however, finite-(\chi) effects produce artificial saturation of (n(t)), so subcritical decay is easier to certify than apparent long-time activity.

A more method-focused study compared three tensor-network routes: direct density-matrix evolution in the double space, Heisenberg-picture operator evolution, and quantum trajectories. In the Lindblad formalism, the vectorized density matrix obeys
[
\frac{d}{dt}\ket{\rho(t)}=\mathbb{L}\ket{\rho(t)},
]
and the relevant complexity measure is the operator-space entanglement entropy
[
\tilde S = -\mathrm{tr}[\tilde \varrho_A \log \tilde \varrho_A].
]
That work found a pronounced operator-space-entanglement “barrier” for both the classical and quantum contact processes, but much higher for the QCP, explaining the larger bond-dimension requirements. It further found that Heisenberg-picture evolution substantially lowers the barrier for observables such as the survival probability, and that quantum trajectories are more effective still, because the entanglement becomes a distribution over pure-state trajectories rather than the entanglement of a single vectorized mixed state [1907.02433].

In the trajectories approach, observables were converged within statistical error up to (\gamma t=10) using (\chi=128)–256, whereas the direct double-space approach required (\chi\sim 1024) and still showed strong deviations by (\gamma t=10) [1907.02433]. The same study therefore identified trajectory-based TEBD as the most reliable of the tested methods for critical QCP dynamics.

A different numerical complication is metastability. In the steady-state analysis, near (\Omega/\Gamma=5.8), the Liouvillian spectrum exhibits one zero mode, one first nonzero mode whose real part approaches zero as (1/L\to0), and a second nonzero mode with
[
\Re(\mu)\approx -0.5.
]
This produces a hierarchy of timescales: the system first relaxes quickly onto a long-lived metastable manifold and only much later reaches the true steady state. For (L=11) and initial state (|\uparrow\uparrow\cdots\uparrow\rangle), the averaged density (\bar n(t)) was shown to remain on a finite-density plateau for
[
t \gtrsim 10^2\,\Gamma^{-1}
]
before ultimately decaying to the absorbing state near the transition [2510.16836]. This is an explicit warning that finite-time simulations can misidentify metastable plateaus as steady active states.

5. Quantum-to-classical crossover, initial conditions, and relation to directed percolation

The relation between QCP criticality and directed percolation remains a central interpretive issue. One line of work argues that the one-dimensional pure QCP is continuous but not DP-like [1902.04515], [1907.02433]. Another identifies a more selective anomaly: the critical behavior depends strongly on the initial condition when incoherent branching/coagulation is added [2004.02672].

In the model with coherent rate (\omega) and incoherent rate (\kappa), the pure quantum limit is (\kappa=0), while (\omega=0) gives the classical contact process. For homogeneous initial conditions, the density decays as
[
n(t)\sim t^{-\alpha}\qquad (\omega=\omega_c).
]
At (\kappa=0) in one dimension, the critical point was estimated as
[
\omega_c \approx 6.04,
]
with
[
\nu_\perp = 1.06 \pm 0.04,
\qquad
\alpha = 0.32 \pm 0.01.
]
The unusual result is that (\alpha) varies continuously over
[
0 \le \kappa < \kappa_,
\qquad
\kappa_ \approx 0.58,
]
with measured values
[
0.32,\ 0.28,\ 0.24,\ 0.22,\ 0.20,\ 0.18
]
for (\kappa=0.0,0.1,0.2,0.3,0.4,0.5), respectively, before approaching the (1d) DP value for (\kappa\gtrsim 0.58) [2004.02672].

Under a single-seed initial condition, however, the same study found DP-like spreading exponents even at (\kappa=0):
[
\delta = 0.16 \pm 0.05,\qquad
\theta = 0.30 \pm 0.05,\qquad
z = 1.55 \pm 0.06,
\qquad
\delta+\theta = 0.32 \pm 0.01.
]
These agree within error bars with the quoted classical (1d) DP values
[
\delta_{\rm DP}\approx 0.159,\quad
\theta_{\rm DP}\approx 0.313,\quad
z_{\rm DP}\approx 1.581.
]
The same work emphasized the identity
[
\rho_d(t)=\rho_{d,s}(t)P(t)\quad \Rightarrow \quad \alpha=\delta+\theta,
]
but found that the classical rapidity-reversal-style identification (\alpha=\delta) does not hold for the one-dimensional QCP at (\kappa=0) [2004.02672].

In two dimensions, that anomalous crossover was not observed. At (\kappa=0), the critical point was estimated as
[
\omega_c \approx 0.94,
]
with exponents
[
\alpha \approx 0.45,\qquad
\nu_\parallel \approx 1.30,\qquad
z \approx 1.76,\qquad
\nu_\perp \approx 0.74,
]
all reported to be consistent with classical (2d) DP. The same study stated that in (2d) classical DP behavior appears in the entire region of (\kappa \ge 0), regardless of initial condition [2004.02672].

A plausible implication is that in low dimensions the QCP is not described by a single universally accepted critical scenario across all protocols presently used in the literature. Some results point to a non-DP quantum critical regime, others to seed-DP scaling with anomalous homogeneous decay, and still others to discontinuous steady-state bistability.

6. Non-Hermitian no-jump QCP and topological coherent extensions

A distinct formulation of the QCP arises from postselecting on trajectories with no quantum jumps. Starting from the Lindblad equation with jump operators
[
\hat{L}^{k}=\sqrt{\Gamma}\,\sigma_{-}^{k},
]
one omits the recycling term and obtains the effective non-Hermitian Hamiltonian
[
\hat H_{\rm eff} = \hat H_{0}-\frac{i}{2}\sum_{k=1}^{L} \hat{L}^{k^\dagger} \hat{L}^{k},
]
with coherent part
[
\hat{H}{0}=\Omega \sum{k=1}^L \left( \hat \sigma_{x}^{k}\hat \sigma_{n}^{k+1}+\hat \sigma_{n}^{k} \hat \sigma_{x}^{k+1} \right),
\qquad
\hat \sigma_n^k=\hat \sigma_+^k \hat \sigma_-^k.
]
The resulting no-jump Hamiltonian is
[
\hat H = \Omega \sum_{k}^{L} \left( \hat \sigma_{x}^{k}\hat \sigma_{n}^{k+1}+\hat \sigma_{n}^{k} \hat \sigma_{x}^{k+1} \right)-\frac{i}{2} \Gamma \sum_{k=1}^{L} \hat \sigma_{+}^{k} \hat \sigma_{-}^{k}.
]
Because (\hat H_0) does not conserve total (z)-polarization, the imaginary onsite term cannot be reduced to a trivial shift of complex energies; instead it generates exceptional points and a many-body “complex-imaginary transition” [2209.10718].

In this non-Hermitian QCP, right and left eigenvectors must be distinguished,
[
\hat H |\phi_{R}^i \rangle = E_i | \phi_{R}^i \rangle,\qquad \langle \phi_{L}^i | \hat H = E_i \langle \phi_{L}^{i}|,
]
with biorthogonal normalization. The “ground state” is defined as the state with minimum real part of the eigenvalue, and observables are evaluated using the right eigenstate,
[
O=\langle \psi_{R}\vert \hat{O} \vert \psi_{R}\rangle.
]
The order parameter is
[
M^x=\left\langle \sum_k \sigma_x^k \right\rangle,
]
and the susceptibility is introduced through a small longitudinal field,
[
\chi= \frac{M^{z}(\delta h)-M^{z}(0)}{2 \delta h}.
]
The reported transition is continuous and induced by non-Hermiticity itself, with
[
M^x\sim -|\Gamma_c(L)-\Gamma|^\beta,\qquad \beta \approx 0.51,
]
and
[
\chi=\chi_0/(\Gamma-\Gamma_c(L))^\gamma,\qquad \gamma \approx 1.52.
]
For (\Omega=1), finite-size critical points drift toward
[
\Gamma_c(\infty)\sim 13.845.
]
A distinctive claim is that both order parameter and susceptibility are singular even at finite (L), because the nonanalyticity is tied to exceptional-point coalescence rather than solely to the thermodynamic limit [2209.10718].

The same study contrasted this with the Hermitian interpolation
[
\hat H=\Omega \sum_{k} \left( \hat \sigma_{x}^{k}\hat \sigma_{n}^{k+1}+\hat \sigma_{n}^{k}\hat \sigma_{x}^{k+1} \right)-\frac{1}{2} (\Gamma_{\rm re}+i\Gamma_{\rm im}) \sum_{k=1}^{L} \hat \sigma_{+}^{k} \hat \sigma_{-}^{k},
]
and reported that the purely Hermitian case shows a first-order level-crossing transition near
[
\Gamma_{\rm re}^{c}\sim -4
]
rather than a continuous EP-driven one. In that sense, the non-Hermitian transition has “no counterpart in the Hermitian case” [2209.10718].

A further extension replaces the dissipative emphasis by a purely coherent, constrained, topological lattice model. In the so-called QXP construction,
[
H_\mathrm{qxp} = \sum_{j=1} \lambda_{j} \, \Bigl( \hat Q_{j-1} \hat X_{j} \hat P_{j+1} + \hat P_{j-1} \hat X_{j} \hat Q_{j+1}\Bigr) + \delta_j \hat Q_j,
]
a site flips only if exactly one of its nearest neighbors is excited. For a single seed at the left boundary and (\lambda_1=\delta_1=0), the many-body dynamics is confined to the domain basis
[
\ket{m}=\ket{\underbrace{\bullet\cdots\bullet}{m}\circ\cdots\circ},
\qquad m=1,\dots,N,
]
and maps exactly to the single-particle Hamiltonian
[
H\mathrm{dom} = \sum_{m=2}^{N} \lambda_{m} \left( \hat c_{m}^\dagger \hat c_{m-1} +\mathrm{h.c.} \right) + \sum_{m=1}^N \eta_m \hat c_m^\dagger \hat c_m.
]
With alternating couplings
[
\lambda_m =
\begin{cases}
\lambda_v=\lambda_0=1, & m\ \text{even},\
\lambda_w, & m\ \text{odd},
\end{cases}
]
the domain-space lattice becomes an SSH chain. The nontrivial phase is
[
\lambda_w>\lambda_v,
]
with localization length
[
\xi = \bigl(\log |\lambda_w|-\log |\lambda_v|\bigr)^{-1}.
]
In this regime, the dynamics is confined to a protected subspace corresponding approximately to a single seed and a fully excited chain, and finite-size hybridization produces oscillations with period
[
T_\mathrm{hyb}=\frac{2\pi}{E_+-E_-},
\qquad
E_\pm = \pm \left| n_1 e^{-(N-1)/\xi}\lambda_v \right|.
]
A time-dependent Aubry-André-Harper realization,
[
\lambda_m(t)=\lambda_0 \sin!\left(\omega t+\frac{4\pi}{3}(m+1)\right),\qquad
\eta_m(t)=\eta_0 \cos!\left(\omega t+\frac{4\pi}{3}(m+1)\right),
]
with (\eta_0=-10\lambda_0) and (\omega\ll\lambda_0), then implements a Thouless pump in domain space, so the growth and shrinkage of the excitation domain occur in quantized steps [2604.03184].

The topological construction is not the standard dissipative QCP of absorbing-state studies. It is a coherent facilitated process motivated by Rydberg tweezer arrays, and its main results concern topology, protected subspaces, and quantized control rather than steady-state Lindblad criticality. Even so, it broadens the meaning of “quantum contact process” by showing that coherent facilitation can support SSH and AAH physics when the many-body spreading problem reduces to an effective single-particle lattice [2604.03184].