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

Abstract

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.

Symmetry breaking and the geometry of reduced density matrices

“…We show the generality of this geometric interpretation by explicitly extending Gibbs' construction to quantum and classical lattice spin models, as well as quantum field theories. Our work considerably extends and clarifies the convex set picture in the context of the N-representability problem in quantum chemistry [5][6][7][8], and provides a very nice connection to the discussions on the monogamy of entanglement and mean-field theory in [9].…”

Lower bounds for ground states of condensed matter systems

“…We initialized the algorithm with α = N for the first step and α N for the remaining steps until convergence. in great detail exploiting different numerical methods in [25]. The Hamiltonian we are considering as a first application of the projected gradient algorithm is given by…”

Joint product numerical range and geometry of reduced density matrices

“…One important idea is to study the properties of 2-RDMs is by looking at the two-and three-dimensional projections [4,5,9,10]. Since these projections are convex sets in R 2 and R 3 respectively, the hope is that the properties of the different quantum phases can be visually available.…”

“…Since 1960s, how to characterize this convex set has been a central topic of research in the field of quantum marginal problem and N -representability problem [1][2][3][4][5]. The recent development in quantum information theory has shown that the characterization of the 2-RDMs is a hard problem even with the existence of a quantum computer [6][7][8].…”