Theoretical research into many-body quantum systems has mostly focused on regular structures which have a small, simple unit cell and where a vanishingly small fraction of the pairs of the constituents directly interact. Motivated by advances in control over the pairwise interactions in many-body simulators, we determine the fate of spin systems on more general, arbitrary graphs. Placing the minimum possible constraints on the underlying graph, we prove how, with certainty in the thermodynamic limit, such systems behave like a single collective spin. We thus understand the emergence of complex many-body physics as dependent on 'exceptional', geometrically constrained structures such as the low-dimensional, regular ones found in nature. Within the space of dense graphs we identify hitherto unknown exceptions via their inhomogeneity and observe how complexity is heralded in these systems by entanglement and highly non-uniform correlation functions. Our work paves the way for the discovery and exploitation of a whole class of geometries which can host uniquely complex phases of matter.
Theoretical research into many-body quantum systems has mostly focused on regular structures which have a small, simple unit cell and where a vanishingly small fraction of the pairs of the constituents directly interact. Motivated by advances in control over the pairwise interactions in many-body simulators, we determine the fate of spin systems on more general, arbitrary graphs. Placing the minimum possible constraints on the underlying graph, we prove how, with certainty in the thermodynamic limit, such systems behave like a single collective spin. We thus understand the emergence of complex many-body physics as dependent on 'exceptional', geometrically constrained structures such as the low-dimensional, regular ones found in nature. Within the space of dense graphs we identify hitherto unknown exceptions via their inhomogeneity and observe how complexity is heralded in these systems by entanglement and highly non-uniform correlation functions. Our work paves the way for the discovery and exploitation of a whole class of geometries which can host uniquely complex phases of matter.
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.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
Quantum entanglement manifests itself in non-local correlations between the constituents of a system. In its simplest realization, a measurement on one subsystem is affected by a prior measurement on its partner, irrespective of their separation. For multiple parties, purely collective types of entanglement exist but their detection, even theoretically, remains an outstanding open question. Here, we argue that all forms of multipartite entanglement entirely disappear during the typical evolution of a physical state as it heats up, evolves in time in a large family of dynamical protocols, or as its parts become separated. We focus on the generic case where the system interacts with an environment. These results mainly follow from the geometry of the entanglement-free continent in the space of physical states, and hold in great generality. We illustrate these phenomena with a frustrated molecular quantum magnet in and out of equilibrium, and a quantum spin chain. In contrast, if the particles are fermions, such as electrons, another notion of entanglement exists that protects bipartite quantum correlations. However, genuinely collective fermionic entanglement disappears during typical evolution, thus sharing the same fate as in bosonic systems. These findings provide fundamental knowledge about the structure of entanglement in quantum matter and architectures, paving the way for its manipulation.
Quantum many-body systems are comprised of complex networks of microscopic interactions that work together to produce novel collective phases and phenomena. \nFor strongly-interacting systems, the connection between the underlying atomic-scale behaviour and the resulting emergent observable phenomena is exponentially (and, in practice, infinitely) complicated such that it becomes impossible to fully understand the connection between the physics at the microscopic and macroscopic levels. \nThe most promising theoretical approaches to addressing this infinite complexity utilize microscopic coarse-graining along with powerful computational simulations and strategies. \nIn this thesis, we apply numerical methods to effective lattice models and field theories with the goal of shedding light onto universal critical behaviour and exotic low-temperature phases in condensed matter physics. \n \nWe start by exploring universal features of critical non-interacting systems, for which several analytical strategies are readily available for calculating certain observables. \nWe study the behaviour of the system's entanglement for various entangling geometries, and utilize numerical techniques in order to isolate new universal features of Gaussian fixed points, which provide insight into the underlying critical theory's renormalization group flow. \nWe then proceed to examine computational strategies and, in particular, Monte Carlo simulations for studying more general interacting models. \nWe focus on the application of such strategies to the field of high-temperature superconductivity, for which we develop a coarse-grained model, simulate macroscopic observables and compare our results with those of recent experiments. \nWe conclude that lattice field theories together with innovative computational methods offer new perspectives on both universality and emergent phenomena in quantum matter.
Projected entangled pair states (PEPS) are a class of quantum many-body states that generalize matrix product states for one-dimensional systems to higher dimensions. In recent years, PEPS have advanced understanding of strongly correlated systems, especially in two dimensions, e.g., quantum spin liquids. Typically described by tensor networks on regular lattices (e.g., square, cubic), PEPS have also been adapted for irregular graphs; however, the computational cost becomes prohibitive for dense graphs with large vertex degrees. In this paper, we present a PEPS algorithm to simulate low-energy states and dynamics defined on arbitrary, fluctuating, and densely connected graphs. We introduce a cut-off, κ ∈ N , to constrain the vertex degree of the PEPS to a set but tunable value, which is enforced in the optimization by applying a simple edge-deletion rule, allowing the geometry of the PEPS to change and adapt dynamically to the system's correlation structure. We benchmark our flexible PEPS algorithm with simulations of classical spin glasses and quantum annealing on densely connected graphs with hundreds of spins, and also study the impact of tuning κ when simulating a uniform quantum spin model on a regular (square) lattice. Our paper opens the way to apply tensor network algorithms to arbitrary, even fluctuating, background geometries.
Quantum mechanics led us to reconsider the scope of physics and its building principles, such as the notions of realism and locality. More recently, quantum theory has changed in an equally dramatic manner our understanding of information processing and computation. On one hand, the fundamental properties of quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure way than possible in the realm of classical physics. On the other hand, the development of systematic procedures to manipulate systems of a large number of particles in the quantum regime, crucial to the implementation of quantum-based information processing, has triggered new possibilities in the exploration of quantum many-body physics and related areas. In this thesis, we present new results relevant to two important problems in quantum information science: the development of a theory of entanglement, intrinsically quantum correlations, and the exploration of the use of controlled quantum systems to the computation and simulation of quantum many-body phenomena. In the first part we introduce a new approach to the study of entanglement