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arXiv:1404.2070 · International Journal of Modern Physics B

Quantum non-Abelian hydrodynamics: Anyonic or spin–orbital entangled liquids, nonunitarity of scattering matrix and charge fractionalization

Tribhuvan Prasad Pareek

Published Sep 30, 2014• 0 citations
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Abstract

In this article, we develop an exact (nonadiabatic, nonperturbative) density matrix scattering theory for a two component quantum liquid which interacts or scatters off from a generic spin-dependent quantum potential. The generic spin dependent quantum potential [Eq. (1)] is a matrix potential, hence, adiabaticity criterion is ill-defined. Therefore the full matrix potential should be treated nonadiabatically. We succeed in doing so using the notion of vectorial matrices which allows us to obtain an exact analytical expression for the scattered density matrix (SDM), ϱsc[Eq. (30)]. We find that the number or charge density in scattered fluid, Tr (ϱsc), expressions in Eqs. (32) depends on nontrivial quantum interference coefficients, [Formula: see text], which arises due to quantum interference between spin-independent and spin-dependent scattering amplitudes and among spin-dependent scattering amplitudes. Further it is shown that Tr (ϱsc) can be expressed in a compact form [Eq. (39)] where the effect of quantum interference coefficients can be included using a vector Qαβ, which allows us to define a vector order parameterQ. Since the number density is obtained using an exact scattered density matrix, therefore, we do not need to prove that Q is non-zero. However, for sake of completeness, we make detailed mathematical analysis for the conditions under which the vector order parameterQ would be zero or nonzero. We find that in presence of spin-dependent interaction the vector order parameterQ is necessarily nonzero and is related to the commutator and anti-commutator of scattering matrix S with its dagger S†[Eq. (78)]. It is further shown that Q≠0, implies four physically equivalent conditions,i.e., spin-orbital entanglement is nonzero, non-Abelian scattering phase, i.e., matrices, scattering matrix is nonunitary and the broken time reversal symmetry for SDM. This also implies that quasi particle excitation are anyonic in nature, hence, charge fractionalization is a natural consequence. This aspect has also been discussed from the perspective of number or charge density conservation, which implies i.e., Tr (ϱ}sc) = Tr (ϱin). On the other hand Q = 0 turns out to be a mathematically forced unphysical solution in presence of spin-dependent potential or scattering which is equivalent to Abelian hydrodynamics, unitary scattering matrix, absence of spin-space entanglement and preserved time reversal symmetry. We have formulated the theory using mesoscopic language, specifically, we have considered two terminal systems connected to spin-dependent scattering region, which is equivalent to having two potential wells separated by a generic spin-dependent potential barrier. The formulation using mesoscopic language is practically useful because it leads directly to the measured quantities such as conductance and spin-polarization density in the leads, however, the presented formulation is not limited to the mesoscopic system only, its generality has been stressed at various places in this article.

DOI
10.1142/s0217979214502415

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In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function $h(k)$ and direct correlation function $c(k)$. The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.

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Exact Deconfined Gauge Structures in the Higher-Spin Yao-Lee Model: A Quantum Spin-Orbital Liquid with Spin Fractionalization and Non-Abelian Anyons

Zhengzhi Wu, J. X. Zhang, Hong Yao

The nonintegrable higher spin Kitaev honeycomb model has an exact Z_{2} gauge structure, which exclusively identifies quantum spin liquid in the half-integer spin Kitaev model. But its constraints for the integer-spin Kitaev model are much limited, and even trivially gapped insulators cannot be excluded. The physical implications of exact Z_{2} gauge structure, especially Z_{2} fluxes, in integer-spin models remain largely unexplored. In this Letter, we theoretically show that a spin-S Yao-Lee model [a spin-orbital model with SU(2) spin-rotation symmetry] possesses a topologically nontrivial quantum spin-orbital liquid ground state for any spin (both integer and half-integer spin) by constructing exact deconfined fermionic Z_{2} gauge charges. We further show that the conserved Z_{2} flux can also demonstrate the intriguing spin fractionalization phenomena in the non-Abelian topological order phase of the spin-1 Yao-Lee model. Its deconfined Z_{2} vortex excitation carries fractionalized spin-1/2 quantum number in the low-energy subspace, which is also a non-Abelian anyon. Our exact manifestation of spin fractionalization in an integer-spin model is rather rare in previous studies, and is absent in the Kitaev honeycomb model.

arXiv:2404.07261•Physical Review Letters•Dec 5, 2024•5 citations

Entanglement dynamics after spatially inhomogeneous quenches.

ANDREA DI PASQUALE

In recent years the out-of-equilibrium dynamics of many-body quantum systems has attracted growing interest, both for its fundamental implications in the understanding of quantum thermalization and for potential applications in quantum information and quantum computation. In particular, quantum quenches provide a natural setting to investigate the mechanisms of correlation propagation and entanglement growth far from equilibrium. Interest in these phenomena stems not only from the possibility of testing theoretical predictions coming from emergent hydrodynamic descriptions, but also from understanding whether, and to what extent, such descriptions remain valid in the presence of spatial gradients or interfaces. Two paradigmatic frameworks in this context are the Generalized Hydrodynamics (GHD) and quasi-particle picture. The former is an effective theory devised to describe the long-time evolution after a quench in the thermodynamic limit. 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The time evolution of Gamma is studied in the Heisenberg picture, where unitary dynamics correspond to an orthogonal rotation of Gamma implemented by an orthogonal matrix determined by the Kitaev Hamiltonian in the Majorana basis. Consequently, the entanglement entropy dynamics following a quench, both homogeneous and inhomogeneous, can be simulated through simple matrix products, making the approach numerically efficient even for sizable systems. The evolution code for Gamma was validated on a homogeneous quench with periodic boundary conditions: subsystem-size (l) curves exhibit the expected finite-size scaling, (Δ \propto 1/l), and numerical growth rates in the linear regime (t < v/2l), extracted by linear fits at fixed l, match the quasi-particle predictions for various values of the new chemical potential. The rate discrepancy behaves asymptotically as (Δ_{rate}} \propto 1/t_{\min}\) for large (t_{\min}\), up to oscillations from sub-extensive terms. 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•Dec 4, 2025•0 citations

Non-Abelian gauge fields

Fabrice Gerbier, Nathan Goldman, Maciej Lewenstein +1 more

Building a universal quantum computer is a central goal of emerging quantum technologies, which has the potential to revolutionize science and technology. Unfortunately, this future does not seem to be very close at hand. However, quantum computers built for a special purpose, i.e. quantum simulators , are currently developed in many leading laboratories. Many schemes for quantum simulation have been proposed and realized using, e.g., ultracold atoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions in arrays of traps, quantum dots, photonic networks, or superconducting circuits. The progress in experimental implementations is more than spectacular. Particularly interesting are those systems that simulate quantum matter evolving in the presence of gauge fields. In the quantum simulation framework, the generated (synthetic) gauge fields may be Abelian, in which case they are the direct analogues of the vector potentials commonly associated with magnetic fields. In condensed matter physics, strong magnetic fields lead to a plethora of fascinating phenomena, among which the most paradigmatic is perhaps the quantum Hall effect. The standard Hall effect consists in the appearance of a transverse current, when a longitudinal voltage difference is applied to a conducting sample. For quasi-two-dimensional semiconductors at low temperatures placed in very strong magnetic fields, the transverse conductivity, the ratio between the transverse current and the applied voltage, exhibits perfect and robust quantization, independent for instance of the material or of its geometry. Such an integer quantum Hall effect, is now understood as a deep consequence of underlying topological order. Although such a system is an insulator in the bulk, it supports topologically robust edge excitations which carry the Hall current. The robustness of these chiral excitations against backscattering explains the universality of the quantum Hall effect. Another interesting and related effect, which arises from the interplay between strong magnetic field and lattice potentials, is the famous Hofstadter butterfly: the energy spectrum of a single particle moving on a lattice and subjected to a strong magnetic field displays a beautiful fractal structure as a function of the magnetic flux penetrating each elementary plaquette of the lattice. When the effects of interparticle interactions become dominant, two-dimensional gases of electrons exhibit even more exotic behaviour leading to the fractional quantum Hall effect. In certain conditions such a strongly interacting electron gas may form a highly correlated state of matter, the prototypical example being the celebrated Laughlin quantum liquid. Even more fascinating is the behaviour of bulk excitations (quasi-hole and quasi-particles): they are neither fermionic nor bosonic, but rather behave as anyons with fractional statistics intermediate between the two. Moreover, for some specific filling factors (ratio between the electronic density and the flux density), these anyons are proven to have an internal structure (several components) and non-Abelian braiding properties. Many of the above statements concern theoretical predictions—they have never been observed in condensed matter systems. For instance, the fractional values of the Hall conductance is seen as a direct consequence of the fractional statistics, but to date direct observation of anyons has not been possible in two-dimensional semiconductors. Realizing these predictions in experiments with atoms, ions, photons etc, which potentially allow the experimentalist to perform measurements complementary to those made in condensed matter systems, is thus highly desirable! Non-Abelian gauge fields couple the motional states of the particles to their internal degrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons, etc). In this sense external non-Abelian fields extend the concept of spin–orbit coupling (Rashba and Dresselhaus couplings), familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating phenomena such as the quantum spin Hall effect, three-dimensional topological insulators, topological superconductors and superfluids of various kinds. One also expects here the appearance of excitations in a form of topological edge states that can support robust transport, or entangled Majorana fermions in the case of topological superconductors or superfluids. Again, while many kinds of topological insulators have been realized in condensed matter systems, a controlled way of creating them in AMO systems and studying quantum phase transitions between various kinds of them is obviously very appealing and challenging. The various systems listed so far correspond to static gauge fields, which are externally imposed by the experimentalists. Even more fascinating is the possibility of generating synthetically dynamical gauge fields, i.e. gauge fields that evolve in time according to an interacting gauge theory, e.g., a full lattice gauge theory (LGT). These dynamical gauge fields can also couple to matter fields, allowing the quantum simulation of such complex systems (notoriously hard to simulate using 'traditional' computers), which are particularly relevant for modern high-energy physics. So far, most of the theoretical proposals concern the simulation of Abelian gauge theories, however, several groups have recently proposed extensions to the non-Abelian scenarios. The scope of the present focused issue of Journal of Physics B is to cover all of these developments, with particular emphasis on the non-Abelian gauge fields. The 14 papers in this issue include contributions from the leading theory groups working in this field; we believe that this collection will provide the reference set for quantum simulations of gauge fields. Although the special issue contains exclusively theoretical proposals and studies, it should be stressed that the progress in experimental studies of artificial Abelian and non-Abelian gauge fields in recent years has been simply spectacular. Multiple leading groups are working on this subject and have already obtained a lot of seminal results. The papers in the special issue are ordered according to the date of acceptance. The issue opens with a review article by Zhou et al [1] on unconventional states of bosons with synthetic spin–orbit coupling. Next, the paper by Maldonado-Mundo et al [2] studies ultracold Fermi gases with artificial Rashba spin–orbit coupling in a 2D gas. Anderson and Charles [3], in contrast, discuss a three-dimensional spin–orbit coupling in a trap. Orth et al [4] investigate correlated topological phases and exotic magnetism with ultracold fermions, again in the presence of artificial gauge fields. The paper of Nascimbène [5] does not address the synthetic gauge fields directly, but describes an experimental proposal for realizing one-dimensional topological superfluids with ultracold atomic gases; obviously, this problem is well situated in the general and growing field of topological superfluids, in particular those realized in the presence of non-Abelian gauge fields/spin–orbit coupling. Graß et al [6] consider in their paper fractional quantum Hall states of a Bose gas with spin–orbit coupling induced by a laser. Particular attention is drawn here to the possibility of realizing states with non-Abelian anyonic excitations. Zheng et al [7] study properties of Bose gases with Raman-induced spin–orbit coupling. Kiffner et al [8] in their paper touch on another kind of system, namely ultracold Rydberg atoms. In particular they study the generation of Abelian and non-Abelian gauge fields in dipole–dipole interacting Rydberg atoms. The behaviour of fermions in synthetic non-Abelian gauge potentials is discussed by Shenoy and Vyasanakere [9]. The paper starts with the study of Rashbon condensates (i.e. Bose condensates in the presence of Rashba coupling) and also introduces novel kinds of exotic Hamiltonians. Goldman et al [10] propose a concrete setup for realizing arbitrary non-Abelian gauge potentials in optical square lattices; they discuss how such synthetic gauge fields can be exploited to generate Chern insulators. Zygelman [11], similarly as Kiffner et al [8], discusses in his paper non-Abelian gauge fields in Rydberg systems. Marchukov et al [12] return to the subject of spin–orbit coupling, and investigate spectral gaps of spin–orbit coupled particles in the realistic situations of deformed traps. The last two papers, in contrast, are devoted to different subjects. Edmonds et al [13] consider a 'dynamical' density-dependent gauge potential, and study the Josephson effect in a Bose–Einstein condensate subject to such a potential. Last, but not least, Mazzucchi et al [14] study the properties of semimetal-superfluid quantum phase transitions in 3D lattices with Dirac points. References [1] Zhou X, Li Y, Cai Z and Wu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001 [2] Maldonado-Mundo D, Öhberg P and Valiente M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134002 [3] Anderson B M and Clark C W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003 [4] Orth P P, Cocks D, Rachel S, Buchhold M, Le Hur K and Hofstetter W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134004 [5] Nascimbène S 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134005 [6] Graß T, Juliá-Díaz B, Burrello M and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134006 [7] Zheng W, Yu Z-Q, Cui X and Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007 [8] Kiffner M, Li W and Jaksch D 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134008 [9] Shenoy V B and Vyasanakere J P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134009 [10] Goldman N, Gerbier F and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134010 [11] Zygelman B 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134011 [12] Marchukov O V, Volosniev A G, Fedorov D V, Jensen A S and Zinner N T 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134012 [13] Edmonds M J, Valiente M and Öhberg P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134013 [14] Mazzucchi G, Lepori L and Trombettoni A 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134014

Journal of Physics B Atomic Molecular and Optical Physics•Jun 24, 2013•11 citations

Fractionalization of Itinerant Anyons in One-Dimensional Chains

Didier Poilblanc, Matthias Troyer, Eddy Ardonne +1 more

We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing, in particular, on itinerant Ising anyon chains, and derive effective anyonic t-J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom--a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization.

arXiv:1112.5950•Physical Review Letters•May 14, 2012•13 citations
In this article, we develop an exact (nonadiabatic, nonperturbative) density matrix scattering theory for a two component quantum liquid which interacts or scatters off from a generic spin-dependent quantum potential. The generic spin dependent quantum potential [Eq. (1)] is a matrix potential, hence, adiabaticity criterion is ill-defined. Therefore the full matrix potential should be treated nonadiabatically. We succeed in doing so using the notion of vectorial matrices which allows us to obtain an exact analytical expression for the scattered density matrix (SDM), ϱsc[Eq. (30)]. We find that the number or charge density in scattered fluid, Tr (ϱsc), expressions in Eqs. (32) depends on nontrivial quantum interference coefficients, [Formula: see text], which arises due to quantum interference between spin-independent and spin-dependent scattering amplitudes and among spin-dependent scattering amplitudes. Further it is shown that Tr (ϱsc) can be expressed in a compact form [Eq. (39)] where the effect of quantum interference coefficients can be included using a vector Qαβ, which allows us to define a vector order parameterQ. Since the number density is obtained using an exact scattered density matrix, therefore, we do not need to prove that Q is non-zero. However, for sake of completeness, we make detailed mathematical analysis for the conditions under which the vector order parameterQ would be zero or nonzero. We find that in presence of spin-dependent interaction the vector order parameterQ is necessarily nonzero and is related to the commutator and anti-commutator of scattering matrix S with its dagger S†[Eq. (78)]. It is further shown that Q≠0, implies four physically equivalent conditions,i.e., spin-orbital entanglement is nonzero, non-Abelian scattering phase, i.e., matrices, scattering matrix is nonunitary and the broken time reversal symmetry for SDM. This also implies that quasi particle excitation are anyonic in nature, hence, charge fractionalization is a natural consequence. This aspect has also been discussed from the perspective of number or charge density conservation, which implies i.e., Tr (ϱ}sc) = Tr (ϱin). On the other hand Q = 0 turns out to be a mathematically forced unphysical solution in presence of spin-dependent potential or scattering which is equivalent to Abelian hydrodynamics, unitary scattering matrix, absence of spin-space entanglement and preserved time reversal symmetry. We have formulated the theory using mesoscopic language, specifically, we have considered two terminal systems connected to spin-dependent scattering region, which is equivalent to having two potential wells separated by a generic spin-dependent potential barrier. The formulation using mesoscopic language is practically useful because it leads directly to the measured quantities such as conductance and spin-polarization density in the leads, however, the presented formulation is not limited to the mesoscopic system only, its generality has been stressed at various places in this article.
In the linear-response regime, a normal Bose or Fermi fluid can be described by an exact kinetic equation whose kernel is nonlocal in space and time. We derive a general expression for this kernel and evaluate it explicitly to second order in the interparticle potential. The result is a wave-vector- and frequency-dependent generalization of the linear Uehling-Uhlenbeck kernel with the Born-approximation cross section. Our theory is developed in terms of a second-quantized form of the Wigner representation. Convenient expressions are obtained for the commutators and anticommutators of the phase-space density operators, and the equilibrium averages of these operators are analyzed in terms of momentum-dependent generalizations of the classical pair distribution function h(k)h(k)h(k) and direct correlation function c(k)c(k)c(k). The central quantity in this study is a two-particle equilibrium correlation function, the phase-space density-density anticommutator, whose Fourier transform $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ gives the symmetrized scattering function $S(k\ensuremath{\omega})$ by integration over the momenta. The kinetic equation is obtained by a formal closure of the quantum BBGKY hierarchy, with the nonlocal kernel expressed in terms of correlation functions involving two, three, and four particles. We show that our method for approximating the kernel and initial condition by a second-order expansion preserves all the sum rules of $S(k\ensuremath{\omega}p{p}^{\ensuremath{'}})$ to the same order and that the result satisfies the appropriate positivity and symmetry conditions.
In non-diffusive conduction regimes of strongly correlated quantum electron systems, electromagnetic perturbations simultaneously probe the electronic dynamics in time and space: the exchanged energy ℏω\hbar ωℏω excites retarded, i.e., frequency-dependent, many-body interactions, while the probing spatial modulation renders the response spatially nonlocal, i.e., dependent on the external wave vector q⃗\vec{q}q​. This work derives the nonlocal electrodynamic response of such dynamical quantum fluids assuming local but frequency-dependent self-energies and particle-hole irreducible two-particle vertex functions, and preserving charge/mass conservation. The latter is ensured by Bethe-Salpeter equations for renormalized interaction vertices, entering the Kubo formalism for two-particle correlation functions (e.g., for density, currents, momentum, stress). Within such a framework, exact symmetry criteria for the absence of vertex corrections are inferred. In particular, vertex corrections vanish at q=0q=0q=0 for single-particle dispersions that are even and bare interaction vertices that are odd with respect to specific momentum-space point group transformations, including inversion for vector vertices, and mirror reflections or two- or higher-fold rotations for tensor vertices. If the dispersion is isotropic, vertex corrections vanish from the transverse part (⊥q⃗\perp\vec{q}⊥q​) of vectorial vertices (such as for electric/particle current) at any finite ℏω\hbar ωℏω and q⃗\vec{q}q​. These cancellations extend to the full current-current correlation function for quadratic isotropic dispersion. Further symmetry breaking, multiband effects, and the additional imposition of momentum conservation, are discussed, with application to the Hall viscosity of Landau levels. Explicit expressions for generic nonlocal correlation functions are derived for Fermi liquids and non-Fermi liquids.
The nonintegrable higher spin Kitaev honeycomb model has an exact Z_{2} gauge structure, which exclusively identifies quantum spin liquid in the half-integer spin Kitaev model. But its constraints for the integer-spin Kitaev model are much limited, and even trivially gapped insulators cannot be excluded. The physical implications of exact Z_{2} gauge structure, especially Z_{2} fluxes, in integer-spin models remain largely unexplored. In this Letter, we theoretically show that a spin-S Yao-Lee model [a spin-orbital model with SU(2) spin-rotation symmetry] possesses a topologically nontrivial quantum spin-orbital liquid ground state for any spin (both integer and half-integer spin) by constructing exact deconfined fermionic Z_{2} gauge charges. We further show that the conserved Z_{2} flux can also demonstrate the intriguing spin fractionalization phenomena in the non-Abelian topological order phase of the spin-1 Yao-Lee model. Its deconfined Z_{2} vortex excitation carries fractionalized spin-1/2 quantum number in the low-energy subspace, which is also a non-Abelian anyon. Our exact manifestation of spin fractionalization in an integer-spin model is rather rare in previous studies, and is absent in the Kitaev honeycomb model.
In recent years the out-of-equilibrium dynamics of many-body quantum systems has attracted growing interest, both for its fundamental implications in the understanding of quantum thermalization and for potential applications in quantum information and quantum computation. In particular, quantum quenches provide a natural setting to investigate the mechanisms of correlation propagation and entanglement growth far from equilibrium. Interest in these phenomena stems not only from the possibility of testing theoretical predictions coming from emergent hydrodynamic descriptions, but also from understanding whether, and to what extent, such descriptions remain valid in the presence of spatial gradients or interfaces. Two paradigmatic frameworks in this context are the Generalized Hydrodynamics (GHD) and quasi-particle picture. The former is an effective theory devised to describe the long-time evolution after a quench in the thermodynamic limit. Within this perspective, inhomogeneous quenches are a crucial testbed for extending hydrodynamic theories to more realistic situations where, for example, scattering at an interface can occur. This is the main goal of the present thesis. The reference physical model is the one-dimensional Kitaev chain, an integrable spinless p-wave superconductor that exhibits both a topological and a trivial phase. The bilinear structure of the Hamiltonian in the fermionic creation and annihilation operators admits an exact rewriting in terms of Majorana fermions, in which the Hamiltonian remains quadratic. This representation allows a compact description of the ground-state properties via the Majorana correlation matrix (Gamma) at time t=0: the entries of Gamma completely encode the quantum information of the Gaussian ground state. The time evolution of Gamma is studied in the Heisenberg picture, where unitary dynamics correspond to an orthogonal rotation of Gamma implemented by an orthogonal matrix determined by the Kitaev Hamiltonian in the Majorana basis. Consequently, the entanglement entropy dynamics following a quench, both homogeneous and inhomogeneous, can be simulated through simple matrix products, making the approach numerically efficient even for sizable systems. The evolution code for Gamma was validated on a homogeneous quench with periodic boundary conditions: subsystem-size (l) curves exhibit the expected finite-size scaling, (Δ \propto 1/l), and numerical growth rates in the linear regime (t < v/2l), extracted by linear fits at fixed l, match the quasi-particle predictions for various values of the new chemical potential. The rate discrepancy behaves asymptotically as (Δ_{rate}} \propto 1/t_{\min}\) for large (t_{\min}\), up to oscillations from sub-extensive terms. The physical problem of interest is then an inhomogeneous quench of the chemical potential on the infinite Kitaev chain ground state with open boundaries, (μ_0 to μ_L, μ_R), producing a central interface. Initial and final Hamiltonians are free in the quasiparticle sense (ballistic propagation), but the presence of two distinct energy bands implies interface scattering. Simulations varying only μ_R (within the same trivial phase) show a nonmonotonic dependence of the entanglement growth rate, with a maximum at a particular μ_R. To explain this, an analytic ansatz for the stationary Majorana wavefunction was constructed: plane waves multiplied by a Majorana spinor, with bulk spinors obtained from the homogeneous PBC problem via the time-independent Schrödinger equation. Imposing matching at the interface yields four nontrivial equations for two unknowns (overdetermined in general). In the special cases (Δ ̸= t) two equations become trivial, the system is determined, and one obtains an explicit transmission coefficient. Combining this result with known quasi-particle formulas for two semi-infinite chains yields a hydrodynamic prediction for entanglement growth with interface scattering. Energy conservation across the interface selects the momenta that can be transmitted and thus contribute to inter-region correlations. Numerical tests for several total lengths (L) confirm the finite-size scaling (Δ_{\mathrm{e.e.}} \propto 1/L\), and show excellent agreement between numerical and theoretical growth rates for both quenches restricted to the right half and quenches applied to both halves. For the right-only case the rate discrepancy again follows (Δ_{rate} \propto 1/t_{min}\) as tmin⁡→∞t_{\min} \to \inftytmin​→∞. Possible future directions include: the study of the general inhomogeneous quench with (Δ ̸= t) via a direct extension of the analytic ansatz; the analysis of quenches starting from the topological ground state; the investigation of universal effects near the quantum critical point (μ=2t); the introduction of multiple interfaces or regions with spatially varying parameters to explore interference phenomena between quasiparticle fronts (e.g. periodic alternation (μ_1,μ_2,μ_1,μ_2,....) and, finally, the extension to interacting models where the Gaussian formalism is no longer applicable.
Building a universal quantum computer is a central goal of emerging quantum technologies, which has the potential to revolutionize science and technology. Unfortunately, this future does not seem to be very close at hand. However, quantum computers built for a special purpose, i.e. quantum simulators , are currently developed in many leading laboratories. Many schemes for quantum simulation have been proposed and realized using, e.g., ultracold atoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions in arrays of traps, quantum dots, photonic networks, or superconducting circuits. The progress in experimental implementations is more than spectacular. Particularly interesting are those systems that simulate quantum matter evolving in the presence of gauge fields. In the quantum simulation framework, the generated (synthetic) gauge fields may be Abelian, in which case they are the direct analogues of the vector potentials commonly associated with magnetic fields. In condensed matter physics, strong magnetic fields lead to a plethora of fascinating phenomena, among which the most paradigmatic is perhaps the quantum Hall effect. The standard Hall effect consists in the appearance of a transverse current, when a longitudinal voltage difference is applied to a conducting sample. For quasi-two-dimensional semiconductors at low temperatures placed in very strong magnetic fields, the transverse conductivity, the ratio between the transverse current and the applied voltage, exhibits perfect and robust quantization, independent for instance of the material or of its geometry. Such an integer quantum Hall effect, is now understood as a deep consequence of underlying topological order. Although such a system is an insulator in the bulk, it supports topologically robust edge excitations which carry the Hall current. The robustness of these chiral excitations against backscattering explains the universality of the quantum Hall effect. Another interesting and related effect, which arises from the interplay between strong magnetic field and lattice potentials, is the famous Hofstadter butterfly: the energy spectrum of a single particle moving on a lattice and subjected to a strong magnetic field displays a beautiful fractal structure as a function of the magnetic flux penetrating each elementary plaquette of the lattice. When the effects of interparticle interactions become dominant, two-dimensional gases of electrons exhibit even more exotic behaviour leading to the fractional quantum Hall effect. In certain conditions such a strongly interacting electron gas may form a highly correlated state of matter, the prototypical example being the celebrated Laughlin quantum liquid. Even more fascinating is the behaviour of bulk excitations (quasi-hole and quasi-particles): they are neither fermionic nor bosonic, but rather behave as anyons with fractional statistics intermediate between the two. Moreover, for some specific filling factors (ratio between the electronic density and the flux density), these anyons are proven to have an internal structure (several components) and non-Abelian braiding properties. Many of the above statements concern theoretical predictions—they have never been observed in condensed matter systems. For instance, the fractional values of the Hall conductance is seen as a direct consequence of the fractional statistics, but to date direct observation of anyons has not been possible in two-dimensional semiconductors. Realizing these predictions in experiments with atoms, ions, photons etc, which potentially allow the experimentalist to perform measurements complementary to those made in condensed matter systems, is thus highly desirable! Non-Abelian gauge fields couple the motional states of the particles to their internal degrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons, etc). In this sense external non-Abelian fields extend the concept of spin–orbit coupling (Rashba and Dresselhaus couplings), familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating phenomena such as the quantum spin Hall effect, three-dimensional topological insulators, topological superconductors and superfluids of various kinds. One also expects here the appearance of excitations in a form of topological edge states that can support robust transport, or entangled Majorana fermions in the case of topological superconductors or superfluids. Again, while many kinds of topological insulators have been realized in condensed matter systems, a controlled way of creating them in AMO systems and studying quantum phase transitions between various kinds of them is obviously very appealing and challenging. The various systems listed so far correspond to static gauge fields, which are externally imposed by the experimentalists. Even more fascinating is the possibility of generating synthetically dynamical gauge fields, i.e. gauge fields that evolve in time according to an interacting gauge theory, e.g., a full lattice gauge theory (LGT). These dynamical gauge fields can also couple to matter fields, allowing the quantum simulation of such complex systems (notoriously hard to simulate using 'traditional' computers), which are particularly relevant for modern high-energy physics. So far, most of the theoretical proposals concern the simulation of Abelian gauge theories, however, several groups have recently proposed extensions to the non-Abelian scenarios. The scope of the present focused issue of Journal of Physics B is to cover all of these developments, with particular emphasis on the non-Abelian gauge fields. The 14 papers in this issue include contributions from the leading theory groups working in this field; we believe that this collection will provide the reference set for quantum simulations of gauge fields. Although the special issue contains exclusively theoretical proposals and studies, it should be stressed that the progress in experimental studies of artificial Abelian and non-Abelian gauge fields in recent years has been simply spectacular. Multiple leading groups are working on this subject and have already obtained a lot of seminal results. The papers in the special issue are ordered according to the date of acceptance. The issue opens with a review article by Zhou et al [1] on unconventional states of bosons with synthetic spin–orbit coupling. Next, the paper by Maldonado-Mundo et al [2] studies ultracold Fermi gases with artificial Rashba spin–orbit coupling in a 2D gas. Anderson and Charles [3], in contrast, discuss a three-dimensional spin–orbit coupling in a trap. Orth et al [4] investigate correlated topological phases and exotic magnetism with ultracold fermions, again in the presence of artificial gauge fields. The paper of Nascimbène [5] does not address the synthetic gauge fields directly, but describes an experimental proposal for realizing one-dimensional topological superfluids with ultracold atomic gases; obviously, this problem is well situated in the general and growing field of topological superfluids, in particular those realized in the presence of non-Abelian gauge fields/spin–orbit coupling. Graß et al [6] consider in their paper fractional quantum Hall states of a Bose gas with spin–orbit coupling induced by a laser. Particular attention is drawn here to the possibility of realizing states with non-Abelian anyonic excitations. Zheng et al [7] study properties of Bose gases with Raman-induced spin–orbit coupling. Kiffner et al [8] in their paper touch on another kind of system, namely ultracold Rydberg atoms. In particular they study the generation of Abelian and non-Abelian gauge fields in dipole–dipole interacting Rydberg atoms. The behaviour of fermions in synthetic non-Abelian gauge potentials is discussed by Shenoy and Vyasanakere [9]. The paper starts with the study of Rashbon condensates (i.e. Bose condensates in the presence of Rashba coupling) and also introduces novel kinds of exotic Hamiltonians. Goldman et al [10] propose a concrete setup for realizing arbitrary non-Abelian gauge potentials in optical square lattices; they discuss how such synthetic gauge fields can be exploited to generate Chern insulators. Zygelman [11], similarly as Kiffner et al [8], discusses in his paper non-Abelian gauge fields in Rydberg systems. Marchukov et al [12] return to the subject of spin–orbit coupling, and investigate spectral gaps of spin–orbit coupled particles in the realistic situations of deformed traps. The last two papers, in contrast, are devoted to different subjects. Edmonds et al [13] consider a 'dynamical' density-dependent gauge potential, and study the Josephson effect in a Bose–Einstein condensate subject to such a potential. Last, but not least, Mazzucchi et al [14] study the properties of semimetal-superfluid quantum phase transitions in 3D lattices with Dirac points. References [1] Zhou X, Li Y, Cai Z and Wu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001 [2] Maldonado-Mundo D, Öhberg P and Valiente M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134002 [3] Anderson B M and Clark C W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003 [4] Orth P P, Cocks D, Rachel S, Buchhold M, Le Hur K and Hofstetter W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134004 [5] Nascimbène S 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134005 [6] Graß T, Juliá-Díaz B, Burrello M and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134006 [7] Zheng W, Yu Z-Q, Cui X and Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007 [8] Kiffner M, Li W and Jaksch D 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134008 [9] Shenoy V B and Vyasanakere J P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134009 [10] Goldman N, Gerbier F and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134010 [11] Zygelman B 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134011 [12] Marchukov O V, Volosniev A G, Fedorov D V, Jensen A S and Zinner N T 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134012 [13] Edmonds M J, Valiente M and Öhberg P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134013 [14] Mazzucchi G, Lepori L and Trombettoni A 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134014
We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing, in particular, on itinerant Ising anyon chains, and derive effective anyonic t-J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom--a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization.