Classification of the phases of 1D spin chains with commuting Hamiltonians
We consider the class of spin Hamiltonians on a 1D chain with periodic boundary conditions that are (i) translational invariant, (ii) commuting and (iii) scale invariant, where by the latter we mean that the ground state degeneracy is independent of the system size. We correspond a directed graph to a Hamiltonian of this form and show that the structure of its ground space can be read from the cycles of the graph. We show that the ground state degeneracy is the only parameter that distinguishes the phases of these Hamiltonians. Our main tool in this paper is the idea of Bravyi and Vyalyi (2005) in using the representation theory of finite dimensional -algebras to study commuting Hamiltonians.
Classification of the phases of matter is a major problem at the heart of recent activities in quantum many-body physics research, especially after the discovery of topologically ordered phases. The quantum double model of Kitaev Kitaev03 () and the string-net condensation model of Levin and Wen Levin-Wen () show that the phase diagram of 2D systems can be very complicated. These two models of commuting Hamiltonians are exactly solvable and exhibit a rich structure in their ground states as well as elementary excitations. In 1D, however, we are not aware of any essential example of a commuting Hamiltonian besides the Ising model. Moreover, simulation of the ground states based on tensor networks has been proven to be efficient in 1D gapped systems Hastings07 (). So we expect to have an easier theory in 1D. Before reviewing known results in this regard let us first describe what we exactly mean by a phase.
Hamiltonians of interest are translational invariant spin Hamiltonians defined on a lattice with periodic boundary conditions. We assume that Hamiltonians are scale invariant, meaning that the ground state degeneracy is independent of the system size, and also have a constant gap above the ground state energy which again is independent of the size of the system. We say that two such Hamiltonians (their associated ground spaces) belong to the same phase if there exists a continuous path of Hamiltonians satisfying the above conditions starting with one of them and ending with the other.
Chen et al. Chen11 () and Schuch et al. Schuch10 () have recently shown that assuming the ground space of such a 1D Hamiltonian is exactly describable by Matrix Product States (MPSs) Fannes (), the only parameter that distinguishes phases is the ground state degeneracy 1111D systems with symmetries have also been studied in these papers.. Although it seems that we can remove the assumption in this result by using the proven area law in 1D Hastings07 (), there are some technical obstacles. First, we are able to approximate the ground state of any gapped 1D Hamiltonian by an MPS, whose bond dimension scales (polynomially) with the system size. In Chen11 () and Schuch10 () however, the bond dimension is assumed to be fixed independent of the system size. Second, we need to show that the true ground state and the approximate MPS are in the same phase which again should be proved with a gap that is independent of the system size.
Yoshida Yoshida11a () has considered the problem of classification of phases in the special case of stabilizer Hamiltonians. He has shown that any translational invariant and scale invariant stabilizer Hamiltonian in 1D is equivalent to some independent copies of the Ising model, and in 2D we essentially have the 2D Ising model and the toric code. He has proved a similar result in 3D assuming some extra conditions Yoshida11b (). Although stabilizer Hamiltonians seem very restrictive, they have been extensively studied in the theory of quantum error correcting codes and quantum memories because of their simple structure of syndrome measurements. The advantage of these results comparing to Chen11 (); Schuch10 () is that Yoshida’s classification gives a characterization of logical operators, and consequently low anergy excitations as well. We emphasis on the latter because for instance in the models of quantum double and string-net condensation the structures of elementary excitations are much richer than that of ground states.
In this paper we generalize Yoshida’s results in 1D. We consider frustration free commuting Hamiltonians that are both translational and scale invariant. Due to the commutativity assumption these Hamiltonians are automatically gapped. We associate a graph with a Hamiltonian of this form and show that cycles of this graph encodes ground space of the Hamiltonian. We also comment that the low energy excited states are described by paths of the graph. Our main result is that the phases of the ground states of these systems are determined only by their degeneracy.
Although our assumption of commutativity comparing to that of Chen11 (); Schuch10 () seems very restrictive, this at least can be easily verified. Even the scale invariance of a Hamiltonians is hard to check in general, but for commuting Hamiltonians we provide a simple criterion to verify that. Moreover, in the last section we show that our results can be applied to a wider class of Hamiltonians than commuting ones.
Ii Some key observations
Let us first fix some notations. Hilbert spaces are denoted by , and is the Hilbert space corresponding to register . is the identity operator of , and is the space of linear operators acting on . We may consider as a Hilbert space equipped with the Hilbert-Schmidt inner product.
In this paper we are interested in the phases of ground spaces of Hamiltonians define on a 1D chain with periodic boundary condisions. We say that ground spaces of two Hamiltonians belong to the same phase if there exists a continuous path of gapped Hamiltonians starting with one of them and ending with the other. Here we often consider translational and scale invariant Hamiltonians in which case we further assume that all Hamiltonians along the path are translational and scale invariant and their gap is independent of the system size. We do not impose this constraint for commuting Hamiltonians, i.e., two commuting Hamiltonians are said to belong to the same phase even if the path connecting them consists of non-commuting ones. This is because we are considering phases of ground spaces, and a phase is usually defined by translational and scale invariant Hamiltonians. So these two properties are essential in the definition of the problem but being commuting is an extra constraint which we impose for simplification.
By this definition when two Hamiltonians and have the same ground spaces, they belong to the same phase. In fact, the linear interpolation , , is the path between them and one can easily show that these Hamiltonians for all are gapped. Local unitaries also do not change the phase of a ground space. That is Hamiltonians and , for unitary belong to the same phase, and the path , , where , connects the two Hamiltonians. Note that translational and scale invariance are preserved in these two interpolations. We may also embed the Hilbert space of local spins into larger Hilbert spaces (and modify the local terms of the Hamiltonian). This does not change the ground space and we again obtain the same phase.
Here we focus only on frustration free Hamiltonians. Thus by a shift of energy we may assume that the local terms of the Hamiltonian are positive semidefinite and the ground space energy is zero. Let be such a Hamiltonian where acts only on sites and . Let be the orthogonal projection onto the positive eigenspaces of , i.e., is a projection and . It is straightforward to see that is gapped if and only if is gapped. Moreover, and have exactly the same ground spaces and by the above observation belong to the same phase. This means that without loss of generality we may assume that the local terms of the Hamiltonians are projections. Note that if is commuting then is commuting as well. This is because belongs to the algebra generated by and the identity operator.
The following lemma due to Bravyi and Vyalyi BravyiV () is a key step towards understanding the ground space of commuting Hamiltonians. For the sake of completeness we also include a sketch of the proof.
Suppose that and are two hermitian operators acting on finite dimensional Hilbert spaces and , respectively. Assume that and commute. Then there exists an index set and decomposition
such that acts nontrivially only on and not (for all ), and similarly acts nontrivially only on . More precisely, if we let be the orthogonal projection onto the subspace , then
for some hermitian operators and acting on and respectively.
Sketch of proof. Using the Schmidt decomposition of and as vectors in bipartite Hilbert spaces and , and writing down the constraint we find that it suffices to prove the following statement.
Let and be two subsets of hermitian matrices acting on such that all operators in commute with elements of . Then there exists a decomposition (1) such that elements of () act trivially on ().
So we need to prove the above statement. Let be the space of all matrices that commute with elements of . Since elements of are hermitian, is a -algebra, and it contains . Using the representation theory of finite dimensional -algebras Takesaki () we find that there exists a decomposition (1) such that
Thus elements of have the desired form. Furthermore, it is not hard to see that a matrix commuting with such has to act trivially on .
Iii Graphs encode ground states of 1D commuting Hamiltonians
Consider a 1D chain of spins of finite dimension (qudits) with periodic boundary conditions. We let the Hamiltonian be , where acts only on sites (Hereafter we adopt indices to be modulo ). By the observation of the previous section, since we let to be frustration free, we further assume that ’s are projections. The extra assumptions that we impose in this section are that is translational invariant, and that is commuting.
According to Lemma II.1 since and commute, there exists a decomposition
of the Hilbert space of the -th qudit such that () acts trivially on (). Using the fact that ’s are translations of each other, we may consider the same decomposition at site . As a result, acts trivially on as well. Therefore, can be written as
where acts on and is a projection. Since ’s are shifts of each other we may ignore the index and represent by . The combined Hilbert space of all qudits then can be written as
We correspond a directed graph to our Hamiltonian. The vertex set of is the index set of decomposition (2). A directed edge is drawn from to if is non-zero. Note that this graph may contain loops, i.e., edges from a vertex to itself. Moreover, is defined based on the local terms and is independent of the system size .
encodes the ground space of the Hamiltonian as follows. Let be a directed cycle of length in which may contain a vertex or edge several times. Since is an edge of , there is a non-zero vector in for . By decomposition (4), is a state in the Hilbert space of spins. Then using (3) we have
is non-zero iff is an edge of . Thus if we let to be the set of ordered cycles of length in , we have
Here we must consider ordered cycles because we have the freedom of choosing which vertex corresponds to the first site, so for example in , and are counted as different cycles if .
As a summary we conclude that the ground space of , for all , is completely characterized by the graph and the subspaces corresponding to each edge . Similarly to the ground states that correspond to cycles of , it is easy to see that excited states with energy 1 correspond to paths of , and eigenstates with eigenvalue 2 correspond to unions of two paths.
Note that the translational invariance assumption in this result is not crucial. If is not translational invariant, instead of a single index set we have index sets ; corresponds to the -th local term. Then the associanted graph would be -partite where there are directed edges from vertices in to vertices in .
An implication of our result is that the ground states of commuting Hamiltonians can be represented by MPSs. An MPS representation consists of a chain of maximally entangled states on which we apply some local linear transformations CiracMps () (see Figure 1). To write the state defined above as an MPS, we need only to map the -th maximally entangled state to . Thus is an MPS whose bond dimension is a constant independent of .
Let us work out an example to clarify the ideas. Let , and using represent each site with two spin-half particles. Define
where and are Pauli matrices. Then is commuting and translational invariant. The index set associated with has four elements and components of decomposition (2) are
Note that in this case since these subspaces are all one dimensional, we do not need to specify subsystems , individually. The corresponding graph has four vertices and is depicted in Figure 2. For for instance, is a cycle of length three and then is a ground state. However, is a path which cannot be completed to a cycle and then is an excited state with energy 1.
Iv Scale invariance
We now apply the assumption that the system is scale invariant, meaning that for sufficiently large the ground state degeneracy is a constant independent of . So we let be commuting, and translational and scale invariant. Our goal in this section is to show that for every
where is the set of loops of .
Let be a cycle of . By going around many times, we may assume with no loss of generality that , the length of this cycle, is sufficiently large. Let and . We have
where by we mean that is an edge in the cycle , and similarly for . For a cycle of length let be the cycle of length which starting from the same vertex as , goes times around it. Then are different cycles in , and similarly are distinct. Therefore,
for every edge in one of the cycles or . (Note that when is an edge of , is non-zero.) We further conclude that and indeed
Thus for some , . By the fact that and are relatively prime, we find that the arbitrarily chosen cycle consists of the repetition of a single loop. We conclude that the scale invariance assumption implies that all cycles of are essentially loops. Then (5) is an immediate consequence of (10).
The structure of the ground space of in even simpler with the scale invariance assumption. Let be a loop of , and let be a vector that spans . Then is a ground state, and the ground space of is spanned by these vectors for all loops of . The vectors are still MPSs and in fact translational invariant MPSs.
The example of Figure 2 is not scale invariant because the graph contains cycles that do not come from loops. For the special case of the 1D Ising model, the graph consists of two vertices with two loops and no other edges. From these two loops and the above construction we obtain the all spin-up and all spin-down states as the ground states of the Ising model.
Elementary excitations are also easier to characterize with the scale invariance assumption. As mentioned in the previous section, elementary excitations correspond to paths of , and excitations with energy 2 come from unions of two paths. A sufficiently long path in a finite graph must contain a cycle. Since loops are the only cycles of such , any elementary excitation of , for sufficiently large , contains several copies of states , for loops , in their subsystems. In fact an energy 1 eigenstate corresponds to a path of the form either or . For example, Ising model does not have any energy 1 eigenstate because its corresponding graph is a union of two loops and there is no edge of the form for .
V Phases are distinguished by the ground state degeneracy
We are now ready to study phases of 1D spin chains with commuting Hamiltonians. Objects of interest are ground spaces of 1D commuting Hamiltonians that are both translational and scale invariant. By the discussion of Section II without loss of generality we can also assume that the Hamiltonian is a summation of local projections. Thus we may use results of the previous two sections which are summarized as
where is one dimensional and is spanned by .
Here since we are interested only in the ground spaces we can replace the other ’s that do not appear in the above expression with the identity operator. More precisely, for every , or where is not a loop, define , and for loops let . Furthermore, define
and . Then is still commuting and translational invariant. Moreover, its corresponding graph consists of the same loops as , but no other edges. In fact, using (11), has the same ground space as and they are in the same phase. So we may replace with , i.e., we assume that and .
For every loop fix arbitrary states and . Let be a unitary operator that acts on in such a way that . Recall that the Hilbert space of two qudits and can be decomposed as
Thus we can define the two-qudit unitary operator that acts as on the subsystem/subspace when is a loop, and acts as the identity operator elsewhere.
Define , and . Since is block-diagonal with respect to the decomposition (12) and acts trivially on and , the new Hamiltonian is commuting. The corresponding graph of is the same as that of and consists of a union of loops. The only difference is that the state corresponding to the loop , is equal to .
Two Hamiltonians and belong to the same phase because they differ only by local unitaries (see Section II). So again without loss of generality we assume and . In this step we turned , which belongs to , into a product state.
By (11) the ground space of is spanned by vectors
for loops , where we set . If we let such that
then is commuting and translational invariant, and has the same ground space as . Then they belong to the same phase. Moreover, since and were arbitrarily chosen, the only parameter that determines is the size of , i.e., the ground space degeneracy. For example for two loops we obtain the Ising model whose local projections are given by
We conclude that the phases of ground spaces of translational and scale invariant commuting Hamiltonians in 1D are characterized by their degeneracy.
Vi Summary and Outlook
In this paper we described the structure of the ground states of translational and scale invariant, 1D commuting Hamiltonians. We associated a graph with a commuting Hamiltonian which encodes the ground space in its cycles and the low energy states in its paths. Our results generalize Yoshida’s work in 1D who considers stabilizer Hamiltonians Yoshida11a (). Comparing to Chen11 (); Schuch10 () instead of assuming that the ground states are described by MPSs, we imposed the assumption that the Hamiltonian is commuting.
In Section IV we argued that ground states of the Hamiltonians under consideration can be exactly written as translational invariant MPSs. Thus we could have skipped the previous section and directly used the result of Chen11 (); Schuch10 () to conclude that the ground state degeneracy is the only parameter that distinguishes phases. Our arguments, however, are based on much simpler techniques and we preferred not to refer to Chen11 (); Schuch10 ().
Our results can be applied on a larger class of Hamiltonians than the commuting ones. Let be an arbitrary translational invariant (frustration free) Hamiltonian. As before without loss of generality we may assume that is positive semidefinite and the ground state energy of is zero. Suppose that there exists a positive definite matrix such that
Then the Hamiltonian with local term
is commuting. These local terms are still positive semidefinite, and one can easily observe that is a zero energy state of if and only if is the ground state of . Thus the new Hamiltonian is frustration free as well. Furthermore, since is assumed to be positive definite and then invertible, the correspondence between ground states is one-to-one. Therefore, is scale invariant if and only if is scale invariant, and in this case results of our paper are applied. For instance, we showed that ground states of commuting Hamiltonians have MPS representations. Moreover, if is an MPS then is an MPS as well (see Figure 1). As a summary, the existence of a positive definite matrix satisfying (13) implies that the ground states of have MPS representations, and results of Chen11 (); Schuch10 () are applied.
With the above technique sometimes we can turn a Hamiltonian whose ground space has MPS representation into a commuting one. In particular the converse of this observation holds for Hamiltonians with a unique injective MPS ground state in the following sense. Consider a translational invariant Hamiltonian with a unique translational invariant MPS ground state . We assume that the bond dimension of this MPS representation is and the corresponding map is . Then we have
where is the maximally entangled state with local dimension . We may also replace with where is a unitary map. In this case is replaced with which by the discussion of Section II belongs to the same phase as .
is injective because we assume that is an injective MPS. Then the map is well-defined on the support of . In fact, we may identify the domain and support of , and by applying an appropriate (replacing with ) assume that is hermitian and positive definite.
We now introduce a commuting Hamiltonian as follows. Consider spins of dimension sitting on sites of a chain. So there are two spins on each site which are denoted by and , and the Hilbert space corresponding to sites and is
Define the local projection
and let . is commuting, translational invariant, and frustration free with the unique ground state . Moreover if we define
then has a unique ground state which is . As a result, the Hamiltonian with ground state can be turned into a commuting one using equation (14) with .
We conclude that MPS ground states and commuting Hamiltonians are in close relation via (14). The advantage of working with commuting Hamiltonians, however, is that verifying commutativity in general is much easier than checking whether the ground space is describable by MPSs.
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