Uncomputability of phase diagrams

Abstract

The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where $$\varphi \in {\mathbb{R}}$$ φ ∈ R , for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian’s parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.

The Complexity of Approximating Critical Points of Quantum Phase Transitions

“…First off, to combine a history state Hamiltonian and a bonus Hamiltonian in a way as to place the energy bonus precisely in between the promise gap of a history state Hamiltonian means we needed to gear together tight spectral bounds on either system. For the history state Hamiltonian, we rely on a result derived in [Wat19]; for the bonus Hamiltonian, we obtain exponentially tight bounds on the bonus energy inflicted, significantly strengthening the bounds derived in both [Bau+20;BCW21].…”

“…Proof Outline. 𝐻 𝑁 (𝜑) is constructed so that its ground state partitions the lattice into checker board grids of varying side length (motivated by the idea in [BCW21]).…”

“…If the ground state 𝜆 min (𝐻 (𝜑)| 𝑆 (𝐿 𝑁 ) ) < 0, then the overall ground state of the lattice becomes a checkerboard of these squares. It can be shown that incomplete squares contribute zero energy, giving a total energy of 𝐿 𝐿 𝑁 2 𝜆 min (𝐻 (𝜑)| 𝑆 (𝐿 𝑁 ) ) (see [BCW21,Corollary F.3]). If 𝜆 min (𝐻 (𝜑)| 𝑆 (𝐿 𝑁 ) ) ≥ 0, then the lattice has ≥ 0.…”

“…Furthermore, several numerical algorithms have been developed to compute spectral gaps, including variational algorithms [HWB19; Jon+19] and using density matrix renormalisation group techniques [CM17] On the other hand, Cubitt et al have shown that the general problem of determining whether a local Hamiltonian is gapped or gapless in the thermodynamic limit is undecidable [CPW15], a result later extended to translationally invariant, nearest neighbour interactions on a 1D spin chain [Bau+20]. Further results show that for a one-parameter Hamiltonian in 2D, determining the phase diagram is uncomputable [BCW21], and that renormalisation group methods provably fail to resolve these systems [WOC21]. Nonetheless, these undecidability and uncomputablity results only imply that it is impossible to study the phase diagrams or spectral gap of models in full generality; no claims are made regarding more restricted sub-types, which may well be solvable.…”

Computational Complexity of the Ground State Energy Density Problem

“…Meanwhile, as a stepping stone to their undecidability result for the spectral gap, [CPGW15b;CPGW15a] proved that deciding whether the ground state energy density is 0 or strictly positive, with no promise gap, is undecidable, Their result holds for quantum, translationally invariant, nearest neighbour Hamiltonians on a 2D square lattice with a fixed local dimension. [Bau+18b] later extended this undecidability result to 1D chains (again as a stepping stone to the spectral gap problem) and [BCW21] extends to to 2D systems for which the local interaction are analytic in the input parameter.…”