Matrix Product States: Symmetries and two-body Hamiltonians
We characterize the conditions under which a translationally invariant matrix product state (MPS) is invariant under local transformations. This allows us to relate the symmetry group of a given state to the symmetry group of a simple tensor. We exploit this result in order to prove and extend a version of the Lieb-Schultz-Mattis theorem, one of the basic results in many-body physics, in the context of MPS. We illustrate the results with an exhaustive search of –invariant two-body Hamiltonians which have such MPS as exact ground states or excitations.
Matrix Product States (MPS) FNW92 (); PVWC07 () encapsulate many of the physical properties of quantum spin chains. Of particular interest in various physical contexts is the subset of translationally invariant (TI) MPS, originally introduced as finitely correlated states FNW92 (). Their importance stems from the fact that with a simple tensor, , one can fully describe relevant states of spins, which, at least in principle, should require to deal with an exponential number of parameters when written in a basis in the corresponding Hilbert space . Thus, all the physical properties of such states are contained in . It is therefore important to obtain methods to extract the physical properties directly from such a tensor, without having to resort to .
An important physical property of a TI state, , is the symmetry group under which it is invariant. That is, the group such that
where and is a unitary representation on . In a recent paper PWSVC08 () we showed that for certain kind of MPS (those fulfilling the so–called injectivity condition FNW92 (); PVWC07 ()), this symmetry group is uniquely determined by the symmetry group of (with a tensor product representation). Roughly speaking this means that by studying the symmetries of we can obtain those for the whole state . This result allows us, for example, to shed a new perspective into string order PWSVC08 (), a key concept in strongly correlated states in many–body quantum systems.
Another relevant property of MPS is that they are all exact ground states of short–range interacting (frustration free) Hamiltonians FNW92 (); PVWC07 (). In particular, for every TIMPS we can always build a (so–called) ’parent’ Hamiltonian for which it is the ground state. Of particular interests are TIMPS with two–body parent Hamiltonians; that is, whose parent Hamiltonian consist of two–body interactions only. And among those, the ones which have a large symmetry group, like . The reason is that those are the ones that naturally appear in condensed matter problems. Two prominent examples are the AKLT AKLT88 () and the Majumdar-Gosh MG68 () states, who have two-body parent Hamiltonians with symmetry. They have served as toy models to understand certain physical behavior in real physical systems, like the existence of a Haldane gap Haldane83 () in spin chains with integer spin, or the phenomenon of dimerization MG68 (), respectively. Despite their key role in the understanding of spin chains, there are very few other examples known of TIMPS with symmetry and with a two-body parent Hamiltonian FNW92 (); KSZ93 (); KM07 ().
In this work we first generalize the results of Ref. PWSVC08 () to arbitrary TIMPS. This enables us to derive some generic properties about those states, as well as to obtain a simple proof for a version of the Lieb-Schultz-Mattis theorem LSM61 (). This celebrated theorem states that all Hamiltonians with symmetry are gapless for semi-integer spin (dim, ). In our case, we can prove that all TIMPS corresponding to systems with semi-integer spins cannot be the unique ground state of a local frustration-free Hamiltonian. Furthermore, we can extend the proof to other groups, like for spin 1/2 systems, and find counterexamples for this last case when the spin is 5/2 or larger.
In the second part of our work we concentrate on MPS that are eigenstates (not necessarily grounds states) of a (so–called ’parent’) Hamiltonian which has symmetry and contains two–body interactions only. We find other families of Hamiltonians beyond the well–known AKLT and Majumdar-Gosh with those features. Furthermore, we find the first examples of MPS that correspond to excited states of -invariant Hamiltonians. There is a new example of state with spin , which is never the ground state of any frustration free -invariant two-body hamiltonian. In order to make a systematic search of all those MPS we develop a simple technique that allows for a numerical systematic search.
This paper is organized as follows. In Section II we review some of the basic properties of TIMPS and establish the notation that will be needed in the following. In Section III we establish the relation between the symmetry group of a TIMPS and that of the tensor defining the MPS. For continuous symmetries, such as , we will see that the set of symmetric TIMPS is intimately related to the set of Clebsch-Gordan coefficients. Section IV then provides an MPS version of the Lieb-Schultz-Mattis theorem and in Section V we give a detailed investigation of symmetric TIMPS which are eigenstates of two-body Hamiltonians.
Ii Matrix Product States
Let us consider a system with periodic boundary conditions of (large but finite) sites, each of them with an associated -dimensional Hilbert space. A translationally invariant MPS on this system can be defined with a valence bond construction in the following way: Let us consider another couple of dimensional ancillary/virtual Hilbert spaces associated to each site and connected to the real/physical dimensional space by a map . Then, by introducing maximally entangled states connecting every pair of neighboring virtual Hilbert spaces (usually called entangled bonds), it is not difficult to prove that the state can be written as
where we call the matrices Kraus operators. A way to work simultaneously with all of them is to define the map
For each MPS there exists a canonical form (PVWC07, , Theorem III.7, Lemma IV.4) which assures that one may choose all matrices with a block diagonal structure 111Eventually after gathering some spins together to neglect the periodic components—something that we will always assume., in such a way that after gathering enough spins together, the Kraus operators fulfil:
Property 1 (Span property).
The set of products , with the collected spins, spans the vector space of all matrices with the same block diagonal structure.
It is an open conjecture stated in PVWC07 () and verified in many particular cases, that an upper bound for the number of sites which have to be gathered to achieve property 1 depends only on the dimension of the Kraus operators. When there is only one block in the above canonical decomposition the MPS is usually called injective, since the linear operator mapping boundary conditions to the resulting states is indeed injective FNW92 (); PVWC07 () when taking sufficiently many particles. The definition reads:
Property 2 (Injectivity).
There exists such that the map is injective.
For each MPS one can construct a Hamiltonian, called parent Hamiltonian, for which is an eigenstate with eigenvalue .
Definition 3 (Parent Hamiltonian).
Let be the reduced density matrix of for particles ( will be called the interaction length of the parent Hamiltonian). Let us suppose that , with , is an orthonormal basis for . Taking any linear combination of projectors , we define , where is the translation operator.
If , then the Hamiltonian is positive semidefinite and is indeed a ground state. Moreover is frustration free, since minimizes the energy locally. Injectivity has now a deep physical significance. If it is reached for particles and every , it ensures that the MPS is the only ground state of its -local parent Hamiltonian, that it is an exponentially clustering state and that there is a gap above the ground state energy FNW92 (); PVWC07 ().
In this work we will focus on symmetries of states instead of Hamiltonians. There is however a close connection between the two approaches. On the one hand, it is clear that the unique ground state of a symmetric Hamiltonian has to keep the symmetry. On the other hand, we have the following
If an MPS is invariant under a representation of a group, one can choose its parent Hamiltonian invariant under the same representation.
To see that it is enough to notice that the symmetry in the state (1) implies the invariance of under the same symmetry. Symmetrizing (i.e., averaging it) w.r.t. the considered group will then yield a symmetric which still constitutes a parent Hamiltonian.
Iii Locally symmetric MPS
In this section we analyze the implications of a given symmetry for a MPS. First, we show that the symmetry transfers to the Kraus operators—generalizing the findings of FNW92 (); PWSVC08 (). In a second step we show that the symmetry in the Kraus operators imposes that they are essentially uniquely defined in terms of Clebsch-Gordan coefficients. Finally, for the special case of one can simplify even further and analyze the qualitative differences between integer and semi-integer spin.
iii.1 Characterization of symmetries
It was demonstrated in PWSVC08 () that the Kraus operators which describe any injective state symmetric under a group fulfil the condition , where and are representations of . We provide in this section a generalization in which injectivity is not required. The appearing in the proof must be sufficiently large to obtain property 1 after collecting spins.
We start by proving the case of discrete symmetries, extending the demonstration to continuous groups below.
Theorem 5 (Discrete symmetries).
Let be the Kraus operators which describe a locally invariant MPS with respect to a single unitary , i.e. . Then, the symmetry in the physical level can be replaced by a local transformation in the virtual level. This means that there exists a unitary – which can be taken block diagonal with the same block structure as the ’s in the MPS and composed with a permutation matrix among blocks, i.e. – such that
We follow here a reasoning as in the proof of (PVWC07, , Lemma IV.4). We collect the spins in five different blocks, each one of them with property 1. Applying gives us the same MPS (we incorporate the global phase in the new matrices) with different matrices ’s, but with the same block diagonal form and also (after gathering) with property 1. We now require the following lemma, which is demonstrated below.
For each block in the ’s, for instance the one given by matrices , there is a block in the ’s, given by matrices , which expands the same MPS.
Since both are now canonical forms of the same injective MPS, by (PVWC07, , Theorem 3.11), 222The condition of (PVWC07, , Theorem 3.11) that the canonical form in the OBC must be unique can be dropped by eq. (3) of [Lamata et al, PRL 101, 180506 (2008)], they must be related by a unitary and a phase: , which finishes the proof of the theorem.
Let us prove now the lemma. By using property 1 and summing with appropriate coefficients, it is possible to show that there exists a block diagonal matrix such that
Since , there exists one block, let us say , different from . Then, summing with appropriate coefficients again we get that there exists a matrix such that
We can now argue as in (PVWC07, , Lemma IV.4) to conclude the proof. ∎
If we have now a symmetry given by a compact connected Lie group , that is, (1) holds for any and a representation , we obtain the following.
Theorem 7 (Continuous symmetries).
The map is a representation of and therefore the trivial one. The maps and are also representations of .
Let us start with the map . From eq. we get
where is the same unitary as but with the blocks permuted according to the permutation . Since and commutes with all other terms appearing in eq. (4), we can multiply successively and use property 1 (with the required block size), to get, for all and all block-diagonal,
By taking for each block , we get that must be . But since we are assuming the group connected, this in turn implies that for all . With this we can split equation (5) into blocks to get, for each , each and each matrix ,
Taking we obtain
In particular, when , we get that and when
that . Gathering both results, the can be
removed and we obtain
Finally, to show that is a representation, it is enough to notice that eq. (6) implies that commutes with every matrix. ∎
A trivial consequence of these theorems is the fact that having an irreducible representation in the virtual level implies that the MPS has to be injective. We give an alternative proof of this fact in the appendix without having to rely on the MPS canonical form. There we analyze also when the reverse implication holds.
iii.2 Uniqueness of the construction method
Once the theorem which provides the condition that the Kraus operators must fulfil in order to generate invariant MPS has been established, the next step is to prove that they can always be constructed by means of Clebsch-Gordan coefficients. To do that, it is more convenient to work with the map defined in (2). From the definition it is clear that the condition reads then . Notice that we have removed the dependence on the phase. By Theorem 7 this can be done for groups with a complex enough structure, as , for which there is no non-trivial one-dimensional representation.
Given a compact group , the tensor product of two irreps –we are choosing a single representative for each class of equivalent irreps– can always be decomposed as a direct sum of irreps
where is a unitary whose elements are called Clebsch-Gordan coefficients. In what follows we will denote by the matrix associated to the restriction of to the -dimensional invariant subspace associated to the irrep , with being the dimensions of the representations and respectively.
We are interested in possible solutions of
where are irreps of a given compact group . It is clear that taking
does the job if we sum over ’s corresponding to equivalent representations . The next lemma guarantees that this is all.
Any verifying eq. gives
which means by Schur’s lemma that and we may assume that, if there is a non-zero solution, it can be taken an isometry. Moreover, introducing , which verifies , one has
From there one gets that is a rank projector ( the dimension of the representation ) that commutes with for all . By Schur’s lemma, it is supported on with ’s such that and in this subspace it is of the form
This implies that for a given unitary . But if we substitute this in (9), since we are assuming a unique fixed representative for each class of equivalent representations, we get and . ∎
From this we can now conclude:
Let us consider a group and two representations (irrep) and . Then, the structure of all possible maps fulfilling is
where is a solution, according to Lemma 8, to .
iii.3 The case of
We consider from now on irreducible representations of the symmetry on the physical spin. Nevertheless, a substantial part of the results can be straightforwardly extended to the reducible case. Hence, we are interested in analyzing the restrictions that impose in the general solution given by Theorem 9 to the equation
where, with some abuse of notation, is the irrep corresponding to spin and is the virtual representation composed of integer irreps and semi-integer irreps. Note that in the Clebsch-Gordan decomposition of all representations appear with multiplicity one. Therefore there is only one term in the sum in (8). At this point one should distinguish the cases of integer or semi-integer. If is integer, zero is the only solution to and for all , and we get in (10) a block diagonal structure:
The paradigmatic example in this case is the AKLT state AKLT88 (), which corresponds to the case of , in (11). In FNW92 (), the authors generalized the AKLT model to arbitrary integer and irreducible. We will call the resulting MPS FNW states. It is shown in FNW92 () how for FNW states are unique ground states of frustration free nearest-neighbor interactions. An alternative construction focused on the restrictions imposed by the symmetry on the density matrix instead of the Kraus operators can be found in DMNS98 ().
If is semi-integer, zero is the only solution to and for all , and we get in (10) an off-diagonal structure:
It is clear that the virtual representations must be reducible now, which is very much related to the Lieb-Schultz-Mattis theorem, as we will show in the following section. The paradigmatic example in this case is the Majumdar-Ghosh model MG68 (), which corresponds to and . A generalization of this model for the case of arbitrary and , was recently proposed in KM07 ().
Iv Lieb-Schultz-Mattis Theorem
The Lieb-Schultz-Mattis theorem states that, for semi-integer spin, a -invariant 1D Hamiltonian cannot have a uniform (independent of the size of the system) energy gap above a unique ground state. That is, symmetry imposes strong restrictions on the possible behaviors of a system. In this section we want to go a step further and analyze which implications one can obtain from having a single symmetric state in a semi-integer spin chain. By restricting our attention to the class of MPS we will show
We consider with , . Then, eq. gives
for a unitary and half-integer. We finish by proving that if is odd, and hence the MPS cannot be injective. From (13) we get unless . The latter is, however, impossible for odd as then the l.h.s. is integer whereas the r.h.s. is half-integer. ∎
From the proof one may get the impression that only symmetry is required, and this is indeed the case if the generator of such symmetry has eigenvalues as above. The next example shows that this is, however, not true for any symmetry, which in turn shows that a larger symmetry like is required for the Lieb-Schultz-Mattis theorem.
Let us consider a local symmetry generated by for a hermitian matrix . Let us choose the physical dimension , which is always even, and the set of Kraus operators . Select such that if and the diagonal matrix (which has in addition only non-zero eigenvalues). With where it is clear that
so the MPS generated by means of the Kraus operators has the local symmetry . Moreover, the MPS is trivially injective when . We can prove this by choosing arbitrary and . Since , we can always find an such that and then .
Let us remark that this counter-example is applicable to spin . Indeed, one can prove Theorem 2 for and spin , which is the content of the following proposition. The case of spin remains an open question.
If is an MPS with physical dimension and invariant under , then cannot be injective.
We will show it by contradiction. By choosing a basis where the physical unitary is diagonal, the condition on the Kraus operators becomes
where is the hermitian generator of the symmetry. Let us expand the expression for infinitesimal angles
which is the equation of eigenvalues for the operator . This can be transformed into an ordinary eigenvalue equation for the matrix operator . The diagonalization can be easily performed by taking the spectral decomposition of , where are orthogonal projectors. It straightforwardly follows that the eigenvalues of are and the corresponding eigenoperators fulfil .
Let us focus now on the case . Then, we have that and for some . If for all and the MPS cannot be injective. The same happens if . So let us assume that and . Now if , we have , and the MPS is block diagonal and hence non-injective. The same happens if . So and and this gives which implies that has dimension . ∎
V General construction of two-body Hamiltonians with MPS eigenstates
We have seen in Definition 3 a way, called the parent Hamiltonian method, to construct local -symmetric Hamiltonians with MPS as eigenstates. In this section we first prove that this method is the most general one to find Hamiltonians having a given MPS as local eigenstate, that is, being an eigenstate of each local term in the Hamiltonian. Then, we show examples (including the AKLT and Majumdar-Ghosh states) of MPS that are excited eigenstates of local two-body translationally invariant -symmetric Hamiltonians. More examples are then provided in the appendix.
v.1 Completeness of the parent Hamiltonian method
Given an MPS , any translational invariant Hamiltonian having it as a local eigenstate is of the form where is a parent Hamiltonian for in the sense of Definition 3.
Let us call the local hamiltonian. By hypothesis of local eigenstate,
for certain . This implies and hence one can find a set of projectors such that we can decompose both and by means of them, i.e. and , where represents the set of projectors which describe the support of . Using eq. (14) with this decomposition gives that for all and hence
Then, the translational invariance hamiltonian is , where is the translation operator. The theorem follows from replacing the result for the local hamiltonian and comparing this with Definition 3 of parent hamiltonian. ∎
This Theorem shows that, given an MPS , looking for all possible parent Hamiltonians of interaction length is equivalent to look for all possible solutions to the equation
with . The next lemma gives yet another equivalent formulation, which is the one we will use in the sequel.
Given a Hermitian matrix and a density matrix , if and only if
One implication is clear. For the other, let us write for . By assumption
So , since it is a positive operator with trace . This implies that and hence . ∎
With this at hand we can systematically search for MPS that are excited local eigenstates of invariant Hamiltonians with two-body interactions. We will proceed as follows. We start with a given symmetric MPS and fix the interaction length . Then we look for possible solutions to Eq. (16) of the form
to ensure symmetry and two body interactions in the Hamiltonian. Finally, to guarantee that the MPS is an excited state, we will find another symmetric MPS with less energy that will act as a witness. In the next section we will illustrate this procedure starting with the AKLT, the Majumdar-Ghosh state, and generalizations. Throughout we work in the thermodynamical limit .
v.2 Examples of two-body Hamiltonians
v.2.1 Spin 1
Let us consider the AKLT state as a first example. Its Kraus operators are , , .
In the case the only solution to Eq. (16) is the AKLT Hamiltonian. In the case , the solutions are given by
where the eigenvalue corresponding to the AKLT state is . The total translational invariant Hamiltonian is then
which contains the usual AKLT model. It is not difficult to check that there is a region in the parameter space where the AKLT state is still the ground state of this Hamiltonian. To find regions where it is an excited eigenstate we will use as a witness the symmetric MPS associated to the virtual representation (see Section III). The result is plotted in fig. 3, where one sees the existence of points in this family of spin Hamiltonians for which the AKLT state is an excited state.
Note that it is possible to perform a change of variables in the total Hamiltonian, for instance and , such that it depends only on two parameters. However, the number of parameters that the local Hamiltonian depends on cannot be reduced, which means that there are non-physical parameters in it. In Fig. 4 we have represented the problem above ( and AKLT state) in terms of the physical parameters. The positive axis corresponds there to the usual AKLT Hamiltonian.
Concerning FNW states, that is integer spin and virtual irrep
, we have performed an exhaustive search and table 1
gathers the main results. The study has been carried out by
increasing and studying the number of parameters which the
family of Hamiltonians depends on (notice that the case of
interaction length contains the case of interaction length
). We have increased until the number of parameters stops
growing. In all the cases considered in the table, a saturation
occurs when , i.e. considering more than
particles does apparently not add new Hamiltonians.
Let us also introduce a new state of spin with virtual spin , given by the Kraus operators
The total translational invariant hamiltonian which has this state as eigenstate is
This state is injective and a local excited state. The fact that this state is an excited state of the global hamiltonian can be checked as above by means of the witness .
Let us consider now the the Majumdar-Ghosh state as an example with semi-integer spin. The Kraus operators are now
As in the previous case, we do not find any solution for and only the Majumdar-Ghosh Hamiltonian for the cases and . For the solutions to Eq. (16) are given by
and the energy associated to the state is . The total Hamiltonian is given by
As in the AKLT case, by means of a change of variables and , the number of physical parameters in the total Hamiltonian is , compared with the four parameters the local Hamiltonian depends on. The Majumdar-Ghosh state is an excited local eigenstate for a region in the space of parameters, which in this case is detected by the witness , as shown in fig. 5. The usual Majumdar-Ghosh Hamiltonian 333The family of hamiltonians constructed in Ku02 () is quite remarkable. In this paper the ground state of the hamiltonian which corresponds to is calculated, and fits perfectly to our results. corresponds to the positive axis .
Let us consider as final example the symmetric MPS corresponding to spin and virtual representation . For , the solutions to Eq. (16) are given by
and the energy associated to the MPS is in this case . The global Hamiltonian reads now