Hydrodynamic relaxation in open quantum many-body systems can be understood in terms of diffusive Nambu--Goldstone (NG) modes associated with strong-to-weak spontaneous symmetry breaking (SWSSB). Here, we find a common structure underlying weak ergodicity breaking in closed quantum systems and diffusive hydrodynamics in open quantum systems. We establish a spectral correspondence between the parent Hamiltonian of rainbow quantum many-body scar (RQMBS) states in closed systems and the effective Lindbladian of strongly dissipative open systems with strong $U(1)$ symmetry and local charge-dephasing Lindblad operators. For broad classes of spin, fermionic, and bosonic systems, we construct asymptotic RQMBS (ARQMBS) states. Within the enlarged scar subspace, the parent Hamiltonian admits a local Rokhsar--Kivelson-type frustration-free decomposition. A single-mode variational construction gives an $O(k^2)$ upper bound on its lowest excitation energy and yields states satisfying the defining ARQMBS criteria: orthogonality, vanishing energy variance, and characteristic entanglement scaling. Under vectorization, RQMBS states are mapped to maximally mixed infinite-temperature states in fixed-charge sectors, which exhibit long-range SWSSB order under finite-density conditions. Assuming uniformly bounded positive transition rates, the effective Lindbladian has the same local projectors as the parent Hamiltonian but with positive transition-dependent weights, yielding two-sided spectral bounds and the same system-size scaling of their gaps. For uniform rates, the two operators are proportional to each other and their eigenmodes coincide. Therefore, an exact quadratic parent-Hamiltonian branch maps directly to a diffusive NG mode. Our results connect different mechanisms of slow relaxation in closed and open quantum many-body systems.
Hydrodynamic relaxation in open quantum many-body systems can be understood in terms of diffusive Nambu--Goldstone (NG) modes associated with strong-to-weak spontaneous symmetry breaking (SWSSB). Here, we find a common structure underlying weak ergodicity breaking in closed quantum systems and diffusive hydrodynamics in open quantum systems. We establish a spectral correspondence between the parent Hamiltonian of rainbow quantum many-body scar (RQMBS) states in closed systems and the effective Lindbladian of strongly dissipative open systems with strong U(1) symmetry and local charge-dephasing Lindblad operators. For broad classes of spin, fermionic, and bosonic systems, we construct asymptotic RQMBS (ARQMBS) states. Within the enlarged scar subspace, the parent Hamiltonian admits a local Rokhsar--Kivelson-type frustration-free decomposition. A single-mode variational construction gives an O(k2) upper bound on its lowest excitation energy and yields states satisfying the defining ARQMBS criteria: orthogonality, vanishing energy variance, and characteristic entanglement scaling. Under vectorization, RQMBS states are mapped to maximally mixed infinite-temperature states in fixed-charge sectors, which exhibit long-range SWSSB order under finite-density conditions. Assuming uniformly bounded positive transition rates, the effective Lindbladian has the same local projectors as the parent Hamiltonian but with positive transition-dependent weights, yielding two-sided spectral bounds and the same system-size scaling of their gaps. For uniform rates, the two operators are proportional to each other and their eigenmodes coincide. Therefore, an exact quadratic parent-Hamiltonian branch maps directly to a diffusive NG mode. Our results connect different mechanisms of slow relaxation in closed and open quantum many-body systems.
Inspired by the hunt for new phases of matter in quantum mixed states, it has recently been proposed that the equivalence of microcanonical and canonical ensembles in statistical mechanics is a manifestation of strong-to-weak spontaneous symmetry breaking (SWSSB) in an underlying many-body quantum description. Here, we build an effective field theory for SWSSB of a global U(1) symmetry; the answer exactly reproduces the Schwinger-Keldysh effective field theory of diffusion for the conserved charge. We conclude that hydrodynamics can be understood as a theory of superfluidity for the broken strong symmetry: a nonvanishing susceptibility is a measurable order parameter for SWSSB, the diffusion mode is the Goldstone boson of the spontaneously broken continuous symmetry, and a generalization of Goldstone's theorem implies that the diffusion mode is always long lived. This perspective provides a transparent physical explanation for the unusual reparameterization symmetries, which are a necessary ingredient of Schwinger-Keldysh effective field theories for normal fluids.
Many-body localisation, Hilbert space fragmentation and quantum many-body scars (QMBS) are well-established mechanisms of ergodicity breaking in quantum systems. Recently, asymptotic quantum many-body scars (AQMBS), non-thermal states whose stability grows with system size, have been proposed theoretically and probed experimentally. Here, we show that AQMBS arise naturally as remnants of certain quantum phase transitions. For a broad class of frustration-free Hamiltonians, we prove that their gapless excitations survive deformation into a scarred Hamiltonian as states whose energy variance is bounded by the excitation gap. Consequently, the gapless excitations at a quantum critical point become AQMBS of the scarred Hamiltonian. Their fidelity relaxation time is then set by the critical scaling of the gap, determined by the dynamic critical exponent. Modelling the critical Fredkin and Motzkin chains and the two-dimensional PVBS model, we obtain AQMBS whose variances directly reflect the dynamic exponents of the underlying transitions. Where the gap of the Motzkin chain closes super-exponentially, we find exponentially stable AQMBS, for which we also observe numerical evidence of volume-law entanglement.
Construction of asymptotic quantum many-body scar states in the SU(N) Hubbard model
We construct asymptotic quantum many-body scars (AQMBS) in one-dimensional SU(N) Hubbard chains (N≥3) by embedding the scar subspace into an auxiliary Hilbert subspace HP and identifying a parent Hamiltonian within it, together with a corresponding extension of the restricted spectrum-generating algebra to the multi-ladder case. Unlike previous applications of the parent-Hamiltonian scheme, we show that the parent Hamiltonian becomes the SU(N) ferromagnetic Heisenberg model rather than the spin-1/2 case, so that its gapless magnons realize explicit AQMBS of the original model. Working in the doublon-holon subspace, we derive this mapping, obtain the one-magnon dispersion for periodic and open boundaries, and prove (i) orthogonality to the scar states, (ii) vanishing energy variance in the thermodynamic limit, and (iii) subvolume entanglement entropy with rigorous MPS/MPO bounds. Our results broaden the parent-Hamiltonian family for AQMBS beyond spin-1/2 and provide analytic, low-entanglement excitations in SU(N)-symmetric systems.
Strong-to-weak spontaneous symmetry breaking (SWSSB) defines a mixed-state phase of matter--without a pure-state counterpart--in which nonlinear observables such as the Rényi-2 correlator develop long-range order while conventional linear correlations remain short-ranged. Here we study the emergence of SWSSB in one-dimensional open quantum systems governed by Lindbladian evolution, where the transition time diverges with system size and SWSSB appears only asymptotically in the steady state. By tracking the late-time growth of the Rényi-2 correlation length, we uncover a universal dynamical regime controlled purely by the symmetry class of the Lindbladian. Contrary to the conventional expectation that late-time dynamics are governed by the low-lying Liouvillian spectrum, we find that the time dependence of the SWSSB transition--exponential versus algebraic--is dictated solely by symmetry, independent of details of the Lindbladian, including whether the Liouvillian spectrum is gapped or gapless. For Z2-symmetric dynamics, the Rényi-2 correlation length grows exponentially in time--even when the spectrum is gapless--yielding an effective transition time tc∝lnL and enabling rapid preparation of the Z2 SWSSB steady state. In contrast, U(1)-symmetric dynamics exhibit algebraic scaling, tc∝Lα, with a filling-dependent dynamical exponent: ballistic growth (α≈1) at finite filling crosses over to diffusive scaling (α=2) in the zero-filling limit. These results establish symmetry--rather than spectral gap structure--as the controlling principle for SWSSB late-time dynamical scaling, and open a new route to nonequilibrium symmetry breaking in open quantum systems.
Abstract Understanding how dissipation affects quantum many-body dynamics is crucial for quantum information processing, yet explicit quantitative predictions remain scarce. While closed quantum systems with spectral gaps exhibit robust dynamical properties, the transition to open systems governed by Lindblad dynamics has lacked constructive bounds relating Hamiltonian gaps to Liouvillian relaxation timescales. Scientific novelty. We derive the first explicit, parameter-dependent bound connecting a constructive Hamiltonian gap to the Liouvillian spectral gap in weakly dissipative quantum lattice systems. Starting from a frustration-free Hamiltonian with Nemirovsky gate gap Δgate = m0 − 2zgVmax > 0 and adding local Lindblad dissipation with strength γ ≪ Δgate, we prove that the Liouvillian gap satisfies ΛL ≥ Δgate − 2π·z·d·γ·Vdiss with explicit constant C = 2π, where z is the lattice coordination number, d is the number of dissipators per site, and Vdiss bounds the dissipator norms. This inequality provides quantitative control over open-system dynamics directly from microscopic parameters, enabling predictions for finite systems without asymptotic limits. We further establish a dissipative spectral weight bound WAdiss(E) ≤ K(A)·min(1, E/Λeff), where Λeff = Δgate − O(γ) characterizes the effective gap reduction. This directly yields the relaxation time lower bound τrelax ≥ ℏ/Λeff, with immediate applications to Rydberg atom arrays, trapped ions, and neutral atom quantum processors. Our results bridge constructive Hamiltonian spectral theory and open quantum dynamics, providing experimentalists with concrete design criteria for quantum gate timing, many-body scar lifetimes, and error correction protocols. Unlike previous existence theorems, our bounds are fully constructive, perturbative, and applicable to experimentally relevant parameter regimes. Keywords: Open quantum systems, Liouvillian gap, spectral bounds, Lindblad dynamics, dissipative quantum many-body systems, relaxation times, quantum simulators, Rydberg atoms, constructive quantum mechanics
Symmetry algebras of quantum many-body systems with locality can be understood using commutant algebras, which are defined as algebras of operators that commute with a given set of local operators. In this work, we show that these symmetry algebras can be expressed as frustration-free ground states of a local superoperator, which we refer to as a “super-Hamiltonian.” We demonstrate this for conventional symmetries such as Z 2 , U ( 1 ) , and S U ( 2 ) , where the symmetry algebras map to various kinds of ferromagnetic ground states, as well as for unconventional ones that lead to weak ergodicity-breaking phenomena of Hilbert-space fragmentation (HSF) and quantum many-body scars. In addition, we show that the low-energy excitations of this super-Hamiltonian can be understood as approximate symmetries, which in turn are related to slowly relaxing hydrodynamic modes in symmetric systems. This connection is made precise by relating the super-Hamiltonian to the superoperator that governs the operator relaxation in noisy symmetric Brownian circuits and this physical interpretation also provides a novel interpretation for Mazur bounds for autocorrelation functions. We find examples of gapped (gapless) super-Hamiltonians indicating the absence (presence) of slow modes, which happens in the presence of discrete (continuous) symmetries. In the gapless cases, we recover hydrodynamic modes such as diffusion, tracer diffusion, and asymptotic scars in the presence of U ( 1 ) symmetry, HSF, and a tower of quantum scars, respectively. In all, this demonstrates the power of the commutant-algebra framework in obtaining a comprehensive understanding of exact symmetries and associated approximate symmetries and hydrodynamic modes, and their dynamical consequences in systems with locality. Published by the American Physical Society 2024