Developing a local description of quantum information is a central problem in quantum many-body physics: the goal is to identify how correlations are organized across positions and scales without choosing specific observables. We introduce a higher-dimensional information lattice, an inclusion-exclusion construction that decomposes the von Neumann information of reduced density matrices into contributions labeled by subsystem position and scale. Above one dimension, overlapping subsystems can form loops, allowing information associated with one subsystem to be inferred nonlocally from the density matrices of others. The sign of the inclusion-exclusion information resolves such overlap redundancies in position and scale: positive values mark positions and scales with net information gain, while negative values occur where redundant contributions dominate new ones. We implement the construction in two dimensions and apply it to many-body ground states with distinct entanglement structures. For localized states, such as disordered-localized states or gapped ground states of local Hamiltonians, the local information decays exponentially along different directions, defining direction-dependent information length scales. In critical states with a Fermi surface, decay becomes algebraic, with dominant directions tied to the velocity of gapless modes. In the toric code, topological order manifests as a negative information value at the largest lattice scale, interpreted as global information redundancy between the smallest loop-enclosing overlapping subsystems. We also isolate the universal information carried by edge states and locate the signatures of non-Abelian fusion channels in position and scale. Our results establish the higher-dimensional information lattice as a framework for resolving universal correlation structures through the spatial and scale distribution of quantum information.
Developing a local description of quantum information is a central problem in quantum many-body physics: the goal is to identify how correlations are organized across positions and scales without choosing specific observables. We introduce a higher-dimensional information lattice, an inclusion-exclusion construction that decomposes the von Neumann information of reduced density matrices into contributions labeled by subsystem position and scale. Above one dimension, overlapping subsystems can form loops, allowing information associated with one subsystem to be inferred nonlocally from the density matrices of others. The sign of the inclusion-exclusion information resolves such overlap redundancies in position and scale: positive values mark positions and scales with net information gain, while negative values occur where redundant contributions dominate new ones. We implement the construction in two dimensions and apply it to many-body ground states with distinct entanglement structures. For localized states, such as disordered-localized states or gapped ground states of local Hamiltonians, the local information decays exponentially along different directions, defining direction-dependent information length scales. In critical states with a Fermi surface, decay becomes algebraic, with dominant directions tied to the velocity of gapless modes. In the toric code, topological order manifests as a negative information value at the largest lattice scale, interpreted as global information redundancy between the smallest loop-enclosing overlapping subsystems. We also isolate the universal information carried by edge states and locate the signatures of non-Abelian fusion channels in position and scale. Our results establish the higher-dimensional information lattice as a framework for resolving universal correlation structures through the spatial and scale distribution of quantum information.
During time evolution of many-body systems entanglement grows rapidly, limiting exact simulations to small-scale systems or small timescales. Quantum information tends, however, to flow towards larger scales without returning to local scales, such that its detailed large-scale structure does not directly affect local observables. This allows for the removal of large-scale quantum information in a way that preserves all local observables and gives access to large-scale and large-time quantum dynamics. To this end, we use the recently introduced to organize quantum information into different scales, allowing us to define and that we employ to systematically discard long-range quantum correlations in a controlled way. Our approach relies on decomposing the system into subsystems up to a maximum scale and time evolving the subsystem density matrices by solving the subsystem von Neumann equations in parallel. Importantly, the information flow needs to be preserved during the discarding of large-scale information. To achieve this without the need to make assumptions about the microscopic details of the information current, we introduce a second scale at which information is discarded, while using the state at the maximum scale to accurately obtain the information flow. The resulting algorithm, which we call local-information time evolution, is highly versatile and suitable for investigating many-body quantum dynamics in both closed and open quantum systems with diverse hydrodynamic behaviors. We present results for the energy transport in the mixed-field Ising model and the magnetization transport in the X X spin chain with onsite dephasing where we accurately determine the power-law exponent and the diffusion coefficients. Furthermore, the information lattice framework employed here promises to offer insightful results about the spatial and temporal behavior of entanglement in many-body systems.
Quantum many-body dynamics generically result in increasing entanglement that eventually leads to thermalization of local observables. This makes the exact description of the dynamics complex despite the apparent simplicity of (high-temperature) thermal states. For accurate but approximate simulations one needs a way to keep track of essential (quantum) information while discarding inessential one. To this end, we first introduce the concept of the information lattice, which supplements the physical spatial lattice with an additional dimension and where a local Hamiltonian gives rise to well-defined locally conserved von Neumann information current. This provides a convenient and insightful way of capturing the flow, through time and space, of information during quantum time-evolution, and gives a distinct signature of when local degrees of freedom decouple from long-range entanglement. As an example, we describe such decoupling of local degrees of freedom for the mixed-field transverse Ising model. Building on this, we secondly construct algorithms to time-evolve sets of local density matrices without any reference to a global state. With the notion of information currents, we motivate algorithms based on the intuition that information for statistical reasons flows from small to large scales. Using this guiding principle, we construct an algorithm that, at worst, shows two-digit convergence in time-evolutions up to very late times for diffusion process governed by the mixed-field transverse Ising Hamiltonian. While we focus on dynamics in 1D with nearest-neighbor Hamiltonians, the algorithms do not essentially rely on these assumptions and can in principle be generalized to higher dimensions and more complicated Hamiltonians.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
Unlike unitary dynamics, measurements of a subsystem can induce long-range entanglement via quantum teleportation. The amount of measurement-induced entanglement or mutual information depends jointly on the measurement basis and the entanglement structure of the state (before measurement), and has operational significance for whether the state is a resource for measurement-based quantum computing, as well as for the computational complexity of simulating the state using quantum or classical computers. In this paper, we examine entropic measures of measurement-induced entanglement (MIE) and information (MII) for the ground states of quantum many-body systems in one and two spatial dimensions. From numerical and analytic analysis of a variety of models encompassing critical points, quantum Hall states, string-net topological orders, and Fermi liquids, we identify universal features of the long-distance structure of MIE and MII that depend only on the underlying phase or critical universality class of the state. We argue that, whereas in 1D the leading contributions to long-range MIE and MII are universal, in 2D, the existence of a teleportation transition for finite-depth circuits implies that trivial 2D states can exhibit long-range MIE, and the universal features lie in subleading corrections. We introduce modified MIE measures that directly extract these universal contributions. As a corollary, we show that the leading contributions to strange correlators, used to numerically identify topological phases, are in fact nonuniversal in two or more dimensions, and explain how our modified constructions enable one to isolate universal components. We discuss the implications of these results for classical- and quantum- computational simulation of quantum materials.
There are many ways to characterize quantum systems, for example, by measuring system-specific observables or probing their response to external perturbations. A complementary approach is based on quantum information measures. Independent of the microscopic details, it offers a universal perspective on quantum systems and enables a unified characterization of diverse quantum systems. Central, in this regard, is the determination of the quantum entanglement within a quantum system. Recently, in the interest of studying genuine nonequilibrium properties of a quantum system, the dynamics of the quantum entanglement have been established as a powerful approach to gain deep insights into the structure of quantum matter. In this Thesis, we investigate the entanglement dynamics in one-dimensional non-interacting fermionic chains following a quantum quench, i.e. after a sudden change of the system parameters. We propose a new and generic setting for the analysis of entanglement spreading in complex inhomogeneous systems, which we call a hybrid quench-probe setup. The key feature of this new approach is a separation between the quenched part of the system and a static probe region, where the quench induced entanglement spreading is measured. This separation allows a spatial and energy-resolved analysis of the entanglement dynamics. We demonstrate the capability of our protocol by identifying fractionally quantized entanglement jumps of log(2)/2 as a signature of fractionalized topological quasi-particles, such as isolated Majorana modes, that appear at the edges of an open-boundary Kitaev chain in the topological phase. In addition, we observe in this setup an interesting nonequilibrium effect: a static and a propagating contribution to the entanglement dynamics. To understand the origin of this effect, a local perspective on the entanglement structure is necessary. For this purpose, we employ the information lattice, which offers a local scale-resolved decomposition of the correlations present in a quantum state. In nonequilibrium systems with unitary dynamics, this framework enables a hydrodynamic description of the information flow through local densities and currents. Using the information lattice, we analyze three quantum quenches in one-dimensional fermionic systems and identify spatiotemporal features of the local information flow, such as information interfaces and long-range transport from topological edge modes. This analysis enables us to derive an analytical description of the nonequilibrium effect observed in the quenched Kitaev chain.
Abstract The nature of the famed long-range entanglement of quantum many-body systems is the paradoxical coexistence of short-range correlation and nonlocal information which cannot be encoded through constant-depth local quantum circuits. This coexistence is only proved in topologically ordered states and their extensions in 2D or higher dimensions, and is in charge of their fundamental importance in fault-tolerant quantum computation. Here, we broaden the knowledge of long-range entangled states beyond the manifold-topology picture by proving the coexistence in a new concrete example in addition to those well-known ones like the toric-code ground state. The new long-range entangled state reveals an intrinsically different paradigm, it describes quantum matter on the newly experimentally realized fractal lattice geometry in 1.58D where the topological framework is negated, but a self-similarity structure shapes how the nonlocal information is encoded. The new paradigm portrays an exotic quantum order and uncovers the potential of fractional-dimension quantum states in studies relevant to quantum computation.