TY - JOUR

T1 - Approximating the ground state of gapped quantum spin systems

AU - Hamza, Eman

AU - Michalakis, Spyridon

AU - Nachtergaele, Bruno

AU - Sims, Robert

N1 - Funding Information:
E.H. would like to acknowledge support through a Junior Research Fellowship at the Erwin Schrödinger Institute in Vienna, where part of this work was done. A part of this work was also supported by the National Science Foundation under Grant Nos. DMS-0605342 and DMS-0757581 (B.N. and S.M.) and DMS-0757424 (R.S.). S.M. further acknowledges support by U.S. DOE under Contract No. DE-AC52-06NA25396.

PY - 2009

Y1 - 2009

N2 - We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Zd. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset XV the ground state projector can be approximated by the product of two projections, one supported on X and one supported on Xc, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. This result generalizes to multidimensional models, a result of Hastings that was an important part of his proof of an area law in one dimension ["An area law for one dimensional quantum systems," J. Stat. Mech.: Theory Exp. 2007, 08024].

AB - We consider quantum spin systems defined on finite sets V equipped with a metric. In typical examples, V is a large, but finite subset of Zd. For finite range Hamiltonians with uniformly bounded interaction terms and a unique, gapped ground state, we demonstrate a locality property of the corresponding ground state projector. In such systems, this ground state projector can be approximated by the product of observables with quantifiable supports. In fact, given any subset XV the ground state projector can be approximated by the product of two projections, one supported on X and one supported on Xc, and a bounded observable supported on a boundary region in such a way that as the boundary region increases, the approximation becomes better. This result generalizes to multidimensional models, a result of Hastings that was an important part of his proof of an area law in one dimension ["An area law for one dimensional quantum systems," J. Stat. Mech.: Theory Exp. 2007, 08024].

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U2 - 10.1063/1.3206662

DO - 10.1063/1.3206662

M3 - Article

AN - SCOPUS:70349675787

VL - 50

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 9

M1 - 095213

ER -