Entanglement spectrum of a topological phase in one dimension

We show that the Haldane phase of S=1 chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\pi$-rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, "topologically trivial", phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

[1]  Thierry Giamarchi,et al.  Rise and fall of hidden string order of lattice bosons , 2008, 0803.2851.

[2]  A. Lefevre,et al.  Entanglement spectrum in one-dimensional systems , 2008, 0806.3059.

[3]  Kennedy,et al.  Rigorous results on valence-bond ground states in antiferromagnets. , 1987, Physical review letters.

[4]  A. Aharony,et al.  Moriya's anisotropic superexchange interaction, frustration, and Dzyaloshinsky's weak ferromagnetism. , 1992, Physical review letters.

[5]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[6]  Hui Li,et al.  Entanglement spectrum as a generalization of entanglement entropy: identification of topological order in non-Abelian fractional quantum Hall effect states. , 2008, Physical review letters.

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

[8]  John Preskill,et al.  Topological entanglement entropy. , 2005, Physical Review Letters.

[9]  Xiao-Gang Wen,et al.  Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order , 2009, 0903.1069.

[10]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[11]  J I Cirac,et al.  String order and symmetries in quantum spin lattices. , 2008, Physical review letters.

[12]  Xiao-Gang Wen,et al.  Detecting topological order in a ground state wave function. , 2005, Physical review letters.

[13]  F. Haldane.  Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model , 1983 .

[14]  G. Vidal,et al.  Infinite time-evolving block decimation algorithm beyond unitary evolution , 2008 .

[15]  F. Anfuso,et al.  Fragility of string orders , 2007 .

[16]  M. Nijs,et al.  Preroughening transitions in crystal surfaces and valence-bond phases in quantum spin chains. , 1989, Physical review. B, Condensed matter.

[17]  M. Hastings,et al.  An area law for one-dimensional quantum systems , 2007, 0705.2024.

[18]  Frank Verstraete,et al.  Matrix product state representations , 2006, Quantum Inf. Comput..

[19]  E. Schmidt.  Zur Theorie der linearen und nichtlinearen Integralgleichungen , 1907 .

[20]  G. Vidal,et al.  Entanglement in quantum critical phenomena. , 2002, Physical review letters.

[21]  H. Katsura,et al.  Topological classification of gapped spin chains: Quantized Berry phase as a local order parameter , 2007, 0710.4198.

[22]  Kennedy,et al.  Hidden Z2 x Z2 symmetry breaking in Haldane-gap antiferromagnets. , 1992, Physical review. B, Condensed matter.

[23]  E. Berg,et al.  Hidden order in 1D bose insulators. , 2006, Physical review letters.

[24]  Norbert Schuch,et al.  Entropy scaling and simulability by matrix product states. , 2007, Physical review letters.

[25]  F. Haldane.  Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State , 1983 .