From effective Hamiltonian to anomaly inflow in topological orders with boundaries

Whether two boundary conditions of a two-dimensional topological order can be continuously connected without a phase transition in between remains a challenging question. We tackle this challenge by constructing an effective Hamiltonian, describing anyon interaction, that realizes such a continuous deformation. At any point along the deformation, the model remains a fixed point model describing a gapped topological order with gapped boundaries. That the deformation retains the gap is due to the anomaly cancelation between the boundary and bulk. Such anomaly inflow is quantitatively studied using our effective Hamiltonian. We apply our method of effective Hamiltonian to the extended twisted quantum double model with boundaries (constructed by two of us in Ref.[1]). We show that for a given gauge group $G$ and a three-cocycle in $H^3[G,U(1)]$ in the bulk, any two gapped boundaries for a fixed subgroup $K\subseteq G$ on the boundary can be continuously connected via an effective Hamiltonian. Our results can be straightforwardly generalized to the extended Levin-Wen model with boundaries (constructed by two of us in Ref.[2].

Introduction: Topologically ordered matter systems have greatly expanded our knowledge of matter phases [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17], may potentially be used as quantum memories [18], and realize topological quantum computation [7,[19][20][21]. Among all the factors that hinder the physical applicability of topological orders, a crucial one is that topological orders have been studied mostly for closed twodimensional systems, whereas experimentally realizable materials mostly have boundaries. When a topological order is placed on an open surface, a boundary is subject to certain gapped boundary condition on which the topological order remains well-defined. It remains however a challenge whether two apparently different gapped boundary conditions of a topological order are physically equivalent. There has been a few constructions of boundary Hamiltonians of topological orders [22][23][24][25][26], which are nevertheless for either restricted cases or in the language of categories. Very recently, in Ref. [1,2,27], we have systematically constructed the boundary Hamiltonians of the Levin-Wen [8] and the twisted quantum double models (TQD) [14] using solely the microscopic degrees of freedom of the models. This allows us to tackle the challenge aforementioned.
that H G,α K,β is connected to H G,α K,1 via an continuous passage that retains the gap. Such an continuous passage can be understood as a unitary transformation relating the Hilbert spaces before and after the continuous passage. It is proposed in Ref. [28] that two topological orders on a closed surface are equivalent if and only if they are related by finite steps of local unitary transformations. In the case with boundaries, however, we find that the unitary transformation associated with an continuous passage is local in the bulk but nonlocal on the boundary. That the system remains gapped throughout the entire continuous passage is due to the anomaly inflow from the boundary to bulk, which is corroborated by the nonlocal unitary transformation on the boundary spectrum. We derive an emergent (effective) HamiltonianH that realizes the continuous passage exp iHt (parameterized by t) between the two models H G,α K,β and H G,α K,1 . Using this emergent Hamiltonian, we quantitatively study the nonlocal unitary transformation and the anomaly inflow.
Our results hold for the extended TQD model with any Abelian finite group G. We accompany our derivation with an explicit example-the extended TQD model with gauge group G = Z 2 × Z 2 . Our results are generic, which can also apply to the extended Levin-Wen model with boundaries systematically constructed in Ref. [2,27].
Extended TQD on a disk: We place the TQD model with gauge group G on a graph that triangulates a disk, as in Fig. 1. That the model is a low-energy fixed point effective theory leads to the topological invariance of the model [1,14], such that the initial arbitrary graph can be reduced by the Pachner moves [1,14,29] into the simple form in Fig. 1(b). The reduced graph consists of N + 1 vertices (one bulk vertex 0 and N boundary vertices) and 2N edges (N bulk edges a 1 , through a N and N boundary edges b 1 through b N ). The bulk edge degrees of freedom a n 's take value in The boundary edge degrees of free- where where δ denotes the 3-coboundary operator. (Mathematically, this defines β as a Frobenius algebra in the category V ect α G .) Deformation class of extended TQD models: Given an extended TQD model H G,α0 K,β0 , we can construct a deformation class of extended TQD models {H G,α t K,β t } for a continuous parameter t, with where ξ t is an arbitrary U (1)-valued function (i.e., a 2cochain) with initial condition ξ t=0 = 1, i.e., α t=0 = α 0 and β t=0 = β 0 . We check that H G,α t K,β t is a well-defined extended TQD model at any t. To see this, one can verify that By the first condition, α t is a 3-cocycle on G, which defines the bulk TQD Hamiltonian. By the second condition, β t is α t -dependent 2-cocycle on K, which defines the boundary Hamiltonian. Hence H G,α t K,β t is an extended TQD model. During the deformation, the energy spectrum of the system remains the same. That is, there is no level crossing and thus no phase transition.
To better understand the above general approach and the physical consequences, let us work on an explicit example hereafter.
Example G = K = Z 2 × Z 2 : This is the simplest example for a nontrivial continuous deformation. The precise form of the matrix elements of the operators in the Hamiltonian (1) in this case are recorded in Appendix A. Since H 2 [Z 2 × Z 2 , U (1)] = Z 2 , the 2-cocycles are grouped into two equivalence classes [1] and [−1]. We then restrict to the case with α 0 = 1 and β 0 = 1 at t = 0, such that the initial extended TQD model reduces to the Z 2 ×Z 2 Kitaev QD model on a disk with a trivial boundary condition. We can then construct the continuous deformation (??) with the one-parameter family which is indexed by a, b = 00, 01, 10, 11 ∈ Z 2 × Z 2 and Consequently, a closed deformation loop forms for t ∈ [0, 4]. See Fig.  2(a). The α t = 1 if and only if t is an integer. In this figure, one can see that only the four big dots correspond to extended QD models, whereas any other point along the deformation loop corresponds to an extended TQD model. That is, the deformation between two extended QD models would have to go into the space of extended TQD models.

Effective 1+1D Hamiltonian of interacting anyons:
To understand the continuous deformation, let us begin with the ground-state wavefunction Φ(t) on the disk, with an explicit t dependence, where and hereafter we let N + 1 = 1. This is the tdeformation of the wavefunction obtained in Ref. [1]. The excitations are characterized by topological quasiparticles, or, anyons, in the bulk and on the boundary. There are two types of quasiparticles: charges identified by A v=0 in the bulk and by A v on all boundary vertices 1 through N ; and flux identified by B p on bulk triangles. By examining the ground state wavefunction, we see no flux will appear duration deformation for all t. Hence we consider excitations with only charges in the bulk and on the boundary. We first express a basis of excitations with charges j 1 through j N residing respectively at the N vertices on the boundary as where ρ j (a) is an irreducible representation of Z 2 × Z 2 . See Fig. 1(c). For j n = j L j R ∈ {00, 01, 10, 11} with j L , j R ∈ {0, 1}, ρ j (a) takes the form Here, the charge 00 is the trivial one or vacuum. In such a basis Ψ j1...j N , however, there is also a charge − N n=1 j n residing at vertex 0 in the bulk, due to a global constraint that the total charge of the system is null.
The basis states Ψ j1...j N t are always the energy eigenstates at time t but not at any other t = t. This deformation in fact defines a one-parameter family of continuous (unitary) transformation on the anyon bases at different t values, which quantifies how anyons recombine and/or shuffle during the deformation. In the following, we will rewrite the ground state Φ(t) at t as a linear combination of excitations at t = 0. Namely, using Eq. (7) and (8), we decompose Φ t as ρ −jn (a n )β t (a n , a −1 n a n+1 ). Using Eq. (10) can be differentiated as with whereh where s = j n − j n and s = j n+1 − j n+1 . In the equations above, the anyon charges in . . . of Ψ remain intact. The interactionh quantifies the exchange of anyon charges between two neighboring anyons. In our example,h reads explicitly as a matrix with matrix indexed by s, s = 00, 01, 10 and 11. Consider a continuous deformation H(t). The parameter can be viewed as a virtual time, while Eq. (14) defines an emergent Hamiltonian describing the interactions of the anyons on boundaries. Such a Hamiltonian determines the adiabatic evolution of the ground state Φ t . The t-dependence of the probability amplitudes |C j1,j2,...,j N | 2 is illustrated in Fig. 2(b).
In the anyon basis, we introduce 4×4 matrix τ x j defined by Pauli matrices Then the effective Hamiltonian becomes a spin chaiñ Charge Conservation: When β t = δγ, i.e., β can be expressed as a 2-coboundary Anyon interactions preserves the total charge. To see this, let γ(a) = exp{itζ(a)}, and define the corresponding Forier transformatioñ We express θ as The two terms ζ(a) and ζ(b) are canceled by the sum in Eq. (14). The remaining term in emergent Hamiltonian is given byh where the delta function implies the total charge conservation during the anyon interaction. This is illustrated in Fig. 3(a). Consequently, the boundary anyons only recombine and shuffle on the boundary. See Fig. 3(c). The effective spin-chain Hamiltonian now reads For example, in the deformation (5), we can define a new path to deform H(t = 0) to H(t = 2), withζ being In general, however, β t = δγ, such that the interactioñ h s,s (16) does not conserve the anyon charge, namely, j 1 + j 2 = j 1 + j 2 because s = −s in general, as in Fig.  3(b). Had the boundary been a stand-alone (1 + 1)-D system, this charge unconservation would cause anomaly. Nonetheless, in our (2 + 1)-D system, the excessive anyon charges does not disappear but leaks into the bulk and cancel the anyon charges in the bulk, as sketched in Fig. 3(c). Such charge unconversation implies anomaly cancelation via anomaly inflow, which we now explain and quantify.
Anomaly inflow: Consider the two extended QD theories in the upper two corners of Fig. 4, where the bulk is restricted to ground states. There exists two stand-along (1+1)-D theories, denoted by TFT t=0 and TFT t=1 in the lower two corners of Fig. 4. Coupling these two (1 + 1)-D theories to a pure gauge theory in the bulk (determined by α t=0 = α t=1 = 1) results in the the two extended QD theories as just mentioned. anomaly G.S. G.S.
FIG. 4: Pure (1 + 1)-D theories defined by β1 and β2 cannot be continuously connected. This anomaly is canceled when β1 and β2 define two boundaries of the same (2 + 1)-D theory.
Now consider a deformation from TFT t=0 to TFT t=1 , not coupled to a bulk. As stand-along (1+1)-D theories, TFT t=0 and TFT t=1 belong to different phases, characterized by two inequivalent 2nd-cohomology classes [1] and [−1] respectively. Hence there must be a phase transition during the deformation. Upon a transition point, the system is gapless, and the corresponding (1 + 1)-D theory is anomalous.
Such an anomaly is a gauge anomaly for the following reason. The anyon charges are gauge charges (with Z 2 × Z 2 viewed as the gauge group). The violation of conservation of boundary anyon charges in the extended QD models implies the violation of gauge invariance of the (1+1)-D TFT theories. Hence the anomaly is a gauge anomaly. The conservation of the total anyon charges in the entire system (bulk plus boundary) implies that the gauge anomaly is canceled by the bulk. Therefore, the inflow of anyon charges from boundary to bulk quantitatively characterizes the gauge anomaly inflow.
We define the total anyon-charge exchange between the boundary and bulk accumulatively from t = 0 to t to be .
We compute Q j t for N = 3, 10, 100, and 1000, using Eq. (5). See Fig. 5. We can see that in the thermodynamic limit N → ∞, Q j t → 1/4 for all j, i.e., evenly distributed. More importantly, Q j t ≡ 0 at integer t for all j = 00, which quantitatively demonstrates the anomaly cancellation by anomaly inflow, as the bulk at integer t is described by the same pure gauge theory.

Appendix A: Vertex and plaquette operators
Here we list the action of the vertex operators in the Hamiltonian (1) for the case with N = 3 in Fig. 1(b).
Here a1 a2 a3 b1 b2 b3 is a shorthand notation for a state on the reduced graph in Fig. 1(b) for N = 3. The plaquette operators B p is defined on triangles. On a triangle p, B p = 1 if the product of the three group elements along the three edges of the triangle clockwise is equal to the identity element of the group, and B p = 0 otherwise.