Duality between disordered nodal semimetals and systems with power-law hopping

TL;DR

This paper reveals a duality between disordered nodal semimetals and systems with power-law hopping, enabling the transfer of theoretical insights and discovering new disorder-driven transitions distinct from Anderson localization.

Contribution

The authors establish an exact mapping between low-energy theories of semimetals and systems with power-law interactions, unveiling novel disorder-driven phase transitions for long-range hopping systems.

Findings

Duality mapping between semimetals and power-law hopping systems

Existence of a new class of disorder-driven transitions for $rac{d}{2}<\gamma<d$

Identification of non-Anderson disorder transitions in long-range systems

Abstract

Nodal semimetals (e.g. Dirac, Weyl and nodal-line semimetals, graphene, etc.) and systems of pinned particles with power-law interactions (trapped ultracold ions, nitrogen defects in diamonds, spins in solids, etc.) are presently at the centre of attention of large communities of researchers working in condensed-matter and atomic, molecular and optical physics. Although seemingly unrelated, both classes of systems are abundant with novel fundamental thermodynamic and transport phenomena. In this paper, we demonstrate that low-energy field theories of quasiparticles in semimetals may be mapped exactly onto those of pinned particles with excitations which exhibit power-law hopping. The duality between the two classes of systems, which we establish, allows one to describe the transport and thermodynamics of each class of systems using the results established for the other class. In particular, using the duality mapping, we establish the existence of a novel class of disorder-driven transitions in systems with the power-law hopping $\propto1/r^\gamma$ of excitations with $d/2<\gamma<d$, different from the conventional Anderson-localisation transition. Non-Anderson disorder-driven transitions have been studied broadly for nodal semimetals, but have been unknown, to our knowledge, for systems with long-range hopping (interactions) with $\gamma<d$.

Equations

$${\mathcal{L}}_{\text{SM}}=-i\int \bar{\psi }\left[E-a|\hat{\mathbf{k}}{|}^{\alpha }+i0\Lambda \right]\psi d\mathbf{r}$$

$$+i\sum _n{\lambda }_n\bar{\psi }({\mathbf{r}}_n)\psi ({\mathbf{r}}_n),$$

$$\mathcal{L}=i\sum _{m,n}\bar{\varphi }({\mathbf{r}}_n)G_E({\mathbf{r}}_n,{\mathbf{r}}_m)\varphi ({\mathbf{r}}_m)-i\sum _n\frac{\bar{\varphi }({\mathbf{r}}_n)\varphi ({\mathbf{r}}_n)}{{\lambda }_n-i0\Lambda },$$

$${\mathcal{L}}_{\text{Hop}}=$$

$$-i\sum _{m,n}\bar{\varphi }({\mathbf{r}}_n)\left[\frac{j}{|{\mathbf{r}}_n-{\mathbf{r}}_m{|}^{d-\alpha }}+{\delta }_{nm}{\lambda }_n^{-1}+i0\Lambda \right]\varphi ({\mathbf{r}}_m),$$

$$\hat{\mathcal{H}}=\sum _n{E}_n{\hat{s}}_n^z-\sum _{n,m}\frac{j}{|{\mathbf{r}}_n-{\mathbf{r}}_m{|}^{\gamma }}\left({\hat{s}}_n^{+}{\hat{s}}_m^{-}+{\hat{s}}_m^{+}{\hat{s}}_n^{-}\right),$$

$${\rho }_{\text{Hop}}(\omega )$$

$${\mathcal{L}}_{\omega }$$

$${\mathcal{L}}_s(\eta )$$

$$\varkappa =N{⟨{\left(E_n-⟨E_n⟩\right)}^{-2}⟩}_{\text{dis}},$$

$${\mathcal{L}}_{\text{SM}}=$$

$$+\frac{1}{2}\varkappa \int {\left[\bar{\psi }\psi \right]}^{2}d\mathbf{r}+i\eta \varkappa \int {\tilde{s}}_R^{\ast }{\tilde{s}}_{R}d\mathbf{r},$$

$${\rho }_{\text{Hop}}(\omega )=\varkappa {\rho }_{\text{SM}}(\omega \varkappa ),$$

$${\rho }_{\text{Hop}}(\omega ,j)={\omega }^{\frac{d}{z}-1}\Phi \left(\omega /|j-j_c{|}^{z\nu }\right)+{\rho }_{\text{smooth}},$$

$$⟨\mathcal{O}⟩=\int D\psi D{\psi }^{†}\dots \exp \left[i\int \bar{\psi }\Lambda \left[E-{\xi }_{\hat{k}}+i0\Lambda \right]\psi d\mathbf{r}-i\int u(\mathbf{r})\bar{\psi }(\mathbf{r})\psi (\mathbf{r})d\mathbf{r}\right],$$

$$\int D\varphi D{\varphi }^{†}\exp \left[i\int \frac{(\bar{\varphi }-u\bar{\psi })(\varphi -u\psi )}{u-i0\Lambda }d\mathbf{r}\right]=1,$$

$$⟨\mathcal{O}⟩=\int D\varphi D{\varphi }^{†}\dots \exp \left[-i\int \bar{\varphi }(\mathbf{r})G(\mathbf{r},{\mathbf{r}}^{\prime })\varphi ({\mathbf{r}}^{\prime })d\mathbf{r}d{\mathbf{r}}^{\prime }+i\int \frac{1}{u(\mathbf{r})-i0\Lambda }\bar{\varphi }(\mathbf{r})\varphi (\mathbf{r})d\mathbf{r}\right],$$

Full text

The last few years have seen an explosion of interest in nodal semimetals, such as Weyl, parabolic and nodal-line semimetals Armitage et al. (2018), owing to their potential applications in future electronic and spintronic devices, in addition to the abundance of novel fundamental phenomena observed in these materials: chiral anomaly Burkov (2018); Parameswaran et al. (2014); Huang et al. (2015); Liang et al. (2018); Burkov (2014), magnetohydrodynamic effects Gorbar et al. (2018); Lucas et al. (2016); Yamamoto (2016); Galitski et al. (2018), topologically protected surface states Wan et al. (2011); Weng et al. (2015); Jia et al. (2016), etc. These systems have also changed the current perspective on phase transitions in disordered systems; Weyl and Dirac semimetals have been demonstrated to display disorder-driven phase transitions (or possibly very sharp crossovers 111 It is still being debated in the literature (see, e.g., Refs. Nandkishore et al., 2014; Sbierski et al., 2016; Pixley et al., 2017; Gurarie, 2017; Buchhold et al., 2018a, b) whether the respective transition in a Weyl semimetal is a genuine phase transitions or a sharp crossover in physical observables. In this paper we do not distinguish between transitions and sharp crossovers, so long as there exists a parametrically large interval of observables where the critical scaling is observed.) in universality classes different from those of the Anderson metal-insulator transitions Fradkin (1986a, b); Syzranov and Radzihovsky (2018). These transitions have been demonstrated Syzranov et al. (2015a) to occur also in a broader class of systems in sufficiently high dimensions.

Another, seemingly unrelated, class of systems, which have recently attracted attention, are systems of pinned particles with power-law interactions. The amplitude of hopping of excitations in these systems displays a power-lay decay with distance, $\propto 1/r^{\gamma}$. Such systems include, but are not limited to, polar molecules Yan et al. (2013); Lahaye et al. (2009) ($\gamma=3$), impurity spins in solids, Rydberg atoms Saffman et al. (2010) ($\gamma=3$ or $\gamma=6$), nitrogen vacancies in diamonds Dutt et al. (2007); Waldherr et al. (2014) ($\gamma=3$) and neutral excitations in strongly disordered electronic systems ($\gamma=3$ in 3D systems and $\gamma=2$ in thin dielectric films Mooij and Schön (1992); Aleiner et al. (2011); Titum et al. (2018)). Furthermore, power-law hopping with a tunable parameter $0<\gamma<3$ has been realised Richerme et al. (2014); Islam et al. (2013); Jurcevic et al. (2014, 2015) recently in 1D and 2D systems of trapped ultracold ions. Similarly to the case of nodal semimetals, in the presence of quenched disorder power-law hopping is expected to lead to unconventional localisation phenomena or disorder-driven criticality (see, e.g., Refs. Levitov (1989, 1999, 1990); Deng et al. (2018); Rodríguez et al. (2003); Malyshev et al. (2004); de Moura et al. (2005); Gärttner et al. (2015); Tikhonov and Mirlin (2018); Yao et al. (2014)) and is also studied often in the context of many-body localisation Basko et al. (2006); Nandkishore and Huse (2015); Abanin and Papić (2017); Abanin et al. (2019); Burin and Maksimov (1989); Burin (2006).

Nodal semimetals and systems with power-law hopping may seem completely unrelated and are usually studied independently by two communities of researchers working, respectively, on condensed-matter systems and in atomic, molecular and optical physics. As it has already been noted Gärttner et al. (2015); Syzranov and Radzihovsky (2018), however, $d$-dimensional systems with power-law hopping, whose amplitude decays with distance as $\propto 1/r^{\gamma}$ with $d<\gamma<3d/2$, display the same type of non-Anderson disorder-driven transitions which take place in nodal semimetals and which are absent for faster decay, corresponding to $\gamma>3d/2$. There is thus a natural connection between the two classes of systems. The phenomenology of systems with slower decay of the hopping, corresponding to $0<\gamma<d$, is understood, in our opinion, much more poorly; the possibility of the unconventional disorder-driven transitions in them has not been investigated until this work.

In this paper, we derive a duality transformation between the field theories of disordered systems with the power-law dispersion $\propto k^{\alpha}$, with $0<\alpha<d$, typical for nodal semimetals, and the field theories of hopping in arrays of random-energy sites with the amplitude $\propto 1/r^{\gamma}$, with $0<\gamma<d$, hereinafter referred to as “long-range hopping” (as opposed to “short-range hopping”, decaying with distance faster than $1/r^{d}$). Since particles displaying the long-range hopping between sites may be considered as having a singular kinetic energy $\propto k^{\gamma-d}$, the duality transformation may be said to establish a mapping between disordered systems with kinetic energies $k^{\alpha}$ and $k^{-\alpha}$, as shown in Fig. 1, where $0<\alpha=\gamma-d<d$. It is of note that this duality exchanges strong and weak disorder in the two phases it connects. This duality mapping may be used to describe phenomena in each of these two classes of systems, by using theoretical descriptions and experimental results available for the other class, or to develop new models and descriptions.

Using the established duality, we predict a novel class of disorder-driven quantum phase transitions in systems with long-range hopping $\propto 1/r^{\gamma}$ with $d/2<\gamma<d$. These transitions are dual to the non-Anderson disorder-driven transitions Syzranov and Radzihovsky (2018) in nodal semimetals and systems with short-range hopping (see Fig. 1). While unconventional disorder-driven transitions have been a target of vigorous studies in the context of Weyl semimetals (see Ref. Syzranov and Radzihovsky (2018) for a review) and of short-range hopping Rodríguez et al. (2003); Malyshev et al. (2004); de Moura et al. (2005); Gärttner et al. (2015); Syzranov et al. (2016) (i.e. decaying faster than $1/r^{d}$), disorder-driven transitions for long-range hopping $\propto 1/r^{\gamma}$, with $\gamma<d$, have not been investigated previously, to the best of our knowledge, and are established here for the first time by means of the duality arguments.

These transitions manifest themselves in the singular behaviour of the density of states (DoS) and other observables. We emphasise that these transitions are not accompanied by localisation or delocalisation of wavefunctions, as all states are expected to be delocalised for long-range hopping Levitov (1989, 1990), and belongs to a universality class different from that Evers and Mirlin (2008); Efetov (1999) of Anderson localisation. These transitions may be realised by means of ultracold ions in optical or magnetic traps Richerme et al. (2014); Islam et al. (2013); Jurcevic et al. (2014, 2015) and also occur for excitations in certain models which are used usually for studying many-body localisation-delocalisation transitions in the presence of power-law hopping Burin and Maksimov (1989); Burin (2006); Gutman et al. (2016).

Duality mapping. To establish the duality, we consider a semimetal with the power-law quasiparticle dispersion $\xi=ak^{\alpha}$ near the node, which has a trivial spin and sublattice structure, in the presence of randomly located short-range impurities, whose potentials may be approximated by delta-functions with amplitudes $\lambda_{n}$; $u({\bf r})=\sum_{n}\lambda_{n}\delta({\bf r}-{\bf r}_{n})$. Our arguments may easily be generalised to the cases of more complicated dispersions, including spin and valley structures, and impurity potentials Sup . In this paper, we focus on single-particle disorder-driven phenomena and use a supersymmetric Efetov (1999) field theory to describe quasiparticle states whose action is given by

[TABLE]

where ${\hat{\bf k}}=-i\partial_{\bf r}$ is the momentum operator; $\psi$ is a four-component supervector with components in the $BF\otimes RA$ (boson-fermion $\otimes$ retarded-advanced) space Efetov (1999); $\Lambda=\mathds{1}_{BF}\otimes\left({\hat{\sigma}}_{z}\right)_{RA}$, and $\bar{\psi}=\psi^{\dagger}\Lambda$.

Values of observables in this semimetal may be represented in the form $\left<{\cal O}\right>=\int D\psi D\psi^{\dagger}\ldots e^{-{\cal L}_{\text{SM}}}$, where $\ldots$ are supersymmetry-breaking terms. Introducing supervectors $\phi({\bf r}_{n})$ at the locations ${\bf r}_{n}$ of the impurities by means of the Habbard-Stratonovich transformation $e^{-i\lambda\bar{\psi}\psi}=\int{\cal D}\phi^{\dagger}{\cal D}\phi e^{i\frac{\bar{\phi}\phi-\lambda\bar{\psi}\phi-\lambda\bar{\phi}\psi}{\lambda-i0\Lambda}}$ and integrating out the fields $\bar{\psi}$ and $\psi$ leads to a field theory with the action

[TABLE]

in terms of the fields $\phi$, where $G_{E}=\int\frac{d{\bf k}}{(2\pi)^{d}}\frac{e^{-i{\bf k}{\bf r}}}{E-a|{\bf k}|^{\alpha}+i0\Lambda}$ is a matrix of the Green’s functions in a disorder-free semimetal.

In the vast majority of nodal semimetals (such as 3D Dirac, Weyl and parabolic semimetals, graphene) the exponent $\alpha$ of the dispersion $\xi_{\bf k}\propto|{\bf k}|^{\alpha}$ is exceeded by the dimensions $d$. Power-law dispersion with $\alpha<d$ may also be realised in 1D and 2D systems of trapped ultracold ions Richerme et al. (2014); Islam et al. (2013); Jurcevic et al. (2014, 2015) and certain other systems, such as 1D plasmons (corresponding to $\alpha=1/2$). In all these systems, the Green’s functions $G_{E}({\bf r})$ at low energies $E$ display power-law decay with distance $r$, which gives the action

[TABLE]

where $j=\frac{\Gamma\left(\frac{d-\alpha}{2}\right)}{2^{\alpha}a\pi^{\frac{d}{2}}\Gamma\left(\frac{\alpha}{2}\right)}$ for the case of the dispersion $\xi_{\bf k}=a|{\bf k}|^{\alpha}$ with a trivial spin and valley structure considered here. For more complicated structures of the dispersion of the semimetal, the first term in the action (3) will still display the power-law decay $\propto 1/|{\bf r}_{n}-{\bf r}_{m}|^{d-\alpha}$, but will involve the respective valley or spin degrees of freedom. For example, in the case of a Weyl semimetal, with the quasiparticle dispersion given by $\xi_{\bf k}=v\hat{\bm{\sigma}}\cdot{\bf k}$, where $\hat{\bm{\sigma}}$ is the pseudospin operator, the system may be described by action (3) with the replacement $j\rightarrow-\frac{1}{8\pi}\frac{\hat{\bm{\sigma}}\cdot({\bf r}_{n}-{\bf r}_{m})}{v|{\bf r}_{n}-{\bf r}_{m}|^{3}}$.

Action (3) describes particle hopping between randomly located sites, with the energies $-\lambda_{n}^{-1}$, where the hopping amplitude decays with distance as the power-law $\propto 1/|{\bf r}_{n}-{\bf r}_{m}|^{d-\alpha}$. The transformation from action (1) to action (3) is, therefore, a duality transformation between the field theories of a disordered nodal semimetal and a systems with power-law hopping in an array of random-energy sites.

This duality may be used to explore novel phenomena in each of these two classes of systems using the theories and experimental results available for the other class. In particular, we predict in what follows a novel disorder-driven quantum phase transition for systems with power-law hopping, which is dual to the disorder-driven transition in semimetals in high dimensions Syzranov and Radzihovsky (2018).

Non-Anderson disorder-driven transitions in systems with long-range hopping. In order to elucidate these novel transitions, we consider a spin-$1/2$ $XY$ model on randomly located sites, where the exchange interaction between the spins depends on the distance between them as the power law. The Hamiltonian of this model, which may be realised by means of trapped ultracold particles, is given by

[TABLE]

where the on-site magnetic fields $-E_{n}$ are random and uncorrelated on different sites and we consider $\gamma<d$.

In general, in the limit of small $j$, spin-flip-type excitations in this model propagate similarly to single particles with the amplitudes which decay with distance as the power law Fleishman and Anderson (1980); Aleiner et al. (2011); Titum et al. (2018); Gutman et al. (2016); Yao et al. (2014). For simplicity, we assume that the on-site field $E_{n}$ fluctuates weakly on top of a large average value $\left<E_{n}\right>_{\text{dis}}$ and, as a result, all the spins are almost entirely polarised in the same direction in the ground state. The propagation of spin-flip excitations may then be mapped exactly onto a single-particle model with the action (3).

In what follows, we demonstrate that the system exhibits a non-Anderson phase transition when changing the coupling amplitude $j$ or the amplitude of the fluctuations of the on-site energies $E_{n}$. This transition may manifest itself in the form of singularities in various physical observables, such as the diffusion coefficient or magnetic susceptibility. In what follows we focus on the behaviour of the DoS of low-energy excitations.

Below, we consider the states of the Hamiltonian (4) with energies $E=\left<E_{n}\right>_{\text{dis}}+\omega$ weakly deviating from the large average on-site field $\left<E_{n}\right>_{\text{dis}}$. The DoS of single spin-flip excitations at energy $\omega$ is given by

[TABLE]

where $s_{R}({\bf r}_{n})$ is the retarded bosonic component of the supervector $\phi({\bf r}_{n})$; $V$ is the volume of the system; ${\cal L}_{\text{Hop}}$ is given by Eq. (3) with $\lambda_{n}^{-1}=-E_{n}+\left<E_{n}\right>_{\text{dis}}$; the term ${\cal L}_{\omega}$ accounts for the effect of the energy $\omega$ on the action; ${\cal L}_{s}(\eta)$ is the supersymmetry-breaking source term; $\eta$ is an infinitesimal parameter.

The full action ${\cal L}={\cal L}_{\text{Hop}}+{\cal L}_{\omega}+{\cal L}_{s}(\eta)$ used to obtain the DoS (5a) thus matches action (3) with the replacement $\lambda_{n}^{-1}\rightarrow-E_{n}+\left<E_{n}\right>_{\text{dis}}+\omega-\eta\cdot\mathds{1}_{BBRR}$, where $\mathds{1}_{BBRR}$ is the projector to the bosonic retarded parts of supervectors. The corresponding dual action of a semimetal is given by Eq. (1) with the same replacement and with $\alpha=d-\gamma$ and $a=\frac{\Gamma\left(\frac{\gamma}{2}\right)}{2^{d-\gamma}\pi^{\frac{d}{2}}\Gamma\left(\frac{d-\gamma}{2}\right)j}$.

Expanding that action to the first order in small parameters $\omega$ and $\eta$ and performing disorder averaging of the DoS, while keeping only the first cumulants in the disorder strength

[TABLE]

gives the effective action of a semimetal

[TABLE]

where $\tilde{s}_{R}^{*}$ and $\tilde{s}_{R}$ are the retarded bosonic components of the supervectors $\bar{\psi}$ and $\psi$. We emphasise that we assume, when deriving Eq. (7), that the variance $\varkappa^{2}/N$ of the quantity $\left(E_{n}-\left<E_{n}\right>\right)^{-1}$ is finite. For example, this quantity may be a random Gaussian variable, in which case the fluctuations of the energies $E_{n}$ are non-Gaussian; the statistics or even the existence of the variance of the energies $E_{n}$ have no bearing on our arguments.

Equation (7) is the action of a disordered semimetal with the quasiparticle dispersion $\xi_{\bf k}=ak^{d-\gamma}$ and with a source term which generates the DoS. According to Eq. (5a) and the action (7), the DoS of low-energy excitations in the model described by the Hamiltonian (4) is given by

[TABLE]

where $\rho_{\text{SM}}(\omega\varkappa)$ is the DoS in the dual semimetal described by the action (7) at energy $\omega\varkappa$ and the disorder strength $\varkappa$ given by Eq. (6).

Semimetals described by the action of the form (7) (Dirac semimetals, high-dimensional semiconductors, quantum kicked rotors, etc.) are known to display phase transitions (or sharp crossovers Note (1)) between strong-disorder and weak-disorder phases at a certain value $\omega_{c}$ of the energy, hereinafter set to zero, and disorder strength.

The relation (8) between the densities of states of a semimetal and a model with power-law hopping indicates that a disordered system with long-range hopping exhibits a critical scaling in the same universality class. However, because the on-site energies $E_{n}$ in the hopping model match the inverse amplitudes $\lambda_{n}^{-1}$ of the impurity potentials in the dual semimetal, the strong- and weak-disorder phases of the semimetal corresponds, respectively, to the weak- and strong-disorder phases of system with long-range hopping. If the disorder strength $\varkappa$ exceeds a critical value $\varkappa_{c}$ in a system with long-range hopping, or the coupling $j$ is smaller than a critical coupling $j_{c}$, the system is in a phase with a suppressed DoS of low-energy excitations. At weaker disorder or larger intersite couplings, the systems exhibits a continuous transition Note (1) to a phase with a larger DoS and better transport of the low-energy excitations.

Using Eq. (8) and the critical scaling proposed in Ref. Kobayashi et al. (2014) for a Dirac semimetal and derived microscopically in Ref. Syzranov et al. (2015b) for a generic semimetal with the power-law quasiparticle dispersion $\xi_{\bf k}\propto k^{\alpha}$, we find that the DoS near the transition in a system with long-range hopping has the form

[TABLE]

where $\Phi(x)$ is a universal scaling function which may be different for different signs of the argument, and $\nu$ and $z$ are the correlation-length and the dynamical critical exponents (matching those of the dual transition in a semimetal Goswami and Chakravarty (2011); Pixley et al. (2016); Sbierski et al. (2016); Liu et al. (2016); Louvet et al. (2017); Malyshev et al. (2004); Syzranov and Radzihovsky (2018)); $\rho_{\text{smooth}}$ is an analytic contribution which comes from the instantons in actions (1) and (3) (“rare-region effects” Wegner (1981); Suslov (1994); Nandkishore et al. (2014); Syzranov et al. (2015b); Pixley et al. (2017); Gurarie (2017); Buchhold et al. (2018a, b)).

In semimetals, the instantonic contribution $\rho_{\text{smooth}}$ to the DoS may be exponentially suppressed by the small deviation $|d-d_{c}|$ of the dimension $d$ from the critical dimension $d_{c}=2\gamma$ of the transition or the number of the particle flavours Syzranov and Radzihovsky (2018). Even in the absence of small parameters, various numerical studies of 3D Weyl and Dirac semimetals have found this contribution to be rather small or unobservable Kobayashi et al. (2014); Sbierski et al. (2014); Pixley et al. (2015); Bera et al. (2016); Liu et al. (2016); Pixley et al. (2016); Sbierski et al. (2016); Wang et al. (2019); Syzranov and Radzihovsky (2018), which allows one to use the DoS, to a good approximation, as an order parameter for the transition. It has also been suggested Note (1) that the instantonic contribution may broaden criticality, thus converting the transition to a sharp crossover. In this paper, however, we do not distinguish between such sharp crossovers and phase transitions. The behaviours of the DoS for various values of the coupling $j$ and the excitation energies $\omega$ near the critical point are summarised in Fig. 2 and follow directly from the scaling form (9) and may also be inferred from the respective results for semimetals Kobayashi et al. (2014); Syzranov and Radzihovsky (2018).

Experimental observation. The DoS of excitations in a spin model with the Hamiltonian 4 may be observed explicitly in experiments on trapped ultracold particles by measuring the spin susceptibility $\chi_{xx}(\omega)=i\sum_{j}\int_{0}^{\infty}\left<\left[{\hat{s}}_{i}^{x}(t),{\hat{s}}_{j}^{x}(0)\right]\right>e^{it(\omega+i0)}dt$. The imaginary part of the susceptibility determines the dissipation in the system due to creating spin-flip excitations and is related to the density of states of the excitations as $\text{Im}\chi_{xx}(\omega)=N\rho(\omega)$.

Describing spin-flip excitations in the spin model with the Hamiltonian (4) by a single-particle model with action (3), considered here, is justified if all the spins are polarised in the ground state by the average value of the on-site magnetic magnetic field. It is also expected often that single-particle descriptions for excitations in the considered spin model are appropriate even if the sign of the magnetic field fluctuates in space, so long as the parameter $j$ is sufficiently small Fleishman and Anderson (1980); Aleiner et al. (2011); Titum et al. (2018); Gutman et al. (2016); Yao et al. (2014). In the latter case, we still expect the existence of disorder-driven phase transitions between the “weak-hopping” (irrelevant ratio of the hopping to disorder amplitudes) phase at small $j$ and a “strong hopping” phase at larger $j$. We leave, however, such models for future studies.

Outlook. Natural further research directions include extending the duality between nodal semimetals and systems with long-range hopping, which we have established here, to interacting systems, more generic and anisotropic dispersions (e.g., in nodal-line and nodal-surface semimetals) and other models of disorder. The new approach of describing semimetals in the dual representation, established here, may also also be used to investigate, for example, rare-region effects and the possibility of other unconventional phase transitions.

Acknowledgements. We are grateful to L. Radzihovsky and B. Sbeirski for useful discussions and feedback on the manuscript. Also, we thank L. Radzihovsky for prior collaboration on related topics. Our work has also been supported by the Hellman Foundation (SVS) and the Faculty Research Grant awarded by the Committee on Research from the University of California, Santa Cruz (SVS).

I Continuous vs. discrete duality transformations

In the main text we focussed, when deriving the duality transformation, on quasiparticles scattered off randomly located $\delta$-impurities and mapped it onto a model of hopping between discrete sites at the locations of the impurities. This duality transformation may be generalised straightforwardly to the case of a random potential $u({\bf r})$ which varies in space continuously.

The value of an observable ${\cal O}$ in a semimetal with the kinetic energy $\xi_{\bf k}$ with a continuous random potential is given by

[TABLE]

where $\ldots$ are the supersymmetry-breaking terms corresponding to this observable. The Hubbard-Stratonovich transformation

[TABLE]

leads to the dual representation

[TABLE]

where for a power-law dispersion $\xi_{\bf k}\sim ak^{\alpha}$ the Green’s function $G({\bf r},{\bf r}^{\prime})$ displays a power-law decay $\propto 1/|{\bf r}-{\bf r}^{\prime}|^{d+\alpha}$ for low energies $E$.

Equations (S1) and (S3) represent a duality mapping between a nodal semimetal in a continuous potential $u({\bf r})$ and a systems with power-lay hopping in the potential $1/u({\bf r})$. In principle, models with continuous random potentials may in general be approximated, respectively, by models with discrete short-ranged impurities and models on discrete sites, considered in the main text, by coarse-graining the potential at sufficiently small length scales.