Frustration-free Systems Research Articles

Order by disorder in frustration-free systems: Quantum Monte Carlo study of a two-dimensional PXP model

In this study, an order by disorder mechanism has been proposed in a two-dimensional PXP model, where the extensive degeneracy of the classical ground-state manifold is due to strict occupation constraints instead of geometrical frustrations. By performing an unbias large-scale quantum monte carlos simulation, we find that local quantum fluctuations, which usually work against long-range ordering, lift the macroscopic classical degeneracy and give rise to a compressible ground state with charge-density-wave long-range order. A simple trial wavefunction has been proposed to capture the essence of the ground-state of the two-dimensional PXP model. The finite temperature properties of this model have also been studied, and we find a thermal phase transition with an universality class of two-dimensional Ising model.

Skyrmions and multisublattice helical states in a frustrated chiral magnet

We investigate the existence and stability of skyrmions in a frustrated chiral ferromagnet by considering the competition between ferromagnetic (FM) nearest-neighbour (NN) interaction ($J_1$) and antiferromagnetic (AFM) next-nearest-neighbour (NNN) interaction ($J_2$). Contrary to the general wisdom that long-range ferromagnetic order is not energy preferable under frustration, the skyrmion lattice not only exists but is even stable for a large field range when $J_2 \leq J_1$ compared with frustration-free systems. We defend that the enlargement of stability window of skyrmions is a consequence of the reduced effective exchange interaction caused by the frustration. A multi-sublattice helical state is found below the skyrmion phase, which results from the competition between AFM coupling that favors a two-sublattice N\'{e}el state and the chiral interaction that prefers a helix. As a byproduct, the hysteresis loop of the frustrated chiral system shrinks as the magnetization goes to zero and then opens up again, known as wasp-waist hysteresis loop. The critical field that separates the narrow and wide part of the wasp-waist loop depends exponentially on the strength of NNN coupling. By measuring the critical field, it is possible to determine the strength of NNN coupling.

Local gap threshold for frustration-free spin systems

We improve Knabe’s spectral gap bound for frustration-free translation-invariant local Hamiltonians in 1D. The bound is based on a relationship between global and local gaps. The global gap is the spectral gap of a size-m chain with periodic boundary conditions, while the local gap is that of a subchain of size n < m with open boundary conditions. Knabe proved that if the local gap is larger than the threshold value 1/(n − 1) for some n > 2, then the global gap is lower bounded by a positive constant in the thermodynamic limit m → ∞. Here we improve the threshold to 6n(n+1), which is better (smaller) for all n > 3 and which is asymptotically optimal. As a corollary we establish a surprising fact about 1D translation-invariant frustration-free systems that are gapless in the thermodynamic limit: for any such system the spectral gap of a size-n chain with open boundary conditions is upper bounded as O(n−2). This contrasts with gapless frustrated systems where the gap can be Θ(n−1). It also limits the extent to which the area law is violated in these frustration-free systems, since it implies that the half-chain entanglement entropy is O(1/ϵ) as a function of spectral gap ϵ. We extend our results to frustration-free systems on a 2D square lattice.

Correlation Length versus Gap in Frustration-Free Systems.

Hastings established exponential decay of correlations for ground states of gapped quantum many-body systems. A ground state of a (geometrically) local Hamiltonian with spectral gap ε has correlation length ξ upper bounded as ξ=O(1/ε). In general this bound cannot be improved. Here we study the scaling of the correlation length as a function of the spectral gap in frustration-free local Hamiltonians, and we prove a tight bound ξ=O(1/sqrt[ε]) in this setting. This highlights a fundamental difference between frustration-free and frustrated systems near criticality. The result is obtained using an improved version of the combinatorial proof of correlation decay due to Aharonov, Arad, Vazirani, and Landau.

Improved one-dimensional area law for frustration-free systems

We present a new proof for the 1D area law for frustration-free systems with a constant gap, which exponentially improves the entropy bound in Hastings\^a 1D area law and which is tight to within a polynomial factor. For particles of dimension $d$, spectral gap $\ensuremath{\epsilon}>0$, and interaction strength at most $J$, our entropy bound is ${S}_{1D}\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$, where $X\stackrel{\mathrm{def}}{=}(J\mathrm{log}d)/\ensuremath{\epsilon}$. Our proof is completely combinatorial, combining the detectability lemma with basic tools from approximation theory. In higher dimensions, when the bipartitioning area is $|\ensuremath{\partial}L|$, we use additional local structure in the proof and show that $S\ensuremath{\le}\mathcal{O}(1)\ifmmode\cdot\else\textperiodcentered\fi{}{|\ensuremath{\partial}L|}^{2}{\mathrm{log}}^{6}|\ensuremath{\partial}L|\ifmmode\cdot\else\textperiodcentered\fi{}{X}^{3}{\mathrm{log}}^{8}X$. This is at the cusp of being nontrivial in the 2D case, in the sense that any further improvement would yield a subvolume law.

Characterizing and Quantifying Frustration in Quantum Many-Body Systems

We present a general scheme for the study of frustration in quantum systems. We introduce a universal measure of frustration for arbitrary quantum systems and we relate it to a class of entanglement monotones via an exact inequality. If all the (pure) ground states of a given Hamiltonian saturate the inequality, then the system is said to be inequality saturating. We introduce sufficient conditions for a quantum spin system to be inequality saturating and confirm them with extensive numerical tests. These conditions provide a generalization to the quantum domain of the Toulouse criteria for classical frustration-free systems. The models satisfying these conditions can be reasonably identified as geometrically unfrustrated and subject to frustration of purely quantum origin. Our results therefore establish a unified framework for studying the intertwining of geometric and quantum contributions to frustration.

The detectability lemma and its applications to quantum Hamiltonian complexity

Quantum Hamiltonian complexity, an emerging area at the intersection of condensed matter physics and quantum complexity theory, studies the properties of local Hamiltonians and their ground states. In this paper we focus on a seemingly specialized technical tool, the detectability lemma (DL), introduced in the context of the quantum PCP challenge (Aharonov et al 2009 arXiv:0811.3412), which is a major open question in quantum Hamiltonian complexity. We show that a reformulated version of the lemma is a versatile tool that can be used in place of the celebrated Lieb–Robinson (LR) bound to prove several important results in quantum Hamiltonian complexity. The resulting proofs are much simpler, more combinatorial and provide a plausible path toward tackling some fundamental open questions in Hamiltonian complexity. We provide an alternative simpler proof of the DL that removes a key restriction in the original statement (Aharonov et al 2009 arXiv:0811.3412), making it more suitable for the broader context of quantum Hamiltonian complexity. Specifically, we first use the DL to provide a one-page proof of Hastings' result that the correlations in the ground states of gapped Hamiltonians decay exponentially with distance (Hastings 2004 Phys. Rev. B 69 104431). We then apply the DL to derive a simpler and more intuitive proof of Hastings' seminal one-dimensional (1D) area law (Hastings 2007 J. Stat. Mech. (2007) P8024) (both these proofs are restricted to frustration-free systems). Proving the area law for two and higher dimensions is one of the most important open questions in the field of Hamiltonian complexity, and the combinatorial nature of the DL-based proof holds out hope for a possible generalization. Indeed, soon after the first publication of the methods presented here, they were applied to derive exponential improvements to Hastings' result (Arad et al 2011, Aharonov et al 2011) in the case of frustration-free 1D systems. Finally, we also provide a more general explanation of how the DL can be used to replace the LR bound.