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Ryan Thorngren

Publications and source records attributed to Ryan Thorngren.

At least 19 recordsLinked to original sources
arXiv2026

Anomalies of TQFTs coupled to parameter spaces and constraints on ground-state degeneracies

Anomalies of global symmetries can arise in long-range entangled quantum many-body systems and provide useful constraints on their dynamics. In this paper, we study anomalies of $d$-dimensional topological quantum field theories (TQFTs) coupled to parameter spaces and their implications for ground-state degeneracies. Specifically, we formulate anomalies as obstructions to constructing parametrized families of TQFTs. We discuss how the classification of these anomalies depends on the choice of topology on the universal $\mathrm{U}(1)$ $(d-1)$-form symmetry. In this framework, we use the projective ground-state bundle obtained by compactifying the TQFT on a spatial manifold to derive a general constraint relating its ground-state degeneracy $N$ to its torsion anomaly $ω$, expressed as $N\int_Wω\in\mathbb{Z}$ for bundles $W$ of spatial manifolds over closed surfaces. We apply this constraint to ground-state degeneracies of (2+1)-dimensional TQFTs with framing and 1-form symmetry anomalies and to obstructions to symmetry-preserving TQFTs. We also discuss implications for bordism generators and propose a conjecture on anomaly detection by manifolds fibered over surfaces.

cond-mat.str-el↗
arXiv2026

Topological Quantum Error-Correcting Thresholds of a Symmetry-Enriched Toric Code

We study the error thresholds of a new topological quantum error-correcting code under the independent bit-flip and phase-flip noisy channel. By mapping to classical spin-glass models, we find that if Pauli X and Z errors are equally probable, the threshold of vertex syndromes is improved by 30% compared to that of the toric code, while the threshold of plaquette syndromes is lowered. Both thresholds increase when the Z component is less likely than the X component, reaching 0.5 if the Z component is absent. An analytical proof justifies this special case of perfect correctability, and a disordered version of Peierls's argument establishes a lower bound for the vertex-syndrome threshold.

quant-ph↗
arXiv2026

Disentangling the Toric Code

Commuting-projector Hamiltonians with integer spectrum such as the 1+1d Ising ferromagnet and the 2+1d toric code may be regarded as the generators of non-on-site $U(1)$ symmetries. In this paper we study whether these symmetries can be made on-site in the presence of finite and infinite-dimensional ancillae. In the finite-dimensional case, we prove that the fractionalized excitations, such as domain walls and anyons, make it impossible to on-site these Hamiltonians. On the other hand, we show that when infinite dimensional ancillae (e.g., rotors) are allowed, both the 1+1d Ising ferromagnet and the 2+1d toric code can be disentangled, yielding on-site symmetries. This is not in contradiction with the non-trivial ground state order of these Hamiltonians, since the ancilla Hamiltonians are no longer bounded from below. Our results indicate agreement between lattice and quantum field theory anomalies in the presence of infinite-dimensional ancillae, while with finite-dimensional ancillae the obstruction to on-siteability may even be non-invertible.

cond-mat.str-el↗
arXiv2026

Quantum cellular automata and invertible phases of matter

We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces $X$ with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over $X\times \mathbb{Z}$. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an $Ω$-spectrum of invertible phases as conjectured by Kitaev. We then show that the $c=\frac{1}{2}$ chiral Majorana fermion net and the $(E_{8})_{1}$ conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on $\mathbb{Z}^{2}$. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.

math-ph↗
arXiv2026

Correlation versus Causation in Quantum Criticality

Correlation functions $\langle O_1(x) O_2(0) \rangle$ reveal scaling dimensions through spatial decay. We instead consider static susceptibility, the change in $\langle O_1(x) \rangle$ from perturbing the Hamiltonian by $O_2(0)$, which we term causation for short. In a conformal field theory (CFT), dimensional analysis predicts decay of $|x|^{-2Δ}$ for correlation and $|x|^{-2Δ+1}$ for causation. Yet we find causation can decay up to fifteen additional orders in $x$ through a general mechanism, which we trace to time-derivative fields being unable to contribute to static response. In higher-dimensional CFTs, this mechanism ensures leading causation arises from primaries, even when descendants dominate correlation, which we leverage with DMRG to identify a previously unresolved corner primary of $Δ\approx 8.8$ and a heavy magnetic line defect primary of $Δ\approx 4.6$ in the $(2+1)$D critical Ising model. Moreover, the same mechanism governs edge-mode localization in $(1+1)$D gapless symmetry-protected topological phases, explaining previously observed anomalously small edge-mode splittings and guiding our construction of spin chains with splittings as small as $1/L^{18}$ and $1/L^{25}$.

cond-mat.str-el↗
arXiv2026

Symmetry Breaking from Monopole Condensation in QED$_3$

QED in three dimensions with an $SU(2)_f$ doublet $ψ^i$ of massless, charge-1 Dirac fermions (and no Chern-Simons term) has a $U(2) = (SU(2)_f \times U(1)_m)/\mathbb{Z}_2$ symmetry that acts on gauge-invariant local operators, including monopole operators charged under $U(1)_m$. We argue that there are only two plausible IR scenarios: either the theory flows to a CFT with $U(2)$ symmetry (a scenario strongly constrained by conformal bootstrap bounds); or it spontaneously breaks $U(2) \to U(1)$ via the condensation of a monopole operator of smallest $U(1)_m$ charge, which is a $U(2)$ doublet. This leads to three Nambu-Goldstone bosons described by a sigma model into a squashed three-sphere $S^3$ with $U(2)$ isometry. The conventional $SU(2)_f$-triplet order parameter $i \bar ψ\vec σ\, ψ$ also gets a vev, exactly aligned with the monopole vev, such that the triplet parametrizes the $\mathbb{CP}^1$ base of the $S^3$ Hopf bundle, with the monopoles providing the $S^1$ fibers. We recall why this scenario is compatible with the Vafa-Witten theorem. These results are obtained by considering the phase diagram as a function of the fermion triplet mass $\vec m$. We argue that for all $\vec m \neq 0$ there is a Coulomb phase with a weakly-coupled photon, which arises from a suitable monopole vev; taking $\vec m \to 0$ leads to the symmetry-breaking scenario above. Throughout, we give a detailed account of anomaly matching, which leads to a $θ=π$ term in the $S^3$ sigma model. In one presentation, it can be understood as a Hopf term in a suitably gauged version of the $\mathbb{CP}^1$ sigma model.

hep-th↗
arXiv2026

Perturbatively Stable Self-Correcting Classical Memory from Gauge Averaging

We show that the self-correcting memory in 3d Wegner gauge theory is stable to arbitrary small enough perturbations of the Hamiltonian. Our proof relies on a new method we dub ``gauge averaging'', which gives conditions under which explicitly broken gauge symmetries are effectively restored by fluctuations. These conditions show that the self-correcting memory phase is fluctuation stabilized, with its robustness to perturbations increasing with increasing temperature, up to some $T_c > 0$.

quant-ph↗
arXiv2026

Chiral Lattice Gauge Theories from Symmetry Disentanglers

We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for $U(1)$ symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the $U(1)$ hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.

hep-th↗
arXiv2026

Translation symmetry-enforced long-range entanglement in mixed states

We show by a counting argument that even though translation symmetry admits symmetric short-range entangled (SRE) eigenstates, there are not enough such SRE eigenstates to span the zero momentum sector. This means that the fixed point strong-to-weak spontaneous symmetry breaking state of translation symmetry is long-range entangled: it cannot be written as a mixture of SRE states. This is a subtle form of long-range entanglement in mixed states that cannot be detected by long-range connected correlation functions.

quant-ph↗
arXiv2026

Parameterized Families of Toric Code Phase: $em$-duality family and higher-order anyon pumping

Within the toric-code phase, we study parameterized families of topologically ordered states. We construct $1$- and $2$-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the $1$-parameter family, we show that the $em$-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the $2$-parameter case, we construct a ``pump of a pump'' that transports an $S^1$-family of a system in one lower spatial dimension. Using similar methods, we also present a $1$-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.

cond-mat.str-el↗
arXiv2026

The Smith Fiber Sequence and Invertible Field Theories

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in a companion paper.

math.AT↗
arXiv2026

From QED$_3$ to Self-Dual Multicriticality in the Fradkin-Shenker Model

We consider the Fradkin-Shenker ${\mathbb Z}_2$ gauge-Higgs lattice model in 2+1 dimensions, i.e. the toric code deformed by an in-plane magnetic field. Its phase diagram contains a multicritical CFT with gapless, mutually non-local electric and magnetic particles, exchanged by a ${\mathbb Z}_2^{\mathsf{D}}$ self-duality symmetry. We introduce a staggered generalization of the model in which these particles carry global $U(1)_e$ and $U(1)_m$ charges, respectively, and we propose a continuum QFT description in terms of QED$_3$ with $N_f = 2$ Dirac fermion flavors and a charge-two Higgs field with Yukawa couplings. The conjectured phase diagram harbors a multicritical CFT with $(O(2)_e \times O(2)_m)\rtimes\mathbb{Z}_2^\mathsf{D}$ symmetry, some of which is emergent in the QFT description. We compute the scaling dimensions of some operators using a large-$N_f$ expansion and find agreement with the emergent selection rules. The staggered model admits a deformation to the original Fradkin-Shenker model, which maps to unit-charge monopole operators in Higgs-Yukawa-QED$_3$ that break the $U(1)_e \times U(1)_m$ symmetry. We show explicitly that this deformation reproduces all features of the Fradkin-Shenker phase diagram. Finally, we propose a multicritical duality between Higgs-Yukawa-QED$_3$ and the easy-plane $\mathbb{ CP}^1$ model (i.e. two-flavor scalar QED$_3$ with a suitable potential), which describes spin-1/2 anti-ferromagnets on a square lattice. This duality implies a first-order line of Néel-VBS transitions ending in a deconfined quantum multicritical point, described by the same $O(2)_e \times O(2)_m$ symmetric CFT that arises in the staggered Fradkin-Shenker model, which separates it from a gapped ${\mathbb Z}_2$ spin liquid phase.

cond-mat.str-el↗
arXiv2025

Exact Chiral Symmetries of 3+1D Hamiltonian Lattice Fermions

We construct Hamiltonian models on a 3+1d cubic lattice for a single Weyl fermion and for a single Weyl doublet protected by exact (as opposed to emergent) chiral symmetries. In the former, we find a not-on-site, non-compact chiral symmetry which can be viewed as a Hamiltonian analog of the Ginsparg-Wilson symmetry in Euclidean lattice models of Weyl fermions. In the latter, we combine an on-site $U(1)$ symmetry with a not-on-site $U(1)$ symmetry, which together generate the $SU(2)$ flavor symmetry of the doublet at low energies, while in the UV they generate an algebra known in integrability as the Onsager algebra. This latter model is in fact the celebrated magnetic Weyl semimetal which is known to have a chiral anomaly from the action of $U(1)$ and crystalline translation, that gives rise to an anomalous Hall response - however reinterpreted in our language, it has two exact $U(1)$ symmetries that gives rise to the global $SU(2)$ anomaly which protects the gaplessness even when crystalline translations are broken. We also construct an exact symmetry-protected single Dirac cone in 2+1d with the $U(1) \rtimes T$ parity anomaly. Our constructions evade both old and recently-proven no-go theorems by using not-on-siteness in a crucial way, showing our results are sharp.

cond-mat.str-el↗
arXiv2025

Charge pumps, pivot Hamiltonians and symmetry-protected topological phases

Generalised charge pumps are topological obstructions to trivialising loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points which must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, `pivot loops', defined by two Hamiltonians, where the first is unitarily evolved by the second `pivot' Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops which pump charge for various symmetry groups, often leading to SPT phases -- including dipole SPTs. Intriguingly, we find examples where non-trivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated to the non-trivial pump to explain the a priori `unnecessary' criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show is equivalent to the Hamiltonians satisfying the Dolan-Grady relation -- known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labelled by spaces that are not manifolds.

cond-mat.str-el↗
arXiv2024

Boundary deconfined quantum criticality at transitions between symmetry-protected topological chains

Decades of research have revealed a deep understanding of topological quantum matter with protected edge modes. We report that even richer physics emerges when tuning between two topological phases of matter whose respective edge modes are incompatible. The frustration at the edge leads to novel boundary physics, such as symmetry-breaking phases with exotic non-Landau transitions -- even when the edge is zero-dimensional. As a minimal case study we consider spin chains with $\mathbb{Z}_3 \times \mathbb{Z}_3$ symmetry, exhibiting two nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D critical transition between these SPT phases, we find two stable 0+1D boundary phases, each spontaneously breaking one of the $\mathbb{Z}_3$ symmetries. Furthermore, we find that a single boundary parameter tunes a non-Landau boundary critical transition between these two phases. This constitutes a 0+1D version of an exotic phenomenon driven by charged vortex condensation known as deconfined quantum criticality. This work highlights the rich unexplored physics of criticality between nontrivial topological phases and provides insights into the burgeoning field of gapless topological phases.

cond-mat.str-el↗
arXiv2024

Long-range entanglement from measuring symmetry-protected topological phases

A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases -- of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the $\mathbb Z_2$ Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by $G \times H$ symmetry gives rise to anomalous LRE upon measuring $G$-charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and uncovers the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way.

cond-mat.str-el↗