# Classification and construction of higher-order symmetry protected topological phases of interacting bosons

###### Abstract

Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: “higher-order symmetry protected topological (HOSPT) phases”. While the usual (1st-order) SPT phases in spatial dimensions support anomalous -dimensional surface states, HOSPT phases in dimensions are characterized by topological boundary states of dimension or smaller, protected by certain global symmetries and robust against disorders. Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries. When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the Künneth formula of group cohomology. Based on a decorated domain wall picture of the Künneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions.

###### pacs:

## I Introduction

The discovery of topological insulatorsHasan2010 ; Hasan2011 ; Qi2011c (TIs) unveiled a large class of symmetry protected topological (SPT) statesChen2013 ; Senthil2015 , which in spatial dimensions feature symmetry-protected surface states on -dimensional open boundaries, such as one-dimensional (1d) helical edge states in two-dimensional (2d) quantum spin Hall insulatorsMaciejko2011 and 2d Dirac fermions on the surface of three-dimensional (3d) topological insulatorsHasan2011 . Recently a new family of “higher-order” topological insulators has been revealedBenalcazar2017 ; Benalcazar2017a ; Song2017 ; Langbehn2017 ; Fang2017 ; Khalaf2017 ; Schindler2018a ; Ezawa2018 ; Yan2018 ; Wang2018b ; Wang2018c ; Trifunovic2018 ; Wang2018a ; Matsugatani2018 ; Franca2018 ; Serra-Garcia2018 ; Schindler2018b ; Peterson2018 ; Zhang2018 ; slager2014 ; calugaru2018 , which do not have gapless surface states, but exhibit gapless modes on hinges and corners of the system. Generally a -th order TI in dimensions hosts robust gapless excitations on -dimensional open boundaries of the system: such as 0-dimensional corner states in 2nd-order 2d TIs and 3rd-order 3d TIs, as well as 1d hinge states in 2nd-order 3d TIs. In this terminology, the usual TIs can be called 1st-order TIs. These lower dimensional boundary excitations are robust against any small perturbations such as disorders and crystal distortions, as long as the global symmetry is protected, analogous to the stability of the TI surface states. It has been shown that the higher-order TIs usually also preserve certain crystalline symmetries in addition to the global symmetryBenalcazar2017 ; Benalcazar2017a ; Song2017 ; Langbehn2017 ; Fang2017 ; Khalaf2017 ; Schindler2018a ; Trifunovic2018 ; Song2018 .
While most of the efforts so far are focused on higher-order topological phases within band theory of non-interacting fermions, little is known about their strongly-interacting counterparts in *e.g. *interacting boson systemsDubinkin2018 ; You2018 . How to understand the higher-order SPT phases in a generic interacting boson system?

The goal of this work is to address this issue. We provide the classification and explicit construction for “strong” higher-order SPT (HOSPT) phases of interacting bosons with various global (*i.e. *onsite) symmetry and crystalline symmetry , whose lower-dimensional boundary excitations are protected only by onsite symmetry and hence robust against disorders and crystal distortions. We show that a -th order SPT phase in spatial dimensions is built by by stacking -dimensional -SPT phases, in a way which preserves crystalline symmetry . In particular for total symmetry as a direct product of onsite symmetry and crystalline symmetry ,, we show that all -th order SPT phases in dimensions are classified within the group cohomology

(1) |

where is isomorphic to crystalline group by regarding each orientation-reversing symmetry operation as an anti-unitary operatorJiang2017 ; Thorngren2018 . The above classification also provides a procedure to construct these HOSPT phases from building blocks of -dimensional SPT phases protected by onsite symmetry only, as illustrated in many examples.

This work is organized as follows. First in section II we discuss the physical picture behind HOSPT phases based on a dimensional reduction point of view. Then we show the general classification of HOSPT phases in section III.1 based on the Kennuth formula of group cohomology, and how to the construct the HOSPT phases using the decorated domain wall picture in section III.3. The classification and construction are demonstrated for 2nd-order SPT phases in two (section IV) and three (section V) dimensions, and 3rd-order SPT phases in three dimensions (VI). We conclude with a few remarks in section VII.

## Ii The physical picture

Before introducing the mathematical classification for higher-order SPT phases, we first discuss an intuitive physical picture which shows how higher-order SPT phases can be built by stacking lower-dimensional SPT phases. Throughout this work, we will focus on the simplest situation where the total symmetry group is a direct product of crystalline symmetry group and onsite (*i.e. *global) symmetry group .

By definition, a -th order SPT phase in dimensions is characterized by symmetry protected gapless states on boundaries of dimensions. For example, as illustrated in FIG. 2, a 2nd-order SPT phases in with 4-fold rotational symmetry hosts gapless zero modes at each corner of a square-shaped system, which are protected by onsite symmetry . Based on this example and without loss of generality, below we present two arguments to establish a dimensional reduction picture for the HOSPT phases: while the 1st argument (section II.1) shows why a -th order SPT phase in dimensions is related to the usual -SPT phases in dimensions, the 2nd argument (section II.2) explicitly demonstrates how to build such a HOSPT phase from lower-dimensional SPT phases. While the 1st argument explains why the classification of HOSPT phases is determined by the classification of -dimensional SPT phases, the 2nd argument shows which of the -dimensional SPT phases can consistently lead to a gapped symmetric -th order SPT phases in dimensions, to be compatible with the crystalline symmetry .

### ii.1 Corner/hinge states as gapless defects on the gapped open surface

We consider a generic HOSPT phase of order on a -dimensional open manifold (such as the square-shaped system in FIG. 1), which is gapped almost everywhere except for a -dimensional submanifold on the boundary (such as the four corners with in FIG. 1). Since by definition the -dimensional boundary is gapped, this is not a “strong” SPT phase protected by onsite symmetry only, and hence there exists a finite-depth quantum circuitChen2013

(2) |

which continuously evolves the HOSPT state into a trivial product state , while preserving onsite symmetry . We label the finite depth of circuit as .

As illustrated in FIG.1, next we divide the total system into two regions: its (simply-connected) interior (both white and gray in FIG.1), and boundary . We can then define a finite-depth () quantum circuit by restricting quantum circuit into region , such that

(3) |

where is the interior (white in FIG. 1) of , differing from only by a “cushion” region (gray in FIG. 1) whose width is of the order . Here denotes the trivial product state on region . In other words, finite-depth quantum circuit can continuously tune the interior region of HOSPT phase into a trivial product state without closing the gap or breaking onsite symmetry , while keeping the boundary states (on ) untouched. As a result, through finite-depth quantum circuit which preserves onsite symmetry , the HOSPT ground state is disentangled into a trivial product state in the bulk , and a state on its -dimensional surface .

Notice that in addition to preserving onsite symmetry , the -dimensional state is mostly gapped except for hosting gapless modes on its -dimensional submanifolds. Therefore, the gapless corner/hinge states in a HOSPT can be viewed as gapless -dimensional defects on a gapped -dimensional surface state with onsite symmetry . As argued in LABEL:Teo2010a,Wen2012, the classification of such a defect falls in the classification of a -dimensional SPT phases protected by the same onsite symmetry .

For example, in a 2nd-order SPT phase in , the gapless corner states in *e.g. *FIG. 1 can be viewed as gapless 0-dimensional domain walls on the gapped 1d edge. Therefore they are reduced to the 1-dimensional -SPT phases. Similarly for a 2nd-order SPT phase in , the gapless hinge states can be viewed as gapless 1d domain walls on a gapped 2d surface, therefore related to 2-dimensional -SPT phases. For a 3rd-order SPT phase in , the gapless corner states should be viewed as gapless 0-dimensional point defects on the gapped 2d surface with symmetry , hence reduced to 1-dimensional -SPT phases.

### ii.2 Building HOSPT phases from lower-dimensional SPT phases

In the previous argument, we have shown that the gapless -dimensional boundary states in a -th order SPT phase in dimensions can be reduced to the classification of -dimensional SPT phases preserved only by onsite symmetry . However, not all of the -SPT phases can lead to a gapped HOSPT phase that preserves crystalline symmetry : certain compatibility conditions must be satisfied to ensure a gapped bulk. Here we provide another argument based on the dimensional reduction approachSong2017a ; Lu2017c ; Huang2017b , which explicitly builds the -th order SPT phases in dimensions out of -dimensional -SPT phases.

Without loss of generality, we demonstrate this dimensional reduction argument using the 2nd-order 2d SPT phase with point group symmetry, as shown in FIG. 2). We first divide the whole open manifold into 4 disconnected shaded regions in FIG. 2 which are related by symmetry, while both the inversion center and 4 gapless corners lie in the rest of the space . Following the same construction as used in the previous argument, we can construct a -preserving finite-depth quantum circuit by restricting circuit in region , such that

(4) |

where represents the trivial product state on region . By symmetrizing circuit w.r.t. rotations, we can construct a symmetric finite-depth quantum circuit

(5) |

which preserves both onsite symmetry and crystalline symmetry , such that

(6) |

In other words, symmetric finite-depth circuit trivializes most of the manifold , except for the four 1d systems connecting the gapless corner to the rotation center. As argued previously, now that each corner state carries a projective representations of onsite symmetry as the boundary state of a 1d -SPT phase, each 1d system connecting the corner to the rotation center must be a 1d -SPT phases with a topological index

(7) |

Note that as a part of the gapped bulk, the rotation center where the ends of the four 1d -SPT chains must form a linear representation of onsite symmetry *i.e.*

(8) |

This compatibility condition comes from the fusion of a number of edges of 1d SPTs dictated by the crystal symmetry :

(9) |

Physically, the fusion map encodes a notion of compatibility between onsite symmetry and crystalline symmetry , so that the bulk of the full system is trivial and gapped. Constructing the map is generally a difficult mathematical problem for an arbitrary symmetry group with both onsite and crystalline symmetries. In this paper we consider the simplest case, where the symmetry group is a direct product of onsite symmetry and global symmetry . As we will show later, this allows a direct reduction via the Künneth formula, where the compatibility conditions between -dimensional -SPT phases and crystalline symmetry in spatial dimensions are captured by group cohomology formula (1).

Finally, we recall that certain SPT phases are beyond the group cohomology classification, such as the 3d time-reversal-SPT phase with ff surface topological ordersVishwanath2013 ; Wang2013a classified by cobordismKapustin2014 ; Kapustin2014a and Kitaev’s chiral 2d stateKitaev2006 ; Lu2012a . We have also considered these beyond-group-cohomology HOSPT phases built from the state, as highlighted in red in TABLE 2.

## Iii Classification and construction from Künneth formula

### iii.1 General classification of HOSPT phases

In this work, we will focus on the cases where the total symmetry group is a direct product of crystalline symmetry and onsite symmetry :

(10) |

In this situation, there is a simple mathematical formula based on group cohomology, which gives the full classification of higher-order SPT phases. It has been shownJiang2017 ; Thorngren2018 within the group cohomology classification of SPT phases, that all -symmetry protected topological phases of interacting bosons in spatial dimensions is given by

(11) |

where is isomorphic to , obtained by replacing each orientation-reversing element of crystalline symmetry group by an anti-unitary operation of the same rank. According to the Künneth formula for group cohomologyWen2014 ; Wen2015 ; Cheng2016 we have

(12) |

The 1st term classifies crystalline SPT phases protected only by crystalline symmetry Song2017a ; Thorngren2018 . The 2nd term vanishes for any finite groupCheng2016 , as in the case considered here where is a point group or magnetic point group.

Therefore we shall focus on the last line of the above Künneth formula (12). Each term in

(13) |

can be interpreted as the classification of -th order SPT phases in spatial dimensions, protected by onsite symmetry and cystalline symmetry . Such a SPT phase is featured by robust gapless states on proper -dimensional open boundaries, which are protected by onsite symmetry alone. For example, the term in (13)

(14) |

corresponds to the 1st-order (*i.e. *the usual “strong”) SPT phases protected by onsite symmetry , featured by gapless modes on -dimensional boundaries.

2nd-order SPT phases in are all captured by term in (13)

(15) |

which host gapless (or anomalous topological orders when ) excitations on -dimensional boundaries protected by onsite symmetry .

Similarly, 3rd-order SPT phases in are all captured by term in (13)

(16) |

which host gapless (or anomalous topological orders when ) excitations on -dimensional boundaries protected by onsite symmetry , such as corner states in .

### iii.2 “Strong” HOSPT phases versus “weak” crystalline SPT phases

As mentioned previously, we define -th order SPT phases in dimensions by the presence of robust -dimensional topological boundary states, protected by onsite (or global) symmetry only. These are “strong” SPT phases, whose boundary excitations do not require protection from the crystalline symmetry . In comparison, there are also “weak” crystalline SPT phases, whose topological boundary excitations are protected by crystalline symmetries (in addition to onsite symmetries)Fu2007 ; Ran2010 ; Song2015 ; Thorngren2018 . In fact, in addition to strong HOSPT phases which are the focus of this paper, certain weak crystalline SPT phases are encoded inside the whole Kunneth formula (12), such as those colored in green in TABLE 3. Before systematically analyzing and constructing HOSPT phases in detail, we briefly discuss the weak crystalline SPT phases.

First of all, the term in Kunneth formula clearly describes weak SPT phases protected only by the crystalline symmetry .

Next, we comment on term in (13):

(17) |

The physics of this term is to assign onsite symmetry charges (linear representation of onsite symmetry ) to defects of the crystalline symmetry . In a simplest example, for the case of 1d insulators () with inversion symmetry (), we have and hence

(18) |

The nontrivial element of the above classification corresponds to assigning an odd number of charges to the inversion center, while the trivial element corresponds to having an even number of charge on the inversion center. There is no gapless boundary excitations for either of the two phases in 1d.

However in , weak SPT phases with boundary states protected by crystalline symmetry generally can appear in the Kunneth formula (12). For example in case with mirror symmetry , corresponds to assigning charges to each domain wall of mirror symmetry on the 1d mirror axis of the 2d system. This leads to gapless boundary states if the boundary of the system preserves mirror symmetry. Similarly in case with -fold rotational symmetry , corresponds to assigning charges to each domain wall of rotational symmetry on the 1d rotation axis. This leads to weak 3d crystalline SPT phases, hosting gapless (or anomalous) boundary states if the boundary preserves symmetry.

Another example is in (13). Take for instance, considering mirror symmetry again, corresponds to assigning 1d -SPT phases classified by to each mirror domain wall on the 2d mirror plane. This can lead to gapless (or anomalous) boundary states protected by both mirror and onsite symmetry, if the boundary preserves mirror symmetry .

As we mentioned before, the boundary states of these weak crystalline SPT phases will generally be destroyed by perturbations that break the crystalline symmetry, such as disorders and crystalline distortions. Meanwhile, their interpretation in the Kunneth formula can be quite tricky, as shown in the above examples. Hereafter we will be focusing on the strong HOSPT phases, whose topological boundary excitations are robust even if crystalline symmetries are broken on the surface.

### iii.3 Decorated domain wall construction

Here we briefly describe how to explicitly construct the higher-order SPT phases in spatial dimensions, using -SPT phases in lower dimensions. In particular, the group cohomology formula (13) provides a clear physical meaning for such a construction, similar to the decorated domain wall constructionChen2014 for the usual (“1st-order”) SPT phases.

First we consider 2nd-order SPT phases in dimensions, classified by 1st group cohomology

(19) |

These are nothing but linear representations of the symmetry group

(20) |

satisfying the 1-cocycle condition

(21) | |||

valued in physically represents a domain wall labeled by symmetry element , decorated by -dimensional -SPT phases labeled by elements in . The above 1-cocycle condition can be viewed as a compatibility condition between the addition rules of -dimensional -SPT phases and the addition rules () of domain walls, in order to ensure a gapped bulk spectrum. To understand this, we see that a domain wall of the symmetry is labeled by a group element . The -dimensional SPT phase associated with this domain wall is labeled by an element . The fusion of two domain walls combines these -dim -SPT’s into . However, the fusion must respect the group structure of , and this consistency condition is exactly captured by Eq. (13). Therefore each element of describes a way to assign -dimensional -SPT phases on the domain walls of crystalline symmetry , which is compatible with a gapped bulk.

Next we consider 2nd-order SPT phases in dimensions, classified by 2nd group cohomology

(22) |

They are nothing but projective representation of symmetry group

(23) | |||

(24) |

satisfying the 2-cocycle (or associativity) condition

(25) |

Since represents the -dimensional domain wall labeled by element of crystalline symmetry , naturally represents the -dimensional manifold where three domain walls , and intersect. The fact that takes values in physically means that these domain wall intersections are decorated by -dimensional -SPT phases, which are classified by group cohomology .

As a simplest example, we consider the -fold rotational symmetry . Each of the domain walls of the symmetry can be decorated by the same -dimensional -SPT phase, such that copies of these -SPT phases intersect at the rotational axis. For the system to be gapped on the rotational axis, these copies of -SPT phases together must fuse to a trivial phase with no gapless boundary states. This exactly corresponds to 2nd-order SPT phases classified by . Meanwhile at the intersection of domain walls of symmetry, the rotational axis itself can also be decorated by a -dimensional -SPT phases, which corresponds to the 3rd-order SPT phases classified by .

Another example is the mirror reflection symmetry , where the orientation-reversing mirror symmetry should be regarded as an anti-unitary symmetry when computing the group cohomology. For , the 2nd-order SPT phases in classified by can be understood as assigning a 1d -SPT phases on each mirror plane.

Below we will classify 2nd-order SPT phases in (TABLE 1,2) and 3rd-order SPT phases in (TABLE 3), for various choices of onsite symmetry and crystalline (and magnetic crystalline) symmetry . We will also explicitly construct these higher-order SPT phases using the decorated domain wall picture as described above, in section IV-VI.

Onsite symmetry | |||||

or | |||||

Crystalline symmetry | or | ||||

-SPTs: |

## Iv 2nd-order SPT phases in two dimensions

As shown in (15), the 2nd-order SPT phases in spatial dimension are classified by , *i.e. *the linear representation of group whose coefficients take value in the -dimensional SPT classification . For case, the building blocks of 2nd-order SPT phases in two dimensions are 1d SPT phases protected by onsite symmetry . Below we provide a full classification for 2nd-order SPT phases in with all possible 2d point group and magnetic point group symmetries, and describe how to use 1d SPT phases to construct these 2nd-order 2d SPT phases.

### iv.1 Classification

To compute , first we need to obtain the group from crystalline symmetry . As mentioned earlier, is isomorphic to , obtained by replacing each orientation-reversing element of by an anti-unitary operation of the same rank. For example we have

(26) | |||

(27) | |||

(28) |

where we use to denote a group generated by an anti-unitary operator of ranking .

After obtaining , the next step is to compute , the coefficient of the desired linear representation . For case, corresponds to the classification of 1d SPT phasesTurner2011 ; Fidkowski2011 ; Chen2011 ; Schuch2011 protected by onsite symmetry : it always forms a finite Abelian group, as summarized in the last line of TABLE 1.

Generally the classification of SPT phases with onsite symmetry always form a discrete Abelian group, which holds for the group cohomology classification and beyond. One important relation for group cohomology is

(29) |

Therefore to compute in (15), we only need to know , and for any finite integer . Since is always a finite Abelian group, making use of relation (29), we can compute purely based on knowledge of for any finite integer . Below we list for all point groups and magnetic point groups :

(30) | |||

(31) | |||

(32) | |||

(33) | |||

(34) | |||

(35) | |||

(36) |

### iv.2 Examples

#### iv.2.1

The simplest examples of 2nd-order SPT phases are protected by -fold rotational symmetry , classified by

(37) |

They can all be built from 1d SPT phases protected by onsite symmetry , where the 1d -SPT phases are aligned in a -symmetric manner as shown in FIG. 3 for case. Since the endpoints of copies of 1d -SPT phases intersect at the center of the system (see FIG. 3), they must form a linear representation of onsite symmetry to ensure a gapped symmetric bulk. This provides a compatibility condition for the 1d -SPT phases, manifested in the group cohomology formula

(38) |

where is the greatest common divisor of integers and .

Formula (38) can be understood as follows. The group cohomology stands for linear representation of group with coefficients in -valued phase factors

(39) |

Denoting the generator of group as with , we have

(40) |

and as a result

(41) |

where denotes the greatest common divisor of integers and . Physically this means the topological index of the 1d -SPT phase decorated on each domain wall must be a multiple of , to ensure the bulk to be gapped at the rotation axis where the domain walls intersect. Hence there are distinct 2nd-order SPT phases with symmetry, characterized by the 1d -SPT phase with classification in formula (38). on each domain wall. This corresponds to the

One immediate physical consequence is the presence of zero-energy corner modes located on each corner of the -symmetric finite system shown in FIG. 3. Each corner mode is nothing but the boundary states of 1d -SPT phases with

#### iv.2.2 or

Consider point group , generated by -fold rotation with along -axis, and mirror reflection whose mirror plane is parallel to -axis. As described earlier, the associated 2nd-order SPT phases are classified by the linear representation (1st group cohomology) of , with coefficients in 1d -SPT phases classified by . The decorated-domain-wall construction of these 2nd-order SPT phases with symmetry can be implied from the following formula

(42) |

The 1st factor labels the 1d -SPT phases assigned on each domain walls and intersected at the rotation center, illustrated by the red lines in FIG. 4. On the other hand, the 2nd factor labels the 1d -SPT phases placed on each of the mirror planes, illustrated by the green lines in FIG. 4. The linear representation corresponding to satisfies the following algebraic conditions:

(43) |

Similar to case discussed earlier, in (42) the linear representation of -fold rotation is given by

(44) |

While is invariant under any gauge transformation on the basis vectors of the linear representation, this is not the case for anti-unitary operator . Specifically under a gauge rotation by phase factor changes as on all basis vectors, the linear representation of anti-unitary operator

(45) |

This indicates that 1d -SPT index on each mirror plane is only well-defined modulo 2, leading to the factor in formula (42). This result has a straightforward physical interpretation: two 1d -SPT phases of the same topological index can be merged from two sides into the mirror plane, hence changing the 1d topological index on the mirror plane by any even integer without closing the bulk gap.

Unlike in the previous case where the copies of 1d -SPT phases can be rotated by an arbitrary angle, here due to the presence of mirror planes (related by rotations), all 1d -SPT phases are assigned to the mirror planes. As a result, as long as the corners of the finite system lie on the mirror planes, they will give rise to zero-energy corner modes protected by onsite symmetry. However as illustrated in FIG. 4, there are two different types of corners, terminating the green lines only versus terminating both green lines. These two types of corners generally support different types of projective representations of onsite symmetry .

Finally, it is straightforward to show that the above classification and construction remain true for magnetic point group