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# Quantum lattice Boltzmann is a quantum walk

- Sauro Succi
^{1}Email author, - François Fillion-Gourdeau
^{2}and - Silvia Palpacelli
^{3}

**2**:12

https://doi.org/10.1140/epjqt/s40507-015-0025-1

© Succi et al.; licensee Springer. 2015

**Received: **1 August 2014

**Accepted: **30 April 2015

**Published: **16 May 2015

## Abstract

Numerical methods for the 1-D Dirac equation based on operator splitting and on the quantum lattice Boltzmann (QLB) schemes are reviewed. It is shown that these discretizations fall within the class of quantum walks, i.e. discrete maps for complex fields, whose continuum limit delivers Dirac-like relativistic quantum wave equations. The correspondence between the quantum walk dynamics and these numerical schemes is given explicitly, allowing a connection between quantum computations, numerical analysis and lattice Boltzmann methods. The QLB method is then extended to the Dirac equation in curved spaces and it is demonstrated that the quantum walk structure is preserved. Finally, it is argued that the existence of this link between the discretized Dirac equation and quantum walks may be employed to simulate relativistic quantum dynamics on quantum computers.

## Keywords

- quantum walks
- Dirac equation
- lattice Boltzmann
- operator splitting

## 1 Introduction

A quantum walk (QW) is defined as the quantum analogue of a classical random walk, where the ‘quantum walker’ is in a superposition of states instead of being described by a probability distribution. One of the earliest realization of this concept was proposed by Feynman as a discrete version of the massive Dirac equation [1]. In recent times, there has been a surge of interest for this topic, due to the conceptual and possible practical import of QW’s as discrete realizations of stochastic quantum processes and because they can solve certain problems with an exponential speedup, i.e. using exponentially less operations than classical computations [2]. Moreover, QW’s are amenable to a number of experimental realizations, such as ion traps [3–5], liquid-state nuclear-magnetic-resonance quantum-information processor [6], photonic devices [7] and other types of optical devices [8]. As a result, they hold promise of playing an important role in many areas of modern physics and quantum technology, such as quantum computing, foundational quantum mechanics and biophysics [9].

One of the most interesting features of discrete QW’s is their continuum limit, which recovers a broad variety of relativistic quantum wave equations [10–12]. As stated earlier, this was first discussed by Feynman and is now known as Feynman’s checkerboard [13]. This was originally formulated for the free Dirac equation but extensions of these ideas, which include the coupling to external fields, have been investigated [10–12]. Pioneering work have been performed in which the Dirac equation is related to cellular automata [14, 15]. Lately, the link between QW’s and the Dirac equation have been discussed extensively [10–12, 16]. In these studies, the starting point is a general QW formulation from which the continuum limit is evaluated and then related to the Dirac equation.

In this article, QW’s and their relations to some known numerical schemes of the Dirac equation are reviewed from a slightly different perspective: it will be demonstrated that the most general QW’s are obtained from lattice discretizations of the relativistic quantum wave equation for spin-1/2 particles. More precisely, starting from the continuum Dirac equation, it is shown that QW’s can be placed in one-to-one correspondence with numerical schemes based on operator splitting and the QLB scheme. These numerical methods have been developed and employed as efficient numerical tools to solve relativistic quantum mechanics problems on classical computers [17–19]. They have a number of interesting properties: they are easy to code, they can be easily parallelized and are very versatile. Moreover, their mathematical structure and the fact that the time discretization is realized by a set of unitary transformations makes the link to QW’s possible. This connection is explored below and represents one of the main purposes of this article. Much of the material discussed here, in particular the numerical simulations, is not completely new, which is in line with the main purpose of this paper, namely an attempt to bridge the techniques utilized in numerical analysis and (quantum) Lattice Boltzmann theory to the language of quantum computing.

This article is organized as follows. In Section 2, a general formulation of QW is presented, where the transfer matrix is time and space dependent. In Section 3, the split operator method for the Dirac equation is presented, along with its exact correspondence with QW. Section 4 is devoted to the QLB

thod and connections with QW. Section 5 is devoted to a qualitative discussion of the link between these numerical schemes and quantum computation. In Section 6, the schemes are casted in the form of a propagation-relaxation process and the notion of quantum equilibrium is introduced. Based on the analogy between QW and QLB, a new QLB scheme for the \((1+1)\) Dirac equation in curved space is proposed in Section 7. Finally, the generalization of these methods to many dimensions is briefly discussed and numerical results are presented in Section 8.

## 2 Quantum walks

*ψ*), obeying a discrete space-time evolution equations described by the following discrete map [10–12, 20]:

The matrix obeys \(B \in SU(2)\) only when \(\xi_{j,n} = k\pi\) for all *j*, *n*, with \(k \in\mathbb{Z}\). Thus, this formulation is slightly more general than QW’s considered in [10, 11, 20] where \(B \in SU(2)\) is studied. As seen in the next section, the choice \(B \in U(2)\) will be important to have a general connection between mass terms of the Dirac equation and QW’s. Finally, the \(U(2)\) QW can also be implemented on quantum computers because the matrix *B* is a unitary transformation: it represents the most general QW consistent with quantum computations.

In the above, the amplitudes \(\psi_{1,2}\) code for the probability of the quantum walker to move up (down) along the lattice site \(j\in\mathbb{Z}\) at the time step \(n \in\mathbb{N}\). This is a very rich structure, which has been shown to recover a variety of important quantum wave equations, as soon as the Euler angles are allowed to acquire a space-time dependence [12]. In addition, it provides a wealth of potential algorithms for quantum computing. This was studied extensively in [10–12] by analysing the continuum limit of these QW, yielding different versions of the Dirac equation. In this work, the opposite path is taken: it is shown that specific discretizations of the Dirac equation, using either a split-operator approach or the lattice Boltzmann technique, naturally lead to a QW formulation.

## 3 Split-operator and quantum walks

*M*is space and time dependent and may include contributions from the physical mass, the coupling to an electromagnetic potential or any other type of coupling. One requirement, however, is that

*M*is a Hermitian local operator without any derivatives. Generally, it can be written as

*c*being the speed of light) such that the translation is exact on the lattice. Equation (7) can then be written as:

*B*and \(B'\). They are expressed in different representation:

*B*uses the Euler angle parametrization while \(B'\) is expressed in the canonical representation obtained by the exponential mapping of the Lie algebra. The latter is given explicitly by

The operator splitting technique presented here also bears a close relationship with the QLB technique, to which we now turn.

## 4 Quantum lattice Boltzmann, operator splitting and quantum walks

The QLB was inspired by a direct analogy between the way the Dirac equation goes to the Schroedinger equation in the limit \(v/c \to0\), and the way that the Navier-Stokes equations of classical fluid-dynamics emerge from the Boltzmann equation in the limit of small Knudsen number, \(Kn \to0\), where \(Kn=l/L\) is the ratio of the molecular mean free path to the typical macroscopic scale. In both cases, the smallness parameter controls the enslaving of the fast modes to the slow ones: *non-equilibrium* to *equilibrium* for the classical case, versus *excited* states to *ground* state in the quantum one. Of course, the quantum case shows no genuine relaxation since its dynamics is reversible. Yet, enslaving can be interpreted in the sense of fast oscillations around a local quantum equilibrium (Zitterbewegung), which average out once time is coarse-grained on a scale larger than the period of the fast oscillations. So, Zitterbewegung may be regarded as the quantum relativistic analogue of classical non-equilibrium fluctuations.

*real*. To obtain the QLB scheme, it is convenient to write the Dirac equation as

*a*,

*b*are spinor indices and the ‘microscopic velocities’ are given by \(v_{1,2} = \pm1\). This is clearly in a ‘Boltzmann-like’ form with two discrete velocities and a collision term

*M*. When QLB is employed in fluid mechanics to solve the continuum equations of motion, there is a family of possible numbers of discrete velocities for a given lattice [23] and each choice yields a different numerical scheme (for instance, the 9 velocities scheme in 2-D and the 27 velocities scheme in 3-D are popular choices on square lattices [24]). For the Dirac equation, this choice is dictated by the mathematical structure of the equation.

### 4.1 An explicit example: the free case

*m*.

## 5 Prospects for quantum simulation

The latter would be particularly useful for the study of relativistic quantum systems where a time-dependent solution of the Dirac equation is required, such as in very high intensity laser physics [29] or graphene physics [30].

Another subject of major interest for future research is the extension of the QLB methodology to quantum many body systems and quantum field theory, two paramount sectors of modern physics which are particularly exposed to the limitations of classical (non-quantum) electronic computing. Progress in this direction depends on the ability to replace the quantum wavefunction by the corresponding second-quantized quantum operators, and show that the dynamics of the second-quantized QLB scheme still preserves the appropriate equal-time commutation relations. Preliminary efforts along this line have been developed in [31] in \(1+1\) dimensions. Extensions to strongly non-linear field theories in \(d>1\) remain to be explored. As to quantum-many body problems, LB-like methods have been recently adapted to electronic structure simulations [32]. In this work, a classical LB scheme is employed to solve the Kohn-Sham equations of density functional theory in the form of diffusion-reaction equations in imaginary time. Allied QLB schemes could prove very useful to solve the corresponding real-time quantum many-body transport problems within the framework of time-dependent density functional theory.

Finally, we wish to point out the intriguing possibility of realizing both quantum and classical LB schemes on quantum analogue simulators, as recently explored in [33].

## 6 Quantum equilibria

In view of quantum computing implementations, it is of interest to cast the Dirac equation in the form of a propagation-relaxation process, where the collision matrix is now interpreted as a scattering process, relaxing the spinor component around a local quantum equilibrium.

*U*is a unitary matrix that depends on

*M*. This transformation is chosen to recast the Dirac equation in relaxation form:

*U*, Ω is not unique but a convenient choice is \(U = e^{iM \tau}\).

*ψ*into the post-collisional state \(\psi'\), and the streaming gate moves the post-collisional spinor to its destination location \(z \pm\Delta z\). Both operations are unitary and can be encoded in logical gates for quantum computing purposes [34, 35].

The expression (40) shows that the local equilibria are linear functions of the actual wavefunction *ψ*, hence itself a function of space and time.

The question is: will the actual wavefunctions ever reach this moving target, i.e., \(\psi= \psi_{\mathrm{eq}}\)?

*M*, namely:

This means that the spinorial wavefunction is a superposition of (both slow and fast) zero-average oscillations around a local equilibrium which, consistently with the reversible nature of quantum mechanics, is actually never attained.

Although this remains to be checked in detail, we conjecture that the same holds true for the case of a massive particle in an external potential, because in this case the Dirac equation is still linear.

Based on (44), the condition for the local equilibrium to depart from a trivial vacuum is that the matrix *M* be singular, i.e. the local equilibrium is a zero-mode of the scattering matrix.

Non-trivial quantum zero-modes, \(\psi_{\mathrm{eq}} \neq0\), may indeed arise for *non-linear* quantum wave equations, such as the Gross-Pitaevski or the mean-field version of the Nambu-Jona-Lasinio model, to be dicussed shortly. A non-trivial local equilibrium would then signal a spontaneously broken symmetry, which is indeed the distinctive trait of the aforementioned non-linear quantum wave equations.

Even though the notion of quantum equilibrium remains purely formal in nature, it is argued that it might nonetheless facilitate quantum computing implementations based on the compact expressions (42) and (43). This stands out a very interesting topic for future research.

## 7 QLB in curved space-time

Quantum walks have been shown to map into Dirac-like equations in curved space as well by evaluating the continuum limit of certain QW’s [12, 36]. Here, in the same spirit as other sections, a QW structure is obtained by discretizing the Dirac equation in curved space time using a QLB-like approach. However, because the wave function propagates on a curved manifold, the structure of the resulting QW is different from Eq. (1) and should include a residency matrix that corrects the streaming step, which is strictly valid only in flat space. This is different from the result obtained in [12, 36].

*j*, to the north and south boundaries at \(j \pm1/2\), respectively. Following a common practice in finite-volume formulations of hyperbolic problems, the flux terms are approximated by [39]:

*c*and the spinors at \((j,n+1)\) are connected to the corresponding spinors at \((j \pm1,n)\) by a local \(2 \times2\) matrix, which can be readily inverted to deliver a fully explicit map. However, when \(A_{j}/c \ne1\), the spinors at \((j,n)\) also enter the map, so that local inversion delivers a slightly more elaborated structure, namely:

*T*is the local \(2 \times2\) transfer matrix including collisions,

*S*is the streaming operator and

*R*the local residency matrix, expressing the fraction of spinors which are left in the cell centered about

*z*as the quantum system advances from

*t*to \(t+\Delta t\). This is depicted in Figure 1.

Clearly, the residency matrix vanishes in the case of a uniform mesh, i.e. no gravity. The mapping (58) represents the ‘gravitational’ QLB. The detailed expressions of the streaming and residency matrices depend on the specific form of the metric tensor and associated vierbeins. Moreover, this analysis concentrates on the mathematical structure (streaming and collision steps) of the resulting scheme rather than on its numerical properties (convergence, stability, etc.). These topics shall make the object of a future publication.

## 8 Multi-dimensions

The discretization presented in this work extends to the \(D+1\) dimensional case by applying the notion of operator splitting. This implies the inclusion of a new dynamic step which is entirely quantum: namely a ‘rotation’, designed so as to keep the spin aligned with the momentum along each of the three spatial directions. Schemes using this strategy can be found in [18, 19] and in [17] for higher order splittings.

It might be that such rotation is not needed by formulating the Dirac equation as a random walk on other lattice with more natural topologies (the diamond) lattice. The QLB-QW equivalence in multi-dimensions will be discussed in a future publication. However, to demonstrate the strength of the numerical schemes presented here and to show some possible applications for quantum computing, numerical results in 2-D are presented in the following.

### 8.1 Numerical results

As an example of possible applications of QLB scheme, we present two representative simulations: Klein tunnelling in the presence of random impurities and Dirac equation with Nambu-Jona-Lasinio (NJL) interactions in \(2+1\) space-time dimensions. Details on the numerical methods used to obtain these results are given in [18, 19, 40]. Also, these results are not completely new as similar systems have been studied in [19, 41].

#### 8.1.1 Graphene with random impurities

From Figure 2, we can see that the wave packet is scattered by the impurities, giving rise to a plane front out of the initial Gaussian configuration. As a consequence of the randomness induced in the wave function by the disordered media, there is a momentum loss and therefore the motion of the wave packet is found to experience a corresponding slow down. It is also found that the wave packet takes more time to regroup as the impurity concentration and impurity potential are increased.

#### 8.1.2 Nambu-Jona-Lasinio interaction

*g*, reads as follows

This model represents a paradigm for dynamic mass acquisition via spontaneous symmetry breaking due to the non-linear interactions.

*σ*. Let \(k_{z}\) and \(k_{y}\) be the initial energy of the wave packet and impose the following initial condition:

A grid size of \(N_{z} \times N_{y} = 1\mbox{,}024^{2}\) elements is used and the initial wave packet spread is set at \(\sigma=48\), a fully relativistic particle (\(m=0\)) is considered.

In these simulations, we impose \(g=0\) and \(g=1\mbox{,}000\) and vary the initial energy of the wave packet \(k\equiv k_{z} = k_{y}\) in order to inspect the effect of this parameter on the wave packet separation, which, in turn, informs on the effective mass acquired by the up and down propagating modes.

*g*. A similar analysis in two spatial dimensions remains to be developed.

## 9 Summary and outlook

Summarizing, we have reviewed discretizations of the Dirac equation and described their mapping into QW’s. These relations may allow the solution of the Dirac equation on quantum computers. In the first part, a general argument is given, using the operator splitting method. Then, the QLB scheme is studied within the same perspective and a similar relation is found. We have also shown that a similar structure remains in curved space, using a scheme based on a finite volume formulation, with the important caveat that the exact nature of the streaming operator, typical of QLB, is no longer preserved. Rather, one sees the appearance of the residency matrix, which characterizes the fraction of spinor which is left in the cell after one step in the time evolution. This scheme, along with its generalization to many dimensions, will be studied in future work.

## Declarations

### Acknowledgements

One of the authors (SS) is very grateful to F Debbasch for introducing him to the notion of quantum walks. He also wishes to express gratitude to Marcelo Alejandro Forets for organizing a very informative workshop on Relativistic Quantum Walks, where the ideas behind this paper have been first drafted out.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Feynman RP, Hibbs AR. Quantum mechanics and path integrals. Int. series in pure and applied physics. New York: McGraw-Hill; 1965. MATHGoogle Scholar
- Childs AM, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman DA. Exponential algorithmic speedup by a quantum walk. In: Proceedings of the thirty-fifth annual ACM symposium on theory of computing. New York: ACM; 2003. p. 59-68. Google Scholar
- Schmitz H, Matjeschk R, Schneider C, Glueckert J, Enderlein M, Huber T, Schaetz T. Quantum walk of a trapped ion in phase space. Phys Rev Lett. 2009;103:090504. ADSView ArticleGoogle Scholar
- Zähringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R, Roos CF. Realization of a quantum walk with one and two trapped ions. Phys Rev Lett. 2010;104:100503. View ArticleGoogle Scholar
- Gerritsma R, Kirchmair G, Zaehringer F, Solano E, Blatt R, Roos C. Quantum simulation of the Dirac equation. Nature. 2010;463(7277):68-71. ADSView ArticleGoogle Scholar
- Ryan CA, Laforest M, Boileau JC, Laflamme R. Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor. Phys Rev A. 2005;72:062317. ADSView ArticleGoogle Scholar
- Aspuru-Guzik A, Walther P. Photonic quantum simulators. Nat Phys. 2012;8(4):285-91. View ArticleGoogle Scholar
- Engel GS, Calhoun TR, Read EL, Ahn T-K, Mančal T, Cheng Y-C, Blankenship RE, Fleming GR. Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature. 2007;446(7137):782-6. ADSView ArticleGoogle Scholar
- Giulini D, Joos E, Kiefer C, Kupsch J, Stamatescu I-O, Zeh HD. Decoherence and the appearance of a classical world in quantum theory. Berlin: Springer; 2003. MATHGoogle Scholar
- Aharonov Y, Davidovich L, Zagury N. Quantum random walks. Phys Rev A. 1993;48:1687-90. ADSView ArticleGoogle Scholar
- Di Molfetta G, Debbasch F. Discrete-time quantum walks: continuous limit and symmetries. J Math Phys. 2012;53(12):123302. ADSMathSciNetView ArticleMATHGoogle Scholar
- Di Molfetta G, Brachet M, Debbasch F. Quantum walks in artificial electric and gravitational fields. Phys A, Stat Mech Appl. 2014;397(0):157-68. MathSciNetView ArticleGoogle Scholar
- Gersch HA. Feynman’s relativistic chessboard as an Ising model. Int J Theor Phys. 1981;20(7):491-501. View ArticleGoogle Scholar
- Bialynicki-Birula I. Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata. Phys Rev D. 1994;49:6920-7. ADSMathSciNetView ArticleGoogle Scholar
- Meyer D. From quantum cellular automata to quantum lattice gases. J Stat Phys. 1996;85(5-6):551-74. ADSMathSciNetView ArticleMATHGoogle Scholar
- Arrighi P, Nesme V, Forets M. The Dirac equation as a quantum walk: higher dimensions, observational convergence. J Phys A, Math Theor. 2014;47(46):465302. ADSMathSciNetView ArticleMATHGoogle Scholar
- Fillion-Gourdeau F, Lorin E, Bandrauk AD. Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling. Comput Phys Commun. 2012;183(7):1403-15. ADSMathSciNetView ArticleMATHGoogle Scholar
- Succi S, Benzi R. Lattice Boltzmann equation for quantum mechanics. Phys D, Nonlinear Phenom. 1993;69(3-4):327-32. ADSMathSciNetView ArticleMATHGoogle Scholar
- Fillion-Gourdeau F, Herrmann HJ, Mendoza M, Palpacelli S, Succi S. Formal analogy between the Dirac equation in its Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. Phys Rev Lett. 2013;111:160602. ADSView ArticleGoogle Scholar
- Chandrashekar CM, Srikanth R, Laflamme R. Optimizing the discrete time quantum walk using a \(\operatorname{su}(2)\) coin. Phys Rev A. 2008;77:032326. ADSView ArticleGoogle Scholar
- Lorin E, Bandrauk A. A simple and accurate mixed solver for the Maxwell-Dirac equations. Nonlinear Anal, Real World Appl. 2011;12(1):190-202. MathSciNetView ArticleMATHGoogle Scholar
- Gilmore R. Lie groups, Lie algebras, and some of their applications. New York: Dover; 2012. MATHGoogle Scholar
- Qian YH, D’Humières D, Lallemand P. Lattice BGK models for Navier-Stokes equation. Europhys Lett. 1992;17(6):479. ADSView ArticleMATHGoogle Scholar
- He X, Luo L-S. Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys Rev E. 1997;56:6811-7. ADSView ArticleGoogle Scholar
- Succi S. The Lattice-Boltzmann equation. Oxford: Oxford University Press; 2001. MATHGoogle Scholar
- Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annu Rev Fluid Mech. 1998;30(1):329-64. ADSMathSciNetView ArticleGoogle Scholar
- Succi S. Kinetic approach to lattice quantum mechanics. In: Cellular automata. Berlin: Springer; 2002. p. 114-22. View ArticleGoogle Scholar
- Kaye P, Mosca M, Laflamme R. An introduction to quantum computing. Oxford: Oxford University Press; 2006. MATHGoogle Scholar
- Di Piazza A, Müller C, Hatsagortsyan KZ, Keitel CH. Extremely high-intensity laser interactions with fundamental quantum systems. Rev Mod Phys. 2012;84:1177-228. ADSView ArticleGoogle Scholar
- Geim AK, Novoselov KS. The rise of graphene. Nat Mater. 2007;6(3):183-91. ADSView ArticleGoogle Scholar
- Succi S. Lattice Boltzmann method for quantum field theory. J Phys A, Math Theor. 2007;40(26):559. ADSMathSciNetView ArticleMATHGoogle Scholar
- Mendoza M, Succi S, Herrmann HJ. Kinetic formulation of the Kohn-Sham equations for ab initio electronic structure calculations. Phys Rev Lett. 2014;113:096402. ADSView ArticleGoogle Scholar
- Mezzacapo A, Sanz M, Lamata L, Egusquiza I, Succi S, Solano E. Quantum simulator for transport phenomena in fluid flows. arXiv preprint, arXiv:1502.00515 (2015).
- Meyer D. From quantum cellular automata to quantum lattice gases. J Stat Phys. 1996;85(5-6):551-74. ADSMathSciNetView ArticleMATHGoogle Scholar
- Boghosian BM, Taylor W IV. Simulating quantum mechanics on a quantum computer. Phys D, Nonlinear Phenom. 1998;120(1-2):30-42. ADSMathSciNetView ArticleMATHGoogle Scholar
- Di Molfetta G, Brachet M, Debbasch F. Quantum walks as massless Dirac fermions in curved space-time. Phys Rev A. 2013;88:042301. ADSView ArticleGoogle Scholar
- Ubertini S, Bella G, Succi S. Lattice Boltzmann method on unstructured grids: further developments. Phys Rev E. 2003;68:016701. ADSMathSciNetView ArticleGoogle Scholar
- Nannelli F, Succi S. The lattice Boltzmann equation on irregular lattices. J Stat Phys. 1992;68(3-4):401-7. ADSMathSciNetView ArticleMATHGoogle Scholar
- LeVeque RJ. Finite volume methods for hyperbolic problems. vol. 31. Cambridge: Cambridge University Press; 2002. View ArticleMATHGoogle Scholar
- Palpacelli S, Romatschke P, Succi S. One-dimensional quantum lattice Boltzmann scheme for the nonlinear Dirac equation. Int J Mod Phys C. 2013;24(12):1340001. ADSMathSciNetView ArticleGoogle Scholar
- Palpacelli S, Mendoza M, Herrmann H, Succi S. Klein tunneling in the presence of random impurities. Int J Mod Phys C. 2012;23(12):1250080. ADSView ArticleGoogle Scholar
- Nambu Y, Jona-Lasinio G. Dynamical model of elementary particles based on an analogy with superconductivity. I. Phys Rev. 1961;122:345-58. ADSView ArticleGoogle Scholar