String-net condensation: A physical mechanism for topological phases

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases—namely topological phases. These phases occur when extended objects, called “string-nets,” become highly fluctuating and condense. We construct a large class of exactly soluble 2D spin Hamiltonians whose ground states are string-net condensed. Each ground state corresponds to a different parity invariant topological phase. The models reveal the mathematical framework underlying topological phases: tensor category theory. One of the Hamiltonians—a spin-$1∕2$ system on the honeycomb lattice—is a simple theoretical realization of a universal fault tolerant quantum computer. The higher dimensional case also yields an interesting result: we find that 3D string-net condensation naturally gives rise to both emergent gauge bosons and emergent fermions. Thus, string-net condensation provides a mechanism for unifying gauge bosons and fermions in 3 and higher dimensions.

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Figure 1

A schematic phase diagram for the generic string-net Hamiltonian (3). When $t∕U$ (the ratio of the kinetic energy to the string tension) is small the system is in the normal phase. The ground state is essentially the vacuum with a few small string-nets. When $t∕U$ is large the string-nets condense and large fluctuating string-nets fill all of space. We expect a phase transition between the two states at some $t∕U$ of order unity. We have omitted string labels and orientations for the sake of clarity.

Figure 2

(Color online) The constraint term ${\prod }_{\text{legs of}\mathit{I}}{\sigma }_{\mathit{i}}^x$ and magnetic term ${\prod }_{\text{edges of}\mathit{p}}{\sigma }_{\mathit{j}}^z$ in $Z_2$ lattice gauge theory. In the dual picture, we regard the links with ${\sigma }^x=-1$ as being occupied by a string, and the links with ${\sigma }^x=+1$ as being unoccupied. The constraint term then requires the strings to be closed—as shown on the right.

Figure 3

Typical string-net configurations in the dual formulation of (a) $Z_2$, (b) $U\left(1\right)$, and (c) $SU\left(2\right)$ gauge theory. In the case of (a) $Z_2$ gauge theory, the string-net configurations consist of closed (nonintersecting) loops. In (b) $U\left(1\right)$ gauge theory, the string-nets are oriented graphs with edges labeled by integers. The string-nets obey the branching rules $E_1+E_2+E_3=0$ for any three edges meeting at a point. In the case of (c) $SU\left(2\right)$ gauge theory, the string-nets consist of (unoriented) graphs with edges labeled by half-integers $1∕2$,1,$3∕2$,…. The branching rules are given by the triangle inequality: $\{E_1$,$E_2$,$E_3\}$ are allowed to meet at a point if and only if $E_1⩽E_2+E_3$, $E_2⩽E_3+E_1$, $E_3⩽E_1+E_2$, and $E_1+E_2+E_3$ is an integer.

Figure 4

The orientation convention for the branching rules.

Figure 5

$i$ and $i^{*}$ label strings with opposite orientations.

Figure 6

Three pairs of string-net configurations that differ only in their short distance structure. We expect string-net wave functions in the same quantum phase to only differ by these short distance details.

Figure 7

A schematic RG flow diagram for a string-net model with 4 string-net condensed phases $a$, $b$, $c$, and $d$. All the states in each phase flow to fixed-points in the long distance limit. The corresponding fixed-point wave functions ${\mathrm{\Phi }}_a$, ${\mathrm{\Phi }}_b$, ${\mathrm{\Phi }}_c$, and ${\mathrm{\Phi }}_d$ capture the universal long distance features of the associated quantum phases. Our ansatz is that the fixed-point wave functions $\mathrm{\Phi }$ are described by local constraints of the form (4, 5, 6, 7).

Figure 8

(Color online) A picture of the lattice spin model (11). The electric charge operator $Q_{\mathit{I}}$ acts on the three spins adjacent to the vertex $\mathit{I}$, while the magnetic energy operator $B_{\mathit{p}}$ acts on the 12 spins adjacent to the hexagonal plaquette $\mathit{p}$. The term $Q_{\mathit{I}}$ constrains the string-nets to obey the branching rules, while $B_{\mathit{p}}$ provides dynamics. A typical state satisfying the low-energy constraints is shown on the right. The empty links have spins in the $i=0$ state.

Figure 9

(Color online) Open and closed string operators for the lattice spin model (11). Open string operators create quasiparticles at the two ends, as shown on the left. Closed string operators, as shown on the right, commute with the Hamiltonian. The closed string operator $W\left(P\right)$ only acts non-trivially on the spins along the path $\mathit{P}={\mathit{I}}_1,{\mathit{I}}_2\mathrm{\dots }$ (thick line), but its action depends on the spin states on the legs (thin lines). The matrix element between an initial state $i_1$,$i_2$,… and a final state $i_1^{\prime }$,$i_2^{\prime }$,… is $W_{i_1{i}_2\dots }^{i_1^{\prime }{i}_2^{\prime }\mathrm{\dots }}\left(e_1{e}_2\mathrm{\dots }\right)=(F_{s^{*}{i}_1^{\prime }{i}_2^{\prime }*}^{e_2{i}_2^{*}{i}_1}{F}_{s^{*}{i}_2^{\prime }{i}_3^{\prime }*}^{e_3{i}_3^{*}{i}_2}\mathrm{\dots })\bullet ((v_{i_1}{v}_{s_1}∕v_{i_1^{\prime }}){\omega }_{i_1}^{i_1^{\prime }}(v_{i_3}{v}_{s_3}∕v_{i_3^{\prime }}){\overline{\omega }}_{i_3}^{i_3^{\prime }}\mathrm{\dots })$ for a type-$s$ simple string and $W_{i_1{i}_2\dots }^{i_1^{\prime }{i}_2^{\prime }\dots }(e_1{e}_2\dots )={\sum }_{\left\{s_k\right\}}(F_{s_2^{*}{i}_1^{\prime }{i}_2^{\prime *}}^{e_2{i}_2^{*}{i}_1}{F}_{s_3^{*}{i}_2^{\prime }{i}_3^{\prime *}}^{e_3{i}_3^{*}{i}_2}\dots )\bullet \mathrm{Tr}((v_{i_1}{v}_{s_1}∕v_{i_1^{\prime }}){\mathrm{\Omega }}_{s_1{s}_2{i}_1}^{i_1^{\prime }}{\delta }_{s_2{s}_3}\mathrm{Id}(v_{i_3}{v}_{s_3}∕v_{i_3^{\prime }}){\overline{\mathrm{\Omega }}}_{s_3{s}_4{i}_3}^{i_3^{\prime }}\dots )$ for a general string.

Figure 10

(Color online) A three-dimensional trivalent lattice, obtained by splitting the sites of the cubic lattice. We replace each vertex of the cubic lattice with 4 other vertices as shown above.

Figure 11

Three plaquettes demonstrating the two fundamental differences between higher dimensional trivalent lattices and the honeycomb lattice. The plaquettes ${\mathit{p}}_1$, ${\mathit{p}}_2$ lie in the $xz$ plane, while ${\mathit{p}}_3$ is oriented in the $xy$ direction. Notice that ${\mathit{p}}_1$ and ${\mathit{p}}_2$ share three vertices, ${\mathit{I}}_1$, ${\mathit{I}}_2$, ${\mathit{I}}_3$—unlike neighboring plaquettes in the honeycomb lattice, which share two vertices. Also, notice that the plaquette boundaries ${\partial \mathit{p}}_1$ and ${\partial \mathit{p}}_3$ intersect only at the line segment ${\mathit{I}}_3{\mathit{I}}_4$. The boundary ${\partial \mathit{p}}_1$ makes a left turn at ${\mathit{I}}_3$, and a right turn at ${\mathit{I}}_4$. Thus, if we shrink the segment ${\mathit{I}}_3{\mathit{I}}_4$ to a point, these two plaquette boundaries intersect exactly once—unlike neighboring plaquettes in the honeycomb lattice, which intersect tangentially when their common boundary is shrunk to a point.

Figure 12

(Color online) The Hamiltonians (39, 40), realizing the two $N=1$ string-condensed phases. Each circle denotes a spin-$1∕2$ spin. The links with ${\sigma }^x=-1$ are thought of as being occupied by a type-1 string, while the links with ${\sigma }^x=+1$ are regarded as empty. The electric charge term acts on the three legs of the vertex $\mathbf{I}$ with ${\sigma }^x$. The magnetic energy term acts on the 6 edges of the plaquette $\mathbf{p}$ with ${\sigma }^z$, and acts on the 6 legs of $\mathbf{p}$ with an operator of the form $f\left({\sigma }^x\right)$. For the $Z_2$ phase, $f=1$, while for the Chern-Simons phase, $f\left(x\right)=i^{\left(1-x\right)∕2}$.

Figure 13

(Color online) A closed string operator $W\left(P\right)$ for the two models (39), (40). The path $P$ is drawn with a thick line, while the legs are drawn with thin lines. The action of the string operators (41), (44) on the legs is different for legs that branch to the right of $P$, “R-legs,” and legs that branch to the left of $P$, “L-legs.” Similarly, we distinguish between “R-vertices” and “L-vertices” which are ends of “R-leg” and “L-leg,” respectively.

Figure 14

The fusion rule (7) can be used to relate the amplitude of (a) to the amplitude of (c) in two different ways. On the one hand, we can apply the fusion rule (7) twice—along the links denoted by solid arrows—to relate (a)→(b)→(c). But we can also apply Eq. (7) three times—along the links denoted by dashed arrows—to relate (a)→(d)→(e)→(c). Self-consistency requires that the two sequences of the operation lead to the same linear relations between the amplitudes of (a) and (c).

Figure 15

Four string-net configurations related by tetrahedral symmetry. In diagram (a), we show the tetrahedron corresponding to $G_{kln}^{ijm}$. In diagrams (b), (c), (d), we show the tetrahedrons $G_{jin}^{lk{m}^{*}}$, $G_{lk{n}^{*}}^{jim}$, $G_{k^{*}nl}^{imj}$, obtained by reflecting (a) in 3 different planes: the plane joining $n$ to the center of $m$, the plane joining $m$ to the center of $n$, and the plane joining $i$ to the center of $k$. The four tetrahedrons correspond to the four terms in the second relation of Eq. (9).

Figure 16

The fattened honeycomb lattice. The strings are forbidden in the shaded region. A string state in the fattened honeycomb lattice (a) can be viewed as a superposition of string states on the links (b).

Figure 17

The action of $B_{\mathit{p}}^s$ is equivalent to adding a loop of type-$s$ string. The resulting string-net state (a) is actually a linear combination of the string-net states (b). The coefficients in this linear relation can be obtained by using the local rules (4, 5, 6, 7) to reduce (a) to (b).

Figure 18

The action of $B_{{\mathit{p}}_1}^{s_1}{B}_{{\mathit{p}}_2}^{s_2}$ on the string-net state (a) can be represented by adding two loops of type-$s_1$ and type-$s_2$ strings as shown in (b). The string-net state (b) is a linear combination of the string-net states (c). The coefficients are obtained by using the rules Eqs. (4, 5, 6, 7) to reduce (b) to (c).

Figure 19

The action of the string operator $W_{\alpha }\left(P\right)$ is equivalent to adding a type-$\alpha$ string along the path $P$. The resulting string-net state can be reduced to a linear combination of states on the honeycomb lattice, using the local rules (4)—(7), Eq. (D1).