Free-fermion descriptions of parafermion chains and string-net models

TL;DR

This paper investigates the extent to which various topological states, including parafermion chains and string-net models, can be described by free fermions, using the interaction distance measure to quantify their complexity.

Contribution

It introduces a method to quantify the role of interactions in topological states across dimensions, revealing which states are exactly free-fermion describable and which are not.

Findings

Some topological states are exactly described by free fermions.

Certain states reach the maximum interaction distance, indicating strong interactions.

The work enables new fermionisation procedures for low-energy physics description.

Abstract

Topological phases of matter remain a focus of interest due to their unique properties -- fractionalisation, ground state degeneracy, and exotic excitations. While some of these properties can occur in systems of free fermions, their emergence is generally associated with interactions between particles. Here we quantify the role of interactions in general classes of topological states of matter in all spatial dimensions, including parafermion chains and string-net models. Using the interaction distance [Nat. Commun. 8, 14926 (2017)], we measure the distinguishability of states of these models from those of free fermions. We find that certain topological states can be exactly described by free fermions, while others saturate the maximum possible interaction distance. Our work opens the door to understanding the complexity of topological models and to applying new types of fermionisation procedures to describe their low-energy physics.

Equations

$$D_{\mathcal{F}}(\rho )=\frac{1}{2}\underset{\{ϵ\}}{\min }\sum _a|{\rho }_a-{\sigma }_a(ϵ)|,$$

$$H_{{\mathbb{Z}}_N}=-\sum _j{\alpha }_{2j}^{†}{\alpha }_{2j+1}-f\sum _j{\alpha }_{2j-1}^{†}{\alpha }_{2j}+\text{h.c.},$$

$$\sigma_{\text{ansatz}}\simeq\mathrm{diag}\big{(}N^{-1},\dots,N^{-1},p,\dots,p\big{)},$$

$$D_{\mathcal{F}}(\bar{\rho }(N))\le 3-\frac{N}{2^n}-\frac{2^{n+1}}{N}.$$

$$U_{j}=\left({\begin{array}[]{cccc}1&0&0&0\\ 0&0&1&0\\ 0&v&0&v^{*}\\ 0&v^{*}&0&v\\ \end{array}}\right),$$

$$H_{{\mathbb{Z}}_4}^{\text{free}}=\mathcal{U}{H}_{{\mathbb{Z}}_2\times {\mathbb{Z}}_2}{\mathcal{U}}^{†},$$

$${\rho }_a=\frac{{\prod }_{j\in a}{d}_{x_j}}{{\mathcal{D}}^{2(|\partial A|-1)}},$$

$$\sigma\simeq\mathrm{diag}\big{(}N^{-1},\dots,N^{-1},p,\dots,p\big{)},$$

$$\sigma \simeq \underset{i}{⨂}\mathrm{diag}\left(\frac{1}{2}+s_i,\frac{1}{2}-s_i\right),$$

$$\sum _{a=0,1}\left|{\rho }_{\pi (a,b)}-{\sigma }_{(a,b)}\right|$$

$$\ge \left|{\rho }_{\pi (b)}^{\prime }-{\sigma }_{(b)}^{\prime }\right|,$$

$$\left|\psi\right\rangle=\sum_{i_{1},\cdots,i_{L}}\left(\eta\right|\Bigg{(}\prod_{j=1}^{L}\Gamma_{i_{j}}\Bigg{)}\left|\rho\right)\left|i_{1}i_{2}\cdots{}i_{L}\right\rangle$$

$$j$$

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$$U\,\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&1\\ 1&-i&-1&i\\ 1&-1&1&-1\\ 1&i&-1&-i\end{array}\right)=\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\end{array}\right)\text{.}$$

$$j$$

$$P(a)=P(\beta )\frac{\mathrm{d}\beta }{\mathrm{d}a}=\frac{\ln 2}{k\left(2^{1/k}-1\right)}{2}^{a/k}\text{.}$$

$$P(D_{\mathcal{F}})=\frac{2/\ln 2}{\sqrt{1+D_{\mathcal{F}}(D_{\mathcal{F}}-6)}},$$

$$P(D_{\mathcal{F}})=\frac{3-D_{\mathcal{F}}}{\sqrt{1+D_{\mathcal{F}}(D_{\mathcal{F}}-6)}}\text{.}$$

$$P(a)=1+\left(a-\frac{1}{2}\right)\frac{\ln 2}{k}+\frac{(a-\frac{1}{2}{)}^2-\frac{1}{12}}{2}{\left(\frac{\ln 2}{k}\right)}^2+O\left(\frac{1}{k^3}\right),$$

$$P(a)+P(1-a)=1+O\left(\frac{1}{k^2}\right)\text{.}$$

Full text

Introduction.– A striking feature of many-body systems is their ability to exhibit collective phenomena without analogue in their constituent particles. Many recent investigations into exotic statistical behaviours focus on topologically ordered systems Wen (1990) that support anyons Leinaas and Myrheim (1977); Wilczek (1982). These systems, such as spin liquids Anderson (1987) and fractional quantum Hall states Tsui et al. (1982), exemplify the non-perturbative effects of interactions in many-electron systems. On the other hand, there are systems such as 2D topological superconductors, which support topological excitations – Majorana zero modes Read and Green (2000); Ivanov (2001). These systems can be modelled by free fermions but lack topological order. Hence, a general question arises: is it possible (and with what accuracy) to describe a given topological state, with anyonic quasiparticles, by a Gaussian state corresponding to some free system?

In this paper we show that a broad class of topological states admit free-fermion descriptions. We use the interaction distance $D_{\mathcal{F}}$ Turner et al. (2017) to measure their distinguishability from free-fermion states. We consider parafermion chains Fendley (2012), which are symmetry-protected topological phases, as well as two- and three-dimensional string-nets Levin and Wen (2005); Walker and Wang (2012), and Kitaev’s honeycomb lattice model Kitaev (2006). These models include RG fixed points of general families of topological systems with excitations that exhibit anyonic and parafermionic statistics. Generally, we find a broad distribution of $D_{\mathcal{F}}$ values for the ground states of these models. For example, we show that some models nearly maximise $D_{\mathcal{F}}$, while others have ground states that are Gaussian states with $D_{\mathcal{F}}=0$, even if their energy spectra cannot be given in terms of free fermions. We support these results by analytical arguments at the fixed points of various models. In addition, we provide strong numerical evidence that the same conclusions hold away from fixed points. Thus, we establish $D_{\mathcal{F}}$ as a new measure of the complexity of topological models. Moreover, as $D_{\cal F}$ is defined for a quantum state instead of the full spectrum of a system (like the well-known Bethe ansatz techniques), our results open the way for investigating new types of fermionisation procedures for describing the low-energy physics of interacting systems with $D_{\mathcal{F}}=0$.

Interaction distance.– To quantify how far a given state is from any Gaussian state, we define the interaction distance Turner et al. (2017) as $D_{\mathcal{F}}(\rho)=\min_{\sigma\in\mathcal{F}}D(\rho,\sigma)$, i.e., the minimal trace distance, $D(\rho,\sigma)$, between the reduced density matrix $\rho$ of a bipartitioned system and the manifold ${\cal F}$, which contains all free-fermion reduced density matrices, $\sigma$. It was proven Turner et al. (2017) that $D_{\cal F}$ can be expressed exclusively in terms of the entanglement spectrum Li and Haldane (2008) as

[TABLE]

where $\rho_{a}$ and $\sigma_{a}$ are the eigenvalues of $\rho$ and $\sigma$, respectively, arranged in decreasing order Turner et al. (2017). For Gaussian states we have $\sigma_{a}=\exp(-\epsilon_{0}-\sum_{j}\epsilon_{j}n_{j}(a))$, where $\{\epsilon_{j}\}$ is the set of variational single-particle energies corresponding to free fermion modes and $n_{j}(a)\in\{0,1\}$ is the modes’ occupation pattern corresponding to the given level $a$ of the entanglement spectrum Li and Haldane (2008); Peschel and Eisler (2009). Intuitively, $D_{\mathcal{F}}$ is dominated by the low-lying part of the entanglement spectrum and it reveals the correlations between the effective quasiparticles emerging from interactions Li and Haldane (2008). Hence, $D_{\mathcal{F}}$ is expected to be stable under perturbations that do not cause phase transitions Turner et al. (2017). We next apply this measure to quantify the distance of various topological states of matter from free fermion states.

Parafermion chains.– The 1D Ising model can be mapped to the Majorana chain by means of a Jordan-Wigner transformation Schultz et al. (1964); Kitaev (2001). Similarly, $\mathbb{Z}_{N>2}$ generalisations of the Ising model known as the clock Potts model can be expressed in terms of parafermions Fradkin and Kadanoff (1980); Alicea and Fendley (2016). Parafermion zero modes may be physically realised at interfaces between 2D topological phases Clarke and Shtengel (2014); Mong et al. (2014). They are described by the Hamiltonian

[TABLE]

where the parafermion operators satisfy the generalised commutation relations $\alpha_{j}\alpha_{k}=\omega\alpha_{k}\alpha_{j}$ for $k>j$, where $\omega=e^{i2\pi/N}$ and $(\alpha_{j})^{N}=1$. Majorana fermions correspond to $N=2$. Recently, phase diagrams of such models have been mapped out numerically Motruk et al. (2013); Li et al. (2015). Here we focus on the gapped regime away from the critical points or critical phases Elitzur et al. (1979), and neglect other possible terms (e.g., chiral phase factor) in the Hamiltonian.

We first consider the system at its fixed point $f=0$ and we place the bipartition between regions $A$ and $B$ at a $(2j,2j+1)$-link, as shown in Fig. 1 (Top). This gives a $N$-fold degenerate spectrum $\bar{\rho}(N)$, with $\bar{\rho}_{a}=1/N$ for all $a$ Fendley (2012), where the overline, $\bar{\rho}$, denotes the density matrices with flat spectrum. We would like to determine the optimal free state corresponding to such a flat probability spectrum. Let $n$ be the greatest integer such that $2^{n}\leq N$. We surmise that the optimal free fermion spectrum is of the form

[TABLE]

where there are $2^{n}$ entries for each value $N^{-1}$ and $p$. Normalisation $\mathrm{tr}(\sigma_{\text{ansatz}})=1$ fixes $p=2^{-n}-N^{-1}$. This ansatz is an element of the variational class $\mathcal{F}$, hence $D(\bar{\rho},\sigma_{\text{ansatz}})$ forms an upper bound for $D_{\mathcal{F}}(\bar{\rho}(N))$:

[TABLE]

To evaluate $D(\bar{\rho},\sigma_{\text{ansatz}})$ we pad the spectrum of $\bar{\rho}(N)$ with zeros, a procedure always viable as it leaves the entropy invariant Turner et al. (2017). We find that the numerically computed $D_{\mathcal{F}}(\bar{\rho}{(N)})$ is in remarkable agreement with this upper bound, as shown in Fig. 1 (Bottom). We analytically proved that the two values coincide for $N\leq 6$, while we numerically verified it for up to $N=2^{8}$ 111Supplemental Online Material.. Hence, we conjecture that the upper bound of Eq. (4) is the exact maximum of $D_{\mathcal{F}}(\bar{\rho})$. This result also applies to the 2D and 3D models presented below.

From Eq. (4) we find that the maximum of the interaction distance is $D_{\mathcal{F}}^{\text{max}}=3-2\sqrt{2}$. This maximum is approached by rational approximations $N/2^{n}$ of $\sqrt{2}$ for increasing $n$, as shown in Fig. 1 (Bottom). By the exhaustive numerical maximisation $\max_{\rho}D_{\mathcal{F}}(\rho)$ for random $\rho$, we have not found states with interaction distance larger than $D_{\mathcal{F}}^{\text{max}}$ ††footnotemark: . Hence, this appears to be the maximum possible value of the interaction distance for any state.

The behaviour of the interaction distance for the flat spectra of parafermion chains, shown in Fig. 1 (Bottom), exhibits a recurring pattern, indicating that $\bar{\rho}$ has exactly the same interaction distance as $\frac{1}{2}(\bar{\rho}\oplus\bar{\rho})$. This doubling of the spectrum is equivalent to adding a zero fermionic mode to $\bar{\rho}$, which is decoupled from the rest of the modes Osborne and Verstraete (2006); Meichanetzidis et al. (2016), and thus it is not expected to change its interaction distance. We conjecture that for a generic $\rho$, i.e., with a non-flat spectrum, we still have $D_{\mathcal{F}}\left(\frac{1}{2}(\rho\oplus\rho)\right)=D_{\mathcal{F}}(\rho)$, which is supported by systematic numerical evidence ††footnotemark: .

In conclusion, we find that $\mathbb{Z}_{N\neq 2^{n}}$ parafermion chains exhibit $D_{\mathcal{F}}\neq 0$, indicating that they are interacting in terms of complex fermions, while the inequality (4) gave $D_{\cal F}=0$ for all $\mathbb{Z}_{2^{n}}$ models. These results have been derived at the fixed point ($f=0$), and now we address their validity away from the fixed point. The entanglement spectrum and, as a consequence, the interaction distance, can distinguish between the universal and non-universal properties of gapped systems Li and Haldane (2008); Turner et al. (2017). When the parafermion chain is away from its fixed point, it acquires a non-zero correlation length, $\xi$. To identify the universal properties of the system through $D_{\cal F}$, the linear size, $L_{A}$, of the partition $A$ should be $L_{A}\gg\xi$. The non-universal part is exponentially suppressed in a gapped phase and $D_{\mathcal{F}}$ predominantly describes the topological properties of the system, as shown in Fig. 2. In this figure we see that $\mathbb{Z}_{4}$ has $D_{\cal F}=0$ for any value of $f$, while the interaction distance for $\mathbb{Z}_{3}$ approaches a step function through the phase transition. Hence, when the parafermion chain is away from criticality, its ground-state $D_{\mathcal{F}}$ is a robust characteristic of the topological phase. The value of $D_{\mathcal{F}}$ is accurately given by the upper bound, Eq. (4), for sufficiently large system and partition sizes.

We have also studied the excited states of parafermion models. In the $\mathbb{Z}_{2^{n}}$ cases, it can be shown that $D_{\mathcal{F}}=0$ for all excited states at the fixed point. However, at any finite $f>0$, the excited states in general have non-zero $D_{\mathcal{F}}$ (with the exception of single quasiparticle excitations above the ground state, which remain approximately free for $f>0$ close to the fixed point). This is consistent with the models, e.g., $\mathbb{Z}_{4}$, being non-integrable for general $f$ Fendley (2012), although its ground state remains Gaussian.

To identify the free fermion description of the $\mathbb{Z}_{4}$ ground state we employ a matrix product state approach ††footnotemark: . For simplicity we restrict to the $f=0$ fixed point. One can express the ground state of $\mathbb{Z}_{4}$ in terms of the Gaussian ground state of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ (which has the same entanglement spectrum) with the help of the rotation, ${\cal U}=\otimes_{j}U_{j}$  ††footnotemark: , where the local unitaries $U_{j}$, acting on each site $j$ of the chain are given by

[TABLE]

where $v=\frac{1+i}{2}$. From this relation between the ground states we can identify the parent free fermion Hamiltonian, $H_{\mathbb{Z}_{4}}^{\text{free}}$, that has the same ground state as $H_{\mathbb{Z}_{4}}$

[TABLE]

where $H_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}$ is defined similarly to the Hamiltonians in (2). This local Hamiltonian gives rise to the same zero modes localised at the end points of a chain like the $\mathbb{Z}_{4}$ model. Nevertheless, their excitation spectra need not coincide. The construction in Eq. (6) can be extended to all $\mathbb{Z}_{2^{n}}$ models with $D_{\cal F}=0$. Note that in the case of $\mathbb{Z}_{2}$ parafermions the corresponding parent Hamiltonian, $H^{\text{free}}_{\mathbb{Z}_{2}}$, is identical to the $H_{\mathbb{Z}_{2}}$, which describes the Majorana chain.

String nets.– We now turn to the string-net models Levin and Wen (2005); Lin and Levin (2014). These are 2D RG fixed-point models that support topological order and anyon excitations. The models are defined in terms of irreducible representations or ‘charges’ of a finite group, $\mathcal{C}=\{1,...,n\}$, that parametrise the edges of a honeycomb lattice, as shown in Fig. 3 (Left). These charges obey the fusion rules $x\times y=\sum_{z}N_{xy}^{z}z$, where $N_{xy}^{z}$ is the multiplicity of each fusion outcome. For each charge, $x$, the quantum dimension $d_{x}$ is defined that satisfies $d_{x}\times d_{y}=\sum_{z}N_{xy}^{z}d_{z}$.

The ground state of a string-net model can be interpreted as a superposition of all configurations of charge loops. The probability spectrum from any bipartition into single component regions $A$ and $B$ is determined by all string configurations, $a$, on the boundary $\partial A$ which fuse to the vacuum

[TABLE]

where $\mathcal{D}=\sqrt{\sum_{x}d_{x}^{2}}$ is the total quantum dimension of the group and $x_{j}$ is an element of the configuration, $a$, of charges at the boundary links, as shown in Fig. 3 (Left). Hence, we can directly evaluate the entanglement spectra of all string-net models, Abelian or non-Abelian, and for any partition Bullivant and Pachos (2016).

We initially consider string-nets defined with an Abelian group $\mathbb{Z}_{N}$. These models have $d_{x}=1$ for all $x\in\mathcal{C}$ and thus $\mathcal{D}=\sqrt{N}$. From Eq. (7) we find that the corresponding probability spectrum for any bipartition is flat with degeneracy $N^{|\partial{A}|-1}$. Hence, the interaction distance $D_{\mathcal{F}}$ is directly determined from Eq. (4) as in the case of parafermion chains.

The cases with $N=2^{n}$ can be exactly described by fermionic zero modes, giving $D_{\mathcal{F}}=0$ for any partition size. Hence, the ground states of these models are Gaussian states. This is a surprising result as anyonic quasiparticles are expected to emerge in interacting systems. Nevertheless, the optimal free states are not necessarily local and their energy spectrum is not necessarily given by filling of single fermion modes. For $N=2$ we obtain the well known Toric Code Kitaev (2003). We now show that the fermionisation of this model is given in terms of free lattice fermions coupled to a $\mathbb{Z}_{2}$ gauge field.

Kitaev’s honeycomb lattice model Kitaev (2006) is an interacting model that supports vortices with Abelian Toric Code or non-Abelian Ising anyonic statistics, depending on its coupling regime. Nevertheless, for fixed vortex configurations its Hamiltonian is reduced to free fermions living on the vertices of the honeycomb lattice coupled to a static $\mathbb{Z}_{2}$ gauge field $u$ that resides on its links Kitaev (2006). When we bipartition the ground state of the system the reduced density matrix splits into a gauge and a fermionic part, i.e. $\rho=\bar{\rho}_{u}\otimes\rho_{\phi}$ Yao and Qi (2010). The gauge part corresponds to a $\mathbb{Z}_{2}$ flat spectrum giving $D_{\mathcal{F}}(\bar{\rho}_{u})=0$ and the fermionic part corresponds to free fermions with $D_{\mathcal{F}}(\rho_{\phi})=0$. This means that $\rho$, as a tensor product of free fermion entanglement spectra, has also $D_{\mathcal{F}}(\rho)=0$ for any partition, rendering the ground state Gaussian. Hence, free fermions coupled to a $\mathbb{Z}_{2}$ gauge field provide the fermionisation prescription of the $\mathbb{Z}_{2}$ string-net model.

For string nets with $N\neq 2^{n}$, $D_{\mathcal{F}}$ is always non-zero. In particular, its value depends on the size $|\partial{A}|$ of the partition boundary. We investigate its behaviour by studying the distribution $P(D_{\mathcal{F}})$ of $D_{\mathcal{F}}$ by varying the size $|\partial{A}|$ of the boundary for a certain model $\mathbb{Z}_{N}$. This distribution can be shown to be given by $P(D_{\mathcal{F}})=\frac{2}{\ln 2}/\sqrt{1+D_{\mathcal{F}}(D_{\mathcal{F}}-6)}$, which, surprisingly, is independent of $N$ ††footnotemark: . Hence, there exist partitions that asymptotically maximise $D_{\cal F}$ for all $N\neq 2^{n}$, as shown in Fig. 3 (Right, Top). Therefore, all $\mathbb{Z}_{N}$ Abelian string-nets either admit a free-fermion description for any partition or they form a class for which the manifestations of interactions are equivalent.

We next consider the non-Abelian string-net models. For concreteness, we take the finite group to be $SU(2)_{k}$ for various levels $k\geq 2$. This group gives rise to string-net models that support a large class of non-Abelian anyons, such as the Ising anyons for $k=2$, with statistics similar to Majorana fermions, or the Fibonacci anyons for $k=3$, that are universal for quantum computation Trebst et al. (2008); Pachos (2012). For simplicity we consider the interaction distance for a single site partition that has $|\partial A|=3$. We find that $D_{\cal F}\neq 0$ for all $k\leq 20$, as shown in Fig. 3 (Right, Bottom). Hence, it is not possible to find a free fermion description of these non-Abelian string-net models. Nevertheless, it is possible to have chiral non-Abelian models that are not RG fixed points, which admit a description of their ground state in terms of free fermions. As we have seen, Kitaev’s honeycomb lattice model falls in this category.

The string-net construction given above for 2D topological models directly generalises to 3D topological systems, with entanglement spectra also given by Eq. (7). A more powerful generalisation is in terms of the Walker-Wang models, which have a rich behaviour in their bulk and at their boundary Walker and Wang (2012). Investigation similar to the string-net SU(2)k models shows that the interaction distance depends not only on the size of $|\partial A|$, but also on the topology of $A$ due to braiding ††footnotemark: .

Conclusions.– We have quantified the effect of interactions in the ground states of broad classes of topological phases of matter in all spatial dimensions. For parafermion chains at the fixed point, we found that the partition size does not affect the value of $D_{\cal F}$. In contrast, for 2D string nets the size of the boundary matters, but not its geometry, while for Walker-Wang 3D models the topology of the boundary also becomes relevant.

Surprisingly, we discovered that the $\mathbb{Z}_{2^{n}}$ parafermion chains, as well as $\mathbb{Z}_{2^{n}}$ string-nets and Walker-Wang models, all have ground states with $D_{\cal F}=0$ for any bipartition at the fixed point. Based on similar arguments, it is possible to show in such cases that $D_{\cal F}=0$ holds also for their excited states. Hence, the exciting possibility arises that the fermionisation procedure we applied to the $\mathbb{Z}_{4}$ parafermion model could be extended to all these states. Moreover, we numerically demonstrated that $D_{\cal F}\approx 0$ continues to hold in the ground state and low-lying excitations, even when the system is away from the fixed point. However, the highly excited states typically have $D_{\cal F}\neq 0$ away from the fixed point.

Identifying Gaussianity in the low-lying energy eigenstates of a model can refine the notion of fermionisation procedures employed to solve quantum Hamiltonians. Such procedures are applicable, in the usual sense, if a system has $D_{\cal F}=0$ for all possible bipartitions and in all its eigenstates, while at the same time the energy spectrum is also free. A more subtle possibility appears when $D_{\cal F}=0$ for some eigenstates (and all cuts), but the energy spectrum is not that of free fermions. We believe integrable systems Sutherland (2004) fall into this category. We also note that $D_{\cal F}=0$ in low-lying eigenstates is in principle compatible with anyon statistics because the latter only emerges when one interpolates adiabatically between different sectors of the conserved charges of the model Lahtinen and Pachos (2009). Finally, in some non-integrable cases like the $\mathbb{Z}_{4}$ parafermion model away from the fixed point, we found that $D_{\cal F}$ can surprisingly be zero in the ground state and the low-lying excited states of the system. This opens up the exciting possibility of describing the low-energy sectors of such interacting systems in terms of new types of free-particle models.

Acknowledgements. We thank Paul Fendley, Alex Bullivant and Jake Southall for inspiring comments. This work was supported by the EPSRC grants EP/I038683/1, EP/M50807X/1 and EP/P009409/1. Statement of compliance with EPSRC policy framework on research data: This publication is theoretical work that does not require supporting research data.