Quasi-particle Statistics and Braiding from Ground State Entanglement

Wave functions of an SU(2) spin symmetric CSL are obtained in the slave particle construction. We write the spins as bilinear in fermions $\vec{S}=\frac{1}{2}f^{\dagger}_{\sigma}[\vec{\sigma}]_{\sigma\sigma^{\prime}}f_{\sigma^{\prime}}$ and assume a chiral d-wave state for the fermions. Operationally, the spin wavefunctions are obtained by Gutzwiller projection of a $d_{x^{2}-y^{2}}+id_{xy}$ superconductor to one fermion per site. More technical details regarding this wave function are in Appendix B. We consider the system on a torus. Before projection, one can write down different fermion states, by choosing periodic or anti-periodic boundary conditions along $\hat{x}$ and $\hat{y}$ directions. These boundary conditions are invisible to the spin degrees of freedom which are bilinear in the fermions and lead to degenerate ground statesRead89 . We denote the ground states by the mean field fluxes in $\hat{x}$ and $\hat{y}$ directions as $\left|\varphi_{1},\varphi_{2}\right\rangle$, $\varphi_{1,2}=0,\pi$. The two fold degeneracy of the CSL implies that only two of the four ground states $\left|0,0\right\rangle$, $\left|\pi,0\right\rangle$, $\left|0,\pi\right\rangle$, $\left|\pi,\pi\right\rangle$ are linearly independent. Here we consider linear combinations of $\left|0,\pi\right\rangle$ and $\left|\pi,0\right\rangle$, which we have numerically checked to be indeed orthogonal for the system sizes that we consider:

We calculated TEE for the state $|\Phi\rangle$ using VMC method and Gutzwiller projected wave functions based on Eqn.32. An efficient VMC algorithm which allows to study a linear combination of Gutzwiller projected wave functions was developed and detailed in Appendix A. To our knowledge, this is the first numerical study to accomplish this.

The geometry and partition of the system are shown in Fig. 2b. The total system size is 12 lattice spacings in both directions with rectangles $A$ and $B$ being $6\times 4$ and rectangle $C$ $12\times 4$. Note that the subsystems $AC$, $BC$, $AB$, $C$ and $ABC$ all wrap around $\hat{y}$ direction so that their TEE will all be equal (and denoted $\gamma^{\prime}$). This is the quantity we wish to access. For contractible subsystems $A$ and $B$ it remains the same as that expected for a region with a single boundary, cut out of a topologically trivial surface (such as a bigger disc) $\gamma$. We use the construction due to Kitaev and Preskill kitaev2006 and effectively isolate the topological contributions in the limit of small correlation length, by evaluating the combination of entropies $S_{A}+S_{B}+S_{C}-S_{AB}-S_{AC}-S_{BC}+S_{ABC}$. This combination is related to the TEE by:

In the second line we have exploited symmetries of the construction to reduce the problem to calculation of the Renyi entropy $S_{2}$ of three regions $A$, $AC$ and $ABC$ for each $\phi$. To measure $S_{2}$ numerically, we calculated the expectation value of a $\rm{Swap_{A}}$ operator, see Reffrank2011 for an elaboration of the method used. Our results for $2\gamma-\gamma^{\prime}\left(\phi\right)$ corresponding to different linear combinations parameterized by $\phi$ are shown in Fig.3. This is one of the main results of this work.

We note the TEE strongly depends on the particular linear combination chosen. The zero of the curve implies that the TEE $\gamma^{\prime}=2\gamma$, intuitive value for an entanglement cut with two boundaries. The corresponding state is the MES. We note that the MES occurs at a nontrivial angle. Understanding this requires connecting the lattice states and the field theory which is done below. We predict this angle to be $0.125\pi$ and the overall TEE dependence to be Eqn. 13, which is plotted as the solid curve Fig. 3, in rather good agreement with the numerical data.

A calculation of ground-state dependence of TEE involves two steps. In the first step, we ask the following question: given a state expressed as a linear combination of MESs, what is the expected TEE? For the CSL, this question has already been answered by Ref. dong2008, that TEE for a state $|\psi\rangle=a_{1}|1\rangle+a_{2}|2\rangle$ is

where $|1\rangle,\,|2\rangle$ are MESs for cutting the torus in the direction in question.

Second, we need to understand the relation between the MES and the physical states that appear in the Gutzwiller wave function. In general it appears that the only way to identify MESs in a generic wave function is by calculating the TEE. However, when the lattice model has additional symmetry, that can also be used to identify MESs. Here, we have a $12\times 12$ system defined on a square lattice and we will exploit the $\pi/2$ rotation symmetry to establish a connection between the flux states $|\varphi_{1},\,\varphi_{2}\rangle$ of the Gutzwiller ansatz and the MESs.

The Gutzwiller projected ground states of the CSL, $\left|0,0\right\rangle$ and $\left|\pi,\pi\right\rangle$ are clearly invariant under a $\pi/2$ rotation symmetry upto a phase factor. A simple calculation shows that the $\left|0,0\right\rangle$ state acquires phase factor $-1$ while the $\left|\pi,\pi\right\rangle$ state acquires no phase under rotation. Similarly, the states $\frac{1}{\sqrt{2}}\left(\left|0,\pi\right\rangle\pm\left|\pi,0\right\rangle\right)$ acquire a phase $\pm 1$ under rotation. Having established the transformation of lattice states under rotation, we now study how the MESs in the field theory respond to rotations. We will see that $\pi/2$ rotation in the basis of the MESs is described by the modular $\mathcal{S}$ matrix. The eigenvectors of the modular $\mathcal{S}$ matrix will then be identified with lattice states that are rotation eigenstates.

The CSL has the same topological order as the half filled Landau level $\nu=1/2$ Laughlin state kalmeyer ; thomale of bosons. The field theory describing the topological order of a $\nu=1/k$ Laughlin state is described by the following Chern-Simons action. Note, here only the very long wavelength degrees of freedom are retained:

One can define the Wilson loop operators $T_{1}=e^{i\theta_{1}}=e^{i\oint a_{x}dx}$ and $T_{2}=e^{i\theta_{2}}=e^{i\oint a_{y}dy}$ around the two distinct cycles of the torus. In terms of $\theta_{i}$, the action is given by

which implies that at the operator level $\left[\theta_{1},\theta_{2}\right]=i\frac{2\pi}{k}$ or

Owing to the above relation, there are $k$ orthogonal ground states $|\psi_{m}\rangle$ that can be chosen to transform under $T_{i}$ as

In the case of a CSL phase, $k=2$. Let us label the two degenerate ground states as $\left(1,0\right)^{T}$ and $\left(0,1\right)^{T}$, which are eigenstates of $T_{2}$:

The last two equations are due to the commutation relation $T_{1}T_{2}=-T_{2}T_{1}$. It follows that the eigenstates of $T_{1}$ are $\left(1,1\right)^{T}/\sqrt{2}$ and $\left(1,-1\right)^{T}/\sqrt{2}$.

The significance of the $T_{1,2}$ eigenstates is that they are MESsdong2008 , for cuts whose boundaries are parallel to the loops used to define $T_{1,2}$. This is because eigenstates of these loop operators have a fixed value of flux enclosed within the relevant cycle of the torus, which minimizes the entanglement entropy for a parallel cut as shown in Fig. 4.

Now consider a $\pi/2$ rotation, under which $\theta_{1}\rightarrow\theta_{2}$ and $\theta_{2}\rightarrow-\theta_{1}$ so $T_{1}\rightarrow T_{2}$ and $T_{2}\rightarrow T_{1}^{-1}=T_{1}$. Thus, the matrix representing the effect of $\pi/2$ rotation for CSL in $T_{2}$ eigenstate basis is:

Note that we have used the symbol $\mathcal{S}$ for the above matrix because it is indeed the modular $\mathcal{S}$ matrix of the Chern-Simons topological quantum field theory corresponding to a CSL. We recall that the modular $\mathcal{S}$ matrix transforms the eigenstates of one Wilson loop operator $T_{2}$ to those of $T_{1}$. We will return to the discussion of deriving $\mathcal{S}$ matrix for CSL state using the entanglement properties of the ground states in Sec. III.1. Here we restrict ourselves to the calculation of TEE for the CSL.

Since we are interested in the entanglement entropy with respect to the cut shown in Fig. 4, let us represent all our states in the basis of the eigenstates of $T_{2}$, i.e. the states $(0,1)$ and $(1,0)$. Then, by matching eigenstates of the $\mathcal{S}$ matrix in the above basis and rotation eigenstates of the lattice problem, we conclude:

We can now expand the general linear combination state $|\Phi(\phi)\rangle$ in MESs:

Then, according to Eqn. (2), theoretically one expects the following expression for TEE:

which is compared with the numerical data in Fig. 3. The MES occur at the value of $\phi=\pi/8$ (mod $\pi/2$).

On a cylinder, the Hamiltonian in Eqn.14 leads to a pair of degenerate ground states (the $A_{s}$ part of the Hamiltonian is suitably modified at the boundary of the cylinder to only include three links). The two normalized ground states $|\xi_{0}\rangle,|\xi_{1}\rangle$, are given by equal superpositions of electric field loop configurations which have an even and odd winding number around the cylinder respectively (see Fig. 6). Consider now partitioning the cylinder into two cylindrical regions $A$ and $B$. Then the Schmidt decomposition of these ground states can be written as:

where the $N_{q}$ distinct configurations represented by $\{q_{l}\}$ denotes the electric field configurations at the cut. The number of field lines crossing the cut is always even, since the ground state is composed of closed loops. For trivial bipartitions, this exhausts all terms in the Schmidt decomposition levin2006 . However, given that the boundary of the cut is non-contractible, the additional index $0,\,1$ appears which counts the parity of electric field winding around the cylinder, within a partition. These are correlated between the two partitions, for the fixed winding number ground states. This is the key difference from a trivial bipartition, leading to the ground-state dependence of TEE.

We now calculate the entanglement entropy associated with such a cut for an arbitrary linear combination of these two ground states $|\Psi\rangle=c_{0}|\xi_{0}\rangle+c_{1}|\xi_{1}\rangle$, with unit norm. Using Eqn. LABEL:eq:decomp one can easily verify:

where $|\Psi^{A(B)}_{\{q_{l}\},\pm}\rangle=\left(|\Psi^{A(B)}_{\{q_{l}\},0}\rangle\pm|\Psi^{A(B)}_{\{q_{l}\},1}\rangle\right)/\sqrt{2}$. For a Schmidt decomposition $|\Psi\rangle=\sum_{a}\sqrt{\lambda_{a}}|\Psi^{A}_{a}\rangle|\Psi^{B}_{a}\rangle$ the $n$th Renyi entropy is given by: $S_{n}=\frac{1}{1-n}\log\left(\sum_{a}\lambda_{a}^{n}\right)$. We arrive at: $S_{n}=\frac{1}{1-n}\log N_{q}^{1-n}[p_{+}^{n}+p_{-}^{n}]$, where $p_{\pm}=|c_{0}\pm c_{1}|^{2}/2$. Recognizing that the closed loop constraint leads to $N_{q}=2^{L-1}$, where $L$ is the length of the cut, and using the definition of TEE in Eqn. 1 we have:

Thus, for the electric field winding eigenstates $|\xi_{0,1}\rangle$ where $p_{\pm}=1/2$, the TEE vanishes. However, for their equal superpositions when one of $p_{+}$ or $p_{-}$ vanishes, the TEE attains its maximal value $\log 2$. These are eigenstates of the Wilson loop operator that encircles the cylinder and measures the Z₂ magnetic flux (vison number) threading it. An example of such a flux operator is $F=\prod_{j\in Q}\sigma_{j}^{z}$, where $Q$ is a closed curve that loops once around the cylinder, such as the boundary $\Gamma$ in Fig. 6. Since TEE reduces the entanglement entropy, the maximum TEE states correspond to MESs. Why these MESs are eigenstates of flux through the cylinder for this particular cut? The number of electric field lines crossing the boundary $\Gamma$ is always even. This constraint carries some information and hence lowers the entropy by bringing in the standard TEE of $\log{2}$. On the other hand, the topology of the cut boundary $\Gamma$ allows for a determination of which magnetic flux sector the cylinder is in. A state that is not an eigenstate of magnetic flux through the cylinder leads to a loss of information and hence a positive contribution to the total entanglement entropy (and reduces TEE). This suggests that the MESs are eigenstates of loop operators which can be defined parallel to the cut $\Gamma$. This is further substantiated by the result for the torus case discussed below, where they are simultaneous eigenstates of magnetic flux enclosed by the cut and electric flux penetrating the cut.

The four degenerate ground states are distinguished by the even-odd parity of the winding number of electric field lines around the two cycles of the torus. The operator $W^{z}(C)$, which generates the set of closed loops $C$ can be used to write the ground states:

where the subscript $a$ ($b$) takes on binary values $0,1$ and denotes whether the loops $C$ belong to the even or odd winding number sectors along the $\hat{x}$ ($\hat{y}$) direction, and $|\mbox{vac}_{x}\rangle$ is a state with $\sigma_{x}=-1$ on every site. The four ground states cannot be mixed by any local operator and hence realize a $Z_{2}$ topological order. Let us consider a ground state as the following linear combination:

We are interested in calculating entanglement entropy for the state $\left|\Psi\right\rangle$ corresponding to the partition shown in the Fig.4 and the dependence of TEE on parameters $c_{a,b}$. After straightforward algebra (see details in Appendix C), one finds the following expression for subsystem A with boundaries of length $L$:

where

and

This is indeed consistent with Eqn.2, given that $\gamma=\log D=\log 2$ and $d_{j}=1$ for an Abelian topological order with $D^{2}=4$ degenerate ground states. Further, Eqn.20 readily leads to the following four MESs:

What is the physical significance of these four states being the MESs? Similar to the cylinder geometry case, these states are the simultaneous eigenstates of Wilson loop operators that encircles the torus and measures the $Z_{2}$ magnetic and electric fluxes threading it, as shown in Table 1 and Fig. We leave more detailed algebra to Appendix C.

When $\gamma^{\prime}_{n}$ is maximized, the corresponding $S_{n}$ is minimized, providing the maximum possible information about a given state. Since the cut is made along $\hat{y}$, it can measure the $Z_{2}$ magnetic and electric fluxes directed parallel to $\hat{x}$. Hence the MES $|\Xi_{\alpha}\rangle$ with definite magnetic and electric flux sectors, maximizes the TEE with $\gamma_{topo}=2\log(2)$, a contribution of $\log(2)$ from each of the two boundaries. As one linear superposes different MESs $|\Xi_{\alpha}\rangle$, the information obtained from measuring magnetic and electric sectors becomes scrambled; especially, at the extreme case of equal superposition of $|\Xi_{\alpha}\rangle$, all information about the global quantum numbers has been lost and we have $\gamma^{\prime}=0$. This offers another example where MESs are the eigenstates of loop operators defined on the cylinder from the entanglement cut.

Numerical Verification in a Generic Z₂ wavefunction: To provide an example of the state dependence of TEE in a $Z_{2}$ topologically ordered beyond the simple toric code model, we employ our VMC method to calculate the TEE of an $SU(2)$ symmetric $Z_{2}$ spin liquid for a cylindrical bipartition of the torus. This is a generic spin wave function, and has a finite correlation length unlike the toric code. It is obtained by Gutzwiller projecting a mean-field BCS state on a square lattice (see Appendix B for details).

For the $Z_{2}$ spin liquid wave function obtained by Gutzwiller projecting the above mean-field ground state, a calculation of TEE corresponding to a disc shaped region $A$ was performed in Ref.frank2011 . It was established that the state is indeed topologically ordered with the value of $\gamma\approx 0.584\pm 0.089$, close to the expected theoretical value of $\gamma=\log 2\approx 0.693$. Here we are concerned with the calculation of TEE for a cylindrical bipartition. We study systems with 12 lattice spacings in the $\hat{x}$ direction and 8 lattice spacings in the $\hat{y}$ direction. The subsystem separation scheme is similar to Fig.2b, where $A$ and $B$ are $4\times 4$ squares and $C$ is a $4\times 8$ rectangle, see Fig. 7. The TEE $2\gamma-\gamma^{\prime}$ can be evaluated in a similar way as Eqn.7.

By employing periodic boundary conditions in both directions, the state we study has zero magnetic flux along both directions, but it is not a MES. Instead, it is an equal superposition of two MESs states that have zero magnetic flux and $0$ or $1$ electric flux respectively through the cylinder. Therefore, according to Eqn.20, with $p_{1}=p_{3}=1/2,p_{2}=p_{4}=0$ and Renyi index $n=2$, one finds:

The VMC simulation yields $2\gamma-\gamma^{\prime}=0.576\pm 0.108$, close to the expected theoretical value $\log(2)\sim 0.693$. The smaller than expected values for both $\gamma$frank2011 ; isakov2011 and $\gamma^{\prime}$ are probably due to quasi-particle excitations across a finite gap, causing breaking of $Z_{2}$ electric field lines over the finite system size we consider. Indeed, spin correlations decay slower for the $Z_{2}$ state, as compared to the CSL, which also arrives closer to its expected value of TEE for both cylindrical bipartition as well as for the disc shaped region $A$ frank2011 .