Concurrence at Quantum Phase Transitions

I investigated pairwise entanglement signatures in a 1D transverse-field Ising chain near its quantum phase transition ($J/h = 1$) under Periodic Boundary Conditions. The goal was to trace out $N-2$ spins from the ground state and calculate Wootters concurrence to locate the inflection point and derivative extremum that mark the phase transition.

To overcome the exponential scaling of the Hilbert space ($k = 2^N$), the codebase evolved through two major phases. The initial exact diagonalization pipeline was memory-bottlenecked at $N \approx 12$ because it required a dense $4096 \times 4096$ complex matrix ($O(2^{2N})$ storage). I refactored the Hamiltonian construction to use Armadillo's sparse representations (`sp_mat`) and offloaded the ground-state search to an ARPACK solver wrapper (`eigs_sym`), shifting to $O(2^N)$ vector storage and extending tractable calculations to $N = 22$ spins.

During testing, compilation dependencies on external libraries (`QIClib`'s partial trace) caused cross-environment compiler mismatches between local workstations and lab cluster setups. To make the solver self-contained, I stripped the third-party dependency and authored a custom native partial trace function to explicitly compute the $4 \times 4$ density matrix directly from the ground-state coefficient vector.

Using the sparse numerical solver, I manually constructed the spin-flipped reduced density matrices $\tilde{\rho}_{12} = (\sigma_y \otimes \sigma_y) \rho_{12}^* (\sigma_y \otimes \sigma_y)$ to calculate Wootters concurrence and its derivative $dC/d\lambda$. The output plots verified critical behavior: the characteristic sharpening of the concurrence inflection and the scaling of the derivative peak exactly at $J/h = 1$, with coordinates correctly shifted using midpoints to align the finite-difference derivatives.

Exact diagonalization and sparse iterative methods still scale exponentially in memory and compute, bounding the simulations to $N \le 22$ on single-machine configurations. This project represents a controlled numerical physics study; it does not claim scalable quantum hardware simulation or tensor-network (DMRG) scaling.