Condensation of non-Abelian anyons in
The color excitations of interacting fermions carrying an color and flavor index in one spatial dimension are studied in the framework of a perturbed Wess-Zumino-Novikov-Witten model. Using Bethe ansatz methods the low energy quasi-particles are found to be massive solitons forming quark and antiquark multiplets. In addition to the color index the solitons carry an internal degree of freedom with non-integer quantum dimension. These zero modes are identified as non-Abelian anyons satisfying fusion rules. Controlling the soliton density by external fields allows to drive the condensation of these anyons into various collective states. The latter are described by parafermionic cosets related to the symmetry of the system. Based on the numerical solution of the thermodynamic Bethe ansatz equations we propose a low temperature phase diagram for this model.
The remarkable properties of topological states of matter which cannot be characterized by a local order parameter but rather through their global entanglement properties have attracted tremendous interest in recent years. One particular consequence of the non-trivial bulk order is the existence of fractionalized quasi-particle excitations with unconventional statistics, so-called non-Abelian anyons. The ongoing search for physical realizations of these objects is driven by the possible utilization of their exotic properties, in particular in the quest for reliable quantum computing where the topological nature of non-Abelian anyons makes them a potentially promising resource Kitaev (2003); Nayak et al. (2008). Candidate systems supporting excitations with fractionalized zero energy degrees of freedom are the topologically ordered phases of two-dimensional quantum matter such as the fractional quantum Hall states or superconductors Moore and Read (1991); Read and Rezayi (1999); Read and Green (2000). In these systems the presence of gapped non-Abelian anyons in the bulk leads to anomalous physics at the edges of the probe or at boundaries between phases of different topological order.
To characterize the latter it is essential to understand the properties of an ensemble of anyons where interactions lift the degeneracies of the anyonic zero modes and correlated many-anyon states are formed. One approach towards a classification of the possible collective states of interacting anyons has been to study effective lattice models Feiguin et al. (2007); Gils et al. (2013); Finch et al. (2014a, b); Braylovskaya et al. (2016); Vernier et al. (2017); Finch et al. (2018). Here the local degrees of freedom are objects in a braided tensor category with operations describing their fusion and braiding. Note that both the Hilbert space of the many-anyon system and the possible local interactions in the lattice model are determined by the fusion rules. These models allow for studies of many interacting anyons in one spatial dimension, both using numerical or, after fine-tuning of couplings to make them integrable, analytical methods thereby providing important insights into the collective behaviour of non-Abelian anyons. However, the question of how these degrees of freedom can be realized in a microscopic physical system is beyond its scope of this approach. In addition, the effective anyon models do not contain parameters, e.g. external fields, which would allow to drive a controlled transition from a phase with isolated anyons into a condensate.
Here we address these questions starting from a particular one-dimensional system of fermions: it is well known that strong correlations together with the quantum fluctuations in such systems may lead to the fractionalization of the elementary degrees of freedom of the constituents, the best-known example being spin-charge separation in correlated electron systems such as the one-dimensional Hubbard model Essler et al. (2005). A similar phenomenon can be observed in a system of spin- electrons with an additional orbital degree of freedom which is integrable by Bethe ansatz methods Tsvelik (2014); Borcherding and Frahm (2018): in the presence of a particularly chosen interaction the elementary excitations in the spin sector of this system are massive solitons (or kinks) connecting the different topological ground states of the model. On these kinks there are localized zero energy modes which, based on the exact low temperature thermodynamics, have been identified as spin- anyons satisfying fusion rules. The mass and density of the kinks (and therefore the anyons) can be controlled by the external magnetic field applied in the underlying electron system. This allows to study the condensation of the anyons into a the collective phase described by a parafermionic conformal field theory.
Below we extend this work by considering fermions forming an ’color’ and an additional flavor multiplet. We focus on the lowest energy excitations in the color subsector of the model in the presence of external fields coupled to the Cartan generators of the global symmetry. The spectrum of quasi-particles and the low temperature thermodynamics of the model are studied using Bethe ansatz methods. Using a combination of analytical methods and numerical solution of the nonlinear thermodynamic Bethe ansatz integral equations we identify anyonic zero modes which are localized on massive solitons in ’quark’ and ’antiquark’ color multiplets or bound states thereof. For sufficiently strong fields the mass gaps of the solitons close and the anyonic modes overlap. The resulting interaction lifts the degeneracy of the zero modes and the anyons condense into a phase with dispersing collective excitations whose low energy behaviour is described by parafermionic cosets. The transitions between the various topological phases realized by these anyons are signaled by singularities in thermodynamic quantities at low temperatures. These are signatures of anyon condensation, complementing similar studies of two-dimensional topological systems Levin and Wen (2005); Bais and Slingerland (2009); Haegeman et al. (2015); Iqbal et al. (2018).
Ii Integrability study of perturbed WZNW model
We consider a system described by fermion fields forming a ’color’ multiplet and an auxiliary ’flavor’ multiplet. In the presence of weak interactions preserving the charge, color, and flavor symmetries separately, conformal embedding can be used to split the Hamiltonian into a sum of three commuting parts describing the fractionalized degrees of freedom in the collective states Francesco et al. (1996). The non-Abelian color degrees of freedom are described in terms of a critical Wess-Zumino-Novikov-Witten model perturbed by current-current interactions with Hamiltonian density
Here and are the right- and left-moving Kac-Moody currents. In terms of the corresponding fermion fields their components are
where are the generators of the Cartan subalgebra and denote the ladder operator for the root in the Cartan-Weyl basis ( and are color and flavor indices, respectively).
The spectrum of (1) can be obtained using Bethe ansatz methods, see e.g. Faddeev and Reshetikhin (1986); Babujian and Tsvelick (1986), based on the observation that the underlying structures coincide with those of an integrable deformation of the magnet with Dynkin label Kulish and Reshetikhin (1981); Perk and Schultz (1981); Kulish et al. (1981); Babelon et al. (1982); Kulish and Reshetikhin (1983); Andrei and Johannesson (1984). Specifically, placing fermions into a box of length with periodic boundary conditions and applying magnetic fields , coupled to the conserved charges the energy eigenvalues in the sector with are
where , , and the parameters and are functions of the coupling constants and . The relativistic invariance of the fermion model is broken by the choice of boundary conditions but will be restored later by considering observables in the scaling limit and such that the mass of the elementary excitations is small compared to the particle density .
The energy eigenvalues (2) are parameterized by complex parameters with and solving the hierarchy of Bethe equations (cf. Refs. Perk and Schultz (1981); Schultz (1983); de Vega and Lopes (1991); Lopes (1992) for the magnet in the fundamental representation, )
where . Based on these equations the thermodynamics of the model can be studied provided that the solutions to the Bethe equations describing the eigenstates in the limit are known. Here we argue that the root configurations corresponding to the ground state and excitations relevant for the low temperature behaviour of (1) can be built based on the string hypothesis for the models Takahashi and Suzuki (1972), i.e. that in the thermodynamic limit the Bethe roots on both levels can be grouped into ’-strings’ of length and with parity
with real centers . The allowed lengths and parities depend on the parameter . In the following we further simplify our discussion by assuming with integer where the following different string configurations contribute to the low temperature thermodynamics:
Within the root density approach the Bethe equations are rewritten as coupled integral equations for the densities of these strings Yang and Yang (1966). For vanishing magnetic fields one finds that the Bethe root configuration corresponding to the lowest energy state is described by finite densities of -strings on both levels . The elementary excitations above this ground state are of three types: similar as in the isotropic magnet Johannesson (1986) there are solitons or ’quarks’ and ’antiquarks’ corresponding to holes in the distributions of -strings on level . They carry quantum numbers in the fundamental representations and of , respectively (independent of the representation used for the construction of the spin chain). The different types of -strings are called ’breathers’. Finally, there are auxiliary modes given by -strings. The densities of these excitations (and for the corresponding vacancies) satisfy the integral equations ( denotes the convolution of and )
see Appendix A. As mentioned above relativistic invariance is restored in the scaling limit where the solitons are massive particles with bare densities and bare energies
Here is the soliton mass while and parameterize the charges of these excitations corresponding to the highest weight states in the quark and antiquark representation. Similarly, breathers have bare densities and energies
Note that the mass of the breathers with coincides with that of the solitons. The magnetic fields, however, couple to these modes in a different way, indicating that they are descendents of the highest weight states in the quark and antiquark multiplet. Therefore excitations of types and will both be labelled solitons (or quarks and antiquarks for solitons on level and , respectively) below. The masses and charges of the auxiliary modes vanish, i.e. .
The energy density of a macro-state with densities given by (5) is
Iii Low Temperature Thermodynamics
Additional insights into the physical properties of the different quasi-particles appearing in the Bethe ansatz solution of the model (1) can be obtained from its low temperature thermodynamics. The equilibrium state at finite temperature is obtained by minimizing the free energy, , with the combinatorial entropy density Yang and Yang (1969)
The resulting thermodynamic Bethe ansatz (TBA) equations for the dressed energies read
It is convenient to rewrite the equations for the auxiliary modes
where , and the kernels are defined in Appendix A. Notice that the integral equations for the auxiliary modes (12) coincide with the integral equations of RSOS models of type up to the driving terms Bazhanov and Reshetikhin (1990). The free energy per particle in terms of the solutions to the TBA equations for the solitons and breathers as
Solving the TBA equations (11) we obtain the spectrum of the model (1) for given temperature and fields. In the following we restrict ourselves to the regime – exchanging corresponds to interchanging the two levels of the Bethe ansatz. From the expressions (6) and (7) for the bare energies of the elementary excitations we can deduce the qualitative behaviour of these modes at low temperatures: As long as solitons and breathers remain gapped. Increasing the fields with the gap of the quarks ( in (6)) closes once . For larger fields they condense into a phase with finite density and the degeneracy of the corresponding zero energy auxiliary modes is lifted. At even larger fields the gap of the highest weight state of the antiquark will close, too, and the systems enters a collective phase with a finite density of quarks and antiquarks. In Figures 1 and 2 the zero temperature mass spectrum for the model with , is shown as function of for and , respectively.111We note that the highest energy soliton levels are not captured by the string hypothesis (4). However, since the gaps of these modes grow with the magnetic field they do not contribute to the low temperature thermodynamics studied in this paper. Note that in the latter case the spectra of elementary excitations on level and coincide, for all .
Based on this picture we now discuss the behaviour of the free energy as function of the fields at temperatures small compared to the relevant energy scales, i.e. the masses or Fermi energies of the solitons, .
iii.1 Non-interacting solitons
For fields solitons and breathers are gapped. At temperatures small compared to the gaps of the solitons the nonlinear integral equations (11) can be solved by iteration: the energies of the massive excitations are well described by their first order approximation Tsvelik (2014) while those of the auxiliary modes are given by the asymptotic solution of (12) for Kirillov (1989)
For solitons and breathers this implies ()
resulting in the free energy
As observed in Refs. Tsvelik (2014); Borcherding and Frahm (2018) each of the terms appearing in this expression is the free energy of an ideal gas of particles with the corresponding mass carrying an internal degree of freedom with non-integer ’quantum dimension’ for the solitons and for the breathers (). Their densities
can be controlled by variation of the temperature and the fields, which act as chemical potentials.
For the interpretation of this observation we consider fields and where the dominant contribution to the free energy is that of the lowest energy quarks, , . Their degeneracy coincides with the quantum dimension of the anyons satisfying fusion rules with topological charge or
see Refs. Bazhanov and Reshetikhin (1990); Frahm and Karaiskos (2015); Schoutens and Wen (2016). Here denotes the Young diagram with nodes in the -th row. Following the discussion in Ref. Tsvelik (2014) we interpret this as a signature for the presence of anyonic zero modes bound to the quarks. The degeneracy of the breather can be understood as a consequence of the breather being a bound state of two quarks, each contributing a factor to the quantum dimension: from the fusion rule for and anyons,
for , the degeneracy of this bound state is obtained to be .
iii.2 Condensate of quarks
For fields the quarks in the highest weight state form a condensate, while the contribution to the free energy of the other solitons and the breathers can be neglected. For large fields the low temperature thermodynamics in this regime can be studied analytically: following Kirillov and Reshetikhin (1987) we observe that the dressed energies and densities can be related as
for with a sufficiently large . is the Fermi function. Inserting this into (10) we get ()
The integrals over can be performed giving
in terms of the Rogers dilogarithm
for . Using the integral equations (5) for simplify in this regime to
we conclude that such that for Consequently, we get for all . Using and the relation with ()
for we get the following low-temperature behavior of the entropy
This is consistent with an effective description of the low energy collective modes in this regime through the coset conformal field theory with central charge
where denotes generalized parafermions Gepner (1987), the collective modes can equivalently be described by a product of a free boson contributing to the central charge and a parafermion coset contributing
To study the transition from the gas of free anyons to the condensate of quarks described by the CFT (29) at intermediate fields we have solved the TBA equations (11) numerically. Similar as in Ref. Borcherding and Frahm (2018) this can be done choosing suitable initial distributions and iterating the integral equations for given fields , temperature and anisotropy parameter .
Using (13) the entropy can be computed from the numerical data as
From the numerical solution of the TBA equations one finds that the low energy behaviour is determined by the quarks and the auxiliary modes on the first level which propagate with Fermi velocities
where is defined by . Note that is the same for all auxiliary modes
The resulting low temperature entropy is the sum of contributions from a boson (quark) and a parafermionic coset (from the auxiliary modes)
This behavior is consistent with the conformal embedding (29). Note that both Fermi velocities depend on the field and approach as