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Existence of a Thermodynamic Limit and Correlations in Quantum Spin Systems

Jaeyoon Cho

Published Aug 28, 2013• 0 citations
OpenAlex

Abstract

We study a class of many-body states enlarged by a series of approximately local unitary transformations, which obey the exponential clustering in any spatial dimension, the area law for entanglement entropy in one dimension, and the logarithmically-corrected area law in higher dimensions. Such states appear as the ground states of local spin Hamiltonians if the system converges sufficiently fast to a thermodynamic limit as the system is enlarged. We prove that nondegenerate ground states of gapped frustration-free Hamiltonians fall into that class and hence exhibit all such correlations.

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We study a class of many-body states enlarged by a series of approximately local unitary transformations, which obey the exponential clustering in any spatial dimension, the area law for entanglement entropy in one dimension, and the logarithmically-corrected area law in higher dimensions. Such states appear as the ground states of local spin Hamiltonians if the system converges sufficiently fast to a thermodynamic limit as the system is enlarged. We prove that nondegenerate ground states of gapped frustration-free Hamiltonians fall into that class and hence exhibit all such correlations.
We introduce the boundary effect on the ground state as an attribute of general local spin systems that restricts the correlations in the ground state. To this end, we introduce what we call a boundary effect function, which characterizes not only the boundary effect, but also the thermodynamic limit of the ground state. We prove various aspects of the boundary effect function to unfold its relationship to other attributes of the system such as a finite spectral gap above the ground state, two-point correlation functions, and entanglement entropies. In particular, it is proven that an exponentially decaying boundary effect function implies the exponential clustering of two-point correlation functions in arbitrary spatial dimension, the entanglement area law in one-dimension, and the logarithmically corrected area law in higher dimension. It is also proven that gapped local spin systems with nondegenerate ground states ordinarily fall into that class. In one-dimension, the area law can also result from a moderately decaying boundary effect function, in which case the system is thermodynamically gapless.
We demonstrate an area law bound on the ground state entanglement entropy of a wide class of gapless quantum states of matter using a strategy called local entanglement thermodynamics. The bound depends only on thermodynamic data, actually a single exponent, the hyperscaling violation exponent $\ensuremath{\theta}$. All systems in ddd spatial dimensions obeying our scaling assumptions and with $\ensuremath{\theta}<d\ensuremath{-}1$ obey the area law, while systems with $\ensuremath{\theta}=d\ensuremath{-}1$ can violate the area law at most logarithmically. We also discuss the case of frustration-free Hamiltonians and show that to violate the area law more than logarithmically these systems must have an unusually large number of low energy states. Finally, we make contact with the recently proposed sss-source framework and argue that $\ensuremath{\theta}$ and sss are related by $s={2}^{\ensuremath{\theta}}$.
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.
The area law for entanglement entropy captures a fundamental constraint on the complexity of quantum many-body ground states and enables their efficient description. While the area law is rigorously established in one dimension, its status in higher-dimensional local systems remains unresolved, and it does not hold in general for geometrically non-local systems. Here, we establish a generalized area law for gapped ground states of fully connected Hamiltonians. We show that the bipartite entanglement entropy grows at most logarithmically with system size despite the absence of geometric locality. In the ground state, each site is only weakly entangled with the rest, while configurations with extensive local fluctuations are strongly suppressed, effectively restricting the accessible Hilbert space. As a consequence, the ground state admits a matrix product state approximation with polynomial bond dimension with respect to the system size at fixed accuracy. In the permutation-invariant setting, we further prove a constant entanglement bound and demonstrate that the gapped ground state can be computed in time polylogarithmic in the system size. These results show that low entanglement complexity need not rely on geometric locality, broadening the conceptual and computational scope of area laws.
We construct upper bounds on entanglement entropies of many-body quantum states that have fixed energy expectation values with respect to local Hamiltonians. Our focus is on entanglement entropies of subsystems that make up approximately half of the full system. The upper bound on the von Neumann entanglement entropy is half the sum of the thermal entropies of two fictitious systems at the same temperature as one another, with an additional area-law contribution in some systems. The effective temperature is chosen such that the sum of the thermal energies of the two fictitious systems matches the constraint on the energy of the state in the original problem; at subextensive energies, this temperature decreases with increasing system size. Our upper bounds on Rényi entanglement entropies take an analogous form. These inequalities show that, in systems that are frustration-free (FF) and whose zero-temperature subsystem thermal entropies scale with surface areas (rather than subsystem volumes), ground states are area-law entangled. For physical models of quantum matter, which have well-defined specific heat capacities (and are not necessarily FF), our bounds provide a way to convert this thermodynamic data into constraints on pure-state entanglement at both subextensive and extensive energies. We also show that our general upper bounds on half-system entanglement entropies are optimal, up to subleading corrections, in systems with reflection symmetry and above a subextensive energy threshold. A key application of our results is to constrain the entanglement of quantum states relevant to simulations of zero-temperature response functions, which have energies of order unity. We prove that, in wide varieties of FF systems in two dimensions and higher, the contributions of excitations to von Neumann entanglement entropy are subleading compared with the contribution from the ground state.