Birgit Kaufmann - Academia.edu (original) (raw)

Papers by Birgit Kaufmann

Aps Meeting Abstracts, Mar 1, 2003

We are investigating the quantum phase transition between the Mott insulating phase and the super... more We are investigating the quantum phase transition between the Mott insulating phase and the superfluid phase in a Bose--Einstein condensate of trapped ultracold atomic gases in an external optical lattice. Experimentally, this phase transition can be studied in great detail due to the fact that all relevant parameters can be continuously varied across the transition, as was shown by Greiner et al. in recent experiments (Nature 415, 39 (2002)). Theoretically, the system can be described by the Bose-Hubbard model. We are focusing on the non-equilibrium aspects by coupling the one-dimensional system to external leads so that particles will flow into the system at one end and out of the system at the other end. We are using numerical diagonalization techniques to study the properties of this system. In this presentation, we will show preliminary results for the case where a stationary flow is established. We will also discuss the applicability of alternative methods to this problem.

Journal of Physics: Conference Series, 2015

We define re-gaugings and enhanced symmetries for graphs with group labels on their edges. These ... more We define re-gaugings and enhanced symmetries for graphs with group labels on their edges. These give rise to interesting projective representations of subgroups of the automorphism groups of the graphs. We furthermore embed this construction into several higher levels of generalization using category theory and show that they are natural in that language. These include projective representations of the re-gauging groupoid and a novel generalization to all symmetries of the graph.

Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and cla... more Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of A n type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an A k singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.

We study a class of graph Hamiltonians given by a type of quiver representation to which we can a... more We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)– commutative geometries. By selecting gauging data these geome-tries are realized by matrices through an explicit construction or a Kan-extension. We describe the changes in gauge via the ac-tion of a regauging groupoid. It acts via matrices that give rise to a noncommutative 2–cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of regaugings. In the commutative case, we deduce that the ex-tended symmetries act via a projective representation. This yields isotypical decompositions and super–selection rules. We apply these results to the PDG and honeycomb wire–networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid...

Abstract. Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac poin... more Abstract. Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of An type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an Ak singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.

We present a study of the two species totally asymmetric diffusion model using the Bethe ansatz. ... more We present a study of the two species totally asymmetric diffusion model using the Bethe ansatz. The Hamiltonian has U q (SU (3)) symmetry. We derive the nested Bethe ansatz equations and obtain the dynamical critical exponent from the finite-size scaling properties of the eigenvalue with the smallest real part. The dynamical critical exponent is 3 2 which is the exponent corresponding to KPZ growth in the single species asymmetric diffusion model.

A new nano–material in the form of a double gyroid has motivated us to study (non)–commutative C ... more A new nano–material in the form of a double gyroid has motivated us to study (non)–commutative C ∗ geometry of periodic wire networks and the associated graph Hamiltonians. Here we present the general abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. In these geometric situations, the non–commutativity is introduced by a constant magnetic field and the theory splits into two pieces: commutative and non–commutative, both of which are governed by a C∗ geometry. In the non–commutative case, we can use tools such as K–theory to make statements about the band structure. In the commutative case, we give geometric and algebraic methods to study band intersections; these methods come from singularity theory and representation theory. We also provide new tools in the study, using K–theory and Chern classes. The latter can be computed using Berry connection in the momentum...

We conjecture the factorized scattering description for OSP(m/2n)/OSP(m − 1/2n) supersphere sigma... more We conjecture the factorized scattering description for OSP(m/2n)/OSP(m − 1/2n) supersphere sigma models and OSP(m/2n) Gross Neveu models. The non unitarity of these field theories translates into a lack of ‘physical unitarity ’ of the S matrices, which are instead unitary with respect to the non-positive scalar product inherited from the orthosymplectic structure. Nevertheless, we find that formal thermodynamic Bethe ansatz calculations appear meaningful, reproduce the correct central charges, and agree with perturbative calculations. This paves the way to a more thorough study of these and other models with supergroup symmetries using the S matrix approach. 1

doi:10.3842/SIGMA.2010.039 Abstract. We present a study of the two species totally asymmetric dif... more doi:10.3842/SIGMA.2010.039 Abstract. We present a study of the two species totally asymmetric diffusion model using the Bethe ansatz. The Hamiltonian has Uq(SU(3)) symmetry. We derive the nested Bethe ansatz equations and obtain the dynamical critical exponent from the finite-size scaling properties of the eigenvalue with the smallest real part. The dynamical critical exponent is 3 2 which is the exponent corresponding to KPZ growth in the single species asymmetric diffusion model. Key words: asymmetric diffusion; nested Uq(SU(3)) Bethe ansatz; dynamical critical exponent 2010 Mathematics Subject Classification: 82C27; 82B20 1

We report on our recent results from a mathematical study of wire network graphs that are complem... more We report on our recent results from a mathematical study of wire network graphs that are complements to triply periodic CMC surfaces and can be synthesized in the lab on the nanoscale. Here, we studied all three cases in which the graphs corresponding to the networks are symmetric and self-dual. These are the cubic, diamond and gyroid surfaces. The gyroid is the most interesting case in its geometry and properties as it exhibits Dirac points (in 3d). It can be seen as a generalization of the honeycomb lattice in 2d that models graphene. Indeed, our theory works in more general cases, such as periodic networks in any dimension and even more abstract settings. After presenting our theoretical results, we aim to invite an experimental study of these Dirac points and a possible quantum Hall effect. The general theory also allows to find local symmetry groups which force degeneracies aka level crossings from a finite graph encoding the elementary cell structure. Vice-versa one could hop...

arXiv: Mathematical Physics, 2016

We generalize the mathbbZ_2\mathbb{Z}_2mathbbZ2 invariant of topological insulators using noncommutative differe... more We generalize the mathbbZ2\mathbb{Z}_2mathbbZ2 invariant of topological insulators using noncommutative differential geometry in two different ways. First, we model Majorana zero modes by KQ-cycles in the framework of analytic K-homology, and we define the noncommutative mathbbZ2\mathbb{Z}_2mathbbZ2 invariant as a topological index in noncommutative topology. Second, we look at the geometric picture of the Pfaffian formalism of the mathbbZ2\mathbb{Z}_2mathbbZ2 invariant, i.e., the Kane--Mele invariant, and we define the noncommutative Kane--Mele invariant over the fixed point algebra of the time reversal symmetry in the noncommutative 2-torus. Finally, we are able to prove the equivalence between the noncommutative topological mathbbZ2\mathbb{Z}_2mathbbZ_2 index and the noncommutative Kane--Mele invariant.

arXiv: Mathematical Physics, 2015

We analyze the topological mathbbZ_2\mathbb{Z}_2mathbbZ2 invariant, which characterizes time reversal invariant ... more We analyze the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant, which characterizes time reversal invariant topological insulators, in the framework of index theory and K-theory. The topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant counts the parity of generalized Majorana zero modes, which can be interpreted as an analytical index. As we show, it fits perfectly into a mod 2 index theorem, and the topological index provides an efficient way to compute the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant. Finally, we give a new version of the bulk-boundary correspondence which yields an alternative explanation of the index theorem and the topological mathbbZ2\mathbb{Z}_2mathbbZ_2 invariant. Here the boundary is not the geometric boundary of a probe, but an effective boundary in the momentum space.

arXiv: Mathematical Physics, 2016

We study the topological band theory of time reversal invariant topological insulators and interp... more We study the topological band theory of time reversal invariant topological insulators and interpret the topological mathbbZ_2\mathbb{Z}_2mathbbZ2 invariant as an obstruction in terms of Stiefel--Whitney classes. The band structure of a topological insulator defines a Pfaffian line bundle over the momentum space, whose structure group can be reduced to mathbbZ2\mathbb{Z}_2mathbbZ2. So the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant will be understood by the Stiefel--Whitney theory, which detects the orientability of a principal mathbbZ2\mathbb{Z}_2mathbbZ2-bundle. Moreover, the relation between weak and strong topological insulators will be understood based on cobordism theory. Finally, the topological mathbbZ2\mathbb{Z}_2mathbbZ_2 invariant gives rise to a fully extended topological quantum field theory (TQFT).

arXiv: Mathematical Physics, 2018

We study topological properties of families of Hamiltonians which may contain degenerate energy l... more We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus, we study local models. These give information about the Chern classes and Berry phases. We then give global constraints for the topological invariants. This is an hitherto relatively unexplored subject. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. The results can also be used in the study of deformations. We furthermore use these constraints to analyse examples which include the Gyroid geometry, which exhibits Weyl points and triple crossings and the honeycomb geometry with its two Dirac points.

We show how using Clifford algebras and their representations can greatly simplify the analysis o... more We show how using Clifford algebras and their representations can greatly simplify the analysis of integrable systems. In particular, we apply this approach to the XX-model with non-diagonal boundaries which is among others related to growing and fluctuating interfaces and stochastic reaction-diffusion systems. Using this approach, we can not only diagonalize the system, but also find new hidden symmetries.

Bulletin of the American Physical Society, 2020

The Journal of Chemical Physics

Quantum annealers are an alternative approach to quantum computing which make use of the adiabati... more Quantum annealers are an alternative approach to quantum computing which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising model Hamiltonian. In this paper we review two different methods for finding the ground state of molecular Hamiltonians using Ising model-based quantum annealers. In addition, we compare the relative effectiveness of each method by calculating the binding energies, bond lengths, and bond angles of the H + 3 and H 2 O molecules and mapping their potential energy curves. We also assess the resource requirements of each method by determining the number of qubits and computation time required to simulate each molecule using various parameter values. While each of these methods is capable of accurately predicting the ground state properties of small molecules, we find that they are still outperformed by modern classical algorithms and that the scaling of the resource requirements remains a challenge.

Physical Review A

We propose experimentally observable signatures of topological Majorana quasiparticles in the few... more We propose experimentally observable signatures of topological Majorana quasiparticles in the few-body limit of the interacting cold-atom model of Iemini et al.

Journal of Statistical Mechanics: Theory and Experiment

A new exponent characterizing the rounding of crystal facets is found by mapping a crystal surfac... more A new exponent characterizing the rounding of crystal facets is found by mapping a crystal surface onto the asymmetric six-vertex model (i.e. with external fields h and v) and using the Bethe ansatz to obtain appropriate expansions of the free energy close to criticality. Leading order exponents in δh, δv are determined along the whole phase boundary and in an arbitrary direction. A possible experimental verification of this result is discussed.

Reviews in Mathematical Physics

This paper is a survey of the [Formula: see text]-valued invariant of topological insulators used... more This paper is a survey of the [Formula: see text]-valued invariant of topological insulators used in condensed matter physics. The [Formula: see text]-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The [Formula: see text] invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.

Aps Meeting Abstracts, Mar 1, 2003

We are investigating the quantum phase transition between the Mott insulating phase and the super... more We are investigating the quantum phase transition between the Mott insulating phase and the superfluid phase in a Bose--Einstein condensate of trapped ultracold atomic gases in an external optical lattice. Experimentally, this phase transition can be studied in great detail due to the fact that all relevant parameters can be continuously varied across the transition, as was shown by Greiner et al. in recent experiments (Nature 415, 39 (2002)). Theoretically, the system can be described by the Bose-Hubbard model. We are focusing on the non-equilibrium aspects by coupling the one-dimensional system to external leads so that particles will flow into the system at one end and out of the system at the other end. We are using numerical diagonalization techniques to study the properties of this system. In this presentation, we will show preliminary results for the case where a stationary flow is established. We will also discuss the applicability of alternative methods to this problem.

Journal of Physics: Conference Series, 2015

We define re-gaugings and enhanced symmetries for graphs with group labels on their edges. These ... more We define re-gaugings and enhanced symmetries for graphs with group labels on their edges. These give rise to interesting projective representations of subgroups of the automorphism groups of the graphs. We furthermore embed this construction into several higher levels of generalization using category theory and show that they are natural in that language. These include projective representations of the re-gauging groupoid and a novel generalization to all symmetries of the graph.

Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and cla... more Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of A n type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an A k singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.

We study a class of graph Hamiltonians given by a type of quiver representation to which we can a... more We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)– commutative geometries. By selecting gauging data these geome-tries are realized by matrices through an explicit construction or a Kan-extension. We describe the changes in gauge via the ac-tion of a regauging groupoid. It acts via matrices that give rise to a noncommutative 2–cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of regaugings. In the commutative case, we deduce that the ex-tended symmetries act via a projective representation. This yields isotypical decompositions and super–selection rules. We apply these results to the PDG and honeycomb wire–networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid...

Abstract. Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac poin... more Abstract. Motivated by the Double Gyroid nanowire network we develop methods to detect Dirac points and classify level crossings, aka. singularities in the spectrum of a family of Hamiltonians. The approach we use is singularity theory. Using this language, we obtain a characterization of Dirac points and also show that the branching behavior of the level crossings is given by an unfolding of An type singularities. Which type of singularity occurs can be read off a characteristic region inside the miniversal unfolding of an Ak singularity. We then apply these methods in the setting of families of graph Hamiltonians, such as those for wire networks. In the particular case of the Double Gyroid we analytically classify its singularities and show that it has Dirac points. This indicates that nanowire systems of this type should have very special physical properties.

We present a study of the two species totally asymmetric diffusion model using the Bethe ansatz. ... more We present a study of the two species totally asymmetric diffusion model using the Bethe ansatz. The Hamiltonian has U q (SU (3)) symmetry. We derive the nested Bethe ansatz equations and obtain the dynamical critical exponent from the finite-size scaling properties of the eigenvalue with the smallest real part. The dynamical critical exponent is 3 2 which is the exponent corresponding to KPZ growth in the single species asymmetric diffusion model.

A new nano–material in the form of a double gyroid has motivated us to study (non)–commutative C ... more A new nano–material in the form of a double gyroid has motivated us to study (non)–commutative C ∗ geometry of periodic wire networks and the associated graph Hamiltonians. Here we present the general abstract framework, which is given by certain quiver representations, with special attention to the original case of the gyroid as well as related cases, such as graphene. In these geometric situations, the non–commutativity is introduced by a constant magnetic field and the theory splits into two pieces: commutative and non–commutative, both of which are governed by a C∗ geometry. In the non–commutative case, we can use tools such as K–theory to make statements about the band structure. In the commutative case, we give geometric and algebraic methods to study band intersections; these methods come from singularity theory and representation theory. We also provide new tools in the study, using K–theory and Chern classes. The latter can be computed using Berry connection in the momentum...

We conjecture the factorized scattering description for OSP(m/2n)/OSP(m − 1/2n) supersphere sigma... more We conjecture the factorized scattering description for OSP(m/2n)/OSP(m − 1/2n) supersphere sigma models and OSP(m/2n) Gross Neveu models. The non unitarity of these field theories translates into a lack of ‘physical unitarity ’ of the S matrices, which are instead unitary with respect to the non-positive scalar product inherited from the orthosymplectic structure. Nevertheless, we find that formal thermodynamic Bethe ansatz calculations appear meaningful, reproduce the correct central charges, and agree with perturbative calculations. This paves the way to a more thorough study of these and other models with supergroup symmetries using the S matrix approach. 1

doi:10.3842/SIGMA.2010.039 Abstract. We present a study of the two species totally asymmetric dif... more doi:10.3842/SIGMA.2010.039 Abstract. We present a study of the two species totally asymmetric diffusion model using the Bethe ansatz. The Hamiltonian has Uq(SU(3)) symmetry. We derive the nested Bethe ansatz equations and obtain the dynamical critical exponent from the finite-size scaling properties of the eigenvalue with the smallest real part. The dynamical critical exponent is 3 2 which is the exponent corresponding to KPZ growth in the single species asymmetric diffusion model. Key words: asymmetric diffusion; nested Uq(SU(3)) Bethe ansatz; dynamical critical exponent 2010 Mathematics Subject Classification: 82C27; 82B20 1

We report on our recent results from a mathematical study of wire network graphs that are complem... more We report on our recent results from a mathematical study of wire network graphs that are complements to triply periodic CMC surfaces and can be synthesized in the lab on the nanoscale. Here, we studied all three cases in which the graphs corresponding to the networks are symmetric and self-dual. These are the cubic, diamond and gyroid surfaces. The gyroid is the most interesting case in its geometry and properties as it exhibits Dirac points (in 3d). It can be seen as a generalization of the honeycomb lattice in 2d that models graphene. Indeed, our theory works in more general cases, such as periodic networks in any dimension and even more abstract settings. After presenting our theoretical results, we aim to invite an experimental study of these Dirac points and a possible quantum Hall effect. The general theory also allows to find local symmetry groups which force degeneracies aka level crossings from a finite graph encoding the elementary cell structure. Vice-versa one could hop...

arXiv: Mathematical Physics, 2016

We generalize the mathbbZ_2\mathbb{Z}_2mathbbZ2 invariant of topological insulators using noncommutative differe... more We generalize the mathbbZ2\mathbb{Z}_2mathbbZ2 invariant of topological insulators using noncommutative differential geometry in two different ways. First, we model Majorana zero modes by KQ-cycles in the framework of analytic K-homology, and we define the noncommutative mathbbZ2\mathbb{Z}_2mathbbZ2 invariant as a topological index in noncommutative topology. Second, we look at the geometric picture of the Pfaffian formalism of the mathbbZ2\mathbb{Z}_2mathbbZ2 invariant, i.e., the Kane--Mele invariant, and we define the noncommutative Kane--Mele invariant over the fixed point algebra of the time reversal symmetry in the noncommutative 2-torus. Finally, we are able to prove the equivalence between the noncommutative topological mathbbZ2\mathbb{Z}_2mathbbZ_2 index and the noncommutative Kane--Mele invariant.

arXiv: Mathematical Physics, 2015

We analyze the topological mathbbZ_2\mathbb{Z}_2mathbbZ2 invariant, which characterizes time reversal invariant ... more We analyze the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant, which characterizes time reversal invariant topological insulators, in the framework of index theory and K-theory. The topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant counts the parity of generalized Majorana zero modes, which can be interpreted as an analytical index. As we show, it fits perfectly into a mod 2 index theorem, and the topological index provides an efficient way to compute the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant. Finally, we give a new version of the bulk-boundary correspondence which yields an alternative explanation of the index theorem and the topological mathbbZ2\mathbb{Z}_2mathbbZ_2 invariant. Here the boundary is not the geometric boundary of a probe, but an effective boundary in the momentum space.

arXiv: Mathematical Physics, 2016

We study the topological band theory of time reversal invariant topological insulators and interp... more We study the topological band theory of time reversal invariant topological insulators and interpret the topological mathbbZ_2\mathbb{Z}_2mathbbZ2 invariant as an obstruction in terms of Stiefel--Whitney classes. The band structure of a topological insulator defines a Pfaffian line bundle over the momentum space, whose structure group can be reduced to mathbbZ2\mathbb{Z}_2mathbbZ2. So the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant will be understood by the Stiefel--Whitney theory, which detects the orientability of a principal mathbbZ2\mathbb{Z}_2mathbbZ2-bundle. Moreover, the relation between weak and strong topological insulators will be understood based on cobordism theory. Finally, the topological mathbbZ2\mathbb{Z}_2mathbbZ_2 invariant gives rise to a fully extended topological quantum field theory (TQFT).

arXiv: Mathematical Physics, 2018

We study topological properties of families of Hamiltonians which may contain degenerate energy l... more We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus, we study local models. These give information about the Chern classes and Berry phases. We then give global constraints for the topological invariants. This is an hitherto relatively unexplored subject. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. The results can also be used in the study of deformations. We furthermore use these constraints to analyse examples which include the Gyroid geometry, which exhibits Weyl points and triple crossings and the honeycomb geometry with its two Dirac points.

We show how using Clifford algebras and their representations can greatly simplify the analysis o... more We show how using Clifford algebras and their representations can greatly simplify the analysis of integrable systems. In particular, we apply this approach to the XX-model with non-diagonal boundaries which is among others related to growing and fluctuating interfaces and stochastic reaction-diffusion systems. Using this approach, we can not only diagonalize the system, but also find new hidden symmetries.

Bulletin of the American Physical Society, 2020

The Journal of Chemical Physics

Quantum annealers are an alternative approach to quantum computing which make use of the adiabati... more Quantum annealers are an alternative approach to quantum computing which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising model Hamiltonian. In this paper we review two different methods for finding the ground state of molecular Hamiltonians using Ising model-based quantum annealers. In addition, we compare the relative effectiveness of each method by calculating the binding energies, bond lengths, and bond angles of the H + 3 and H 2 O molecules and mapping their potential energy curves. We also assess the resource requirements of each method by determining the number of qubits and computation time required to simulate each molecule using various parameter values. While each of these methods is capable of accurately predicting the ground state properties of small molecules, we find that they are still outperformed by modern classical algorithms and that the scaling of the resource requirements remains a challenge.

Physical Review A

We propose experimentally observable signatures of topological Majorana quasiparticles in the few... more We propose experimentally observable signatures of topological Majorana quasiparticles in the few-body limit of the interacting cold-atom model of Iemini et al.

Journal of Statistical Mechanics: Theory and Experiment

A new exponent characterizing the rounding of crystal facets is found by mapping a crystal surfac... more A new exponent characterizing the rounding of crystal facets is found by mapping a crystal surface onto the asymmetric six-vertex model (i.e. with external fields h and v) and using the Bethe ansatz to obtain appropriate expansions of the free energy close to criticality. Leading order exponents in δh, δv are determined along the whole phase boundary and in an arbitrary direction. A possible experimental verification of this result is discussed.

Reviews in Mathematical Physics

This paper is a survey of the [Formula: see text]-valued invariant of topological insulators used... more This paper is a survey of the [Formula: see text]-valued invariant of topological insulators used in condensed matter physics. The [Formula: see text]-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The [Formula: see text] invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.