Recent advances in emergent geometry have identified a new class of models that represent spacetime as the graph obtained as the ground state of interacting Ising spins.These models have many desirable features, including stable excitations possessing many of the characteristics of a quantum particle.We analyze the dynamics of such excitations, including a detailed treatment of the edge states not previously addressed.Using a minimal prescription for the interaction of defects we numerically investigate approximate bounds to the speed of propagation of such a "particle".We discover, using numerical simulations, that there may be a Lieb-Robinson bound to propagation that could point the way to how a causal structure could be accommodated in this class of emergent geometry models.
Recent advances in emergent geometry have identified a new class of models that represent spacetime as the graph obtained as the ground state of interacting Ising spins.These models have many desirable features, including stable excitations possessing many of the characteristics of a quantum particle.We analyze the dynamics of such excitations, including a detailed treatment of the edge states not previously addressed.Using a minimal prescription for the interaction of defects we numerically investigate approximate bounds to the speed of propagation of such a "particle".We discover, using numerical simulations, that there may be a Lieb-Robinson bound to propagation that could point the way to how a causal structure could be accommodated in this class of emergent geometry models.
Many quantum lattice models have an emergent relativistic description in their continuum limit. The celebrated example is graphene, whose continuum limit is described by the Dirac equation on a Minkowski spacetime. Not only does the continuum limit provide us with a dictionary of geometric observables to describe the models with, but it also allows one to solve models that were otherwise analytically intractable. In this thesis, we investigate novel features of this relativistic description for a range of quantum lattice models. In particular, we demonstrate how to generate emergent curved spacetimes and identify observables at the lattice level which reveal this emergent behaviour, allowing one to simulate relativistic effects in the laboratory. We first study carbon nanotubes, a system with an edge, which allows us to test the interesting feature of the Dirac equation that it allows for bulk states with support on the edges of the system. We then study Kitaev's honeycomb model which has a continuum limit describing Majorana spinors on a Minkowski spacetime. We show how to generate a non-trivial metric in the continuum limit of this model and how to observe the effects of this metric and its corresponding curvature in the lattice observables, such as Majorana correlators, Majorana zero modes and the spin densities. We also discuss how lattice defects and Z2 gauge fields at the lattice level can generate chiral gauge fields in the continuum limit and we reveal their adiabatic equivalence. Finally, we discuss a chiral modification of the 1D XY model which makes the model interacting and introduces a non-trivial phase diagram. We see that this generates a black hole metric in the continuum limit, where the inside and outside of the black hole are in different phases. We then demonstrate that by quenching this model we can simulate Hawking radiation.
The conventional description of quantum mechanics treats the state of a system as a point in an abstract Hilbert space, and the evolution of that point is governed by the Schrödinger equation. In this work we propose that the geometry of the quantum state space itself possesses a nontrivial boundary structure that supports distinct dynamical laws. We introduce the concept of the Quantum State Edge Dynamics (QSED) and show that the boundary, denoted ∂S for a state space S, can host states that evolve according to modified equations of motion. By constructing a simple effective model, we demonstrate that the edge dynamics can give rise to protected, robust excitations that are insensitive to local perturbations, analogous to topological edge states in condensed matter systems. The existence of such edge dynamics suggests a new class of quantum states with potential applications in fault‑tolerant quantum computing and quantum simulation. Theoretical analysis indicates that the QSED framework may reveal previously unrecognized structures in quantum theory and could have profound implications for the interpretation of quantum mechanics. This paper provides a detailed mathematical formulation of QSED, explores its physical consequences, and outlines possible routes for experimental observation in engineered quantum systems.
This paper explores the fundamental nature of spacetime from the perspective of emergent causal structures, rooted in quantum mechanical principles. We propose a theoretical framework that treats spacetime not as a pre-existing background, but as a collective phenomenon arising from the interaction of quantum causal relations. By analyzing the constraints of causality and quantum coherence, we derive key implications for the emergence of classical spacetime geometry and the limits of local realism. Our results suggest a new way to bridge quantum mechanics and gravitational theory, providing a foundation for future studies in quantum gravity and spacetime physics.
Title:Interactions and Effective Forces Between Topological Defects in Emergent Schrödinger Dynamics Publication type:Preprint / Working Paper Description: This paper investigates the emergence of particle interactions in a classical wave system through topological defects. Building on previous work establishing an emergent Schrödinger equation in classical elastic media, we show how multiple particle-like excitations—identified as topological phase singularities—interact via phase-mediated mechanisms without introducing fundamental force laws. Key findings: Defect interactions arise automatically from the shared global wavefield, with no explicit interaction terms in the underlying wave equation. The interaction is long-ranged and geometry-driven, with like topological charges repelling and opposite charges attracting in a Coulomb-like manner. Inertia emerges from the distributed kinetic energy of the wavefield, yielding an effective mass that depends on the defect structure and system geometry. The dynamics exhibits deterministic scattering and bound-state behavior governed entirely by wave-mediated topology. Despite structural similarities to quantum mechanics, the framework remains fully classical and deterministic, with no superposition of trajectories, entanglement, or quantum statistics. Theoretical framework: The analysis is formulated within classical continuum mechanics, where an effective Schrödinger equation governs the envelope dynamics of elastic waves. Particle-like excitations correspond to phase singularities whose motion is constrained by a guidance equation. Multi-defect configurations are encoded within a single global wavefield whose phase topology captures all interactions. Significance: This work demonstrates how concepts typically regarded as fundamental—particles, mass, forces, and interactions—can emerge from continuous classical fields through topological constraints. The model provides a concrete and physically transparent example of emergence in physics, with implications for understanding: the relationship between wave and particle descriptions, geometric origins of effective forces, classical analogs of quantum-like behavior, and the role of topology in dynamical systems. Mathematical approach: The paper employs tools from: topological field theory (phase singularities, winding numbers), differential geometry (phase gradient and velocity fields), classical mechanics (effective mass, scattering dynamics), and continuum mechanics (elastic wave equations). Relation to existing work: This study completes a self-consistent framework connecting wave dynamics, particle motion, and interactions within classical physics. It relates to vortex dynamics in fluids, pilot-wave hydrodynamics, and emergent phenomena in condensed matter systems, while maintaining clear conceptual and physical distinctions from quantum mechanics.
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.
We introduce a framework in which spacetime locality and geometric causal structure are not assumed as fundamental, but instead emerge conditionally from quantum dynamics under restricted observational access. We define depth as a restriction on the set of observables accessible to an observer and show that, un- der appropriate conditions of redundancy and decoherence, effective locality and finite propagation speed arise from intrinsically nonlocal quantum systems. Con- versely, locality degrades when these conditions fail. Using simple toy models, we demonstrate threshold behavior separating regimes with and without stable causal structure. This approach reframes spacetime geometry as an emergent description rather than a primitive input, while remaining compatible with standard quantum mechanics and general relativity within their respective domains of validity.