Symmetry Protected Topological Order by Folding a One-Dimensional Spin-1/2 Chain

Symmetry Protected Topological Order by Folding a One-Dimensional Spin- Chain

Pejman Jouzdani Department of Physics, University of Central Florida, Orlando, Florida 32816, USA
July 7, 2019
Abstract

We present a toy model with a Hamiltonian on a folded one-dimensional spin chain. The non-trivial ground states of are separated by a gap from the excited states. By analyzing the symmetries in the model, we find that the topological order is protected by a global symmetry. However, by using perturbation series and excluding thermal effects, we show that the symmetry is stable in comparison to a standard nearest-neighbor Ising model with a Hamiltonian . We find that is a member of a family of Hamiltonians that are adiabatically connected to . Furthermore, the generalizations of this class of Hamiltonians, their adiabatic connection to , and the relation to quantum error-correcting codes are discussed. Finally, we show the correspondence between the two ground states of and the unpaired Majorana modes, and provide numerical examples.

pacs:
03.67.Lx, 03.67.Pp, 03.65.Yz, 05.50.+q

Intro.– A large effort has been made to construct robust protocols for quantum computation by exploiting topological properties of many-body systems Sarma (); KitaevA (). Models such as the toric and surface codes Dennis (); kitaevT (); bravyi1998 () are proposed. A particular attention has been given to detect and employ exotic non-Abelian excitations in quantum computation Nayak (), with a special attention to the Majorana fermions Jason (); Fu (); ginossar (); Buhler (); Kraus (). In this order, a crucial step is to detect and characterize the topological orders. It has been shown by Levin and Wen LevinWen () that the ground state of quantum many-body systems with non-trivial orders can be seen as a condensate of fluctuating string-like objects. In particular, quantum phases are commonly studied by the projective symmetry groups (PSG) tool PSGWen () which has been used to identify symmetry protected topological orders (SPT) Xie (); Afflek (); Chen (); Son (); Pollmann (), and with focus to quantum computation and quantum error correction Akimasa (); Else ().

Consider a one-dimensional spin- Ising model with the Hamiltonian

(1)

on a chain with spins () where is the -th component of the Pauli matrices acting on site . Ignoring thermal excitation for a moment, a longitudinal field perturbation lifts the degeneracy of the ground states for any non-zero . However, in the absence of a longitudinal field, the degeneracy is topologically protected aginst a perturbation .

In comparison to the toric code kitaevT () where the system is defined on a lattice and the protection against perturbation is of the order of , we could think of the one-dimensional Ising chain as a two-dimensional lattice. The lattice has a width of only “one” lattice site. Therefore, any (longitudinal) single-spin perturbation already reaches the size of the lattice. Thus, the Ising chain has a topological phase, but the phase can not be realized due to the short width of the lattice. Although in different words, this claim was originally expressed in a footnote in Ref. Yu ().

In this Letter we introduce an adiabatic transformation that does not change the topological characteristic of the Ising Hamiltonian , but effectively folds the spin chain to a lattice. As a result, we obtain a Hamiltonian and the topological protection of the one-dimensional Ising chain tremendously improves, unexpectedly. In order to do so, the steps bellow are followed.

First, a Hamiltonian on an open chain of spin- is defined. is a sum of four-spin operators. The set of these operators form a group that is denoted by . In addition, we find two symmetry groups and that commute with all the elements of . We find that which indicates the non-trivial order of the model is protected as long as the global symmetry is preserved. Next, we show through a degenerate perturbation analysis that the global symmetry is robust. Then, the extension to models with wider width and with symmetry groups is discussed. Finally, we show the Hamiltonian is adiabatically connected to in Eq. (1),

(2)

where is the unitary transformation

(3)

Here is a sum of two-spin interactions and is a scalar parameter. The relation between Majorana modes of the Kitaev toy model Yu () and the two ground states of are explained as well.

The Model.

Figure 1: (a) The standard one-dimensional Ising model with nearest-neighbor interactions. (b) The reshaped Ising model into a ladder after the unitary transformation in spin space. The diagonal dashed and dash-dot lines show the two-spin terms used in the transformation. (c) The Hamiltonian resulting from the transformation has four-spin interactions, associated to plaquettes. The bulk has two different types of plaquettes, and . There are three operators of type and one operator of type . The two left and right plaquettes on the boundaries are named and , respectively. There are two operators acting on each of these boundary plaquettes.

Consider a one-dimensional spin-1/2 system of a length , with a positive and integer number . We define as

(4)

where we have introduced the following operators (the stabilizers ):

(5a)
(5b)
(5c)
(5d)
(5e)
(5f)
(5g)
(5h)

with and .

All the terms on the r.h.s. of Eq. (4) commute with each other. As shown in Fig. 1c, one can identify and as bulk plaquettes operators, while and act as boundary plaquette operators. For there are plaquette operators of type , plaquette operators of type , and two plaquette operators of types and each. Therfore, there are overall plaquette operators - that is . Without proof, the set of the stabilizers defined in Eq. (5) generates a group that we denote by .

Separately, consider the set of the operators , on pairs . Unless otherwise mentioned, we use the notation for the two sites and that are connected with a dashed or a dot-dash line in Fig. 1b. We define the group generated by these operators as . All elements in commute with all elements in .

Furthermore, consider the set of operators on pairs . We define the group generated by these operators as . All elements in commute with all elements in and all elements in . Especially, we find (there are generators in , generators in , and generators in ; ). A comprehensive classification of SPT orders in one-dimensional spin systems is given in Ref. Xie ().

The physical implication of the above statements is the following. If we manage to have a fixed “gauge” for the ground state subspace of , as long as the symmetry is not broken, the ground state is doubly degenerate and we can have such that for any generator of note1 (). The phase is a global phase and it is equal to for defined in Eq. (4).

To see this, consider the states and where by definition and . Next, for every element we have (), and thus the gauge is set to +1. Then, by applying the group elements of on each of the states and we obtain

(6)

and

(7)

where the product is over the stabilizers . Notice that is the number of distinguishable plaquettes in Fig. 1c. Thus, is the number of “loops” that can be constructed on the folded chain using the plaquettes as the unit blocks. The basis states appear as condensates of string-like configurations LevinWen (). Interestingly, for a generator of

(8)

since . The equations (6), (7), and the property in Eq. (8) can be examined in the examples given at the end of this Letter.

In fact, the property in Eq. (8) is not a coincidence if one remembers that by a Jordan-Wigner transformation the Ising model maps to the unpaired Majorana problem Yu (). Thus, the generators in the group act equivalently as a logical operation for the ground states of .

Stability of the symmetry.– In a one-dimensional Ising spin on an open chain a non-zero longitudinal field opens a gap between and . The gap stimulates topological excitations (a propagating domain wall) from a false vacum to a true vacum and eventually destroys the symmetry Simons ().

Similar to , the Hamiltonian has a discrete energy spectrum and at low temperature () thermal excitations are energetically costly and can be considered forbidden. By a degenerate perturbation series approach we see that a transverse field perturbation such as (or ) has vanishing matrix elements for all the powers , in the ground states supspace (). Therefore, we have a topological protection with respect to this transversal field. Especially, the only non-vanishing term is .

Considering the bases , we find and any generator as the bit-flip and phase-flip logical operations, respectively.

In contrast to , in the presence of two transversal fields (), the first non-vanishing term appears at the second order of the perturbation series and only on pairs . This means that, out of number of possible terms in the second order, only of them are non-zero. Thus, by increasing the length, the second order non-vanishing terms are suppressed by a factor of . This is opposite to the case where in the second order of perturbation there are non-vanishing terms (). This pattern continues in all the orders. The odd orders vanish. In the fourth order there is a suppression factor of , etc.

Thus, as long as the perturbation (noise) affects single spins (no correlated noise) we should expect that the interplay between multiplicity and energy cost in the perturbation series (the statistical ground for a phase transition) to be substantially suppressed in our model with Hamiltonian. Therefore, we expect a stable global symmetry . Whether the symmetry will still be destroyed through other mechanisms is a question to be answered. For a ladder of the toric code this has been studied Karimpour ().

Translational symmetry breaking and generalization.– By a close look at Fig. 1a and Fig. 1c, we notice that the periodicity changes as we move from the Ising chain in Fig. 1a to in Fig. 1c. The unit cell in is the two-plaques and the periodicity goes as in the bulk, while in the Ising chain it goes as . The emergence of non-trivial topological orders in one dimensional “organic” polymers by breaking translational symmetry has an old history Su ().

Also, notice that the width of the folded chain in Fig. 1c contains two sites. It should be now clear to the reader why we chose . There are sites () in each unit cell and the last sites are added to keep the inversion symmetry and have . Although, it is not clear whether the inversion symmetry is necessary.

Similarly, we can construct a Hamiltonian with a width of “three” sites and define stabilizers with “six” operators note2 (). In this case, the length of the chain is chosen to be with a positive integer . That is, unit cells in the bulk. There are five different -type operators, defined on plaquettes, and one -type stabilizer (in a hexagonal shape), with two boundary stabilizers on each edge. Therefore, there are number of stabilizers. The non-trivial part is to show that there are two independent symmetry groups and that commute with the group of stabilizers of , . That is, they satisfy .

is the first member of the quantum error-correcting codes that can be constructed by folding a line and has a code distance of ; That is roughly half of the width () of the folded configuration. In principle, it should be possible to construct quantum error-correcting codes with longer code distance and larger stabilizers with Hamiltonian . As we will see for the case of bellow, there is an adiabatic connection between this class of Hamiltonians and in Eq. (1).

The Hamiltonian is adiabatically connected to the one-dimensional Ising Hamiltonian .– We begin with the standard nearest-neighbor Ising Hamiltonian in Eq. (1) with . Next, the unitary transformation defined in Eq. (3) with

(9)

is used to map to according to Eq. (2). The result is

(10)

where and involve three-body and four-body interaction terms, respectively. For , the contribution of to the total Hamiltonian drops out and we obtain .

Since we obtain by a unitary transformation from , the energy spectrum must stay unchanged and the ground state subspace of must be two-fold degenerate. However, if two gapped states are connected by a set of local unitary transformations they belong to the same phase Chen2 (). This can be seen by applying on a state . One obtains

(11)

which is a product state. As it can be checked, clearly a second degenerate state is not accessible by using Eq. (8). That is, for a generator of the group , we have while . This means . This is not surprising if one thinks of the Majorana counterpart of the problem where the two unpaired Majorana modes ( and ) are paired up and experimentally undetectable.

Then, how can we observe the phase that corresponds to (or )? One quick answer is to find a way to implement logical quantum gates. We are seeking an operation such that . Since is a logical phase-flip operation and is the logical bit-flip operation, we should be able to decompose and have

(12)

Notice that the logical operation is unitary but non-local. Implementing , if ever possible experimentally, would change the quantum phase from a product state in Eq. (11) to a Majorana mode . This can be checked for a chain with in the examples bellow.

Notice that we can obtain and , corresponding to the logical operations of , by transforming the and the order parameter corresponding to under . In general, to obtain the logical operations corresponding to one needs to know the adiabatic transformation from and the knowledge of the symmetry groups of the Hamiltonian becomes irrelevant.

Figure 2: Chains with (a) and (b) spins. The solid lines represent the initial Ising nearest-neighbor interaction between the spins, . The dashed and dotted lines indicate the interaction terms present in (Eq. (9)) which are used to generate the transformation .

Examples and numerical results.– Let us consider two examples. The first example is a chain of . Following Eq. (4), we have (see Fig. 2b)

(13)

By exact diagonalization we obtain the two basis states

The second example is a chain with . It does not exactly follow the prescription defined in Eq. (4). However, we can define (see Fig. 2a)

(15)

and the inversion symmetry in this case is still preserved. It has the degenerate basis states

(16)

Notice that by applying on () one obtains ().

Figure 3: The energy splitting between the two low-lying states at (, circles) and (, squares) as a function of external field for a spin chain. (a) The splitting as a function of a longitudinal field is shown for the two and . For any non-zero longitudinal magnetic field , a gap opens linearly for . The situation is visibly different (quadratic) for as discussed by perturbation analysis. (b) The splitting as function of a transverse field is the same for both and .

We use the dependence of the splitting of the ground states, , on the global external magnetic field as a criterion to numerically verify the enhanced protection in . The numerical calculation of the splitting for at the points () and () is shown in Figs. 3a and 3b as a function of (longitudinal) and (transverse) external magnetic fields, respectively. One can see that the dependence of the splitting goes from linear for to quadratic for , indicating increased protection. Figure 3b shows the gap which behaves topologically protected for both and , as expected.

Summary.– The standard one-dimensional Ising chain with a Hamiltonian could theoretically be in a topological phase if the global symmetry were stable. The symmetry is not stable since the order parameter of the system is just a single spin. We showed how to adiabatically obtain a Hamiltonian by a local unitary transformation with only two-spin interactions from . We showed that the protection against single-spin errors in the transformed Hamiltonian scales with the length of the chain. We discussed a family of Hamiltonians that are adiabatically connected to .

Acknowledgments.– The author thanks Eduardo Mucciolo for his advise and support and Alioscia Hamma for his critical comments. This work was supported in part by the National Science Foundation grant CCF-1117241.

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