In strongly correlated quantum systems, the presence of conserved quantities beyond the obvious symmetries often leads to unexpected robustness of certain phases and excitations. We propose a systematic framework to search for hidden conserved operators, denoted Q_h, that commute with the full Hamiltonian H of a complex material. The core idea is that such hidden conservation laws may underpin anomalously stable states that defy conventional classification, potentially opening new avenues for topological quantum computation and material design. We formulate the problem as a constrained eigenvalue optimization, derive analytic expressions for candidate Q_h in representative lattice models, and introduce an iterative operator construction algorithm that exploits the algebraic structure of the spin, fermionic, and bosonic operators involved. By applying the method to a set of benchmark systems—spin chains with next‑nearest‑neighbour interactions, Hubbard‑type models with spin–orbit coupling, and coupled spin‑boson lattices—we demonstrate the emergence of non‑trivial conserved operators that are not linear combinations of known symmetries. The discovery of these hidden laws not only clarifies the stability of exotic phases such as quantum spin liquids and fractional topological insulators but also suggests that a deeper symmetry principle may underlie many unexplained experimental observations. Finally, we discuss the implications of our findings for future theoretical developments and the pursuit of a Nobel‑level breakthrough in the field of quantum materials.
In strongly correlated quantum systems, the presence of conserved quantities beyond the obvious symmetries often leads to unexpected robustness of certain phases and excitations. We propose a systematic framework to search for hidden conserved operators, denoted Q_h, that commute with the full Hamiltonian H of a complex material. The core idea is that such hidden conservation laws may underpin anomalously stable states that defy conventional classification, potentially opening new avenues for topological quantum computation and material design. We formulate the problem as a constrained eigenvalue optimization, derive analytic expressions for candidate Q_h in representative lattice models, and introduce an iterative operator construction algorithm that exploits the algebraic structure of the spin, fermionic, and bosonic operators involved. By applying the method to a set of benchmark systems—spin chains with next‑nearest‑neighbour interactions, Hubbard‑type models with spin–orbit coupling, and coupled spin‑boson lattices—we demonstrate the emergence of non‑trivial conserved operators that are not linear combinations of known symmetries. The discovery of these hidden laws not only clarifies the stability of exotic phases such as quantum spin liquids and fractional topological insulators but also suggests that a deeper symmetry principle may underlie many unexplained experimental observations. Finally, we discuss the implications of our findings for future theoretical developments and the pursuit of a Nobel‑level breakthrough in the field of quantum materials.
Quantum many‑body systems often display emergent stability that cannot be explained by any obvious symmetry of the microscopic Hamiltonian. In this work we propose a systematic framework, the Hidden Conservation Law Search in Quantum Matter (HCLQM), for uncovering unknown conserved quantities Qh that may underlie such anomalous behavior. We develop an algebraic criterion that any operator Qh must satisfy in order to commute with the full Hamiltonian H while not belonging to the algebra generated by known symmetries. Using this criterion we design a numerical search algorithm based on tensor‑network representations of the Hilbert space, enabling us to explore large Hilbert spaces efficiently. We apply HCLQM to three representative systems: a one‑dimensional spin‑1/2 XXZ chain, a two‑dimensional Chern insulator with disorder, and a Hubbard‑type strongly correlated lattice model. In each case we discover non‑trivial conserved operators that explain the presence of long‑lived excited states, protected edge modes, and anomalous transport properties. These findings demonstrate that hidden conserved quantities can be systematically identified and may provide a route to discovering new symmetries in quantum materials, potentially leading to Nobel‑level breakthroughs.
Quantum materials exhibit a wealth of emergent phenomena that arise from the complex interplay of electron correlations, spin dynamics, lattice vibrations, and external perturbations. Conventional analyses attribute observed responses to changes in the conventional spatial dimensions (x, y, z) and to well‑known order parameters such as magnetization, polarization, or superconducting gap. In this work we propose a new conceptual framework in which the response functions of quantum matter are expanded to include a hidden internal variable, denoted xi, that encapsulates an additional degree of freedom not captured by the spatial coordinates. The central hypothesis is that many observed anomalies, including unconventional superconductivity, quantum spin liquids, and topological phase transitions, can be understood as manifestations of a hidden dimensional response (HDR) mechanism. We formulate the response function R as a function of (x, y, z, xi), derive its implications for symmetry properties, and discuss possible experimental signatures that could confirm the existence of xi. This framework offers a unified viewpoint for interpreting a broad class of phenomena and suggests new avenues for material design and discovery.
This paper introduces the concept of Hidden Causal Order in Quantum Matter (HCOQM), a novel framework proposing that within strongly correlated quantum materials, a previously unrecognized causal structure exists, operating beneath the conventional, empirically-derived descriptions. We posit that this 'hidden' causality isn't simply a consequence of complex interactions, but a fundamental organizing principle. This framework seeks to provide a new classification of quantum phases, moving beyond traditional approaches based solely on measured properties. The core concept is represented by the notation [C(x)], where C(x) denotes the causal relationship at position x. While experimental validation remains a significant challenge, HCOQM offers a theoretical lens through which to interpret complex quantum phenomena and potentially unlock the discovery of entirely new quantum phases. The paper explores the theoretical underpinnings of HCOQM and outlines potential avenues for future investigation.
In many-body quantum systems the conventional picture of energy transport assumes that excitations propagate through well‑defined quasiparticle channels governed by the system Hamiltonian. Recent theoretical investigations suggest that in strongly correlated materials there exist additional, hidden pathways for energy exchange that are not captured by standard quasiparticle descriptions. These hidden energy channels (HECs) arise from collective reconfigurations of the entanglement structure, mediated by emergent gauge fields and topological constraints. In this paper we formulate a unified framework to identify and characterize HECs in generic lattice models. We derive explicit expressions for the energy current operator that include both conventional and hidden contributions, and demonstrate how HECs modify the temperature dependence of thermal conductivity in prototypical models such as the two‑dimensional Hubbard model and the Kitaev honeycomb model. Our analysis predicts anomalous scaling laws and non‑monotonic temperature profiles that can serve as experimental signatures of HECs. The existence of these channels opens a new avenue for engineering efficient energy transport in quantum materials, with implications for quantum information processing and low‑temperature thermoelectric devices.
Symmetry is foundational in theoretical physics. In this thesis, we show how recent advances in symmetry have reinforced its power to classify, characterize, and constrain quantum systems non-perturbatively. We study symmetries, ranging from spatial symmetries to generalized symmetries, in both quantum field theory and quantum lattice systems. In particular, this thesis develops five sets of results. First, we show that ordinary ordered phases generically host emergent generalized symmetries in the infrared, associated with homotopy defects of the order-parameter space. We use these emergent generalized symmetries to characterize and classify defect-suppressed disorder transitions to phases with topological order and emergent gauge structures. Second, we turn to lattice anomalies and show how the chiral anomaly of a massless Dirac fermion in two spacetime dimensions can be realized on the lattice. In particular, we identify lattice realizations of the vector and axial symmetries and demonstrate how their anomaly is matched by the non-abelian Onsager algebra in the staggered fermion Hamiltonian. Third, we give a microscopic lattice understanding of T-duality in quantum spin chains using non-invertible symmetry. This non-invertible symmetry, together with lattice momentum and winding symmetries, also enforces gaplessness, hence providing a novel way to realize perturbative anomalies in quantum spin-chain models. Fourth, we show that in local lattice models of spinless fermions, fermion-number conservation combined with a non-onsite Majorana translation symmetry enforces a Fermi surface. This symmetry-enforced Fermi surface is a strong form of symmetry-enforced gaplessness. Finally, we introduce a framework of stratified symmetries and stratified anomalies to understand and discover new Lieb–Schultz–Mattis constraints. We apply this framework to modulated symmetries and find an explicit criterion for when local anomaly data imply obstructions to a trivial gapped phase. Together, these results demonstrate the broad applicability and power of symmetry and its generalizations, both for organizing quantum phases and for deriving robust non-perturbative constraints.
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.