Soft symmetries of topological orders

Physical review. B./Physical review. B · 2026-03-04 · 1 citations

(2+1)D topological orders possess emergent symmetries given by a group $\text{Aut}(\mathcal{C})$, which consists of the braided tensor autoequivalences of the modular tensor category $\mathcal{C}$ that describes the anyons. In this paper we discuss cases where $\text{Aut}(\mathcal{C})$ has elements that neither permute anyons nor are associated to any symmetry fractionalization but are still non-trivial, which we refer to as soft symmetries. We point out that one can construct topological defects corresponding to such exotic symmetry actions by decorating with a certain class of gauged SPT states that cannot be distinguished by their torus partition function. This gives a physical interpretation to work by Davydov on soft braided tensor autoequivalences. This has a number of important implications for the classification of gapped boundaries, non-invertible spontaneous symmetry breaking, and the general classification of symmetry-enriched topological phases of matter. We also demonstrate analogous phenomena in higher dimensions, such as (3+1)D gauge theory with gauge group given by the quaternion group $Q_8$.

Towards a Science of AI: Scaling laws and synthetic data

PIRSA · 2026-05-05

Disclinations, dislocations, and emanant flux at Dirac criticality

arXiv (Cornell University) · 2025-01-23 · 1 citations

What happens when fermions hop on a lattice with crystalline defects? The answer depends on topological quantum numbers which specify the action of lattice rotations and translations in the low energy theory. One can understand the topological quantum numbers as a twist of continuum gauge fields in terms of crystalline gauge fields. We find that disclinations and dislocations -- defects of crystalline symmetries -- generally lead in the continuum to a certain ``emanant'' quantized magnetic flux. To demonstrate these facts, we study in detail tight-binding models whose low-energy descriptions are (2+1)D Dirac cones. Our map from lattice to continuum defects explains the crystalline topological response to disclinations and dislocations, and motivates the fermion crystalline equivalence principle used in the classification of crystalline topological phases. When the gap closes, the presence of emanant flux leads to pair creation from the vacuum with the particles and anti-particles swirling around the defect. We compute the associated currents and energy density using the tools of defect conformal field theory. There is a rich set of renormalization group fixed points, depending on how particles scatter from the defect. At half flux, there is a defect conformal manifold leading to a continuum of possible low-energy theories. We present extensive numerical evidence supporting the emanant magnetic flux at lattice defects and we test our map between lattice and continuum defects in detail. We also point out a no-go result, which implies that a single (2+1)D Dirac cone in symmetry class AII is incompatible with a commuting $C_M$ rotational symmetry with $(C_M)^M = +1$.

Crystalline invariants of fractional Chern insulators

Physical review. B./Physical review. B · 2025-06-27 · 3 citations

In the presence of crystalline symmetry, topologically ordered states can acquire a host of symmetry-protected invariants. These determine the patterns of crystalline symmetry fractionalization of the anyons in addition to fractionally quantized responses to lattice defects. Here we show how ground state expectation values of partial rotations centered at high symmetry points can be used to extract crystalline invariants. Using methods from conformal field theory and G-crossed braided tensor categories, we develop a theory of invariants obtained from partial rotations, which apply to both Abelian and non-Abelian topological orders. We then perform numerical Monte Carlo calculations for projected parton wave functions of fractional Chern insulators, demonstrating remarkable agreement between theory and numerics. For the topological orders we consider, we show that the Hall conductivity, filling fraction, and partial rotation invariants fully characterize the crystalline invariants of the system. Our results also yield invariants of continuum fractional quantum Hall states protected by spatial rotational symmetry.

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Physical review. B./Physical review. B · 2025-11-17

It is common in condensed matter systems for reflection ($R$) and time-reversal ($T$) symmetry to both be broken while the combination $RT$ is preserved. In this paper, we study invariants that arise due to $RT$ symmetry. We consider many-body systems of interacting fermions with fermionic symmetry groups ${G}_{f}={\mathbb{Z}}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$, $\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, and $\mathrm{U}{(1)}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$, ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$, and ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ classification, respectively, which we compute using the framework of $G$-crossed braided tensor categories. We provide a many-body $RT$ invariant in terms of a tripartite entanglement measure, and which we show can be understood using an edge conformal field theory computation in terms of vertex states. For ${G}_{f}=\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, which applies to charged fermions in a magnetic field, the nontrivial value of the ${\mathbb{Z}}_{2}$ invariant requires strong interactions. For symmetry-preserving boundaries, the phases are distinguished by zero modes at the intersection of the reflection axis and the boundary. Additional invariants arise in the presence of translation or rotation symmetry.

Fractionally quantized electric polarization and discrete shift of crystalline fractional Chern insulators

Physical review. B./Physical review. B · 2025-06-13 · 1 citations

Fractional Chern insulators (FCI) with crystalline symmetry possess topological invariants that fundamentally have no analog in continuum fractional quantum Hall (FQH) states. Here we demonstrate through numerical calculations on model wave functions that FCIs possess a fractionally quantized electric polarization, $\vec{\mathscr{P}}_{\text{o}}$, where $\text{o}$ is a high symmetry point. $\vec{\mathscr{P}}_{\text{o}}$ takes fractional values as compared to the allowed values for integer Chern insulators because of the possibility that anyons carry fractional quantum numbers under lattice translation symmetries. $\vec{\mathscr{P}}_{\text{o}}$, together with the discrete shift $\mathscr{S}_{\text{o}}$, determine fractionally quantized universal contributions to electric charge in regions containing lattice disclinations, dislocations, boundaries, and/or corners, and which are fractions of the minimal anyon charge. We demonstrate how these invariants can be extracted using Monte Carlo computations on model wave functions with lattice defects for 1/2-Laughlin and 1/3-Laughlin FCIs on the square and honeycomb lattice, respectively, obtained using the parton construction. These results comprise a class of fractionally quantized response properties of topologically ordered states that go beyond the known ones discovered over thirty years ago.

Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

Physical review. B./Physical review. B · 2025-02-27 · 3 citations

Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift ${\mathcal{S}}_{\mathrm{o}}$ and electric polarization ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$, where $\mathrm{o}$ labels a high-symmetry point. Crucially, these can be defined even with nonzero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of ${\mathcal{S}}_{\mathrm{o}}$ and ${\stackrel{P\vec}{\mathcal{P}}}_{\mathrm{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.

Electric polarization in Chern insulators: Unifying many-body and single-particle approaches

Physical review. B./Physical review. B · 2025-09-11 · 1 citations

Recently, it has been established that Chern insulators possess an intrinsic two-dimensional electric polarization, despite having gapless edge states and nonlocalizable Wannier orbitals. This polarization, ${\stackrel{P\vec}{P}}_{\text{o}}$, can be defined in a many-body setting from various physical quantities, including dislocation charges, boundary charge distributions, and linear momentum. Importantly, there is a dependence on a choice of real-space origin o within the unit cell. In contrast, Coh and Vanderbilt extended the single-particle Berry phase definition of polarization to Chern insulators by choosing an arbitrary point in momentum space, ${\stackrel{P\vec}{k}}_{0}$. In this paper, we unify these two approaches and show that when the real-space origin o and momentum-space point ${\stackrel{P\vec}{k}}_{0}$ are appropriately chosen in relation to each other, the Berry phase and many-body definitions of polarization are equal.

(How) Can Transformers Predict Pseudo-Random Numbers?

ArXiv.org · 2025-02-14

Transformers excel at discovering patterns in sequential data, yet their fundamental limitations and learning mechanisms remain crucial topics of investigation. In this paper, we study the ability of Transformers to learn pseudo-random number sequences from linear congruential generators (LCGs), defined by the recurrence relation $x_{t+1} = a x_t + c \;\mathrm{mod}\; m$. We find that with sufficient architectural capacity and training data variety, Transformers can perform in-context prediction of LCG sequences with unseen moduli ($m$) and parameters ($a,c$). By analyzing the embedding layers and attention patterns, we uncover how Transformers develop algorithmic structures to learn these sequences in two scenarios of increasing complexity. First, we investigate how Transformers learn LCG sequences with unseen ($a, c$) but fixed modulus; and demonstrate successful learning up to $m = 2^{32}$. We find that models learn to factorize $m$ and utilize digit-wise number representations to make sequential predictions. In the second, more challenging scenario of unseen moduli, we show that Transformers can generalize to unseen moduli up to $m_{\text{test}} = 2^{16}$. In this case, the model employs a two-step strategy: first estimating the unknown modulus from the context, then utilizing prime factorizations to generate predictions. For this task, we observe a sharp transition in the accuracy at a critical depth $d= 3$. We also find that the number of in-context sequence elements needed to reach high accuracy scales sublinearly with the modulus.

Electric polarization and discrete shift from boundary and corner charge in crystalline Chern insulators

arXiv (Cornell University) · 2024-10-04

Recently, it has been shown how topological phases of matter with crystalline symmetry and $U(1)$ charge conservation can be partially characterized by a set of many-body invariants, the discrete shift $\mathscr{S}_{\text{o}}$ and electric polarization $\vec{\mathscr{P}}_{\text{o}}$, where $\text{o}$ labels a high symmetry point. Crucially, these can be defined even with non-zero Chern number and/or magnetic field. One manifestation of these invariants is through quantized fractional contributions to the charge in the vicinity of a lattice disclination or dislocation. In this paper, we show that these invariants can also be extracted from the length and corner dependence of the total charge (mod 1) on the boundary of the system. We provide a general formula in terms of $\mathscr{S}_{\text{o}}$ and $\vec{\mathscr{P}}_{\text{o}}$ for the total charge of any subregion of the system which can include full boundaries or bulk lattice defects, unifying boundary, corner, disclination, and dislocation charge responses into a single general theory. These results hold for Chern insulators, despite their gapless chiral edge modes, and for which an unambiguous definition of an intrinsically two-dimensional electric polarization has been unclear until recently. We also discuss how our theory can fully characterize the topological response of quadrupole insulators.