Quasiparticles as Detector of Topological Quantum Phase Transitions

Phases and phase transitions provide an important framework to understand the physics of strongly correlated quantum many-body systems. Topologically ordered phases of matter are particularly challenging in this context, because they are characterized by long-range entanglement and go beyond the Landau-Ginzburg theory. A few tools have been developed to study topological phase transitions, but the needed computations are generally demanding, they typically require the system to have particular boundary conditions, and they often provide only partial information. There is hence a high demand for developing further probes. Here, we propose to use the study of quasiparticle properties to detect phase transitions. Topologically ordered states support anyonic quasiparticles with special braiding properties and fractional charge. Being able to generate a given type of anyons in a system is a direct method to detect the topology, and the approach is independent from the choice of boundary conditions. We provide three examples, and for all of them we find that it is sufficient to study the anyon charge to detect the phase transition point. This makes the method numerically cheap.


I. INTRODUCTION
Describing physical systems in terms of phases allows us to focus on key properties rather than the full set of microscopic details. Quantum phase transitions take place at zero temperature, when a control parameter, such as pressure or magnetic field strength, is varied [1,2]. In conventionally ordered phases, quantum phase transitions can be characterized by a local order parameter, arising from the broken symmetry of the system. Often the choice of the order parameter is obvious, for example in the ferromagnetic-paramagnetic transition, the total magnetization can be used.
The idea of detecting phase transitions using local order parameters does, however, break down for the case of topologically ordered systems [3]. These incompressible phases cannot be described by local order parameters. Instead they are characterized by global order parameters. Therefore different types of probes are essential to distinguish the topological phases and to detect topological quantum phase transitions (TQPTs). A further complication arises for strongly correlated quantum manybody systems, since these systems are demanding to study numerically. Density matrix renormalization group investigations are usually limited to one-dimensional systems or quasi two-dimensional systems such as ladders or thin cylinders [4]. Many systems that may harbor topologically ordered phases are afflicted with the so-called "sign problem", precluding studies by large-scale quantum Monte Carlo simulations [5].
In this article, we show that quasiparticles are a powerful tool to detect TQPTs. It is well-known that topologically ordered systems can host anyonic quasiparticles. The anyons are neither fermions nor bosons, which can be revealed from the braiding properties of the anyons. In addition, they can have fractional charge. Generating anyons in a system and demonstrating their properties is hence a direct way to demonstrate that a system is topological. Our starting point is to modify the Hamiltonian to allow for quasiparticles to be generated in the ground state. In the simplest case, this can be done by adding a potential. We then study the properties of the quasiparticles as a function of a parameter. One could, for instance, add a term to the Hamiltonian, which is expected to drive the system out of the topological phase. The method works for all types of anyons, and it does not require a particular choice of boundary conditions. We test the method on three concrete examples. In Sec. II, we consider a lattice Moore-Read state, which undergoes a TQPT as a function of the lattice filling factor. In Sec. III, we investigate a Bose-Hubbard type model in a magnetic field, which has a Laughlin type ground state, and which also undergoes a TQPT as a function of the lattice filling. In Sec. IV, we study Kitaev's toric code, which undergoes a TQPT when a sufficiently strong magnetic field is added. For all these quite different examples, we find that it is sufficient to compute the charge of the anyons to determine the phase transition point, which means that the method is numerically cheap.  1. Topologically ordered systems have a ground state degeneracy that depends on the topology of the surface on which the system is defined. The entanglement entropy SA of a subsystem A typically follows an area law SA = αLA − γ + . . ., where . . . are the subleading terms which vanish in the thermodynamic limit, α is a constant, LA is the length of the boundary of A, and γ is the topological entanglement entropy. Another probe for topology is the many-body Chern number on the torus. Topologically ordered systems can host anyonic quasiparticles, which have unusual braiding properties and possibly fractional charge. The idea presented in the present paper is that one can detect TQPTs by creating quasiparticles and studying their properties.

II. DETECTION OF A TOPOLOGICAL QUANTUM PHASE TRANSITION IN A LATTICE MOORE-READ STATE
In this section we investigate a particular type of lattice Moore-Read state. The properties of the state can be investigated with Monte Carlo simulations, and earlier studies [21] have shown, by computation of the topological entanglement entropy γ, that this state exhibits a TQPT as a function of the filling factor of the lattice. Specifically, the transition point was found to be between a lattice filling of 1/8 and 1/2. Here, we show that the position of the critical point can be found based on computations of the charge of quasiparticles in the system. Our computation is significantly cheaper numerically than the γ computation for more reasons. First, the γ computation involves computations of entanglement entropies. These are done using the replica trick, and therefore the simulated system has twice as many particles and sites as the physical system. To compute the quasiparticle charge, on the other hand, no doubling is needed. Second, the needed size of the physical system is also smaller for the computation of the charges than for the topological entanglement entropy. These advantages enable us to obtain a much more accurate value for the transition point than in [21].
FIG. 2. Interpolation between the continuum limit (a) and the lattice limit (b). We define the lattice on a 2D complex plane (Re(zj), Im(zj)). Each lattice site may be either empty (black circle), singly occupied (red circle), or doubly occupied (two blue circles). We mark the area a of a lattice site with a green square and define the parameter η = a/(2π). In the continuum limit η → 0 + , and in the lattice limit η → 1. The number of particles per area is independent of η.

A. Family of the lattice Moore-Read states
We commence by introducing a family of lattice Moore-Read states. We consider an arbitrary lattice in the 2D complex plane with lattice site positions z 1 , . . . , z N and with anyon positions w 1 , . . . , w Q , where N and Q are the number of lattice sites and the number of anyons, respectively. Let us take n j ∈ {0, 1, . . . , p} to be the number of particles at site j and define a local basis |n j at the jth lattice site. Henceforth the Hilbert space dimension is (p + 1) N . We set the magnetic length to unity in the following. We define a parameter η = a 2π where a is the area per lattice site. The parameter η serves as the interpolation parameter between the lattice limit (η −→ 1) and the continuum limit (η −→ 0 + , N −→ ∞). We keep ηN fixed while doing the interpolation. Therefore the number of particles per area remains the same and the number of lattice sites per particle changes from q in the lattice limit to infinite in the continuum limit. We display this scenario in Fig. 2 for q = 2 and n j ∈ {0, 1, 2}.
The state is defined as where C is a normalization constant. The explicit form of Ψ is derived by using conformal field correlators of the underlying conformal field theory. We inscribe the analytical forms of the states for different cases as follows [22][23][24].
In the presence of Abelian type anyons where δ n = 1 if the total number of particles is and otherwise δ n = 0. The charge of the kth anyon is p k /q, z are the positions of the M singly occupied lattice sites, and Pf(. . .) is the Pfaffian, which, to be nonzero, requires M to be even. The Pfaffian is antisymmetric and therefore we have fermionic (bosonic) states for q even (odd). We get the state without anyons from (2) by taking Q = 0 and hence the lattice filling fraction is η/q. In the presence of the non-Abelian type anyons where δ n = 1 if the total number of particles is and otherwise δ n = 0. The charge of the kth anyon is P k /q. For the case of two anyons (Q = 2), we have One can derive exact, few-body parent Hamiltonians for these states [24].

B. Anyon charges and topological quantum phase transition detection
In Ref. [21] it was shown that the Moore-Read state defined by q = 2 and n j ∈ {0, 1, 2} undergoes a TQPT while changing η. The finite topological entanglement entropy γ, in the topological phase, is related to the total quantum dimension D of the anyons in the system as γ = ln D, where D = i d 2 i and d i is the quantum dimension of the ith anyon. For the Moore-Read family of states we have [25] D = √ 4q. In Ref. [21] it was shown that for η = 1/4 the value of γ is close to 1 2 ln (8), indicating that the system is in the topological phase (the same as the continuum Moore-Read state), and for η = 1 the value of γ 0 which defines the non-topological phase. This shows that there is a phase transition in the interval η ∈ {1/4, 1} and η serves as a parameter that drives the system back and forth between a topological phase and a non-topological phase. Following Ref. [24] we find that in the topological region both types of anyons have the right braiding statistics. We have done this computation for one value of η, but we show below that the anyon charges are enough to detect the transition.
For any state |Φ we define n(z i ) = Φ|n(z i )|Φ as the particle density at the ith lattice site. We define the density profile of the anyons as where n(z i ) Q is the particle density at the ith lattice site in the presence of Q anyons. In the fractional quantum Hall effect, we take the standard fermionic particle charge as −1. We define the excess charge of the kth anyon to be the sum of minus the density profile ρ(z i ) over a circular region of radius R around the kth anyon. The charge C of the kth anyon is defined as the value that the total excess charge Q k converges to for large R, when the region is far from the edge and far from the other anyons in the system. We incorporate two anyons in our system, one positively charged and one negatively charged. We concentrate on the absolute values of the anyon charges. We use the symbol Q + to denote the absolute value of the charge of the positively charged anyon for the non-Abelian case, andQ + denotes the same quantity for the Abelian case.
Similarly we write Q − andQ − for the negatively charged anyons. Q + and Q − are computed using the state in (4) whileQ + andQ − are computed using the state in (2).
We present the results in Fig. 3 for both the Abelian type and the non-Abelian type anyons in the system. We fix the number of particles to be M = 40 and vary the number of lattice sites, on a square lattice, from N = 316 to N = 80 to achieve different η values ranging from η 1/4 to η = 1 respectively. We keep ηN fixed throughout this interpolation. The expected charges of the non-Abelian type and of the Abelian type anyons, in the topological phase, are ±0.25 and ±0.5 respectively. Therefore Fig. 3(a) and 3(d) show that the anyon charges detect the phase transition and we capture the critical value to be η c 0.45 at which the transition takes place. We identify the region η < η c to be in the topological phase and η > η c to be in the non-topological one. We show the density profiles of the non-Abelian type anyons, near the critical point for η 0.44 (topological phase) and for η 0.46 (non-topological phase) respectively in Fig. 3(b) and 3(c). We display the similar scenario for the Abelian type anyons in Fig. 3(e) and 3(f). We notice that the anyons exhibit proper charges and hence are well screened for η 0.44 but for η 0.46 the anyon charges are deviating from the expected values and hence the anyons are not screened. We note that both the Abelian type and the non-Abelian type anyons capture the phase transition with the same critical point.

III. DETECTION OF A TOPOLOGICAL QUANTUM PHASE TRANSITION IN A BOSE-HUBBARD TYPE MODEL
We study a Bose-Hubbard type model in a uniform background magnetic field on a square lattice in the limit of hardcore interactions. Earlier studies [13,26] have shown that this model exhibits a TQPT as a function of the amount of flux per plaquette in the system. This was concluded by computing the many-body Chern number. The system sizes that can be reached are too small to allow for a computation of the topological entanglement entropy. We show that we can detect the TQPT from the anyon charges. We perform the computations for open boundary conditions using exact diagonalization. In our case, a single diagonalization per flux value is sufficient to determine the phase transition point. This is significantly faster than the Chern number computation, where the eigenstates are needed for a grid of twisted boundary conditions, which means that a large number of diagonalizations are needed for each value of the flux.

A. Model
We consider a Bose-Hubbard type model on a 2D square lattice. Hopping is allowed between nearest neighbor sites, and we assume that the on-site interactions are so strong that states with more than one particle per site can be ignored. We incorporate a uniform background magnetic field by using the Peierls substitution, and the resulting Hamiltonian takes the form Here, c j is the hardcore boson annihilation operator on site j, the sums are over nearest neighbor sites, J is the hopping strength, which we set to unity, and φ 0 = h/e is the unit of quantum magnetic flux. We denote h as Planck's constant and e as the charge of an electron. This model describes the behavior of a charged particle on a lattice system with a magnetic flux αφ 0 going through each unit cell. Thus we have an effective magnetic field in the lattice and 2πα is the phase acquired by a particle going around a lattice plaquette. We choose the vector potential A = B 2 (−y, x, 0) in the symmetric gauge where B is the magnetic field. One can switch to a different gauge in a straightforward way. We take the path of the integration as the straight line connecting the two nearest neighboring sites. We take N to be the number of lattice sites, M to be the number of atoms in the system, and N φ = N α to be the number of magnetic fluxes in units of φ 0 . The filling factor is then M/N φ . In the following, we study the case of M/N φ = 1/2.

B. Anyon charges and topological quantum phase transition detection
As pointed out in Ref. [26], this model shows a TQPT as a function of α with the critical point of α t 0.4. In the region α < α t the system is in the topological phase and for α > α t the phase is non-topological. Also for α 0.3 the Laughlin fractional quantum Hall state provides a reasonable description for the ground state of the model. We use a potential to trap the anyons, where n k = c † k c k is the number operator at the kth lattice site and V is the strength of the potential, sufficiently larger than the hopping strength [27,28]. The potential in (10) traps one positively charged anyon and one negatively charged anyon in the system by giving an energy penalty to the lattice site to be occupied and to be unoccupied respectively.
We define the density profile of the anyons as where n(z i ) H0+H V and n(z i ) H0 are the particle densities at the ith lattice site in the presence and in the absence of the anyons respectively. The excess charges and the charges of the anyons are computed by using (8). The quantities of interest are the absolute values of the anyon charges which are sufficient to show the screening of anyons. We use the symbols Q + and Q − to denote the absolute values of the charge of the positively and the negatively charged anyon, respectively. We take suitable values of M and N , numerically accessible for exact diagonalization, and keep M/(N α) = 1/2 fixed. Hence we obtain different values of α ∈ {0.1, 0.6}. We present the results in Fig. 4 where we plot Q + and Q − as a function of α in (a). The expected charges of the anyons, in the topological phase, are ±0.5. Therefore the data in (a) show that the anyon charges detect the phase transition and we capture the critical value to be α c 0.38 at which the transition takes place. We identify the region α < α c to be in the topological phase and the region α > α c to be in the non-topological phase. We show the different choices of M and N we take and the values of Q + and Q − in Table I (topological phase) and for α 0.39 (non-topological phase) respectively in Fig. 4(b) and 4(c). We notice that the anyons exhibit proper charges for α 0.37, but for α 0.39 the charges are deviating from the expected values.

IV. DETECTION OF A TOPOLOGICAL QUANTUM PHASE TRANSITION IN KITAEV'S TORIC CODE MODEL
We study Kitaev's toric code [29,30] on a square lattice with periodic boundary conditions, which is known to exhibit a Z 2 topologically ordered phase. We create anyons in the system and investigate their properties to analyze the stability of the topological phase with respect to an added uniform magnetic field. We use exact diagonalization and show that the anyons can detect the TQPT. The computations we do are numerically cheaper than the earlier investigations in Refs. [31,32] which used quantum Monte Carlo simulations.

A. Kitaev's toric code
We here consider Kitaev's toric code [29,30] constructed on an N x × N y square lattice with periodic boundary conditions. On each edge there is a spin-1/2 as illustrated in Fig. 5(a). The total number of spins is N = 2N x N y and therefore the Hilbert space dimension is 2 N . We write the Hamiltonian as where the sums are over all plaquettes p and over all stars s of the lattice. The plaquette operator B p acts on the four spins on bonds which surround the plaquette p, and the star operator A s acts on the four spins on bonds which surround the star s (see Fig. 5(a)). We write the operators as There are N/2 of the B p operators and N/2 of the A s operators. The A s and B p operators commute with each other for all p, p , s, s . This makes H TC exactly solvable, and the ground states are those states for which each eigenvalue of B p is equal to 1 and each eigenvalue of A s is equal to 1. The ground states are four-fold degenerate on the torus and exhibit a Z 2 topological order with Abelian anyonic excitations [29,30].

B. Creation of anyons in the ground state
In the toric code, one creates states containing anyons by applying certain string operators to the ground state. The string operator either changes the eigenvalue of two A s operators to −1 or the eigenvalue of two B p operators to −1. In the former case, two electric excitations e s are created, and in the latter case two magnetic excitations m p are created. The wavefunction acquires a minus sign if one m p is moved around one e s , and the excitations are hence Abelian anyons.
In our case, we instead modify the Hamiltonian, such that anyons are created in the ground states. The idea is to add suitable operators to H TC such that the sign of two of the B p or two of the A s operators is inverted. In particular, the ground states of the Hamiltonian H TC + H m , where has one m p on each of the plaquettes p 1 and p 2 . Similarly, the ground states of the Hamiltonian H TC + H e , where has one e s on each of the stars s 1 and s 2 .

C. Detection of the topological quantum phase transition
We aspire to drive the system through a phase transition. We do this by turning on a uniform magnetic field in the z-direction, which amounts to adding the term (17) to the Hamiltonian. Here λ > 0 is proportional to the strength of the field. Owing to the term H λ , which does not commute with H TC in (12), the model H TC + H λ is not exactly solvable. For sufficiently large λ, the term H λ will drive the system into a ferromagnetic phase, as it is energetically favourable to polarize all the spins.
Previous investigations of the Hamiltonian H TC + H λ have shown, based on Monte Carlo simulations, that the critical strength of the magnetic field at which the transition takes place is 0.22 in Ref. [31] and 0.58 in Ref. [32]. One can also determine the phase transition point from exact diagonalization using an order parameter for the magnetic phase. Specifically, we plot the magnetization per unit spin M s and the magnetic susceptibility χ s per unit spin, We denote the plaquette operator by Bp (red square) and the star operator by As (blue star), respectively. Two types of anyons can be created, namely mp (red circles) in the plaquettes p1 and p2 and es (blue crosses) in the stars s1 and s2 by using (15) and (16), respectively. In (b) we plot the quantities Es (blue squares) from (19) as the measure of the anyons and Ms (red triangles) from (18) as the magnetization of the system as a function of the external magnetic field strength λ. We note that Es gradually tends to zero from the value of −1 and that Ms gradually approaches the value of −1 from zero. In (c) we show χe (blue squares) and χs (red triangles) from (19) as a function λ. We note that both the χe and χs display peaks at around λc 0.4. In the inset we show the energy gap ∆E (magenta crosses) between the fourth and the fifth energy states as a function of λ. We note that ∆E 0 at around λ 0.7. We expect that the deviation between this λ value and λc is the finite size effects. Both in (b) and (c) we show the color gradient from yellow (for small λ and hence the topological phase) to green (for large λ and hence the non-topological phase) as plotted according to the value of Es.
look at the following quantities to detect the anyons in the system, where s ∈ {s 1 , s 2 }. We note that E s = −1 signify the presence of the anyons in those stars. In the fully polarized phase, on the other hand, E s vanishes. We could also have chosen to study the expectation value i∈p σ z i for the m p particles, where p ∈ {p 1 , p 2 }. This expectation value is, however, not a good choice to detect the TQPT for the following reason. The operator B p = i∈p σ z i is itself one of the plaquette operators, and it commutes with all terms in H TC + H m + H λ . This means that all eigenstates of the Hamiltonian are also eigenstates of B p , and half of these eigenstates have B p eigenvalue +1 and the other half have eigenvalue −1. The value of i∈p σ z i is hence always either +1 or −1, and it only measures whether the ground state has B p eigenvalue +1 or −1. We find that the first transition to a ground state with B p eigenvalue +1 happens around λ 2.08, but this does not exclude gap closings at smaller λ values. On the contrary, H e does not commute with H λ , and the above problems are hence not encountered for E s .
We use exact diagonalization to compute E s and χ e for the ground states of the Hamiltonian H TC + H e + H λ as a function of λ, as shown in Fig. 5. We note that E s changes from −1 in the topological phase to 0 in the ferromagnetic phase as expected. The critical point is around λ c 0.4. Both the prediction for the critical point and the width of the transition region, which is due to finite size effects, are comparable to the same quantities obtained from the magnetic order parameter.

V. CONCLUSIONS
We have shown that quasiparticles can be used to detect topological quantum phase transitions. Topologically ordered systems can host anyonic quasiparticles with particular braiding properties. The anyons can be trapped in the ground state of the system by adding suitable trapping operators to the Hamiltonian. As long as the quasiparticles are anyonic, we know that the system is topologically ordered. We now change a parameter in the Hamiltonian, which could drive the system into a non-topological phase or into another topological phase with a different set of anyons. When looking at a type of anyons that is only supported on one side of the transition, the anyons must undergo a change, when the phase transition is crossed. The main idea of the present paper is to use this change to detect the topological quantum phase transition. We have tested this approach for three different cases, namely a model with an analytical lattice Moore-Read state as ground state, a Bose-Hubbard type model with a Laughlin type ground state, and Kitaev's toric code with a magnetic field added. In all cases, we have found that the quasiparticles can accurately detect the quantum phase transition.
If we are able to do robust braiding in a system with results corresponding to a particular topological order, then we know that the system allows these braiding properties. For the studied examples, we have seen that it is, in fact, enough to look just at the charge of the anyons to obtain the phase transition point. This makes the computations numerically cheap.
There is currently a high demand for finding appropriate methods to detect topological quantum phase transitions, both for numerical investigations and for experimental implementations. The approach suggested here is particularly direct. To fully exploit the interesting properties of topologically ordered systems, one needs to be able to create anyons in the systems, move the anyons around in controlled ways, and measure their properties. The suggested approach finds the phase transitions based directly on the anyons, and hence does not need additional work. In the Bose-Hubbard type model, the anyons can be created by adding a local potential, and the charge of the anyons used to detect the phase transition can be measured by measuring the expectation value of the number of particles on each site. Both of these can be done in experiments with ultracold atoms in optical lattices. For the toric code model, the anyons are harder to create. It is interesting to note, however, that the op-erators needed to create the anyons are of the same type as the operators already appearing in the Hamiltonian, so if one finds a method to implement the Hamiltonian experimentally, one could also create the anyons.
The ideas presented in this work can be applied to any type of topological system with anyonic quasiparticles. In the future, it would be interesting to also test the ideas for other types of systems. One could, e.g., study what quasiparticles tell us about transitions between different non-topological phases. It would also be interesting to apply similar ideas to gapless phases.