arXiv:0910.1130v1 [cond-mat.str-el] 6 Oct 2009
Renormalization and tensor product states in spin
chains and lattices
J. Ignacio Cirac
Max-Planck-Institut f¨ur Qua ntenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching,
Germany.
E-mail: ignacio.cirac@mpq.mpg.de
Frank Verstraete
Fakult¨at f¨ur Physik, Universit¨at Wien, Boltzmanngass e 5, A-1090 Wien, Austria.
E-mail: frank.verstraete@univie.ac.at
Abstract. We review different descriptions of many–body quantum systems in terms
of tenso r product states. We introduce several families of such states in terms of known
renormaliz ation procedures, and show that they naturally aris e in that context. We
concentrate o n Matr ix Product States, Tree Tensor Sta tes, Multiscale Entanglement
Renormalization Ansatz, and Projected Entangled Pair States. We highlight some of
their properties, and show how they can be used to desc ribe a variety of sys tems .
PACS numbers: 02.70.-c,03.67.-a,05.30.-d,03.65.Ud
Renormalization an d tensor product states in spin chains an d lattices 2
1. Introduction
Many-body quantum states appear in many contexts in Physics and other areas of
Science. They are very hard to describe, even computationally, due to the number
of parameters required to express them, which typically grows exponentially with the
number of particles. Let us consider a spin chain of N spin s systems. If we write
an arbitrary state in the basis |n
1
, . . . , n
N
i, where n
k
= 1, . . . , (2s + 1 ), we will have to
specify (2s+1)
N
coefficients. Even if s = 1/2, and N ∼ 50, it is impossible to store such
a number of coefficients in a computer. Furthermore, even if that would be possible,
whenever we want to make any prediction, like the expectation value of an observable,
we will have to operate with those coefficients, and thus the number of operations (and
therefore the computational time) will inevitably grow exponentially with N.
In many important situations, one can circumvent this problem by using certain
approximations. For example, sometimes it is possible to describe the state in t he so-
called mean field approximation, where we write |Ψi = |φ
1
, . . . , φ
N
i, ie. as a pr oduct
state. Here, we just have to specify each of the |φ
M
i, and thus we need only (2s + 1)N
coefficients. This method and its extensions, even though it has a restricted validity
in general, has been very successfully used to describe many of t he phenomena that
appear in quantum many-body systems. This indicates that, among all possible states,
the ones that are relevant for many practical situations possess the same properties as
product states. Another very successful method is renormalization [1, 2] which, in the
context o f spin chains and lattices, tries to obtain the physics of the low energy states by
grouping degrees of freedom and defining new ones that are simple to ha ndle, so that at
the end we can cope with very large systems and using very few parameters. There exist
very successful methods to uncover the physical properties of many-body systems which
exploit numerical approaches. The first one is Quantum Monte Carlo, which samples
product states in order to get the expectation values of physical observa bles. Another
one is Density Matrix Renormalization Group (DMRG) [3, 4], which is specially suited
for 1D lattices, and is based on renormalization group ideas.
1.1. DMRG and Tensor Product States
Wilson’s renormalization method provided a practical way of qualitatively determining
the low energy behavior of some of those systems. However, it was by no means sufficient
to describe them quantitatively. In 1991, Steve White [3, 4] proposed a new way of
performing the renormalization procedure in 1D systems, which gave extraordinary
precise results. He developed t he DMRG algorithm, in which the renormalizatio n
procedure takes explicitly into account the whole system at each step. This is done by
keeping the states of subsystems which are relevant to describe the whole wavefunction,
and not those that minimize the energy o n that subsystem. The algorithm was rapidly
extended and adapted to different situations [5, 6], becoming the method of choice for
1D systems. In 1995, Ostlund a nd Rommer [7] realized that the state resulting from the
DMRG algorithm could be written as a so–called Matrix Product State (MPS), ie, in
Renormalization an d tensor product states in spin chains an d lattices 3
terms of products of certain matrices (see also [8, 9, 10]). They proposed to use this set
of states as a variational family for infinite homogeneous systems, where one could stat e
the problem without the language of DMRG, although the results did not look as precise
as with the finite version of that method. Those states had appeared in the literature in
many different contexts and with different names. First, as a variational Ansatz for the
transfer matrix in the estimation of the pa r t ition function of a classical model [11]. Later
on, in the AKLT model in 1D [12, 13], where the ground state has t he form of a valence
bond solid (VBS) which can be exactly written as a MPS. Translationally invariant
MPS in infinite chains were thoroughly studied and characterized mathematically in full
generality in Ref. [14], where they called such family finitely correlated states (FCS).
The name MPS was coined later on by Kl¨umper et al [15, 16], who introduced different
models extending the AKLT where the gr ound state had the explicit FCS form. All
those studies were carried out for translationa lly invariant systems, where the matrices
associated to each spin do not depend on the position of the spin. An extension of
the work of Fannes et al [14] to general MPS (ie finite and non–homogeneous states)
appeared much later on [17 ].
Given the success of DMRG in 1D, several authors tried to extend it to higher
dimensions. The first a t tempts considered a 2D system as a chain, and used DMRG
directly on the chain [18, 1 9, 20], obtaining much less precise results than in 1D. Another
attempt considered homogeneous 3D classical system and used ideas taken from DMRG
to estimate t he partition function of the Ising Model [21]. A different approach was
first suggested by Sierra and Mar t in-Delgado [22] inspired by the ideas of Ostlund and
Rommer [7]. They introduced two families of translationally invariant states, the vertex-
and face- matrix product state Ans¨atze, and proposed to use t hem variationally fo r 2D
systems, in the same way as Ostlund and Rommer used FCS in 1D. The first family
generalized the AKLT 2D VBS state [12] in as much the same way as FCS did it in
1D. The second one was inspired by interaction-ro und-the-face models in (classical)
Statistical Mechanics. The inclusion of few parameters in the VBS wavefunction of
AKLT to extend that model was first suggested in Ref. [23]. The authors also showed
that the calculation of expectation values in those VBS could be thought of as evaluating
a classical partition function, something they did using Monte Carlo methods [23], and
Hieida et al using ideas taken from DMRG [24]. Later on, Nishino and collaborators
used the representations proposed by Sierra and Martin-Delgado, as well as ano ther one
they called interaction-round-a-face (inspired by t he specific structure of the tra nsfer
matrix of the classical Ising model) to determine t he par tition function of the classical
Ising Model in 3D variationally [25, 26, 27, 28, 29]. For instance, in [28] a vertical density
matrix algorithm was introduced to calculate thermodynamical properties of that model
based on the t he interaction-round-a-face r epresentatio n and in [27] a perturbation
approach was taken for the vertex matrix product state, which turn out to be numerically
unstable. Eventually, quantum systems at zero temp era t ure were considered by direct
minimization of trial wavefunctions of the VBS type with few variational par ameters
[30, 31]. In summary, most of the attempts in 2D quantum systems (and 3D classical
Renormalization an d tensor product states in spin chains an d lattices 4
ones) tried to generalize the method of Ostlund and Rommer [7] by using families of
states that extended FCS to higher dimensions, and dealing with infinite homogenous
systems. FCS and their extensions were based on tensors contracted in some special
ways, and thus all of them were called tensor product states (TPS). Nevertheless, no
DMRG–like algorithm for 2D or higher dimensions was put forward (except for the
direct application of DMRG by considering the 2D system as a 1D chain).
The success of DMRG for 1D systems indicated that the family of states on which
it based, namely MPS, may provide an efficient and accurate description of spin chain
systems. In higher dimensions, however, the situation was much less clear since only
infinite systems were considered and the numerical results were not entirely satisfactory.
1.2. The corner of Hilbert space
One can look at the pro blem of describing many–body quantum systems from a different
perspective. The fact that product states in some occasions may capture the physics of a
many-body problem may look very surprising at first sight: if we choose a random state
in the Hilbert space (say, according to the Haar measure) the overlap with a product
state will be exponentially small with N. This apparent contradiction is resolved by
the fact that the states that appear in Nature are not random states, but they have
very peculiar forms. This is so because of the following reason. If we consider states
in thermal equilibrium, each state of a system, described by the density operator ρ,
is completely characterized by the Hamiltonian describing that system, H, and the
temperature, T , ρ ∝ e
−H/T
. In all systems we know, the Hamiltonian contains terms
with at most k–body interactions, where k is a fixed number independent of N which
typically equals 2. We can thus parameterize all possible Hamiltonians in Nature in
terms of (N, k) ×(2s + 1)
2k
. The first term is the number of groups of k spins, whereas
the second one gives the number of parameters of a general Hamiltonian acting on k
spins. This number scales only polynomially with N, and thus all possible density
operators will also depend on a polynomial number of parameters. In practice, if we
just have 2– body interactions (ie, k=2), and short–range interactions, the number will
be linear in N. If we additionally have translatio nal symmetry, the number will even be
independent o f N. This shows that even though we just need an exponential number of
parameters to describe a general state, we need very few to describe the relevant states
that appear in Nature. In this sense, the relevant stat es are contained in ”a corner of
the Hilbert space”. This representation is, however, not satisfactory, since it does not
allow one to calculate expectation values.
These facts define a new challenge in many-body quantum physics, namely, to find
good and economic descriptions of that corner of Hilbert space. That is, a family of
states depending on few para meters (which increase only polynomially with N), such
that all relevant states in Nature can be approximated by members of such family. If we
are able to do that, as well as to characterize and study the pro perties of such f amily,
we would have a new language to describe many-body quantum systems which may
Renormalization an d tensor product states in spin chains an d lattices 5
be more appropriate t han the one we use ba sed on Hilbert space expansions. Apart
from that, if we find algorithms which, for any given problem (say, a Hamilto nian and a
temperature), allows us to determine the state in t he family which approaches the exact
one, we will have a very powerful numerical method to describe quantum many-body
systems. It is clear that product states are not enough for that task, since they do
not posses correlations ( nor entanglement), something that is crucial in many physical
phenomena. So, the question is how to extend product states in a way that they cover
the relevant corner of Hilbert space.
One possible strategy to follow is to determine a property that all the states on that
corner have, at least for a set of important problems. If we then find a family of states
which includes all states with that particular property, we will have succeeded in our
challenge. But, what property could that be? Here, the answer may come from ideas
developed in the context, among others, of quantum information theory. One of such
ideas appears for the subclass of problems with Hamiltonians that contain finite-range
interactions (ie, two spins interact if they are at a distance smaller than some constant),
have a gap (ie, for all N, E ≥ E
0
+ ∆, where E is t he energy of any excited state, E
0
that of the ground state, and ∆ > 0), and are at zero temperature. In that case, the
so–called area law naturally emerges [32, 33, 3 4, 35, 36, 37]. It states that for the ground
state of such a Hamiltonian, |Ψ
0
i, if we consider a block, A, of neighboring spins, the
von Neumann entropy of the reduced density operator of such a region, ρ
A
, scales with
the number of particles at the border of that region. This is quite remarka ble, since the
von Neumann entropy being an extensive quantity, for a random state it will scale with
the numb er of particles in A, and not in the border. The area law has been proven in
1D spin chains [38], and it is fulfilled for all Hamiltonians we know in higher dimensions.
Even for critical systems, where the ga p condition is not fulfilled, o nly a slight violation
occurs (namely that it is proportional to the number of spins at the border, L, t imes
log L) [36, 39, 40]. What happens at finite temperature? In that case one can also find
a global property o f all states in the corner of Hilbert space which extends the area
law. In contrast to the zero temperature case, now t his can be rigorously proven for
any Hamiltonian po ssessing finite-range interactions in a rbitrary dimensions [41]. The
property is the following: given a region A a s before, the mutual information between
the spins in that region and the rest of the spins (in region B) is bounded by a constant
times the number of spins at the border divided by the temperature. Here, the mutual
information is defined as I(A : B) = S(ρ
A
) + S(ρ
B
) − S(ρ
AB
), where S is the von
Neumann entropy, and ρ
X
is t he density operator corresponding to region X.
There is a way of constructing families o f states explicitly fulfilling the area law.
Let us first consider a 1D chain with two spins per site in which we entangle each of
them with the nearest neighbor spins (to the right and to the left, respectively). If we
now consider a block of neighboring sites, only the outermost spins with contribute to
the entropy of that block, and thus the area law will be fulfilled. Furthermore, if we
project in each site the stat e of the two spins onto a subspace of lower dimension, the
resulting state will also fulfill the area law. This construction can be straightforwardly
