# Dr Sam Carr

Lecturer in Physics

Dr Sam Carr graduated from the University of Oxford in 1999, before continuing, under the supervision of Professor Alexei Tsvelik, with his DPhil research which was split between Oxford and Brookhaven National Laboratory, NY, USA. He graduated in 2003 with a thesis entitled 'Non-perturbative solutions to quasi-one-dimensional strongly correlated systems'.  Sam then held various postdoctoral positions in the Abdus Salam International Center for Theoretical Physics, Trieste, Italy; the theoretical physics group at the University of Birmingham; and the Theorie der Kondensierten Materie group at the University of Karlsruhe, Germany. In 2013, he moved to the University of Kent to take up a Lecturer position.

## Research interests

Sam’s research interest is broadly in the field of strongly correlated materials – materials where the single particle picture breaks down due to interactions. His current interests can be divided into a few more specific areas. Low-dimensional materialsIn low dimensions, correlation effects are always more pronounced due to the lack of 'room' for particles to avoid each other. On the other hand, there are powerful non-perturbative techniques in one dimension such as bosonization, integrability and conformal field theory. This allows us to make progress in constructing the phase diagram and understanding the exotic correlated phases seen in low-dimensional models.Sam is particularly interested in ladder models, where it is possible to make progress on the old question of what happens when you try to combine orbital effects of magnetic field, a lattice, and interactions all within the same framework.  Another interesting example of a ladder model is the low-energy effective theory of carbon nanotubes, one of nature’s most perfect examples of a real one-dimensional system.Quasi-one-dimensional materials and dimensional crossoverWhile there is the occasional experimental example of a real one-dimensional system, such as carbon nanotubes, most experiments are done on real three-dimensional materials.  However, if the material is sufficiently anisotropic, one may consider it as weakly coupled lower dimensional units.  One can then ask the effect of this ‘inter-chain coupling’ on the phase diagram of the model.Real materials that can be modelled by such a scheme re Sr14Cu24O41, which has a spin gap, and under calcium doping and pressure has a superconducting transition; and the Bechgaard salts, a large class of organic quasi-one-dimensional crystals.Ultracold atomic systemsThe recent advances in atom trap technology and the development of optical lattices (also known as ‘crystals of light’) have allowed the creation and measurement of quantum systems with an unprecedented level of control. This allows one to think of them as ‘quantum analogue simulations’ of models of strongly correlated electrons. Along with these experiments, there is much theoretical work that can be done: among Sam’s current interests are exploitation of the dipole-dipole interaction to engineer interesting interaction geometries, and the effect of the harmonic trapping potential on the phases of the system.Transport and non-equilibrium noise through quantum dotsWhen materials become small enough in some dimensions (which is where interaction effects become strongest), standard scattering experiments become unfeasible due to the lack of scattering cross-section. For many true zero-dimensional or one-dimensionalsystems (as opposed to quasi-one-dimensional mentioned above), the only reasonable experiments that can be performed are transport. While transport measurements themselves will not usually well probe the
correlation properties of the system, the current-noise often will. These non-equilibrium properties, however, are still not completely understood, even for systems as simple as the Coulomb blockaded quantum dot.

## Publications

### Article

• Santos, R., Gutman, D. and Carr, S. (2019). Interplay between intrinsic and emergent topological protection on interacting helical modes. Physical Review B: Condensed Matter and Materials Physics [Online] 99:75129. Available at: https://doi.org/10.1103/PhysRevB.99.075129.
The interplay between topology and interactions on the edge of a two-dimensional topological insulator with time-reversal symmetry is studied. We consider a simple noninteracting system of three helical channels with an inherent
Z_2 topological protection and hence a zero-temperature conductance of G=e^2/h. We show that when interactions are added to the model, the ground state exhibits two different phases as a function of the interaction parameters. One of these phases is a trivial insulator at zero temperature, as the symmetry protecting the noninteracting topological phase is spontaneously broken. In this phase there is zero conductance (G=0) at zero temperature. The other phase displays enhanced topological properties, with a topologically protected zero-temperature conductance of G=3e^2/h and an emergent Z_3 symmetry not present in the lattice model. The neutral sector in this phase is described by a massive version of Z_3
parafermions. This state is an example of a dynamically enhanced symmetry-protected topological state.
• Camacho, G., Schmitteckert, P. and Carr, S. (2019). Exact equilibrium results in the interacting resonant level model. Physical Review B: Condensed Matter and Materials Physics [Online] 99:85122. Available at: https://doi.org/10.1103/PhysRevB.99.085122.
We present exact results for the susceptibility of the interacting resonant level model in equilibrium. Detailed simulations using both the numerical renormalization group and density matrix renormalization group were performed in order to compare with closed analytical expressions. By first bosonizing the model and then utilizing the integrability of the resulting boundary sine-Gordon model, one finds an analytic expression for the relevant energy scale T_K with excellent agreement to the numerical results. On the other hand, direct application of the Bethe ansatz of the interacting resonant level mode does not correctly reproduce T_K —however, if the bare parameters in the model are renormalized, then quantities obtained via the direct Bethe ansatz such as the occupation of the resonant level as a function of the local chemical potential do match the numerical results. The case of one lead is studied in the most detail, with many results also extending to multiple leads, although there still remain open questions in this case.
• Kainaris, N., Carr, S. and Mirlin, A. (2018). Transmission through a potential barrier in Luttinger liquids with a topological spin gap. Physical Review B [Online] 97:115107. Available at: https://doi.org/10.1103/PhysRevB.97.115107.
We study theoretically the transport of the one-dimensional single-channel interacting electron gas through a strong potential barrier in the parameter regime where the spin sector of the low-energy theory is gapped by interaction (Luther-Emery liquid). There are two distinct phases of this nature, of which one is of particular interest as it exhibits nontrivial interaction-induced topological properties. Focusing on this phase and using bosonization and an expansion in the tunneling strength we calculate the conductance through the barrier as a function of the temperature as well as the local density of states (LDOS) at the barrier. Our main result concerns the mechanism of bound-state-mediated tunneling. The characteristic feature of the topological phase is the emergence of protected zero-energy bound states with fractional spin located at the impurity position. By flipping this fractional spin, single electrons can tunnel across the impurity even though the bulk spectrum for spin excitations is gapped. This results in a finite LDOS below the bulk gap and in a nonmonotonic behavior of the conductance. The system represents an important physical example of an interacting symmetry-protected topological phase, which combines features of a topological spin insulator and a topological charge metal, in which the topology can be probed by measuring transport properties.
• Kainaris, N., Santos, R., Gutman, D. and Carr, S. (2017). Interaction induced topological protection in one-dimensional conductors. Fortschritte der Physik [Online] 65:1600054. Available at: https://doi.org/10.1002/prop.201600054.
We discuss two one?dimensional model systems – the first is a single channel quantum wire with Ising anisotropy, while the second is two coupled helical edge states. We show that the two models are governed by the same low energy effective field theory, and interactions drive both systems to exhibit phases which are metallic, but with all single particle excitations gapped. We show that such states may be either topological or trivial; in the former case, the system demonstrates gapless end states, and insensitivity to disorder.
• Carr, S., Santos, R. and Gutman, D. (2016). Phase diagram of two interacting helical states. Physical Review B: Condensed Matter and Materials Physics [Online] 93. Available at: http://dx.doi.org/10.1103/PhysRevB.93.235436.
We consider two coupled time-reversal-invariant helical edge modes of the same helicity, such as would occur on two stacked quantum spin Hall insulators. In the presence of interaction, the low-energy physics is described by two collective modes, one corresponding to the total current flowing around the edge and the other one describing relative fluctuations between the two edges.We find that quite generically, the relative mode becomes gapped at low temperatures, but only when tunneling between the two helical modes is nonzero. There are two distinct possibilities for the gapped state depending on the relative size of different interactions. If the intraedge interaction is stronger than the interedge interaction, the state is characterized as a spin-nematic phase. However,
in the opposite limit, when the interaction between the helical edge modes is strong compared to the interaction within each mode, a spin-density wave forms, with emergent topological properties. First, the gap protects the conducting phase against localization by weak nonmagnetic impurities; second, the protected phase hosts localized zero modes on the ends of the edge that may be created by sufficiently strong nonmagnetic impurities.
• Kainaris, N. and Carr, S. (2015). Emergent topological properties in interacting one-dimensional systems with spin-orbit coupling. PHYSICAL REVIEW B [Online] 92. Available at: https://doi.org/10.1103/PhysRevB.92.035139.
• Slizovskiy, S., Betouras, J., Carr, S. and Quintanilla, J. (2014). Effect of paramagnetic fluctuations on a Fermi-surface topological transition in two dimensions. Physical Review B [Online] 90:165110. Available at: http://dx.doi.org/10.1103/PhysRevB.90.165110.
We study the Fermi-surface topological transition of the pocket-opening type in a two-dimensional Fermi liquid. We find that the paramagnetic fluctuations in an interacting Fermi liquid typically drive the transition first order at zero temperature. We first gain insight from a calculation using second-order perturbation theory in the self-energy. This is valid for weak interaction and far from instabilities. We then extend the results to stronger interaction, using the self-consistent fluctuation approximation. Experimental signatures are given in light of our results.
• Kainaris, N., Gornyi, I., Carr, S. and Mirlin, A. (2014). Conductivity of a generic helical liquid. PHYSICAL REVIEW B [Online] 90. Available at: https://doi.org/10.1103/PhysRevB.90.075118.
• Mazo, V., Huang, C., Shimshoni, E., Carr, S. and Fertig, H. (2014). Superfluid-insulator transition of quantum Hall domain walls in bilayer graphene. Physical Review B - Condensed Matter and Materials Physics [Online] 89:121411-121416. Available at: http://www.dx.doi.org/10.1103/PhysRevB.89.121411.
We consider the Î½=0 quantum Hall ferromagnetic state of bilayer graphene subject to a kinklike perpendicular electric field, which generates domain walls in the electronic state and low-energy collective modes confined to move along them. In particular, it is shown that two pairs of collective helical modes are formed at opposite sides of the kink, each pair consisting of modes with identical helicities. We derive an effective field-theoretical model of these modes in terms of two weakly coupled anisotropic quantum spin ladders, with parameters tunable through control of the electric and magnetic fields. This yields a rich phase diagram, where, due to the helical nature of the modes, distinct phases possess very different charge conduction properties. Most notably, this system can potentially exhibit a transition from a superfluid to an insulating phase.
• Schmitteckert, P., Carr, S. and Saleur, H. (2014). Transport through nanostructures: Finite time versus finite size. Physical Review B - Condensed Matter and Materials Physics [Online] 89:81401-81406. Available at: http://www.dx.doi.org/10.1103/PhysRevB.89.081401.
Numerical simulations and experiments on nanostructures out of equilibrium usually exhibit strong finite size and finite measuring time tm effects. We discuss how these affect the determination of the full counting statistics for a general quantum impurity problem. We find that, while there are many methods available to improve upon finite-size effects, any real-time simulation or experiment will still be subject to finite-time effects: In short size matters, but time is limiting. We show that the leading correction to the cumulant generating function (CGF) at zero temperature for single-channel quantum impurity problems is proportional to lntm, where the constant of proportionality is universally related to the steady state CGF itself for non-interacting systems; universal in this context means independent of details of the quench procedure, i.e., independent of the switching on of both voltage and counting field. We give detailed numerical evidence for the case of the self-dual interacting resonant level model that this relation survives the addition of interactions. This allows the extrapolation of finite measuring time in our numerics to the long-time limit, in excellent agreement with Bethe-ansatz results.
• Carr, S., Narozhny, B. and Nersesyan, A. (2013). Spinful fermionic ladders at incommensurate filling: Phase diagram, local perturbations, and ionic potentials. Annals of Physics [Online] 339:22-80. Available at: http://www.dx.doi.org/10.1016/j.aop.2013.08.007.
We study the effect of external potential on transport properties of the fermionic two-leg ladder model. The response of the system to a local perturbation is strongly dependent on the ground state properties of the system and especially on the dominant correlations. We categorize all phases and transitions in the model (for incommensurate filling) and introduce "hopping-driven transitions" that the system undergoes as the inter-chain hopping is increased from zero. We also describe the response of the system to an ionic potential. The physics of this effect is similar to that of the single impurity, except that the ionic potential can affect the bulk properties of the system and in particular induce true long range order.
• Huang, C., Carr, S., Gutman, D., Shimshoni, E. and Mirlin, A. (2013). Transport via double constrictions in integer and fractional topological insulators. Physical Review B - Condensed Matter and Materials Physics [Online] 88:125134-125134. Available at: http://www.scopus.com/inward/record.url?eid=2-s2.0-84884869078&partnerID=40&md5=fdb35fde5df225a7b51b827083e83a4f.
We study transport properties of the helical edge states of two-dimensional integer and fractional topological insulators via double constrictions. Such constrictions couple the upper and lower edges of the sample and can be made and tuned by adding side gates to the system. Using renormalization group and duality mapping, we analyze phase diagrams and transport properties in each of these cases. Most interesting is the case of two constrictions tuned to resonance, where we obtain Kondo behavior, with a tunable Kondo temperature. Moving away from resonance gives the possibility of a metal-insulator transition at some finite detuning. For integer topological insulators, this physics is predicted to occur for realistic interaction strengths and gives a conductance G with two temperature T scales where the sign of dG/dT changes, one being related to the Kondo temperature while the other is related to the detuning. Â© 2013 American Physical Society.
• Carr, S., Fellows, J., Hooley, C. and Schmalian, J. (2012). Unbinding of Giant Vortices in States of Competing Order. Physical Review Letters: Moving Physics Forward [Online] 109:155703. Available at: http://dx.doi.org/10.1103/PhysRevLett.109.155703.
We consider a two-dimensional system with two order parameters, one with O(2) symmetry and one with O(M), near a point in parameter space where they couple to become a single O(2+M) order. While the O(2) sector supports vortex excitations, these vortices must somehow disappear as the high symmetry point is approached. We develop a variational argument which shows that the size of the vortex cores diverges as 1/???? and the Berezinskii-Kosterlitz-Thouless transition temperature of the O(2) order vanishes as 1/ln(1/?), where ? denotes the distance from the high-symmetry point. Our physical picture is confirmed by a renormalization group analysis which gives further logarithmic corrections, and demonstrates full symmetry restoration within the cores.
• Schneider, M., Carr, S., Gornyi, I. and Mirlin, A. (2012). Weak localization and magnetoresistance in a two-leg ladder model. Physical Review B - condensed matter and materials physics [Online] 86:155141-155158. Available at: http://www.dx.doi.org/10.1103/PhysRevB.86.155141.
We analyze the weak-localization correction to the conductivity of a spinless two-leg ladder model in the limit of strong dephasing ? ?? tr,, paying particular attention to the presence of a magnetic field, which leads to an unconventional magnetoresistance behavior. We find that the magnetic field leads to three different effects: (i) negative magnetoresistance due to the regular weak-localization correction, (ii) effective decoupling of the two chains, leading to positive magnetoresistance, and (iii) oscillations in the magnetoresistance originating from the nature of the low-energy collective excitations. All three effects can be observed depending on the parameter range, but it turns out that large magnetic fields always decouple the chains and thus lead to the curious effect of magnetic-field-enhanced localization.
• Carr, S., Bagrets, D. and Schmitteckert, P. (2011). Full Counting Statistics in the Self-Dual Interacting Resonant Level Model. Physical Review Letters [Online] 107:206801-206801. Available at: http://dx.doi.org/10.1103/PhysRevLett.107.206801.
We present a general technique to obtain the zero temperature cumulant generating function of the full counting statistics of charge transfer in interacting impurity models out of equilibrium from time-dependent simulations on a lattice. We demonstrate the technique with application to the self-dual interacting resonant level model, where very good agreement between numerical simulations using the density matrix renormalization group and those obtained analytically from the thermodynamic Bethe ansatz is found. We show from the exact form of counting statistics that the quasiparticles involved in transport carry charge 2e in the low bias regime and e/2 in the high bias regime.
• Fellows, J. and Carr, S. (2011). Superfluid, solid, and supersolid phases of dipolar bosons in a quasi-one-dimensional optical lattice. Physical Review A: Atomic, Molecular and Optical Physics [Online] 84:51602-51607. Available at: http://www.dx.doi.org/10.1103/PhysRevA.84.051602.
We discuss a model of dipolar bosons trapped in a weakly coupled planar array of one-dimensional tubes. We consider the situation where the dipolar moments are aligned by an external field, and we find a rich phase diagram as a function of the angle of this field exhibiting quantum phase transitions between solid, superfluid, and supersolid phases. In the low energy limit, the model turns out to be identical to one describing quasi-one-dimensional superconductivity in condensed matter systems. This opens the possibility of using bosons as a quantum analog simulator of electronic systems, a scenario arising from the intricate relation between statistics and interactions in quasi-one-dimensional systems.
• Carr, S., Narozhny, B. and Nersesyan, A. (2011). Effect of a local perturbation in a fermionic ladder. Physical Review Letters: Moving Physics Forward [Online] 106:126805-126809. Available at: http://www.dx.doi.org/10.1103/PhysRevLett.106.126805.
We study the effect of a local external potential on a system of two parallel spin-polarized nanowires placed close to each other. For single-channel nanowires with repulsive interaction we find that transport properties of the system are highly sensitive to the transverse gradient of the perturbation: the asymmetric part completely reflects the electrons leading to vanishing conductance at zero temperature, while the flat potential remains transparent. We envisage a possible application of this unusual property in the sensitive measurement of local potential field gradients.
• Carr, S., Quintanilla, J. and Betouras, J. (2010). Lifshitz transitions and crystallization of fully polarized dipolar fermions in an anisotropic two-dimensional lattice. Physical Review B: Condensed Matter and Materials Physics [Online] 82:045110 -1. Available at: http://dx.doi.org/10.1103/PhysRevB.82.045110.
We consider a two-dimensional model of noninteracting chains of spinless fermions weakly coupled via a small interchain hopping and a repulsive interchain interaction. The phase diagram of this model has a surprising feature: an abrupt change in the Fermi surface as the interaction is increased. We study in detail this metanematic transition and show that the well-known 21/2-order Lifshitz transition is the critical end point of this first-order quantum phase transition. Furthermore, in the vicinity of the end point, the order parameter has a nonperturbative BCS-type form. We also study a competing crystallization transition in this model and derive the full phase diagram. This physics can be demonstrated experimentally in dipolar ultracold atomic or molecular gases. In the presence of a harmonic trap, it manifests itself as a sharp jump in the density profile.
• Carr, S., Quintanilla, J. and Betouras, J. (2009). Deconfinement and Quantum Liquid Crystalline States of Dipolar Fermions in Optical Lattices. International Journal of Modern Physics B [Online] 23:4074-4086. Available at: http://dx.doi.org/10.1142/S0217979209063262.
We describe a simple model of fermions in quasi-one dimension that features interaction-induced deconfinement (a phase transition where the effective dimensionality of the system increases as interactions are turned on) and which can be realised using dipolar fermions in an optical lattice(1). The model provides a relisation of a "soft quantum matter" phase diagram of strongly-correlated fermions, featuring meta-nematic, smectic and crystalline states, in addition to the normal Fermi liquid. In this paper we review the model and discuss in detail the mechanism behind each of these transitions on the basis of bosonization and detailed analysis of the RPA susceptibility.
• Quintanilla, J., Carr, S. and Betouras, J. (2009). Metanematic, smectic, and crystalline phases of dipolar fermions in an optical lattice. Physical Review A: Atomic, Molecular and Optical Physics [Online] 79. Available at: http://dx.doi.org/10.1103/PhysRevA.79.031601.
It has been suggested that some strongly correlated matter might be understood qualitatively in terms of liquid crystalline phases intervening between the Fermi gas and the Wigner crystal or Mott insulator. We propose a tunable realization of this soft quantum matter physics in an ultracold gas. It uses optical lattices and dipolar interactions to realize a particularly simple model. Our analysis reveals a rich phase diagram featuring a metanematic transition where the Fermi liquid changes dimensionality; a smectic phase (stripes) and a crystalline "checkerboard" phase.
• Carr, S. (2008). Strong correlation effects in single-wall carbon nanotubes. International Journal of Modern Physics B [Online] 22:5235-5260. Available at: http://www.dx.doi.org/10.1142/S0217979208049455.
We present an overview of strong correlations in single-wall carbon nanotubes, and an introduction to the techniques used to study them theoretically. We concentrate on zigzag nanotubes, although universality dictates that much of the theory can also be applied to armchair or chiral nanotubes. We show how interaction effects lead to exotic low energy properties and discuss future directions for studies on correlation effects in nanotubes.
• Carr, S., Gogolin, A. and Nersesyan, A. (2007). Interaction induced dimerization in zigzag single wall carbon nanotubes. Physical Review B - Condensed Matter and Materials Physics [Online] 76:245121-245130. Available at: http://www.dx.doi.org/10.1103/PhysRevB.76.245121.
We derive a low-energy effective model of metallic zigzag carbon nanotubes at half filling. We show that there are three important features characterizing the low-energy properties of these systems: the long-range Coulomb interaction, umklapp scattering, and an explicit dimerization generated by interactions. The ratio of the dimerization induced gap and the Mott gap induced by the umklapp interactions is dependent on the radius of the nanotube and can drive the system upon increasing dimerization strength from a Haldane spin-liquid phase through a quantum phase transition with SU (2)1 quantum symmetry to a dimerized phase. We consider the physical properties of the phases on either side of this transition, which should be relevant for realistic nanotubes.
• Carr, S., Narozhny, B. and Nersesyan, A. (2006). Spinless fermionic ladders in a magnetic field: Phase diagram. Physical Review B - Condensed Matter and Materials Physics [Online] 73:195114-195128. Available at: http://www.dx.doi.org/10.1103/PhysRevB.73.195114.
The system of interacting spinless fermions hopping on a two-leg ladder exhibits a series of quantum phase transitions when subjected to an external magnetic field. At half-filling, these are either U(1) Gaussian phase transitions between two phases with distinct types of long-range order or Berezinskii-Kosterlitz-Thouless transitions between ordered and gapless phases.
• Narozhny, B., Carr, S. and Nersesyan, A. (2005). Fractional charge excitations in fermionic ladders. Physical Review B: Condensed Matter and Materials Physics [Online] 71:161101-161105. Available at: http://www.dx.doi.org/10.1103/PhysRevB.71.161101.
The system of interacting spinless fermions hopping on a two-leg ladder in the presence of an external magnetic field is shown to possess a long-range order: the bond density wave or the staggered flux phase. In both cases the elementary excitations are Z2 kinks and carry one half the charge of an electron. Â© 2005 The American Physical Society.
• Carr, S. and Tsvelik, A. (2003). Spectrum and correlation functions of a quasi-one-dimensional quantum Ising model. Physical Review Letters: Moving Physics Forward [Online] 90:177206-177210. Available at: http://www.dx.doi.org/10.1103/PhysRevLett.90.177206.
The dynamical susceptibility was studied, focusing on the region of the phase diagram well below the transition line where new nonuniversal physics can be found. It was shown that some of the beautiful physics of the quantum Ising model in a magnetic field with a hidden E8 symmetry may be observed even in a realistic quasi-one-dimensional model in its ordered phase far from the transition line.
• Carr, S. and Tsvelik, A. (2002). Superconductivity and charge-density waves in a quasi-one-dimensional spin-gap system. Physical Review B: Condensed Matter and Materials Physics [Online] 65:195121-195131. Available at: http://www.dx.doi.org/10.1103/PhysRevB.65.195121.
We consider a model of spin-gapped chains weakly coupled by Josephson and Coulomb interactions. Combining such nonperturbative methods as bosonization and the Bethe ansatz to treat the intrachain interactions with the random phase approximation for the interchain couplings and the first corrections to this, we investigate the phase diagram of this model. The phase diagram shows both charge-density wave ordering and superconductivity. These phases are separated by line of critical points which exhibits an approximate SU(2) symmetry. We consider the effects of a magnetic field on the system. We apply the theory to the material Sr2Ca12Cu24O41 and suggest further experiments.

### Conference or workshop item

• Mazo, V., Shimshoni, E., Huang, C., Carr, S. and Fertig, H. (2015). Helical quantum Hall edge modes in bilayer graphene: a realization of quantum spin-ladders. In: Frontiers of Quantum and Mesoscopic Thermodynamics (FQMT13). IOP Science, p. 14019. Available at: http://dx.doi.org/10.1088/0031-8949/2015/T165/014019.
The rich phase diagram of quantum spin-ladder systems has attracted much attention in the theoretical literature. The progress in experimental realizations of this fascinating physics however has been much slower. While materials with a ladder-like structure exist, one always has coupling between the ladders to muddy the waters. In addition, such materials exhibit limited (if any) tunability in terms of the magnetic exchange parameters, and experimental probing of the different phases is a great challenge. In this work, we show that a realization of spin-ladder physics can occur in an engineered nanostructure made out of bilayer graphene in the $\nu =0$ quantum Hall state. Specifically, we describe a split-double-gated setup in which a domain wall (DW) is explicitly induced in the middle of the sample, and show that an effective spin-ladder forms along this DW. The interaction strengths of the ladder are tunable by adjusting magnetic and electric fields as well as the spacing between the gates. Furthermore, we demonstrate that the effective spin ladder has a helical nature, meaning that the spin-correlations may be probed rather simply with charge transport experiments. We describe the phase diagram of this system, and show that certain transport measurements are very sensitive to the phase.
• Carr, S., Schmitteckert, P. and Saleur, H. (2015). Full counting statistics in the not-so-long-time limit. In: Frontiers of Quantum and Mesoscopic Thermodynamics (FQMT13). IOP Science, p. 14009. Available at: http://doi.org/10.1088/0031-8949/2015/T165/014009.
The full counting statistics of charge transport is the probability distribution ${p}_{n}({t}_{m})$ that n electrons have flown through the system in measuring time tm. The cumulant generating function (CGF) of this distribution $F(\chi ,{t}_{m})$ has been well studied in the long time limit ${t}_{m}\to \infty$, however there are relatively few results on the finite measuring time corrections to this. In this work, we study the leading finite time corrections to the CGF of interacting Fermi systems with a single transmission channel at zero temperature but driven out of equilibrium by a bias voltage. We conjecture that the leading finite time corrections are logarithmic in tm with a coefficient universally related to the long time limit. We provide detailed numerical evidence for this with reference to the self-dual interacting resonant level model. This model further contains a phase transition associated with the fractionalization of charge at a critical bias voltage. This transition manifests itself technically as branch points in the CGF. We provide numerical results of the dependence of the CGF on measuring time for model parameters in the vicinity of this transition, and thus identify features in the time evolution associated with the phase transition itself.

### Thesis

• Hewitt, T. (2017). Phase Diagram of the Anisotropic Heisenberg Spin Ladder.
In this thesis, we deal with 1-dimensional anisotropic spin models more specifically with the chain and ladder geometries. These models are fundamental to the study of condensed matter theory and quantum magnetism because of their simplicity. Research into the chain model dates back to 1930s with Hans Bethe. One method to study these systems is through numerical analysis including renormalization group techniques. These allow for the diagonalization of the Hamiltonian without sacrificing computation to a large virtual Hilbert space.

In this thesis, we present results for both the spin chain and spin ladder geometries in open boundary conditions. This work uses the density matrix renormalization group technique to calculate system energy. Initially we will present a study into the spin energy gap of the spin-\sfrac{1}{2} anisotropic (XXZ) Heisenberg chain in open boundary conditions (OBCs). The energy gap is shown to be reduced to half in the ground sector when compared to periodic boundary conditions (PBCs) due to edge effects. Secondly, a full phase diagram for the spin-\sfrac{1}{2} anisotropic Heisenberg ladder is presented which shows the emergence of a rich schematic with a variety of phases. Lastly, weak coupling limit maps are compared to quantum field theory phase transition predictions. This research was motivated by the lack of comprehensive phase diagram results for ladders and by real materials being investigated at the University of Kent.
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