Dissipative Transverse Ising Model

• The dissipative transverse Ising model is a quantum Ising system with environmental couplings that induce dissipation, altering dynamic critical behavior and universality classes.
• Methodologies including Monte Carlo simulations and strong-coupling expansions reveal modified scaling exponents and the smearing of phase transitions under disorder and driven conditions.
• The topic has practical implications for quantum simulation and experimental platforms, illustrating how decoherence and dissipation impact quantum coherence and criticality.

The dissipative transverse Ising model encompasses quantum Ising systems subjected to environmental couplings that induce dissipation or decoherence. This class of models is central to the paper of quantum phase transitions in open systems, quantum critical scaling under non-unitary dynamics, and the emergence of non-equilibrium steady states with fundamentally altered universality properties. Dissipation can act in multiple forms, including site-local (each spin coupled independently to a bath), bond (coupling gradients to a bath), and more general Markovian or non-Markovian baths. The dynamical critical exponent $z$ and correlation length exponent $\nu$ in the presence of dissipation are sharply distinguished from their values in the coherent transverse Ising model, with implications for the universality class, nonequilibrium phase transitions, and the fate of quantum coherence.

1. Quantum Criticality and Dynamical Exponents in the Presence of Dissipation

In the non-dissipative transverse Ising model, the quadratic action yields an inverse propagator of the form $q^2 + \omega^2$, resulting in isotropic scaling with $z = 1$. Upon introducing Ohmic dissipation at the site level, the propagator is modified to $q^2 + \omega^2 + (\alpha/2)|\omega|$. In the long-wavelength, low-frequency limit, the $|\omega|$ term dominates over $\omega^2$, shifting the dynamical scaling to $q^2 \sim | \omega | \implies z = 2$ (Sperstad et al., 2010). This implies a strong anisotropy between space and time, mapping a $d$-dimensional quantum dissipative Ising model to a $(d+2)$-dimensional classical model.

For Ohmic bond dissipation, the dissipative term assumes the form $(\alpha/2)|\omega| q^2$, which is subleading compared to $q^2$, so the dynamic exponent remains $z \approx 1$, maintaining the universality class of the coherent transverse Ising model.

Monte Carlo simulations confirm:

• For site dissipation in (2+1)D, $z \approx 1.97(3)$ (Sperstad et al., 2010).
• For bond dissipation in (1+1)D, $z \approx 1$.

The precise value of $z$ is critical for scaling relations such as $\xi_\tau \sim \xi^z$, for quantum-to-classical mappings, and for finite-size scaling in numerical studies (with the imaginary-time extent $L_\tau \propto L^z$).

2. Smearing of Quantum Phase Transitions and Griffiths Effects

In random (disordered) dissipative Ising chains, coupling each spin to a local bath with an Ohmic or sub-Ohmic spectrum causes sufficiently large spatial clusters ("rare regions") to become frozen, ceasing to tunnel and ordering independently (Hoyos et al., 2012, Al-Ali et al., 2013). This destroys the sharp infinite-randomness critical point found in the dissipationless case and smears the quantum phase transition. Signatures include:

• The magnetization acquires an exponential temperature tail, $m = m_0 \exp[-(T_c^0-T)^{-\nu}]$.
• The susceptibility exhibits non-universal power-law scaling with system size in the Griffiths regime.
• The dynamical exponent $z'$ diverges as the endpoint of the smeared transition is approached.

Under strong disorder renormalization group (SDRG), the correlation length diverges as $\xi \sim h_0^{-\nu_h}$ with $\nu_h \approx 1$, and the logarithmic energy scale obeys $\ln \epsilon \sim L^{1/2}$ at the infinite disorder fixed point (Petö et al., 5 Jan 2025). For Ohmic dissipation, the transition is replaced by an "inhomogeneously ordered" phase, with no singularity at the global critical point.

Super-Ohmic dissipation is irrelevant in the RG sense; the quantum critical properties revert to those of the clean (dissipationless) model.

3. Non-Equilibrium Steady States, Dynamical Phase Transitions, and Liouvillian Spectra

Driven-dissipative extensions of the transverse Ising model, specifically those described by Lindblad master equations, exhibit rich non-equilibrium phase diagrams (Ates et al., 2011, Lee et al., 2013, Overbeck et al., 2016, Song et al., 2023, Roberts et al., 2023). Examples:

• A dynamical first-order phase transition between "active" (high emission) and "inactive" (dark) dynamical phases, manifested in the non-analyticity of the dynamical free energy $\theta(s)$ of quantum jump trajectories (Ates et al., 2011).
• Bistability and intermittency in emission statistics, associated with coexisting steady states and a vanishing Liouvillian gap.
• The emergence of multicritical points with non-mean-field exponents in systems where dissipation drives the quartic coefficient negative in an effective Landau expansion (Overbeck et al., 2016).
• In all-to-all (infinite-range) models, the exact steady-state solution can be constructed for arbitrary site-dependent transverse fields and local dissipation, permitting the characterization of both first- and second-order dissipative phase transitions via the structure of an effective free energy landscape. "Spin blockade" phenomena arise at resonant settings of the system parameters, where higher order correlations are suppressed (Roberts et al., 2023).

Table 1: Qualitative Forms of Dissipation and Effects

Dissipation Type	Critical Dynamics ($z$)	Transition Character	Notable Physics
Ohmic site (local)	$z\approx 2$	Sharp or smeared (w/disorder)	Strong anisotropy, frozen clusters
Ohmic bond (gradient)	$z\approx 1$	Conventional Ising transition	Dissipation irrelevant to universality
Markovian (Lindblad)	Model-dependent	First-, second-, or multi-critical	Activity/intermittency, Liouvillian gap
Collective (global)	Model-dependent	Discontinuous to continuous crossover	Bistability, spin blockade, phase mixing

4. Decoherence, Topological Excitations, and the Fate of Quantum Information

In the presence of a transverse dissipative interaction (dissipation parallel to the field and perpendicular to the Ising axis), the quantum dynamics can be understood via a strong-coupling expansion (Weisbrich et al., 2018). The system supports a relaxation-free subspace consisting of quantum delocalized domain wall (kink) excitations that are protected against decay to the ground state due to parity selection rules and symmetry. The Lindblad equation for weak dissipation shows that only a subset of excitations with specific center-of-mass quantum numbers couple to the ground state; others remain "dark" or relaxation-free. This nontrivial subspace structure has implications for coherence and decoherence management in quantum information protocols.

5. Entanglement, Spin Squeezing, and Correlations in Dissipative Steady States

Despite decoherence from local spontaneous emission, infinite-range dissipative transverse Ising models can sustain steady-state spin-squeezing—an entanglement witness—due to the presence of a transverse field. Phase-space techniques (Wigner function, Fokker-Planck formalism) yield analytic predictions for collective spin fluctuations and squeezing parameters (Lee et al., 2013). In regimes of bistability, the covariance matrix structure implies macroscopic quantum jumps between distinct steady states, and at criticality fluctuation amplitudes diverge.

In driven-dissipative XY models, the range of bipartite entanglement can diverge as one tunes to the isotropic limit, but the actual magnitude of negativity vanishes—a singular limiting behavior well-captured by spin-wave theory (Joshi et al., 2013). This clarifies how non-equilibrium dissipation limits critical entanglement even as correlation lengths diverge.

6. Kibble-Zurek Scaling, Non-Hermitian Extensions, and Quantum Simulation

Under linear ramps through the quantum phase transition in a dissipative transverse Ising chain, defects (kinks) are generated according to Kibble-Zurek scaling, but the scaling exponent is modified relative to the isolated system. With Ohmic dissipation ($\alpha$), exponents shift from $\nu = 1$, $z = 1$ ($b=0.5$) to $\nu \approx 0.63$, $z \approx 2$ ($b \approx 0.28$) (Oshiyama et al., 2020). This slowdown due to the environment has direct experimental relevance for quantum annealers.

Non-Hermitian extensions, where dissipation is represented as a complex longitudinal or transverse field, reveal new classes of quantum dynamical phase transitions (Gustafson et al., 2023, Yan et al., 12 Dec 2024):

• The Lee-Yang edge singularity emerges as the locus of exceptional points in the complex field plane.
• In complex transverse-field Ising models (cTFIM), the interplay between unitary and dissipative dynamics can induce two-sided transitions with both first- and second-order characteristics, depending on the direction from which criticality is approached.
• The steady-state is maximally mixed, but nonlocal observables can show oscillatory exponential decay (gapless, long-range correlated phase) or pure exponential decay (gapped, ferromagnetic phase), with the transition point set by the balance of coherent and dissipative rates.

Digital quantum simulation protocols, employing Kraus channels and Trotterized evolution, demonstrate the physical reality of these non-Hermitian phase transitions on present-day quantum devices.

7. Experimental Realization and Outlook

Realistic platforms for the dissipative transverse Ising model include:

• Rydberg atom arrays and cold atoms with engineered loss and drive,
• Trapped ion simulators with all-to-all coupling and controlled dissipation,
• Superconducting qubit networks with tunable transverse fields and dissipative channels,
• Cavity and circuit QED systems, where the quantized transverse field is implemented via a global light-matter coupling, mapping onto a Dicke-Ising or quantum Rabi Hamiltonian (Rohn et al., 2020).

The impact of dissipation is manifest in features such as: phase boundary shifting, smearing of phase transitions, emergence of bistability and hysteresis, suppression or enhancement of critical exponents ($z$, $\nu$), onset of new universality classes, and the realization of non-Hermitian physics.

Key signatures—diverging relaxation times (Liouvillian gap closing), dynamical phase coexistence (intermittency in quantum-jump records), and the collapse of entanglement or coherence—are now observable in multiple platforms.

Further investigative directions include:

• The classification of multicritical and tricritical points in combined coherent/dissipative Ising-like systems,
• The integration of spatial inhomogeneity and disorder, leading to robust analytic solutions in infinite-range models (Roberts et al., 2023),
• Exploration of quantum information concepts (relaxation-free subspaces, topological protection) in open system settings.

The dissipative transverse Ising model thus remains a central touchstone for the paper of open-system quantum criticality, non-equilibrium universality, and the convergence of quantum simulation and quantum statistical mechanics.