We provide the first lattice-QCD estimate of the crossover line down to $T\simeq108$~MeV. We introduce a new method that combines the Lee-Yang edge in the complex plane of baryon chemical potential $\mu_B$ with universal chiral scaling to determine the $\mu_B$ dependence of the QCD chiral critical and pseudo-critical temperatures. By performing $(2\!+\!1)$-flavor lattice QCD simulations at $T\simeq108$~MeV and purely imaginary $\mu_B$ with a single lattice spacing and two volumes, we compute $\mu_B$-dependent baryon-number susceptibilities and extract the location of the Lee-Yang edge. Together with universal scaling near the QCD chiral transition, it constrains the mapping function between $\{T,\mu_B\}$ and the scaling variable (\textit{i.e.}\ the argument of the universal scaling functions). This mapping function then yields the $\mu_B$ dependence of the critical and pseudo-critical temperatures for $T\gtrsim108$~MeV. While our calculation is performed only at a single value of low temperature without explicit input from small-$\mu_B$ expansion, the resulting $\mu_B$ dependence of the pseudo-critical temperature is consistent with established lattice-QCD determinations at small $\mu_B$ and compatible with chemical freeze-out parameters of heavy-ion collisions down to low temperatures, demonstrating the validity and robustness of the method. Application of this method can be systematically extended to additional temperatures and finer discretizations, opening a pathway to charting the QCD phase diagram in the low-$T$, high-$\mu_B$ regime.

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.

The dynamics of Wilson loops is governed by an infinite set of Schwinger-Dyson equations and trace relations. In the context of the lattice positivity bootstrap, a central challenge is determining a dynamically independent basis of these operators within a truncated space. We present a systematic framework to solve this problem, utilizing a geometric plaquette-cut and subloop-cut strategy to efficiently generate all (local) direct equations. Furthermore, we identify and analyze ``indirect equations", which arise from the elimination of higher-length intermediate loops. We elucidate the origin of these subtle relations and propose a vertex-filtering strategy to construct them. Applying the above framework to SU(2) lattice Yang-Mills theory, we provide explicit counting of independent canonical loops and equations in 2, 3, and 4 dimensions, along with a statistical analysis of their asymptotic growth.

Framing anomaly is a key property of $(2+1)d$ chiral topological orders, for it reveals that the chirality is an intrinsic bulk property of the system, rather than a property of the boundary between two systems. Understanding framing anomaly in lattice models is particularly interesting, as concrete, solvable lattice models of chiral topological orders are rare. In a recent work, we defined and solved the $U(1)$ Chern-Simons-Maxwell theory on spacetime lattice, showing its chiral edge mode and the associated gravitational anomaly on boundary. In this work, we show its framing anomaly in the absence of boundary, by computing the expectation of a lattice version of the modular $T$ operator in the ground subspace on a spatial torus, from which we extract that $\langle T \rangle$ has a universal phase of $-2\pi/12$ as expected: $-2\pi/8$ from the Gauss-Milgram sum of the topological spins of the ground states, and $2\pi/24$ from the framing anomaly; we can also extract the $2\pi/24$ framing anomaly phase alone from the full spectrum of $T$ in the ground subspace by computing $\langle T^m \rangle$. This pins down the last and most crucial property required for a valid lattice definition of $U(1)$ Chern-Simons theory.

In this work, we scrutinize local gauge-invariant vector operators of dimension four in the adjoint $SU(2)$ Higgs model, which are candidates for interpolating fields of the fundamental excitations of the model due to the so-called FMS mechanism. We use the equations of motion and the properties of the BRST operator to derive a Ward identity that allows us to determine whether a given operator can propagate. To corroborate this analysis, we explicitly compute the two-point function of the non-propagating operator at the one-loop level.