ABSTRACT: For quantum lattice systems with local interactions, the Lieb-Robinson bound
acts as an alternative for the strict causality of relativistic systems and
allows to prove many interesting results, in particular when the energy
spectrum exhibits an energy gap. In this Letter, we show that for translation
invariant systems, simultaneous eigenstates of energy and momentum with an
eigenvalue that is separated from the rest of the spectrum in that momentum
sector, can be arbitrarily well approximated by building a momentum
superposition of a local operator acting on the ground state. The error
decreases in the size of the support of the local operator, with a rate that is
set by the gap below and above the targeted eigenvalue. We show this explicitly
for the AKLT model and discuss generalizations and applications of our result.
ABSTRACT: We develop in full detail the formalism of tangent states to the manifold of
matrix product states, and show how they naturally appear in studying
time-evolution, excitations and spectral functions. We focus on the case of
systems with translation invariance in the thermodynamic limit, where momentum
is a well defined quantum number. We present some new illustrative results and
discuss analogous constructions for other variational classes. We also discuss
generalizations and extensions beyond the tangent space, and give a general
outlook towards post matrix product methods.
ABSTRACT: We show how to construct renormalization group (RG) flows of quantum field theories in real space, as opposed to the usual Wilsonian approach in momentum space. This is achieved by generalizing the multiscale entanglement renormalization ansatz to continuum theories. The variational class of wave functions arising from this RG flow are translation invariant and exhibits an entropy-area law. We illustrate the construction for a free nonrelativistic boson model, and argue that the full power of the construction should emerge in the case of interacting theories.
Physical Review Letters 03/2013; 110(10):100402. · 7.37 Impact Factor
ABSTRACT: We study the second-order quantum phase-transition of massive real scalar
field theory with a quartic interaction in (1+1) dimensions on an infinite
spatial lattice using matrix product states (MPS). We introduce and apply a
naive variational conjugate gradient method, based on the time-dependent
variational principle (TDVP) for imaginary time, to obtain approximate ground
states, using a related ansatz for excitations to calculate the particle and
soliton masses and to obtain the spectral density. We also estimate the central
charge using finite-entanglement scaling. Our value for the critical parameter
agrees well with recent Monte Carlo results, improving on an earlier study
which used the related DMRG method, verifying that these techniques are
well-suited to studying critical field systems. We also obtain critical
exponents that agree, as expected, with those of the transverse Ising model.
Additionally, we treat the special case of uniform product states (mean field
theory) separately, showing that they may be used to investigate non-critical
quantum field theories under certain conditions.
ABSTRACT: 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 continuous 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.
ABSTRACT: We introduce a variational method for calculating dispersion relations of
translation invariant (1+1)-dimensional quantum field theories. The method is
based on continuous matrix product states and can be implemented efficiently.
We study the Lieb-Liniger model as a benchmark where, despite criticality,
excellent agreement with the exact solution is found, including, clear
solitonic effects in Lieb's Type II excitation. In addition, a non-integrable
model is studied where a U(1)-symmetry breaking term is added to the
Lieb-Liniger Hamiltonian. For this model we find evidence of a non-trivial
bound-state excitation in the dispersion relation.
ABSTRACT: We discuss various properties of the variational class of continuous matrix
product states, a class of ansatz states for one-dimensional quantum fields
that was recently introduced as the direct continuum limit of the highly
successful class of matrix product states. We discuss both attributes of the
physical states, e.g. by showing in detail how to compute expectation values,
as well as properties intrinsic to the representation itself, such as the gauge
freedom. We consider general translation non-invariant systems made of several
particle species and derive certain regularity properties that need to be
satisfied by the variational parameters. We also devote a section to the
translation invariant setting in the thermodynamic limit and show how
continuous matrix product states possess an intrinsic ultraviolet cutoff.
Finally, we introduce a new set of states which are tangent to the original set
of continuous matrix product states. For the case of matrix product states,
this construction has recently proven relevant in the development of new
algorithms for studying time evolution and elementary excitations of quantum
spin chains. We thus lay the foundation for similar developments for
one-dimensional quantum fields.
ABSTRACT: We study the geometric properties of the manifold of states described as
(uniform) matrix product states. Due to the parameter redundancy in the matrix
product state representation, matrix product states have the mathematical
structure of a (principal) fiber bundle. The total space or bundle space
corresponds to the parameter space, i.e. the space of tensors associated to
every physical site. The base manifold is embedded in Hilbert space and can be
given the structure of a K\"ahler manifold by inducing the Hilbert space
metric. Our main interest is in the states living in the tangent space to the
base manifold, which have recently been shown to be interesting in relation to
time dependence and elementary excitations. By lifting these tangent vectors to
the (tangent space) of the bundle space using a well-chosen prescription (a
principal bundle connection), we can define and efficiently compute an inverse
metric, and introduce differential geometric concepts such as parallel
transport (related to the Levi-Civita connection) and the Riemann curvature
ABSTRACT: We argue that the natural way to generalise a tensor network variational
class to a continuous quantum system is to use the Feynman path integral to
implement a continuous tensor contraction. This approach is illustrated for the
case of a recently introduced class of quantum field states known as continuous
matrix-product states (cMPS). As an example of the utility of the path-integral
representation we argue that the state of a dynamically evolving quantum field
admits a natural representation as a cMPS. An argument that all states in Fock
space admit a cMPS representation when the number of variational parameters
tends to infinity is also provided.
ABSTRACT: We describe how to implement the time-dependent variational principle for
matrix product states in the thermodynamic limit for nonuniform lattice
systems. This is achieved by confining the nonuniformity to a (dynamically
growable) finite region with fixed boundary conditions. The suppression of
unphysical quasiparticle reflections from the boundary of the nonuniform region
is also discussed. Using this algorithm we study the dynamics of localized
excitations in infinite systems, which we illustrate in the case of the spin-1
anti-ferromagnetic Heisenberg model and the phi^4 model.
ABSTRACT: We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary-time dynamics for infinite one-dimensional quantum lattices. This procedure (i) is argued to be optimal, (ii) does not rely on the Trotter decomposition and thus has no Trotter error, (iii) preserves all symmetries and conservation laws, and (iv) has low computational complexity. The algorithm is illustrated by using both an imaginary-time and a real-time example.
Physical Review Letters 08/2011; 107(7):070601. · 7.37 Impact Factor
ABSTRACT: A variational ansatz for momentum eigenstates of translation invariant
quantum spin chains is formulated. The matrix product state ansatz works
directly in the thermodynamic limit and allows for an efficient implementation
(cubic scaling in the bond dimension) of the variational principle. Unlike
previous approaches, the ansatz includes topologically non-trivial states
(kinks, domain walls) for systems with symmetry breaking. The method is
benchmarked using the spin-1/2 XXZ antiferromagnet and the spin-1 Heisenberg
antiferromagnet and we obtain surprisingly accurate results.
ABSTRACT: We show how continuous matrix product states of quantum fields can be described in terms of the dissipative nonequilibrium dynamics of a lower-dimensional auxiliary boundary field by demonstrating that the spatial correlation functions of the bulk field correspond to the temporal statistics of the boundary field. This equivalence (1) illustrates an intimate connection between the theory of continuous quantum measurement and quantum field theory, (2) gives an explicit construction of the boundary field allowing the extension of real-space renormalization group methods to arbitrary dimensional quantum field theories without the introduction of a lattice parameter, and (3) yields a novel interpretation of recent cavity QED experiments in terms of quantum field theory, and hence paves the way toward observing genuine quantum phase transitions in such zero-dimensional driven quantum systems.
Physical Review Letters 12/2010; 105(26):260401. · 7.37 Impact Factor
ABSTRACT: We extend the recently introduced continuous matrix product state variational class to the setting of (1+1)-dimensional relativistic quantum field theories. This allows one to overcome the difficulties highlighted by Feynman concerning the application of the variational procedure to relativistic theories, and provides a new way to regularize quantum field theories. A fermionic version of the continuous matrix product state is introduced which is manifestly free of fermion doubling and sign problems. We illustrate the power of the formalism by studying the momentum occupation for free massive Dirac fermions and the chiral symmetry breaking in the Gross-Neveu model.
Physical Review Letters 12/2010; 105(25):251601. · 7.37 Impact Factor
ABSTRACT: We show that excitations of interacting quantum particles in lattice models
always propagate with a finite speed of sound. Our argument is simple yet
general and shows that by focusing on the physically relevant observables one
can generally expect a bounded speed of information propagation. The argument
applies equally to quantum spins, bosons such as in the Bose-Hubbard model,
fermions, anyons, and general mixtures thereof, on arbitrary lattices of any
dimension. It also pertains to dissipative dynamics on the lattice, and
generalizes to the continuum for quantum fields. Our result can be seen as a
meaningful analogue of the Lieb-Robinson bound for strongly correlated models.
ABSTRACT: We show that ground states of unfrustrated quantum spin-1/2 systems on general lattices satisfy an entanglement area law, provided that the Hamiltonian can be decomposed into nearest-neighbor interaction terms which have entangled excited states. The ground state manifold can be efficiently described as the image of a low-dimensional subspace of low Schmidt measure, under an efficiently contractible tree-tensor network. This structure gives rise to the possibility of efficiently simulating the complete ground space (which is in general degenerate). We briefly discuss "non-generic" cases, including highly degenerate interactions with product eigenbases, using a relationship to percolation theory. We finally assess the possibility of using such tree tensor networks to simulate almost frustration-free spin models. Comment: 14 pages, 4 figures
ABSTRACT: We investigate the propagation of information through one-dimensional quantum chains in fluctuating external fields. We find that information propagation is suppressed, but in a quite different way compared to the situation with static disorder. We study two settings: (i) a general model where an unobservable fluctuating field acts as a source of decoherence; (ii) the XX model with both observable and unobservable fluctuating fields. In the first setting we establish a noise threshold below which information can propagate ballistically and above which information is localised. In the second setting we find localisation for all levels of unobservable noise, whilst an observable field can yield diffusive propagation of information. Comment: 5 pages, 2 figures
ABSTRACT: We investigate the propagation of information through the disordered XY model. We find that all correlations, both classical and quantum, are exponentially suppressed outside of an effective light cone whose radius grows at most logarithmically with |t|.
Physical Review Letters 11/2007; 99(16):167201. · 7.37 Impact Factor
ABSTRACT: We investigate the propagation of information through the disordered XY
model. We find, with a probability that increases with the size of the system,
that all correlations, both classical and quantum, are suppressed outside of an
effective lightcone whose radius grows at most polylogarithmically with |t|.
ABSTRACT: In this Letter we show that an arbitrarily good approximation to the propagator e(itH) for a 1D lattice of n quantum spins with Hamiltonian H may be obtained with polynomial computational resources in n and the error epsilon and exponential resources in |t|. Our proof makes use of the finitely correlated state or matrix product state formalism exploited by numerical renormalization group algorithms like the density matrix renormalization group. There are two immediate consequences of this result. The first is that Vidal's time-dependent density matrix renormalization group will require only polynomial resources to simulate 1D quantum spin systems for logarithmic |t|. The second consequence is that continuous-time 1D quantum circuits with logarithmic |t| can be simulated efficiently on a classical computer, despite the fact that, after discretization, such circuits are of polynomial depth.
Physical Review Letters 11/2006; 97(15):157202. · 7.37 Impact Factor