Twisted quantum doubles are sign problem-free

Abstract

The sign problem is one of the central obstacles to efficiently simulating quantum many-body systems. It is commonly believed that some phases of matter, such as the double semion model, have an intrinsic sign problem and can never be realized in a local sign problem-free Hamiltonian due to the non-positivity of the wavefunction. We show that this is not the case. Despite failing to be stoquastic - the standard criteria for the existence of a sign problem - the double semion model as well as all twisted quantum double phases of matter for finite groups $\mathcal{G}$ can be realized in local Hamiltonians that are sign problem-free within a stochastic series expansion. The lack of a sign problem is not fine-tuned and does not require the Hamiltonian to be exactly solvable, with sign problem-free perturbations allowing access to a variety of topological phase transitions.

Figures (4)

• Figure 1: The condensation of semion/anti-semion pairs in the double semion model can be measured with a sign problem-free stochastic series expansion. Simulations were conducted on a $16 \times 16$ lattice at inverse temperature $\beta = 20$. Due to known ergodicity issues in simulations of gauge theories, we run separate QMC simulations in sectors with different magnetic fluxes, both global (specified by $(\pm, \pm)$ indicating absence or presence of flux in the $x$ and $y$ direction) as well as local magnetic flux excitations. Note that the three sectors $(+, -)$, $(-, +)$, and $(-, -)$ are related on the triangular lattice by a $C_3$ rotation, and the expectation values in these three sectors are nearly identical.
• Figure 2: The action of $A_v^g$ on a vertex ($v_3$ in the figure) can be represented by inserting a new vertex $v_3'$ with $g_{v_3' v_3} = g$ and computing the phases associated to three of the plaquettes using the 3-cocycle $\omega$.
• Figure 3: We illustrate the action of $A_v^g$ on neighboring sites. The action on neighboring vertices generates face-sharing tetrahedra whose contributions cancel out, leading to a trivial total action on a lattice without a boundary provided we choose our $A_v^g$'s such that the lattice is mapped back to itself under the transformation.
• Figure 4: Across various magnetic flux sectors as well as with both twisted and untwisted boundary conditions, the double semion Hamiltonian in Eq. \ref{['eq:ds_app']} remains sign problem-free and undergoes a phase transition into a trivial state at a critical value of Ising coupling $J_c \approx 0.44$. To confirm the simulation's validity, we also include DMRG simulations of the $(+, +)$ sector on an $8 \times 4$ open cylinder.