In quantum many-body systems subjected to generic perturbations, eigenstates are typically fragile and prone to strong mixing, leading to rapid thermalization in dynamics to an equilibrium state described by statistical mechanics. However, there exist robust counterexamples that defy this typical expectation, exhibiting novel non-equilibrium phenomena and offering potential applications in (quantum) information processing. This dissertation explores such systems and establishes their stability against perturbations with mathematical rigor. First, by generalizing the exponentially slow tunneling effect in single-particle quantum mechanics, we develop a theory of metastability in quantum many-body systems, demonstrating their nonperturbatively long lifetimes. As a canonical application, we provide the first general and tight bounds on the slow decay of false vacua. Second, it is known that ground states in finite spatial dimensions can resist arbitrary local perturbations through topologically ordered phases of matter. We extend this notion of stability to quantum error-correcting codes in infinite dimensions, specifically quantum low-density parity-check (LDPC) codes. These codes are of significant interest due to their promise of lower fault-tolerance overhead compared to finite-dimensional codes. Lastly, we analyze classical counterparts of LDPC codes and prove that their eigenstates remain localized in the many-body Hilbert space under generic perturbations. This finding presents the first unambiguous example of a robust violation of the eigenstate thermalization hypothesis. To achieve these results, we introduce advanced techniques to control operator locality, leveraging and generalizing methods such as Lieb-Robinson bounds, cluster expansions, and Schrieffer-Wolff transformations. These tools not only underpin our specific findings but may also prove valuable for broader challenges in mathematical physics of quantum many-body systems.
In quantum many-body systems subjected to generic perturbations, eigenstates are typically fragile and prone to strong mixing, leading to rapid thermalization in dynamics to an equilibrium state described by statistical mechanics. However, there exist robust counterexamples that defy this typical expectation, exhibiting novel non-equilibrium phenomena and offering potential applications in (quantum) information processing. This dissertation explores such systems and establishes their stability against perturbations with mathematical rigor. First, by generalizing the exponentially slow tunneling effect in single-particle quantum mechanics, we develop a theory of metastability in quantum many-body systems, demonstrating their nonperturbatively long lifetimes. As a canonical application, we provide the first general and tight bounds on the slow decay of false vacua. Second, it is known that ground states in finite spatial dimensions can resist arbitrary local perturbations through topologically ordered phases of matter. We extend this notion of stability to quantum error-correcting codes in infinite dimensions, specifically quantum low-density parity-check (LDPC) codes. These codes are of significant interest due to their promise of lower fault-tolerance overhead compared to finite-dimensional codes. Lastly, we analyze classical counterparts of LDPC codes and prove that their eigenstates remain localized in the many-body Hilbert space under generic perturbations. This finding presents the first unambiguous example of a robust violation of the eigenstate thermalization hypothesis. To achieve these results, we introduce advanced techniques to control operator locality, leveraging and generalizing methods such as Lieb-Robinson bounds, cluster expansions, and Schrieffer-Wolff transformations. These tools not only underpin our specific findings but may also prove valuable for broader challenges in mathematical physics of quantum many-body systems.
We present a mathematical theory of metastable pure states in closed many-body quantum systems with finite-dimensional Hilbert space. Given a Hamiltonian, a pure state is defined to be metastable when all sufficiently local operators either stabilize the state or raise its average energy. We prove that short-range-entangled metastable states are necessarily eigenstates (scars) of a perturbatively close Hamiltonian. Given any metastable eigenstate of a Hamiltonian, in the presence of perturbations, we prove the presence of prethermal behavior: Local correlation functions decay at a rate bounded by a timescale nonperturbatively long in the inverse metastability radius, rather than Fermi’s golden rule. Inspired by this general theory, we prove that the lifetime of the false vacuum in certain d -dimensional quantum models grows at least as fast as exp ( ε − d ) , where ε → 0 is the relative energy density of the false vacuum; this lower bound matches, for the first time, explicit calculations using quantum field theory. We identify metastable states at finite energy density in the PXP model, along with exponentially many metastable states in “helical” spin chains and the two-dimensional Ising model. Our inherently quantum formalism reveals precise connections between many problems, including prethermalization, robust quantum scars, and quantum nucleation theory, and applies to systems without known semiclassical and/or field-theoretic limits.
Quantum many body physics is an exciting research area, involving novel phases of matter with fundamentally new properties, but is also notoriously hard due to complexity of interacting quantum systems. Some popular approaches involve approximation techniques and numerical simulations, which are known to fail in several important cases. In contrast, rigorous mathematical tools, such as exact solutions and operator inequalities, have a narrower range of applicability, but provide guaranteed results and insights into the underlying physical mechanisms. This research aims to develop new techniques in this direction and use them to explore novel phases of matter. My first direction is to construct toy models where exact solutions are possible. Such models are important as they prove that certain physical phenomena are theoretically possible in nature, and often lead to the discovery of new phases. Such insights are provided by three exactly solvable models I have discovered: (1) a family of 1D quantum spin models hosting free parastatistical quasiparticles (an exotic type of identical particles beyond fermions, bosons, and anyons), proving for the first time that parastatistics is theoretically possible as an emergent phenomenon; (2) a 3D classical Ising model whose phases are characterized by topological features of certain loop observables, suggesting existence of previously unknown classical phases and phase transitions with topological order parameters; and (3) a family of models with exact p-wave superconducting ground states demonstrating the existence of Majorana quasiparticles and non-Abelian statistics in particle number-conserving systems. My second direction is to derive rigorous bounds and exact constraints on physical observables, which are applicable to large families of quantum many-body systems. I present three directions of progress: (1) a method that dramatically improves the upper bounds on the speed of information propagation in locally-interacting systems, which significantly extends the scope of these bounds and enables new applications; (2) bounds on finite-size errors in numerical simulations of lattice systems, including quench dynamics and gapped ground states; and (3) a locality bound on gapped ground states of power-law interacting systems, which leads to a generalization of the aforementioned error bounds to such systems. These error bounds have important theoretical implications such as proving the existence of the thermodynamic limit and stability of phases, and are practically useful in determining the validity of finite-size numerical simulations.
We generalize the proof of stability of topological order, due to Bravyi, Hastings, and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity-check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians H 0 defined by ⟦ N , K , d ⟧ LDPC codes, which obey certain topological quantum order conditions: (i) code distance d ≥ c log ( N ) , implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states—these include good quantum LDPC codes and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasilocal on the interaction graph defined by H 0 and that can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the O ( 1 ) smallest eigenvalues of H 0 . The band originating from the smallest eigenvalue has 2 K states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth δ = C N e − Θ ( d ) , which is tighter than the best-known bounds even in the Euclidean case. We also obtain that the new ground-state subspace is related to the initial-code subspace by a quasilocal unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we analytically demonstrate that systems satisfying the weak ETH exhibit thermalization for two very natural classes of far-from-equilibrium initial conditions: the overwhelming majority of all pure states with a preset nonequilibrium expectation value of some given local observable, and the Gibbs states of a Hamiltonian which subsequently is subject to a quantum quench in the form of a sudden change of some local system properties.
In recent decades, the rapid development of quantum technologies has led to a new era of programmable platforms, enabling the realizations of quantum simulation and quantum computation. This dissertation is motivated by recent experimental progress on controlling individual quantum degrees of freedom in systems such as trapped ions and Rydberg atom arrays. By tailoring the interactions in these quantum systems, we study analog quantum simulations of various physical phenomena, including non-equilibrium quantum dynamics and nontrivial topological physics. In the first part of the dissertation, we study slow quantum many-body dynamics in trapped-ion systems and Rydberg atom arrays. We first show that either the long-range interactions or an additional symmetry-breaking field can give rise to a confining potential. Such a potential can couple domain wall quasiparticles into mesonic or baryonic bound states. These confined quasiparticles strongly suppress the quantum information dynamics and lead to slow thermalization. In thelimit of strict domain-wall confinement, the full Hilbert space is fragmented into exponentially many disconnected subspaces. Furthermore, we demonstrate that thermalization can be halted by quantum engineering a uniformly increasing field in the trapped-ion quantum simulator. The second part of the dissertation focuses on topologically relevant phenomena in quantum simulators. We first study the effect of experimentally relevant disorder in 2D Rydberg atom arrays. We find that there are three distinct localization regimes due to the presence of nontrivial topological bands. We further study the non-equilibrium dynamics of Abelian anyons in a one-dimensional system. We show that the interplay of anyonic statistics and interactions can give rise to spatially asymmetric quantum dynamics. Finally, we use Nielsen’s geometric approach to quantify circuit complexity in topological models. We find that the circuit complexities of ground states and non-equilibrium steady states both exhibit nonanalytical behavior at topological transition points.
Perturbation theories provide valuable insights on quantum many-body systems. Systems of interacting particles, like electrons, are often treated perturbatively around exactly solvable Gaussian points. Systems of interacting qubits have gained increasing prominence as another class of models for quantum systems thanks to the recent advances in experimentally realizing mesoscopic quantum devices. Stabilizer states, innately defined on systems of qudits, have correspondingly emerged as another class of classically simulatable starting point for the study of quantum error-correcting codes and topological phases of matter in such devices. As a step towards analyzing more general quantum many-body problems on these platforms, we develop a systematic stabilizer perturbation theory in qubit systems. Our approach relies on the local Schrieffer-Wolff transformation, which we show can be efficiently performed through the binary encoding the Pauli algebra. As demonstrations, we first benchmark the stabilizer perturbation theory on the transverse field Ising chain in one dimension. The method is then further applied to Z2 toric code on square lattice and kagome lattice to probe the tendency toward confinement for anyonic excitations.