Nonequilibrium steady state in Lindblad dynamics for infinite quantum spin systems

Technical Summary: Nonequilibrium Steady State in Lindblad Dynamics for Infinite Quantum Spin Systems

Problem Statement
The paper addresses the rigorous mathematical formulation of nonequilibrium steady states (NESS) and time-averaged nonequilibrium steady states (TANESS) for open quantum spin systems defined on infinite lattices. While NESS phases of matter are a subject of intense physical interest, their operator-algebraic definition in infinite systems remains underdeveloped compared to equilibrium states. A central difficulty arises from the non-commutativity of two fundamental limits: the thermodynamic limit (system size $\Lambda \to \Gamma$) and the long-time limit ($t \to \infty$ or time averaging). In finite systems, the NESS is typically defined as the kernel of the Liouvillian generator. However, on infinite lattices, the Liouvillian is an unbounded operator, making the definition of its kernel ambiguous. Furthermore, it is generally unclear whether the NESS of the infinite system coincides with the thermodynamic limit of the NESSs of finite subsystems.

Methodology
The authors employ a $C^*$-algebraic framework to describe quantum spin systems on a countable lattice $\Gamma$.

1. Dynamics: They utilize Lindblad dynamics, defined by a generator $\mathcal{L}$ acting on the quasi-local algebra $\mathcal{A}$. The existence of the thermodynamic limit for these dynamics is guaranteed by Lieb-Robinson bounds under specific locality assumptions (Assumption 2).
2. Definitions:
   • NESS ($\omega_{ss}$): Defined as a weak-$*$ cluster point of the net of evolved states $(\omega \circ \gamma^\Gamma_t)_{t>0}$.
   • TANESS ($\omega_{ss}^{ave}$): Defined as a weak-$*$ cluster point of the time-averaged states $(\frac{1}{T}\int_0^T \omega \circ \gamma^\Gamma_t dt)_{T>0}$.
3. Spectral Analysis: The core of the analysis involves studying the spectral properties of the Liouvillian $\mathcal{L}_\Lambda$ on finite subsystems $\Lambda$. The authors introduce three types of spectral gaps:
   • $\Delta_\Lambda$: The "line gap," measuring the distance of the non-zero spectrum from the imaginary axis.
   • $\Delta^p_\Lambda$: The "point gap," measuring the distance of the non-zero spectrum from the origin.
   • $\Delta^{ex}_\Lambda$: The "exceptional gap," measuring the distance of non-semisimple eigenvalues from the imaginary axis.
4. Condition Number: A critical component of the methodology is the introduction of a condition number $\kappa_{\Lambda'}(V^\epsilon_\Lambda)$. This quantifies the departure from normality of the Liouvillian by measuring the stability of the linear map $V^\epsilon_\Lambda$ that transforms the normalized Liouvillian into its Jordan canonical form.

Key Contributions and Results

1. Sufficient Conditions for Commutativity:
   The primary theoretical contribution is the establishment of sufficient conditions under which the thermodynamic limit and the long-time limit (or time average) commute.

   • Theorem 8 (NESS): If the Liouvillians on finite subsystems possess a uniform line gap ($\Delta_\Lambda \ge \Delta > 0$) and a uniform exceptional gap ($\Delta^{ex}_\Lambda \ge \Delta^{ex} > 0$), and the condition number $\kappa_{\Lambda'}$ is uniformly bounded from above, then the NESS of the infinite system is unique and equals the thermodynamic limit of the finite-system NESSs.
   • Theorem 10 (TANESS): Similarly, if a uniform point gap ($\Delta^p_\Lambda \ge \Delta^p > 0$) and a uniform exceptional gap exist, and the condition number is uniformly bounded, the TANESS is unique and commutes with the thermodynamic limit.

2. Role of the Condition Number:
   The paper demonstrates that spectral gaps alone are insufficient to guarantee the commutativity of limits. The condition number is shown to be a necessary constraint.

   • Counter-Example (Section 5): The authors construct a specific model (a spin chain with specific dissipative couplings) where the spectral gaps ($\Delta_\Lambda, \Delta^{ex}_\Lambda$) remain uniformly bounded away from zero for all finite subsystems. However, the condition number diverges as the system size increases. In this model, the thermodynamic limit and the long-time limit do not commute; the NESS obtained by taking the infinite limit first differs from the limit of the finite-system NESSs. This highlights that non-normality (captured by the condition number) can destabilize the steady state in the thermodynamic limit even when spectral gaps suggest stability.

3. Application to Onsite Liouvillians:
   The authors prove that for diagonalizable onsite Liouvillians (where the generator is a sum of single-site terms), the condition number is automatically uniformly bounded (Theorem 14). Consequently, for such systems, the commutativity of limits relies solely on the existence of spectral gaps (Corollary 18).

Significance
The paper provides a rigorous operator-algebraic definition of NESS and TANESS for infinite open quantum systems, bridging the gap between finite-system numerical/analytical approaches and infinite-system theoretical physics.

• It clarifies that the "thermodynamic limit of the NESS" is not a priori equivalent to the "NESS of the thermodynamic limit."
• It identifies the condition number as a central physical quantity governing the stability of steady states in the thermodynamic limit, alongside traditional spectral gaps. This is particularly relevant for non-Hermitian physics and open quantum systems where Liouvillians are often non-normal.
• The work establishes a framework for characterizing phases of matter in NESSs, noting that future work (cited as [46]) will investigate spontaneous symmetry breaking within this formalism.

The authors maintain a modest scope, focusing on the mathematical existence and uniqueness of these states under specific spectral and stability conditions, rather than proposing new experimental setups or broad physical applications beyond the theoretical characterization of the steady states themselves.