VARIATIONAL QUANTUM FIELD STATES WITH SYMMETRY: THE CONTINUUM LIMIT OF A PROJECTED ENTANGLED PAIR STATE

We obtain a well-behaved continuum limit of projected entangled pair states (PEPS) that provides an abstract class of quantum field states with natural symmetries. Making use of the recently introduced path integral representation of one-dimensional con- tinuous matrix product states (cMPS) for quantum fields, we demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of euclidean symmetries on the physical system requires that the auxiliary system involved in the class' definition must be Lorentz invariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of the lower dimensional auxiliary field system.

[1]  R. Feynman,et al.  Quantum Mechanics and Path Integrals , 1965 .

[2]  R. Schrader,et al.  Feynman-kac formula for euclidean fermi and bose fields , 1972 .

[3]  H. Nielsen,et al.  Absence of neutrinos on a lattice: (I). Proof by homotopy theory , 1981 .

[4]  Wu-Ki Tung,et al.  Group Theory in Physics , 1985 .

[5]  S. Schweber.  Feynman and the visualization of space-time processes , 1986 .

[6]  R. Feynman.  Difficulties in Applying the Variational Principle to Quantum Field Theories , 1988 .

[7]  Mehta.  Euclidean continuation of the Dirac fermion. , 1990, Physical review letters.

[8]  M. Fannes,et al.  Finitely correlated states on quantum spin chains , 1992 .

[9]  F. Verstraete,et al.  Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions , 2004, cond-mat/0407066.

[10]  Matthew B. Hastings,et al.  Spectral Gap and Exponential Decay of Correlations , 2005 .

[11]  F. Verstraete,et al.  Matrix product states represent ground states faithfully , 2005, cond-mat/0505140.

[12]  Frank Verstraete,et al.  Quantum phase transitions in matrix product systems. , 2006, Physical review letters.

[13]  J. I. Cirac,et al.  Variational study of hard-core bosons in a two-dimensional optical lattice using projected entangled pair states , 2007 .

[14]  R. Jackiw,et al.  Chiral gauge theory for graphene. , 2007, Physical review letters.

[15]  F. Verstraete,et al.  Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems , 2008, 0907.2796.

[16]  Frank Verstraete,et al.  Applying the variational principle to (1+1)-dimensional quantum field theories. , 2010, Physical review letters.

[17]  J. Eisert,et al.  Area laws for the entanglement entropy - a review , 2008, 0808.3773.

[18]  J. Eisert,et al.  Holographic quantum states. , 2010, Physical review letters.

[19]  C. Wetterich.  Spinors in euclidean field theory, complex structures and discrete symmetries , 2011 .

[20]  J. García-Ripoll,et al.  Simulating the Dirac field with cold atoms in a honeycomb lattice , 2011 .

[21]  F. Verstraete,et al.  Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space. , 2011, Physical review letters.

[22]  David Jennings,et al.  The continuum limit of a tensor network: a path integral representation , 2012, 1210.5401.

[23]  F. Verstraete,et al.  Entanglement renormalization for quantum fields in real space. , 2011, Physical review letters.