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)

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

• Variational manifolds for quantum many-body systems

• Time dependent variational principle (TDVP)

– Creating effective quadratic Hamiltonians on the tangent plane

• Manifold of matrix products states

– TDVP and excitations

• MPS for the continuum

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

• 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 D2 :

• Reduced density matrix on a block is supported by vectors of the kind

– If we create a projector orthogonal to all those D2 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

=

=

• 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

=

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:

0 ; 1

logTrS

Schuch, Cirac, FV ‘07

• 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

• 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

• 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

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

ii

i

ii

i

ii LLLLLLHidt

d ***

2

1],[

Entanglement spectrum for the Lieb-Liniger model:

Projected entangled pair states (PEPS)

• Same construction in higher dimensions: PEPS

– Similar properties like area laws, injectivity, fixed points of CP-

maps, …(cfr. talks of Poilblanc and Cirac)

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, …)

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):

• 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

+ eik

+ ei2k

J. Haegeman, S. Michelakis, T. Osborne, N. Schuch, FV ‘12

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

Example: spin 1 AKLT model

One particle band

Two particle band

Three particle band

• 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

Excitations in the tangent plane

Spin 1 Heisenberg model

Haegeman et al., 2011

• In the case of symmetry breaking, elementary excitations are

typically domain walls between the two phases: topological nontrivial

excitations (cfr. Mandelstam ansatz)

Spin 1 XXZ

Geben Sie hier eine Formel ein.

Hsegeman et al., ‘11

Lieb-Liniger model: excitations

Cfr. Poster of D. Draxler

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

• 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!

+ eik

+ …

+ eink

• This holds more generally for non-MPS systems: e.g. XY model

Pizorn ‘12

Conclusion

• Gapped quantum spin systems share crucial properties with non-

equilibrium 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?