arXiv subjects

Daniel Sheinbaum

Publications and source records attributed to Daniel Sheinbaum.

8 recordsLinked to original sources

arXiv2026

Free to Interacting Map for Crystalline SPT Phases: Equivariance vs Crystalline Equivalence

Freed and Hopkins developed an ansatz for classifying interacting SPT phases using invertible field theories with a natural Free to Interacting (FTI) map from free fermion phases. This ansatz has been generalized to include crystalline phases and a crystalline equivalence principle (CEP). However, motivated by failure of the CEP for weak free fermions and the FTI, here we generalize the original Freed and Hopkins ansatz to a fully equivariant version for symmorphic crystallographic symmetries and show there is a natural equivariant FTI map from symmorphic crystalline weak free fermions. We further discuss why this equivariant ansatz is both mathematically and physically more natural than the spatial symmetry extension by Freed and Hopkins and why full equivariance should hold over the CEP.

math-ph↗

arXiv2025

Weak in the boundary: How weak SPT phases spoil anomaly matching

We show how weak symmetry protected topological (SPT) phases on systems with a boundary are not in 1-to-1 correspondence with weak SPT phases on fully periodic systems, breaking the standard anomaly inflow interpretation of SPT phases. We further discuss the implications for the crystalline equivalence principle (CEP).

cond-mat.str-el↗

arXiv2024

Weak topological phases in the presence of interactions

We study weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions. By comparing homotopical free and interacting classifications of these SPTs, we predict their stability under interactions as well as identify potential intrinsically-interacting phases. We mathematically compute the groups of weak phases in dimensions zero through three for all tenfold-way symmetry types using homotopy theory; specifically, we use Atiyah's Real $\mathit{KR}$-theory and the low-energy invertible field theory ansatz of Freed--Hopkins for the free and interacting cases, resp. Our computational techniques involve T-duality, which relates $K$-theory of the spatial torus with $K$-theory of the Brillouin torus, and a binomial formula for computing generalized cohomology of a torus. Our results carry potential implications for theoretical and experimental studies of weak phases.

math-ph↗

arXiv2024

Failure of the Crystalline Equivalence Principle for Weak Free Fermions

Interacting crystalline SPT phases were first classified by Thorngren and Else through the crystalline equivalence principle, suggesting that a spatial group symmetry can be treated as if it were an internal symmetry group. Using techniques from topology we elucidate how this principle holds for one of the proposals for the interacting bosonic case and yet fails for weak free fermion phases. Last we show how a variant of the principle does hold for strong crystalline free fermion phases, showing that it is not necessary for the cohomology theory to be of Borel type.

cond-mat.str-el↗

arXiv2020

Solitons in Weakly Non-linear Topological Systems: Linearization, Equivariant Cohomology and K-theory

There is a lack of knowledge about the topological invariants of non-linear $d$-dimensional systems with a periodic potential. We study these systems through a classification of the linearized NLS/GP equation around their soliton solutions. Stability conditions under linearized (mode) adiabatic evolution can be interpreted topologically and we can use equivariant $\mathit{K}$-theory and cohomology for their classification. On a lattice with crystallographic point group $P$, modes around stable, $P$-symmetric solitons are coarsely classified by the groups $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)\oplus H^2(BP;\mathbb{Z})$. Similarly, for $P$-symmetric gap solitons that are oscillatory stable, we have $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)\oplus \tilde{R}(P)$ instead. If we include a boundary, we can replace $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)$ with $\mathit{K}^{-1,\tau}_{P}(\mathbb{T}^{d-1})$. Finally, we mention how to use these, and the spaces of soliton solutions $M_{D}(E_{gap})$ and $M_{O}(E_{gap})$ to provide global invariants for the system.

nlin.PS↗

arXiv2020

Crystallographic Interacting Topological Phases and Equivariant Cohomology: To assume or not to assume

For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional torus gives rise to different interacting phases, where $P$ denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.

math-ph↗

arXiv2019

Classifying space for quantum contextuality

We construct a topological space to study contextuality in quantum mechanics. The resulting space is a classifying space in the sense of algebraic topology. Cohomological invariants of our space correspond to physical quantities relevant to the study of contextuality. Within this framework the Wigner function of a quantum state can be interpreted as a class in the twisted $K$-theory of the classifying space.

quant-ph↗

arXiv2015

Topology of Fermi Surfaces and anomaly inflows

We derive a rigorous classification of topologically stable Fermi surfaces of non-interacting, discrete translation-invariant systems from electronic band theory, adiabatic evolution and their topological interpretations. For systems on an infinite crystal it is shown that there can only be topologically unstable Fermi surfaces. For systems on a half- space and with a gapped bulk, our derivation naturally yields a $\mathit{K}$-theory classification. Given the $d-1$-dimensional surface Brillouin zone $\mathrm{X}_{s}$ of a $d$-dimensional half-space, our result implies that different classes of globally stable Fermi surfaces belong in $\mathit{K^{-1}}\mathrm{(X_{s})}$ for systems with only discrete translation-invariance. This result has a chiral anomaly inflow interpretation, as it reduces to the spectral flow for $d = 2$. Through equivariant homotopy methods we extend these results for symmetry classes $AI,\,AII,\, C$ and $D$ and discuss their corresponding anomaly inflow interpretation.

cond-mat.mes-hall↗