Entanglement spectrum and Matrix Product States

Transcription

1 Entanglement spectrum and Matrix Product States Frank Verstraete J. Haegeman, D. Draxler, B. Pirvu, V. Stojevic, V. Zauner, I. Pizorn I. Cirac (MPQ), T. Osborne (Hannover), N. Schuch (Aachen)

2 Outline Valence bond construction for MPS / PEPS Area laws, entanglement spectrum Parent Hamiltonians relation to completely positive maps Low energy excitations for MPS parent Hamiltonians Localized particles Entanglement spectrum tangent planes on the manifold of MPS

3 Valence bond picture of MPS What is the easiest way of constructing translational invariant state? Start with collection of virtual entangled pairs project complementary pairs of them to the local physical space

4 This tensor A maps the physical space to the virtual space The MPS is called injective if every virtual state can be created by acting on a finite number of physical sites: = If an MPS is injective, we can prove that it is the unique ground state of a gapped local frustration free Hamiltonian, and also that the Schmidt rank is D Conversely, this implies that the rank of every reduced density matrix of a large enough contiguous block of spins will be D 2 : Reduced density matrix on a block is supported by vectors of the kind = If we create a projector orthogonal to all those D 2 states, then the MPS is anihilated by it. The sum of all those projectors (obtained by translating them) is the parent Hamiltonian, with the MPS the zero-energy ground state of it

5 In the case of a half-infinite block: reduced density matrix is supported by D states : = By choosing the right gauge, those can be chosen to form an orthonormal set. This defines an isometry that maps all physical many-body degrees of freedom to virtual ones Dimensional reduction: map an exponentially large system to a D states (zero-dimensional!) The Schmidt decomposition of a state is written in this basis gives rise to the entanglement spectrum

6 Interludum: Renyi Entropy vs. approximatable with MPS: Matrix product states can be obtained for any state by cutting the Schmidt spectrum with respect to all bipartitions in 2 halve-chains The overlap with the original state is still large if the entanglement entropy is small: S logtr ; 1 0 Schuch, Cirac, FV 07

7 The reduced density matrix of one halve of an MPS can be written as is an operator that lives in the virtual space (D-dimensional Hilbert space). this operator inherits all symmetry properties of the original state (forms basis for classification of MPS under symmetry protected adiabatic transformations, cfr. Tayor, Pollmann, Schuch, Cirac, Wen, ) What is the meaning of? It is the (unique) fixed point of the trace-preserving completely positive map (TPCP-map) defined by the Kraus operators

8 A TPCP map is the most general linear map that maps positive operators to positive operators (describes dissipative dynamics) It is the quantum analogue of a stochastic matrix Perron-Frobenius proves the uniqueness and positivity of the stationary distribution for a stochastic map if it is primitive Similar arguments allow to prove the uniqueness of the fixed point of a TPCP map if it is injective The eigenvalues of a TPCP map determine the convergence rate to the stationary distribution (fixed point). Note that these eigenvalues do not have to be real Translated to the many-body state: the gap of the TPCP map determines the correlation length Temme et al., 11

9 This is a manifestation of the holographic principle: description of 0- dimensional D-level nonequilibrium / dissipative dynamics is equivalent to the description of the static ground state properties of a 1 dimensional quantum spin system Temporal correlation functions for non-equilibrium system are in 1 to 1 correspondence with spatial correlation functions of the ground state The Schmidt spectrum is hence determined by the eigenvalues of the stationary distribution of a TPCP map Note that from this point of view, there is no relation between the Schmidt spectrum and the correlation length Taking the logarithm of this (positive) stationary state yields a Hamiltonian (the entanglement Hamiltonian ). This makes complete sense if the dissipative dynamics generates Gibb s states; it is unknown however why this makes sense in practically all relevant cases

10 Generalizations: continuous MPS Here Q and R are matrices acting on the D-dimensional virtual (auxiliary) space Same properties as MPS: ground state of local Hamiltonians, entanglement spectrum bounded by D, Instead of TPCP-map, the density matrix is related to the stationary distribution determined by the Lindblad equation: i i i i i i i i i L L L L L L H i dt d * * * 2 1 ], [

11 Entanglement spectrum for the Lieb-Liniger model:

12 Projected entangled pair states (PEPS) Same construction in higher dimensions: PEPS Similar properties like area laws, injectivity, fixed points of CPmaps, (cfr. talks of Poilblanc and Cirac)

13 Elementary excitations for MPS Is it possible to make any general statements about the low-lying excitations of gapped quantum spin Hamiltonians? We can draw inspiration from the consequences of locality in relativistic quantum field theories: Spin-Statistics theorem (Fierz, Pauli 1940) CPT-theorem (Bell, Luders, Pauli 1954) existence of a mass (energy) gap implies exponential decay of spatial correlations (Ruelle 1962) Emergence of the particle picture in axiomatic field theory (Zimmerman, Haag, Fredenhagen,...) Crucial element in those proofs: existence of space-like separated intervals (light cone) Related concept in quantum spin systems: Lieb-Robinson bound; this gave rise to Hastings theorems (area law, stability of topological order, )

14 What about analogue of particle-like excitations? For MPS, it is possible to prove that the elementary excitations (eigenstates!) on an isolated branch can be obtained by acting locally with an operator on the ground state (similar to Feynman-Bijl ansatz): + e ik + e i2k The number of sites on which those blocks have to act is proportional to the gap between the 1-particle band and the continuum band above it J. Haegeman, S. Michelakis, T. Osborne, N. Schuch, FV 12

15 Sketch of proof for parent Hamiltonians of MPS Consider the vector space generated by acting with all possible local operators O (acting on L sites) on the ground state We can project the full Hamiltonian on that space; the resulting Hamiltonian is equal to the original Hamiltonian on L sites plus 2 boundary terms acting on D-level systems: The gap etc. converge to the one of the thermodynamic limit, so the wavefunction corresponding to this first excited state is a wavepacket built up of plane wave excited states with similar energies Filtering the momenta using Lieb-Robinson type of ideas shows that the extent of those local operators is related to the gap above the isolated branch

16 Example: spin 1 AKLT model Three particle band Two particle band One particle band

17 So the lowest lying excitations for those gapped systems can be described by the Feynman-Bijl type ansatz: all the information is in the ground state I find this pretty amazing: in principle, knowledge of 1 eigenvector of a matrix does not contain much information abouth the structure of the other eigenvectors From the point of view of matrix product states, those excitations are MPS with bond dimension 2.D: This MPS has a Jordan block structure, and is actually a state that lives in the tangent plane of the manifold of translational invariant MPS: an effective Hamiltonian on top of a strongly correlated vacuum state is obtained by projecting the full Hamiltonian on this tangent plane Haegeman et al., 12

18 Excitations in the tangent plane Spin 1 Heisenberg model Haegeman et al., 2011

19 In the case of symmetry breaking, elementary excitations are typically domain walls between the two phases: topological nontrivial excitations (cfr. Mandelstam ansatz)

20 Spin 1 XXZ Geben Sie hier eine Formel ein. Hsegeman et al., 11

21 Lieb-Liniger model: excitations Cfr. Poster of D. Draxler

22 Entanglement spectrum for elementary excitations As the bond dimension of the MPS is doubled, the entanglement spectrum is also exactly doubled: the set of new Schmidt coefficients is a direct sum of two copies of the original one + e ik + + e ink This is a simple consequence of the orthogonality of the halve-infinite wavefunctions where the particle is to the left or to the right In the case of more quasi-particles, the entanglement spectrum is just summed over once more: 2 quasiparticles changes the entropy with a factor of 2, n quasiparticles with a factor of n Except if you have bound states of quasi-particles!

23 This holds more generally for non-mps systems: e.g. XY model Pizorn 12

24 Conclusion Gapped quantum spin systems share crucial properties with nonequilibrium processes in 1 dimension lower Entanglement spectrum is related to the eigenvalues of the stationary distribution of a CP-map (or Lindblad in case of continuum systems) Low lying excited states for gapped quantum spin systems in 1D are particle-like. As a consequence, the entanglement entropy is increased by 1 (or n in case of n unbounded quasi-particles ) What about 2-dimensional systems?