We study frustrated quantum systems from a quantum information perspective. Within this approach, we find that highly frustrated systems do not follow any general "area law" of block entanglement, while weakly frustrated ones have area laws similar to those of nonfrustrated systems away from criticality. To calculate the block entanglement in systems with degenerate ground states, typical in frustrated systems, we define a "cooling" procedure of the ground state manifold and propose a frustration degree and a method to quantify constructive and destructive interference effects of entanglement.
We study frustrated quantum systems from a quantum information perspective. Within this approach, we find that highly frustrated systems do not follow any general "area law" of block entanglement, while weakly frustrated ones have area laws similar to those of nonfrustrated systems away from criticality. To calculate the block entanglement in systems with degenerate ground states, typical in frustrated systems, we define a "cooling" procedure of the ground state manifold and propose a frustration degree and a method to quantify constructive and destructive interference effects of entanglement.
We show that ground states of unfrustrated quantum spin-1/2 systems on general lattices satisfy an entanglement area law, provided that the Hamiltonian can be decomposed into nearest-neighbor interaction terms that have entangled excited states. The ground state manifold can be efficiently described as the image of a low-dimensional subspace of low Schmidt measure, under an efficiently contractible tree-tensor network. This structure gives rise to the possibility of efficiently simulating the complete ground space (which is in general degenerate). We briefly discuss 'non-generic' cases, including highly degenerate interactions with product eigenbases, using a relationship to percolation theory. We finally assess the possibility of using such tree tensor networks to simulate almost frustration-free spin models.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.
Abstract At the core of every frustrated system, one can identify the existence of frustrated rings that are usually interpreted in terms of single–particle physics. We check this point of view through a careful analysis of the entanglement entropy of both models that admit an exact single–particle decomposition of their Hilbert space due to integrability and those for which the latter is supposed to hold only as a low energy approximation. In particular, we study generic spin chains made by an odd number of sites with short-range antiferromagnetic interactions and periodic boundary conditions, thus characterized by a weak , i.e. nonextensive, frustration. While for distances of the order of the correlation length the phenomenology of these chains is similar to that of the non-frustrated cases, we find that correlation functions involving a number of sites scaling like the system size follow different rules. We quantify the long-range correlations through the von Neumann entanglement entropy, finding that indeed it violates the area law, while not diverging with the system size. This behavior is well fitted by a universal law that we derive from the conjectured single–particle picture.
We identify a large class of quantum many-body systems that can be solved exactly: natural frustration-free spin-1/2 nearest-neighbor Hamiltonians on arbitrary lattices. We show that the entire ground-state manifold of such models can be found exactly by a tensor network of isometries acting on a space locally isomorphic to the symmetric subspace. Thus, for this wide class of models, real-space renormalization can be made exact. Our findings also imply that every such frustration-free spin model satisfies an area law for the entanglement entropy of the ground state, establishing a novel large class of models for which an area law is known. Finally, we show that our approach gives rise to an ansatz class useful for the simulation of almost frustration-free models in a simple fashion, outperforming mean-field theory.
Defects in frustrated antiferromagnetic spin chains are universally present in geometrically frustrated systems. We consider the defects of the one-dimensional, spin-s XXZ chain with single-ion anisotropy on a periodic chain with N sites that was famously studied by Haldane. For N odd, the antiferromagnetic model is frustrated, and the ground state must include a soliton defect. We consider the Heisenberg interaction perturbatively and determine the corresponding perturbative solitonic ground state. Then we compute the entanglement spectrum, entanglement entropy (EE), capacity of entanglement (CE), and spin correlations in the solitonic ground state. For weak frustration, we find an algebraic violation of the area law for the EE consistent with recent results on weakly frustrated chains. Our analysis then moves beyond the weak frustration regime, and we obtain a novel extensive scaling law for the EE when strong frustration prevails, signaling large entanglement, and failure of the quasiparticle interpretation in this regime. Enhanced frustration results in less total correlations but relatively more nonlocal correlations.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).