cristina diamantini | Istituto Nazionale di Fisica Nucleare (original) (raw)

Papers by cristina diamantini

Nuclear Physics B, Oct 4, 1993

We examine the strong coupling limit of both compact and non-compact quantum electro-dynamics (QE... more We examine the strong coupling limit of both compact and non-compact quantum electro-dynamics (QED) on a lattice with staggered fermions. We show that every SU ( NL) quantum antiferromagnet with spins in a particular fundamental representation of the SU ( NL) Lie algebra and with nearest neighbor couplings on a bipartite lattice is exactly equivalent to the infinite coupling limit of lattice QED with the number of flavors of electrons related to NL and the dimension of space-time, D + 1. We find that, for both compact and non-compact QED, when NL is odd the ground state of the strong coupling limit breaks chiral symmetry if any dimensions and for any NL and the condensate is an isoscalar mass operator. When NL is even, chiral symmetry is broken if D ⩾ 2 and if NL is small enough; in this case the order parameter is an isovector mass operator. We also find the exact ground state of the lattice Coulomb gas as well as a variety of related lattice statistical systems with long-range interactions.

We consider several aspects of 'confining strings', recently proposed to describe the confining p... more We consider several aspects of 'confining strings', recently proposed to describe the confining phase of gauge field theories. We perform the exact duality transformation that leads to the confining string action and show that it reduces to the Polyakov action in the semiclassical approximation. In 4D we introduce a 'θ-term' and compute the low-energy effective action for the confining string in a derivative expansion. We find that the coefficient of the extrinsic curvature (stiffness) is negative, confirming previous proposals. In the absence of a θ-term, the effective string action is only a cut-off theory for finite values of the coupling e, whereas for generic values of θ, the action can be renormalized and to leading order we obtain the Nambu-Goto action plus a topological 'spin' term that could stabilize the system.

J Phys a Math Theor, 2011

We show that different classes of topological order can be distinguished by the dynamical symmetr... more We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. A fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W1 + ∞. We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern-Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU(m) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the nu = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.

Physical Review Letters, 2006

Qubit networks with long-range interactions inspired by the Hebb rule can be used as quantum asso... more Qubit networks with long-range interactions inspired by the Hebb rule can be used as quantum associative memories. Starting from a uniform superposition, the unitary evolution generated by these interactions drives the network through a quantum phase transition at a critical computation time, after which ferromagnetic order guarantees that a measurement retrieves the stored pattern. The maximum memory capacity of these qubit networks is reached at a memory density alpha=p/n=1.

Physical Review Letters, Nov 1, 1995

We investigate nonperturbative features of a planar lattice Chern-Simons gauge theory modeling th... more We investigate nonperturbative features of a planar lattice Chern-Simons gauge theory modeling the physics of Josephson junction arrays. By identifying the relevant topological configurations and their interactions, we determine the phase structure of the model. Our results match observed phase transitions in Josephson junction arrays and suggest also the possibility of oblique confining ground states corresponding to purely planar quantum Hall regimes for either charges or vortices.

Journal of High Energy Physics, 2015

We propose a spin gauge field theory in which the curl of a Dirac fermion current density plays t... more We propose a spin gauge field theory in which the curl of a Dirac fermion current density plays the role of the pseudovector charge density. In this field-theoretic model, spin interactions are mediated by a single scalar gauge boson in its antisymmetric tensor formulation. We show that these long range spin interactions induce a gauge invariant photon mass in the one-loop effective action. The fermion loop generates a coupling between photons and the spin gauge boson, which acquires thus charge. This coupling represents also an induced, gauge invariant, topological mass for the photons, leading to the Meissner effect. The one-loop effective equations of motion for the charged spin gauge boson are the London equations. We propose thus spin gauge interactions as an alternative, topological mechanism for superconductivity in which no spontaneous symmetry breaking is involved. PACS numbers: 11.15.Wx,74.20.Mn

We review our models of quantum associative memories that represent the “quantization” of fully c... more We review our models of quantum associative memories that represent the “quantization” of fully coupled neural networks like the Hopfield model. The idea is to replace the classical irreversible attractor dynamics driven by an Ising model with pattern-dependent weights by the reversible rotation of an input quantum state onto an output quantum state consisting of a linear superpo- sition with probability amplitudes peaked on the stored pattern closest to the input in Hamming distance, resulting in a high probability of measuring a memory pattern very similar to the input. The unitary operator implementing this transformation can be formulated as a sequence of one- qubit and two-qubit elementary quantum gates and is thus the exponential of an ordered quantum Ising model with sequential operations and with pattern-dependent interactions, exactly as in the classical case. Probabilistic quantum memories, that make use of postselection of the measurement result of control qubits, overcom...

Physical Review D, 2014

Spin-charge separation, a crucial ingredient in 2D models of strongly correlated systems, in most... more Spin-charge separation, a crucial ingredient in 2D models of strongly correlated systems, in mostly considered in condensed matter applications. In this paper we present a relativistic field-theoretic model in which charged particles of spin 1/2 emerge by soldering spinless charges and magnetic vortices in a confinement quantum phase transition modelled as a tensor Higgs mechanism. The model involves two gauge fields, a vector one and a two-form gauge field interacting by the topological BF term. When this tensor gauge symmetry is spontaneously broken charges are soldered to the ends of magnetic vortices and thus confined by a linear potential. If the vector potential has a topological θ-term with value θ = π, the constituents of this "meson" acquire spin 1/2 in this transition.

Physical Review E, 2014

The classic Landauer bound can be lowered when erasure errors are permitted. Here we point out th... more The classic Landauer bound can be lowered when erasure errors are permitted. Here we point out that continuous phase transitions characterized by an order parameter can also be viewed as information erasure by resetting a certain number of bits to a standard value. The informationtheoretic expression for the generalized Landauer bound in terms of error probability implies thus a universal form for the thermodynamic entropy in the partially ordered phase. We explicitly show that the thermodynamic entropy as a function of interaction parameters and temperature is identical to the information-theoretic expression in terms of error probability alone in the specific example of the Hopfield neural network model of associative memory, a distributed information-processing system of many interacting stochastic bits. In this framework the Landauer bound sets a lower limit for the work associated with "remembering" rather than "forgetting".

Nuclear Physics B, 2015

We present a new Higgsless model of superconductivity, inspired from anyon superconductivity but ... more We present a new Higgsless model of superconductivity, inspired from anyon superconductivity but P-and T-invariant and generalizable to any dimension. While the original anyon superconductivity mechanism was based on incompressible quantum Hall fluids as average field states, our mechanism involves topological insulators as average field states. In D space dimensions it involves a (D-1)-form fictitious pseudovector gauge field which originates from the condensation of topological defects in compact low-energy effective BF theories. In the average field approximation, the corresponding uniform emergent charge creates a gap for the (D-2)-dimensional branes via the Magnus force, the dual of the Lorentz force. One particular combination of intrinsic and emergent charge fluctuations that leaves the total charge distribution invariant constitutes an isolated gapless mode leading to superfluidity. The remaining massive modes organise themselves into a D-dimensional charged, massive vector. There is no massive Higgs scalar as there is no local order parameter. When electromagnetism is switched on, the photon acquires mass by the topological BF mechanism. Although the charge of the gapless mode (2) and the topological order (4) are the same as those of the standard Higgs model, the two models of superconductivity are clearly different since the origins of the gap, reflected in the high-energy sectors are totally different. In 2D this type of superconductivity is explicitly realized as global superconductivity in Josephson junction arrays. In 3D this model predicts a possible phase transition from topological insulators to Higgsless superconductors.

Journal of High Energy Physics, 2002

We show that the high-temperature behaviour of the recently proposed confining strings reproduces... more We show that the high-temperature behaviour of the recently proposed confining strings reproduces exactly the correct large-N QCD result, for a large class of truncations of the long-range interaction between surface elements.

Physical Review B, 2011

Topological matter is characterized by the presence of a topological BF term in its long-distance... more Topological matter is characterized by the presence of a topological BF term in its long-distance effective action. Topological defects due to the compactness of the U(1) gauge fields induce quantum phase transitions between topological insulators, topological superconductors and topological confinement. In conventional superconductivity, due to spontaneous symmetry breaking, the photon acquires a mass due to the Anderson-Higgs mechanism. In this paper we derive the corresponding effective actions for the electromagnetic field in topological superconductors and topological confinement phases. In topological superconductors magnetic flux is confined and the photon acquires a topological mass through the BF mechanism: no symmetry breaking is involved, the ground state has topological order and the transition is induced by quantum fluctuations. In topological confinement, instead, electric charge is linearly confined and the photon becomes a massive antisymmetric tensor via the Stückelberg mechanism. Oblique confinement phases arise when the string condensate carries both magnetic and electric flux (dyonic strings). Such phases are characterized by a vortex quantum Hall effect potentially relevant for the dissipationless transport of information stored on vortices.

New Journal of Physics, 2012

Topological matter in 3D is characterized by the presence of a topological BF term in its longdis... more Topological matter in 3D is characterized by the presence of a topological BF term in its longdistance effective action. We show that, in 3D, there is another marginal term that must be added to the action in order to fully determine the physical content of the model. The quantum phase structure is governed by three parameters that drive the condensation of topological defects: the BF coupling, the electric permittivity and the magnetic permeability of the material. For intermediate levels of electric permittivity and magnetic permeability the material is a topological insulator. We predict, however, new states of matter when these parameters cross critical values: a topological superconductor when electric permittivity is increased and magnetic permeability is lowered and a charge confinement phase in the opposite case of low electric permittivity and high magnetic permeability. Synthetic topological matter may be fabricated as 3D arrays of Josephson junctions.

Journal of Physics A: Mathematical and Theoretical, 2014

Journal of Physics A: Mathematical and Theoretical, 2011

We show that different classes of topological order can be distinguished by the dynamical symmetr... more We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. Fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W1+∞. We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern-Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU(m) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the ν = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.

Journal of Physics A: Mathematical and Theoretical, 2013

Conformal field theories do not only classify 2D classical critical behavior but they also govern... more Conformal field theories do not only classify 2D classical critical behavior but they also govern a certain class of 2D quantum critical behavior. In this latter case it is the ground state wave functional of the quantum theory that is conformally invariant, rather than the classical action. We show that the superconducting-insulating (SI) quantum phase transition in 2D Josephson junction arrays (JJAs) is a (doubled) c = 1 Gaussian conformal quantum critical point. The quantum action describing this system is a doubled Maxwell-Chern-Simons model in the strong coupling limit. We also argue that the SI quantum transitions in frustrated JJAs realize the other possible universality classes of conformal quantum critical behavior, corresponding to the unitary minimal models at central charge c = 1 − 6/m(m + 1).

Journal of Physics A: Mathematical and General, 2006

We argue that frustrated Josephson junction arrays may support a topologically ordered supercondu... more We argue that frustrated Josephson junction arrays may support a topologically ordered superconducting ground state, characterized by a non-trivial groundstate degeneracy on the torus. This superconducting quantum fluid provides an explicit example of a system in which superconductivity arises from a topological mechanism rather than from the usual Landau-Ginzburg mechanism.

International Journal of Modern Physics B, 1992

We investigate the low energy second quantized theory of non-relativistic planar electrons in ext... more We investigate the low energy second quantized theory of non-relativistic planar electrons in external magnetic fields. We show that a U(1) Chern-Simons term appears in the effective action, for both the continuum and the lattice theories, as a result of a nontrivial phase holonomy. We also show, using the Jordan-Wigner transformation for planar spin systems, that the quantum frustrated XY

Nuclear Physics B, Oct 4, 1993

We examine the strong coupling limit of both compact and non-compact quantum electro-dynamics (QE... more We examine the strong coupling limit of both compact and non-compact quantum electro-dynamics (QED) on a lattice with staggered fermions. We show that every SU ( NL) quantum antiferromagnet with spins in a particular fundamental representation of the SU ( NL) Lie algebra and with nearest neighbor couplings on a bipartite lattice is exactly equivalent to the infinite coupling limit of lattice QED with the number of flavors of electrons related to NL and the dimension of space-time, D + 1. We find that, for both compact and non-compact QED, when NL is odd the ground state of the strong coupling limit breaks chiral symmetry if any dimensions and for any NL and the condensate is an isoscalar mass operator. When NL is even, chiral symmetry is broken if D ⩾ 2 and if NL is small enough; in this case the order parameter is an isovector mass operator. We also find the exact ground state of the lattice Coulomb gas as well as a variety of related lattice statistical systems with long-range interactions.

We consider several aspects of 'confining strings', recently proposed to describe the confining p... more We consider several aspects of 'confining strings', recently proposed to describe the confining phase of gauge field theories. We perform the exact duality transformation that leads to the confining string action and show that it reduces to the Polyakov action in the semiclassical approximation. In 4D we introduce a 'θ-term' and compute the low-energy effective action for the confining string in a derivative expansion. We find that the coefficient of the extrinsic curvature (stiffness) is negative, confirming previous proposals. In the absence of a θ-term, the effective string action is only a cut-off theory for finite values of the coupling e, whereas for generic values of θ, the action can be renormalized and to leading order we obtain the Nambu-Goto action plus a topological 'spin' term that could stabilize the system.

J Phys a Math Theor, 2011

We show that different classes of topological order can be distinguished by the dynamical symmetr... more We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. A fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W1 + ∞. We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern-Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU(m) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the nu = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.

Physical Review Letters, 2006

Qubit networks with long-range interactions inspired by the Hebb rule can be used as quantum asso... more Qubit networks with long-range interactions inspired by the Hebb rule can be used as quantum associative memories. Starting from a uniform superposition, the unitary evolution generated by these interactions drives the network through a quantum phase transition at a critical computation time, after which ferromagnetic order guarantees that a measurement retrieves the stored pattern. The maximum memory capacity of these qubit networks is reached at a memory density alpha=p/n=1.

Physical Review Letters, Nov 1, 1995

We investigate nonperturbative features of a planar lattice Chern-Simons gauge theory modeling th... more We investigate nonperturbative features of a planar lattice Chern-Simons gauge theory modeling the physics of Josephson junction arrays. By identifying the relevant topological configurations and their interactions, we determine the phase structure of the model. Our results match observed phase transitions in Josephson junction arrays and suggest also the possibility of oblique confining ground states corresponding to purely planar quantum Hall regimes for either charges or vortices.

Journal of High Energy Physics, 2015

We propose a spin gauge field theory in which the curl of a Dirac fermion current density plays t... more We propose a spin gauge field theory in which the curl of a Dirac fermion current density plays the role of the pseudovector charge density. In this field-theoretic model, spin interactions are mediated by a single scalar gauge boson in its antisymmetric tensor formulation. We show that these long range spin interactions induce a gauge invariant photon mass in the one-loop effective action. The fermion loop generates a coupling between photons and the spin gauge boson, which acquires thus charge. This coupling represents also an induced, gauge invariant, topological mass for the photons, leading to the Meissner effect. The one-loop effective equations of motion for the charged spin gauge boson are the London equations. We propose thus spin gauge interactions as an alternative, topological mechanism for superconductivity in which no spontaneous symmetry breaking is involved. PACS numbers: 11.15.Wx,74.20.Mn

We review our models of quantum associative memories that represent the “quantization” of fully c... more We review our models of quantum associative memories that represent the “quantization” of fully coupled neural networks like the Hopfield model. The idea is to replace the classical irreversible attractor dynamics driven by an Ising model with pattern-dependent weights by the reversible rotation of an input quantum state onto an output quantum state consisting of a linear superpo- sition with probability amplitudes peaked on the stored pattern closest to the input in Hamming distance, resulting in a high probability of measuring a memory pattern very similar to the input. The unitary operator implementing this transformation can be formulated as a sequence of one- qubit and two-qubit elementary quantum gates and is thus the exponential of an ordered quantum Ising model with sequential operations and with pattern-dependent interactions, exactly as in the classical case. Probabilistic quantum memories, that make use of postselection of the measurement result of control qubits, overcom...

Physical Review D, 2014

Spin-charge separation, a crucial ingredient in 2D models of strongly correlated systems, in most... more Spin-charge separation, a crucial ingredient in 2D models of strongly correlated systems, in mostly considered in condensed matter applications. In this paper we present a relativistic field-theoretic model in which charged particles of spin 1/2 emerge by soldering spinless charges and magnetic vortices in a confinement quantum phase transition modelled as a tensor Higgs mechanism. The model involves two gauge fields, a vector one and a two-form gauge field interacting by the topological BF term. When this tensor gauge symmetry is spontaneously broken charges are soldered to the ends of magnetic vortices and thus confined by a linear potential. If the vector potential has a topological θ-term with value θ = π, the constituents of this "meson" acquire spin 1/2 in this transition.

Physical Review E, 2014

The classic Landauer bound can be lowered when erasure errors are permitted. Here we point out th... more The classic Landauer bound can be lowered when erasure errors are permitted. Here we point out that continuous phase transitions characterized by an order parameter can also be viewed as information erasure by resetting a certain number of bits to a standard value. The informationtheoretic expression for the generalized Landauer bound in terms of error probability implies thus a universal form for the thermodynamic entropy in the partially ordered phase. We explicitly show that the thermodynamic entropy as a function of interaction parameters and temperature is identical to the information-theoretic expression in terms of error probability alone in the specific example of the Hopfield neural network model of associative memory, a distributed information-processing system of many interacting stochastic bits. In this framework the Landauer bound sets a lower limit for the work associated with "remembering" rather than "forgetting".

Nuclear Physics B, 2015

We present a new Higgsless model of superconductivity, inspired from anyon superconductivity but ... more We present a new Higgsless model of superconductivity, inspired from anyon superconductivity but P-and T-invariant and generalizable to any dimension. While the original anyon superconductivity mechanism was based on incompressible quantum Hall fluids as average field states, our mechanism involves topological insulators as average field states. In D space dimensions it involves a (D-1)-form fictitious pseudovector gauge field which originates from the condensation of topological defects in compact low-energy effective BF theories. In the average field approximation, the corresponding uniform emergent charge creates a gap for the (D-2)-dimensional branes via the Magnus force, the dual of the Lorentz force. One particular combination of intrinsic and emergent charge fluctuations that leaves the total charge distribution invariant constitutes an isolated gapless mode leading to superfluidity. The remaining massive modes organise themselves into a D-dimensional charged, massive vector. There is no massive Higgs scalar as there is no local order parameter. When electromagnetism is switched on, the photon acquires mass by the topological BF mechanism. Although the charge of the gapless mode (2) and the topological order (4) are the same as those of the standard Higgs model, the two models of superconductivity are clearly different since the origins of the gap, reflected in the high-energy sectors are totally different. In 2D this type of superconductivity is explicitly realized as global superconductivity in Josephson junction arrays. In 3D this model predicts a possible phase transition from topological insulators to Higgsless superconductors.

Journal of High Energy Physics, 2002

We show that the high-temperature behaviour of the recently proposed confining strings reproduces... more We show that the high-temperature behaviour of the recently proposed confining strings reproduces exactly the correct large-N QCD result, for a large class of truncations of the long-range interaction between surface elements.

Physical Review B, 2011

Topological matter is characterized by the presence of a topological BF term in its long-distance... more Topological matter is characterized by the presence of a topological BF term in its long-distance effective action. Topological defects due to the compactness of the U(1) gauge fields induce quantum phase transitions between topological insulators, topological superconductors and topological confinement. In conventional superconductivity, due to spontaneous symmetry breaking, the photon acquires a mass due to the Anderson-Higgs mechanism. In this paper we derive the corresponding effective actions for the electromagnetic field in topological superconductors and topological confinement phases. In topological superconductors magnetic flux is confined and the photon acquires a topological mass through the BF mechanism: no symmetry breaking is involved, the ground state has topological order and the transition is induced by quantum fluctuations. In topological confinement, instead, electric charge is linearly confined and the photon becomes a massive antisymmetric tensor via the Stückelberg mechanism. Oblique confinement phases arise when the string condensate carries both magnetic and electric flux (dyonic strings). Such phases are characterized by a vortex quantum Hall effect potentially relevant for the dissipationless transport of information stored on vortices.

New Journal of Physics, 2012

Topological matter in 3D is characterized by the presence of a topological BF term in its longdis... more Topological matter in 3D is characterized by the presence of a topological BF term in its longdistance effective action. We show that, in 3D, there is another marginal term that must be added to the action in order to fully determine the physical content of the model. The quantum phase structure is governed by three parameters that drive the condensation of topological defects: the BF coupling, the electric permittivity and the magnetic permeability of the material. For intermediate levels of electric permittivity and magnetic permeability the material is a topological insulator. We predict, however, new states of matter when these parameters cross critical values: a topological superconductor when electric permittivity is increased and magnetic permeability is lowered and a charge confinement phase in the opposite case of low electric permittivity and high magnetic permeability. Synthetic topological matter may be fabricated as 3D arrays of Josephson junctions.

Journal of Physics A: Mathematical and Theoretical, 2014

Journal of Physics A: Mathematical and Theoretical, 2011

We show that different classes of topological order can be distinguished by the dynamical symmetr... more We show that different classes of topological order can be distinguished by the dynamical symmetry algebra of edge excitations. Fundamental topological order is realized when this algebra is the largest possible, the algebra of quantum area-preserving diffeomorphisms, called W1+∞. We argue that this order is realized in the Jain hierarchy of fractional quantum Hall states and show that it is more robust than the standard Abelian Chern-Simons order since it has a lower entanglement entropy due to the non-Abelian character of the quasi-particle anyon excitations. These behave as SU(m) quarks, where m is the number of components in the hierarchy. We propose the topological entanglement entropy as the experimental measure to detect the existence of these quantum Hall quarks. Non-Abelian anyons in the ν = 2/5 fractional quantum Hall states could be the primary candidates to realize qbits for topological quantum computation.

Journal of Physics A: Mathematical and Theoretical, 2013

Conformal field theories do not only classify 2D classical critical behavior but they also govern... more Conformal field theories do not only classify 2D classical critical behavior but they also govern a certain class of 2D quantum critical behavior. In this latter case it is the ground state wave functional of the quantum theory that is conformally invariant, rather than the classical action. We show that the superconducting-insulating (SI) quantum phase transition in 2D Josephson junction arrays (JJAs) is a (doubled) c = 1 Gaussian conformal quantum critical point. The quantum action describing this system is a doubled Maxwell-Chern-Simons model in the strong coupling limit. We also argue that the SI quantum transitions in frustrated JJAs realize the other possible universality classes of conformal quantum critical behavior, corresponding to the unitary minimal models at central charge c = 1 − 6/m(m + 1).

Journal of Physics A: Mathematical and General, 2006

We argue that frustrated Josephson junction arrays may support a topologically ordered supercondu... more We argue that frustrated Josephson junction arrays may support a topologically ordered superconducting ground state, characterized by a non-trivial groundstate degeneracy on the torus. This superconducting quantum fluid provides an explicit example of a system in which superconductivity arises from a topological mechanism rather than from the usual Landau-Ginzburg mechanism.

International Journal of Modern Physics B, 1992

We investigate the low energy second quantized theory of non-relativistic planar electrons in ext... more We investigate the low energy second quantized theory of non-relativistic planar electrons in external magnetic fields. We show that a U(1) Chern-Simons term appears in the effective action, for both the continuum and the lattice theories, as a result of a nontrivial phase holonomy. We also show, using the Jordan-Wigner transformation for planar spin systems, that the quantum frustrated XY