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Convex-set description of quantum phase transitions in the transverse Ising model using reduced-density-matrix theory

J. Chem. Phys. 130, 224102 (2009); doi:10.1063/1.3143403

Published 9 June 2009

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Christine A. Schwerdtfeger and David A. Mazziotti
Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
Quantum phase transitions in N-particle systems can be identified and characterized by the movement of the two-particle reduced density matrix (2-RDM) along the boundary of its N-representable convex set as a function of the Hamiltonian parameter controlling the phase transition [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 74, 012501 (2006)]. For the one-dimensional transverse Ising model quantum phase transitions as well as their finite-lattice analogs are computed and characterized by the 2-RDM movement with respect to the transverse magnetic field strength g. The definition of a 2-RDM “speed” quantifies the movement of the 2-RDM per unit of g, which reaches its maximum at the critical point of the phase transition. For the infinite lattice the convex set of 2-RDMs and the 2-RDM speed are computed from the exact solution of the 2-RDM in the thermodynamic limit of infinite N [P. Pfeuty, Ann. Phys. 57, 79 (1970)]. For the finite lattices we compute the 2-RDM convex set and its speed by the variational 2-RDM method [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] in which approximate ground-state 2-RDMs are calculated without N-particle wave functions by using constraints, known as N-representability conditions, to restrict the 2-RDMs to represent quantum system of N fermions. Advantages of the method include: (i) rigorous lower bounds on the ground-state energies, (ii) polynomial scaling of the calculation with N, and (iii) independence of the N-representability conditions from a reference wave function, which enables the modeling of multiple quantum phases. Comparing the 2-RDM convex sets for the finite- and infinite-site lattices reveals that the variational 2-RDM method accurately captures the shape of the convex set and the signature of the phase transition in the 2-RDM movement. From the 2-RDM all one- and two-particle expectation values (or order parameters) of the quantum Ising model can also be computed including the pair correlation function, which decays rapidly around the critical field strength g. ©2009 American Institute of Physics
History: Received 22 October 2008; accepted 7 May 2009; published 9 June 2009
Permalink: http://link.aip.org/link/?JCPSA6/130/224102/1
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KEYWORDS and PACS

Keywords
PACS
  • 05.50.+q
    Lattice theory and statistics
  • 05.30.Fk
    Fermion systems and electron gas (quantum statistical mechanics)
  • 05.70.Fh
    Phase transitions: general studies
  • 05.70.Jk
    Critical point phenomena in thermodynamics
  • 02.30.Xx
    Calculus of variations
  • YEAR: 2009

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ISSN:
0021-9606 (print)   1089-7690 (online)
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