# Dr Jorge Quintanilla

Senior Lecturer in Condensed Matter Theory
Group Leader of the Physics of Quantum Materials Research Group

Dr Jorge Quintanilla studied Physics at the University of Salamanca in his native Spain (Licenciado, 1992-1997) and at the University of Bristol (PhD, 1997-2001) where he completed a thesis on unconventional superconductors under the supervision of the late Balazs L. Gyorffy. He then moved to Brazil, where he held a postdoctoral position at the University of Sao Paulo at Sao Carlos (2002) and worked with Luiz Oliveira and Klaus Capelle on proximity effects in superconductors and with Vivaldo L. Campo on electronic statates in small-world networks.Moving back to the UK to take up a research fellowship at the University of Birmingham (2003-2005), Jorge began new collaborations with his supervisor Andy Schofield, with Chris Hooley and, later, with Sam Carr. During this time his research interests broadened further to include quantum liquid crystal states and the Pomeranchuk instability and ultra-cold atoms. After Birmingham, Jorge took up the Atlas Research Fellowship in Condensed Matter Theory at the ISIS Facility in the STFC Rutherford-Appleton Laboratory, in Harwell, and at St Catherine's College, Oxford (2005-2010). At RAL, he developed a keen interest in extending, through his theoretical research, the capabilities of large-scale facilities employed in condensed matter research, including the muon and neutron instruments at the ISIS spallation neutron source.During the course of his career Jorge has been an Individual Marie Curie Fellow; a Fellow of St Catherine’s College, Oxford; and an honorary research associate of the London Centre for Nanotechnology. He also chaired SEPnet's Atomic and Condensed Matter research theme (2010-2012) and is a founding member of the Hubbard Theory Consortium.

## Research interests

Dr Jorge Quintanilla is a theorist working on quantum condensed matter and materials physics. His main interests are in the area of strongly correlated quantum matter and include unconventional superconductors, frustrated and quantum magnets, and other materials with strongly-correlated electrons. Jorge's current interests include broken time-reversal symmetry in centrosymmetric and non-centrosymmetric superconductors, topological defects and excitations in condensed matter and experimental signatures of quantum entanglement in magnetic materials. Past interests also include ultra-cold gases and complex networks.

## Teaching

Jorge is involved in the teaching of superconductivity and magnetism, quantum mechanics and electricity and light. He is one of the lecturers in the biannual SEPnet GRADnet workshop on strong correlations.

## Supervision

Jorge can offer PhD and MSc (R) projects in a number of areas, including unconventional superconductors, entanglement in quantum magnets and magnetic monopoles in spin ice. Every year he also offers one or more MPhys research projects. Visit his blog for more information on student research opportunities within his group.

## Publications

### Article

• Csire, G., Annett, J., Quintanilla, J. and Újfalussy, B. (2020). First-principles based theory of spin-orbit coupling induced triplet pairing: Application to the superconducting ground state of rhenium. arXiv.org [Online]. Available at: https://arxiv.org/abs/2005.05702.
Recent μSR measurements revealed that spontaneous magnetism exists in the superconducting state of rhenium and it also appears in other rhenium based materials like Re6Zr, Re6Hf, Re6Ti. The superconducting state of these materials show s-wave-like properties and the pairing mechanism is most likely driven by electron-phonon coupling. In this paper we take elemental rhenium as a testbed and investigate its ground state. By developing an LCAO formalism for the solution of the spin-generalized Bogoliubov-de Gennes equation we use every details of the first-principles band-structure together with spin-orbit coupling. In this paper we provide a possible explanation of the spontaneous time-reverseal symmetry breaking in the superconducting ground state of rhenium by arguing that taking into account the orbital degrees of freedom, spin-orbit coupling is inducing even-parity odd-orbital spin triplet Cooper pairs, and Cooper pairs' migration between the equal-spin triplet states may lower the total energy. We show how magnetism emerges and the structure of the gap changes as a function of the triplet component of the interaction strength.
• Santos, D., Faust, J. and Quintanilla, J. (2020). Feedback: A brief history of kite physics. Physics World [Online] 33:20-21. Available at: http://dx.doi.org/10.1088/2058-7058/33/2/28.
In reply to Margaret Harris's feature on the challenges and opportunities for airborne wind energy ("Harnessing the wind" December 2019, pp26–30).
• Tomasello, B., Castelnovo, C., Moessner, R. and Quintanilla, J. (2019). Correlated Quantum Tunneling of Monopoles in Spin Ice. Physical Review Letters [Online] 123. Available at: https://dx.doi.org/10.1103/PhysRevLett.123.067204.
The spin ice materials Ho$$_2$$Ti$$_2$$O$$_7$$ and Dy$$_2$$Ti$$_2$$O$$_7$$ are by now perhaps the best-studied classical frustrated magnets. A crucial step towards the understanding of their low temperature behaviour -- both regarding their unusual dynamical properties and the possibility of observing their quantum coherent time evolution -- is a quantitative understanding of the spin-flip processes which underpin the hopping of magnetic monopoles. We attack this problem in the framework of a quantum treatment of a single-ion subject to the crystal, exchange and dipolar fields from neighbouring ions. By studying the fundamental quantum mechanical mechanisms, we discover a bimodal distribution of hopping rates which depends on the local spin configuration, in broad agreement with rates extracted from experiment. Applying the same analysis to Pr$$_2$$Sn$$_2$$O$$_7$$ and Pr$$_2$$Zr$$_2$$O$$_7$$, we find an even more pronounced separation of timescales signalling the likelihood of coherent many-body dynamics.
• Shang, T., Smidman, M., Ghosh, S., Baines, C., Chang, L., Gawryluk, D., Barker, J., Singh, R., Paul, D., Balakrishnan, G., Pomjakushina, E., Shi, M., Medarde, M., Hillier, A., Yuan, H., Quintanilla, J., Mesot, J. and Shiroka, T. (2018). Time-reversal symmetry breaking in Re-based superconductors. Physical Review Letters [Online] 121. Available at: https://dx.doi.org/10.1103/PhysRevLett.121.257002.
To trace the origin of time-reversal symmetry breaking (TRSB) in Re-based superconductors, we performed comparative muon-spin rotation/relaxation (µSR) studies of superconducting noncentrosymmetric Re$$_{0.82}$$Nb$$_{0.18}$$ ($$T$$$$_c$$ = 8.8 K) and centrosymmetric Re ($$T$$$$_c$$ = 2.7 K). In Re$$_{0.82}$$Nb$$_{0.18}$$, the low-temperature superfluid density and the electronic specific heat evidence a fully-gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling. In both Re$$_{0.82}$$Nb$$_{0.18}$$ and pure Re, the spontaneous magnetic fields revealed by zero-field µSR below $$T$$$$_c$$ indicate time-reversal symmetry breaking and thus unconventional superconductivity. The concomitant occurrence of TRSB in centrosymmetric Re and noncentrosymmetric Re$$T$$ ($$T$$ = transition metal), yet its preservation in the isostructural noncentrosymmetric superconductors Mg$$_{10}$$Ir$$_{19}$$B$$_{16}$$ and Nb$$_{0.5}$$Os$$_{0.5}$$, strongly suggests that the local electronic structure of Re is crucial for understanding the TRSB superconducting state in Re and Re$$T$$. We discuss the superconducting order parameter symmetries that are compatible with the observations.
• Ghosh, S., Annett, J. and Quintanilla, J. (2018). Time-reversal symmetry breaking in superconductors through loop Josephson-current order. arXiv [Online]. Available at: https://arxiv.org/abs/1803.02618.
We propose a superconducting instability where loop Josephson-currents form spontaneously
within a unit cell at the critical temperature, T c . Such currents break time-reversal symmetry
(TRS) without needing an unconventional pairing mechanism. Using Ginzburg-Landau theory we
show how they emerge in a toy model and estimate the size of the resulting magnetization, which
is consistent with recent muon-spin relaxation experiments. We discuss the crystal symmetry re-
quirements and show that they are met by the Re 6 X (X=Zr, Hf, Ti) family of TRS-breaking, but
otherwise seemingly conventional, superconductors.
• Whittlesea, P., Quintanilla, J., Annett, J., Hillier, A. and Hooley, C. (2018). Can topological transitions be exploited to engineer intrinsically quench-resistant wires?. IEEE Transactions on Applied Superconductivity [Online] 28. Available at: https://doi.org/10.1109/TASC.2018.2791515.
We investigate whether by synthesising superconductors that are tuned to a topological, node-reconstruction transition point we could create superconducting wires that are intrinsically resilient to quenches. Recent work shows that the exponent characterising the temperature dependence of the specific heat of a nodal superconductor is lowered over a region of the phase diagram near topological transitions where nodal lines form or reconnect. Our idea is that the resulting enhancement of the low-temperature specific heat could have potential application in the prevention of superconductor quenches. We perform numerical simulations of a simplified superconductor quench model. Results show that decreasing the specific heat exponent can prevent a quench from occurring and improve quench resilience, though in our simple model the effects are small. Further work will be necessary to establish the practical feasibility of this approach.
• Irons, H., Quintanilla, J., Perring, T., Amico, L. and Aeppli, G. (2017). Control of entanglement transitions in quantum spin clusters. Physical Review B: Condensed Matter and Materials Physics [Online] 96:224408-224408. Available at: http://dx.doi.org/10.1103/PhysRevB.96.224408.
Quantum spin clusters provide a new platform for the experimental study of many-body entanglement. Here we address a simple model of a single-molecule nano-magnet featuring N interacting spins in a transverse field. The field can control an entanglement transition (ET). We calculate the magnetisation, low-energy gap and neutron-scattering cross-section and find that the ET has distinct signatures, detectable at temperatures as high as 5% of the interaction strength. The signatures are stronger for smaller clusters.
• Weng, Z., Zhang, J., Smidman, M., Shang, T., Quintanilla, J., Annett, J., Nicklas, M., Pang, G., Jiao, L., Jiang, W., Chen, Y., Stelgich, F. and Yuan, H. (2016). Two-gap superconductivity in LaNiGa$$_2$$ with nonunitary triplet pairing and even parity gap symmetry. Physical Review Letters [Online] 117. Available at: http://dx.doi.org/10.1103/PhysRevLett.117.027001.
The nature of the pairing states of superconducting LaNiC$$_2$$ and LaNiGa$$_2$$ has to date remained a puzzling question. Broken time reversal symmetry has been observed in both compounds and a group theoretical analysis implies a non-unitary triplet pairing state. However all the allowed non-unitary triplet states have nodal gap functions but most thermodynamic and NMR measurements indicate fully gapped superconductivity in LaNiC$$_2$$. Here we probe the gap symmetry of LaNiGa$$_2$$ by measuring the London penetration depth, specific heat and upper critical field. These measurements demonstrate two-gap nodeless superconductivity in LaNiGa$$_2$$, suggesting that this is a common feature of both compounds. These results allow us to propose a novel triplet superconducting state, where the pairing occurs between electrons of the same spin, but on different orbitals. In this case the superconducting wavefunction has a triplet spin component but isotropic even parity gap symmetry, yet the overall wavefunction remains antisymmetric under particle exchange. This model leads to a nodeless two-gap superconducting state which breaks time reversal symmetry, and therefore accounts well for the seemingly contradictory experimental results.
• Tomasello, B., Castelnovo, C., Moessner, R. and Quintanilla, J. (2015). Single-ion anisotropy and magnetic field response in the spin-ice materials Ho2Ti2O7 and Dy2Ti2O7. Physical Review B: Condensed Matter and Materials Physics [Online] 92:155120-1. Available at: http://dx.doi.org/10.1103/PhysRevB.92.155120.
Motivated by its role as a central pillar of current theories of the dynamics of spin ice in and out of equilibrium, we study the single-ion dynamics of the magnetic rare-earth ions in their local environments, subject to the effective fields set up by the magnetic moments with which they interact. This effective field has a transverse component with respect to the local easy axis of the crystal electric field, which can induce quantum tunneling. We go beyond the projective spin-1/2 picture and use instead the full crystal-field Hamiltonian. We find that the Kramers versus non-Kramers nature, as well as the symmetries of the crystal-field Hamiltonian, result in different perturbative behavior at small fields (?1T), with transverse field effects being more pronounced in Ho2Ti2O7 than in Dy2Ti2O7. Remarkably, the energy splitting range we find is consistent with time scales extracted from experiments. We also present a study of the static magnetic response, which highlights the anisotropy of the system in the form of an off-diagonal g tensor, and we investigate the effects of thermal fluctuations in the temperature regime of relevance to experiments. We show that there is a narrow but accessible window of experimental parameters where the anisotropic response can be observed.
• 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.
• Singh, R., Hillier, A., Mazidian, B., Quintanilla, J., Annett, J., Paul, D., Balakrishnan, G. and Lees, M. (2014). Detection of Time-Reversal Symmetry Breaking in the Noncentrosymmetric Superconductor Re$$_6$$Zr Using Muon-Spin Spectroscopy. Physical Review Letters [Online] 112. Available at: http://dx.doi.org/10.1103/PhysRevLett.112.107002.
We have investigated the superconducting state of the non-centrosymmetric compound Re$$_6$$Zr using magnetization, heat capacity, and muon-spin relaxation/rotation ($$\mu$$SR) measurements. Re$$_6$$Zr has a superconducting transition temperature, T$$_c$$ = 6.75±0.05 K. Transverse-field $$\mu$$SR experiments, used to probe the superfluid density, suggest an $$s$$-wave character for the superconducting gap. However, zero and longitudinal-field $$\mu$$SR data reveal the presence of spontaneous static magnetic fields below $$T$$$$_c$$ indicating that time-reversal symmetry is broken in the superconducting state and an unconventional pairing mechanism. An analysis of the pairing symmetries identifies the ground states compatible with time-reversal symmetry breaking.
• Bhattacharyya, A., Adroja, D., Quintanilla, J., Hillier, A., Kase, N., Strydom, A. and Akimitsu, J. (2014). Broken time-reversal symmetry probed by muon spin relaxation in the caged type superconductor Lu5Rh6Sn18. Physical Review B: Condensed Matter and Materials Physics [Online] 91:060503-1. Available at: http://link.aps.org/doi/10.1103/PhysRevB.91.060503.
The superconducting state of the caged type compound Lu5Rh6Sn18 has been investigated by using magnetization, heat capacity, and muon spin relaxation or rotation (?SR) measurements, and the results interpreted on the basis of the group theoretical classifications of the possible pairing symmetries and a simple model of the resulting quasiparticle spectra. Our zero-field ?SR measurements clearly reveal the spontaneous appearance of an internal magnetic field below the transition temperature, which indicates that the superconducting state in this material is characterized by broken time-reversal symmetry. Further, the analysis of the temperature dependence of the magnetic penetration depth measured using the transverse-field ?SR measurements suggests an isotropic s?wave character for the superconducting gap. This is in agreement with the heat capacity behavior, and we show that it can be interpreted in terms of a nonunitary triplet state with point nodes and an open Fermi surface.
• Mazidian, B., Quintanilla, J., Hillier, A. and Annett, J. (2013). Anomalous thermodynamic power laws near topological transitions in nodal superconductors. Physical Review B: Condensed Matter and Materials Physics [Online] 88:224504. Available at: http://dx.doi.org/10.1103/PhysRevB.88.224504.
Unconventional superconductors are most frequently identified by the observation of power-law behavior on low-temperature thermodynamic or transport properties, such as specific heat. Here, we show that, in addition to the usual point and line nodes, a much wider class of different nodal types can occur. These new types of nodes typically occur when there are transitions between different types of gap node topology, for example, when point or line nodes first appear as a function of some physical parameter. We identify anomalous, noninteger thermodynamic power laws associated with these new nodal types, and give physical examples of superconductors in which they might be observed experimentally, including the noncentrosymmetric superconductor Li2Pd3?xPtxB.
• Campo, V., Capelle, K., Hooley, C., Quintanilla, J. and Scarola, V. (2012). Thermal versus quantum fluctuations of optical-lattice fermions. Physical Review A: Atomic, Molecular and Optical Physics [Online] 85:33644. Available at: http://dx.doi.org/10.1103/PhysRevA.85.033644.
We show that, for fermionic atoms in a one-dimensional optical lattice, the fraction of atoms in doubly occupied sites is a highly nonmonotonic function of temperature. We demonstrate that this property persists even in the presence of realistic harmonic confinement, and that it leads to a suppression of entropy at intermediate temperatures that offers a route to adiabatic cooling. Our interpretation of the suppression is that such intermediate temperatures are simultaneously too high for quantum coherence and too low for significant thermal excitation of double occupancy thus offering a clear indicator of the onset of quantum fluctuations.
• Hillier, A., Quintanilla, J., Mazidian, B., Annett, J. and Cywinski, R. (2012). Nonunitary Triplet Pairing in the Centrosymmetric Superconductor LaNiGa2. Physical Review Letters: Moving Physics Forward [Online] 109:0-0. Available at: http://link.aps.org/doi/10.1103/PhysRevLett.109.097001.
Muon spin rotation and relaxation experiments on the centrosymmetric intermetallic superconductor LaNiGa2 are reported. The appearance of spontaneous magnetic fields coincides with the onset of superconductivity, implying that the superconducting state breaks time reversal symmetry, similarly to noncentrosymmetric LaNiC2. Only four triplet states are compatible with this observation, all of which are nonunitary triplets. This suggests that LaNiGa2 is the centrosymmetric analogue of LaNiC2. We argue that these materials are representatives of a new family of paramagnetic nonunitary superconductors.
• Quintanilla, J., Hillier, A., Annett, J. and Cywinski, R. (2010). Relativistic analysis of the pairing symmetry of the noncentrosymmetric superconductor LaNiC2. Physical Review B: Condensed Matter and Materials Physics [Online] 82:174511. Available at: http://dx.doi.org/10.1103/PhysRevB.82.174511.
We present a relativistic symmetry analysis of the allowed pairing states in the noncentroymmetric superconductor LaNiC2. The case of zero spin-orbit coupling (SOC) is discussed first and then the evolution of the symmetry-allowed superconducting instabilities as SOC is adiabatically turned on is described. In addition to mixing singlet with triplet pairing, SOC splits some triplet pairing states with degenerate order-parameter spaces into nondegenerate pairing states with different critical temperatures. We address the breaking of time-reversal symmetry detected in recent muon spin-relaxation experiments and show that it is only compatible with such nonunitary triplet pairing states. In particular, an alternative scenario featuring conventional singlet pairing with a small admixture of triplet pairing is shown to be incompatible with the experimental data.
• 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.
• Ciftja, O. and Quintanilla, J. (2010). Effective Interaction Potentials in the Uppermost Landau Level. Journal of Low Temperature Physics [Online] 159:189-192. Available at: http://dx.doi.org/10.1007/s10909-009-0123-5.
We consider a quantum Hall system of electrons confined to the uppermost Landau level and assume that the lower Landau levels are full and inert causing no Landau level mixing. While it is known that the problem of electrons interacting with the Coulomb interaction in a higher Landau level is mathematically equivalent to the problem of electrons in the lowest Landau level interacting with an effective interaction, the way the effective interaction can be calculated is not unique. We focus on the details of two different calculations of such effective interaction potentials in the uppermost Landau level and discuss the influence of one or another form of the effective potential on the stability of various correlated electronic phases in the quantum Hall regime.
• Campo, V., Quintanilla, J. and Hooley, C. (2009). Possible critical behavior driven by the confining potential in optical lattices with ultra-cold fermions. Physica B: Condensed Matter [Online] 404:3328-3331. Available at: http://dx.doi.org/10.1016/j.physb.2009.07.089.
A recent paper [V.L. Campo, et al., Phys. Rev. Lett. 99 (2007) 240403] has proposed a two-parameter scaling method to determine the phase diagram of the fermionic Hubbard model from optical lattice experiments. Motivated by this proposal, we investigate in more detail the behavior of the ground-state energy per site as a function of trap size (L) and confining potential (V(x) = t(x/L)(alpha)) in the one-dimensional case. Using the BALDA-DFT method, we find signatures of critical behavior as alpha ->infinity.
• 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. and Hooley, C. (2009). The strong-correlations puzzle. Physics World 22:32-37.
A solution to one of the most famous problems in theoretical physics, formulated almost 50 years ago, may at last be within reach. But as Jorge Quintanilla and Chris Hooley explain, it relies not on theory, but on experiments with ultracold atoms trapped by beams of light
• 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.
• Hillier, A., Quintanilla, J. and Cywinski, R. (2009). Evidence for Time-Reversal Symmetry Breaking in the Noncentrosymmetric Superconductor LaNiC2. Physical Review Letters: Moving Physics Forward [Online] 102:117007 -1. Available at: http://dx.doi.org/10.1103/PhysRevLett.102.117007.
Muon spin relaxation experiments on the noncentrosymmetric intermetallic superconductor LaNiC2 are reported. We find that the onset of superconductivity coincides with the appearance of spontaneous magnetic fields, implying that in the superconducting state time-reversal symmetry is broken. An analysis of the possible pairing symmetries suggests only four triplet states compatible with this observation, all of them nonunitary. They include the intriguing possibility of triplet pairing with the full point group symmetry of the crystal, which is possible only in a noncentrosymmetric superconductor.
• Quintanilla, J., Capelle, K. and Oliveira, L. (2008). Density-functional description of superconducting and magnetic proximity effects across a tunneling barrier. Physical Review B: Condensed Matter and Materials Physics [Online] 78:1-10. Available at: http://dx.doi.org/10.1103/PhysRevB.78.205426.
A density-functional formalism for superconductivity and magnetism is presented. The resulting relations unify previously derived Kohn-Sham equations for superconductors and for noncollinear magnetism. The formalism, which discriminates Cooper-pair singlets from triplets, is applied to two quantum liquids coupled by tunneling through a barrier. An exact expression is derived, relating the eigenstates and eigenvalues of the Kohn-Sham equations, unperturbed by tunneling, on one side of the barrier to the proximity-induced ordering potential on the other.
• Quintanilla, J., Haque, M. and Schofield, A. (2008). Symmetry-breaking Fermi surface deformations from central interactions in two dimensions. Physical Review B: Condensed Matter and Materials Physics [Online] 78:35131. Available at: http://dx.doi.org/10.1103/PhysRevB.78.035131.
We present a mean-field theory of the Pomeranchuk instability in two dimensions, starting from a generic central interaction potential described in terms of a few microscopic parameters. For a significant range of parameters, the instability is found to be pre-empted by a first-order quantum phase transition. We provide the ground-state phase diagram in terms of our generic parameters.
• Quintanilla, J., Hooley, C., Powell, B., Schofield, A. and Haque, M. (2008). Pomeranchuk instability: Symmetry-breaking and experimental signatures. Physica B: Condensed Matter [Online] 403:1279-1281. Available at: http://dx.doi.org/10.1016/j.physb.2007.10.126.
We discuss the emergence of symmetry-breaking via the Pomeranchuk instability from interactions that respect the underlying point-group symmetry. We use a variational mean-field theory to consider a 2D continuum and a square lattice. We describe two experimental signatures: a symmetry-breaking pattern of Friedel oscillations around an impurity; and a structural transition.
• Campo, V., Capelle, K., Quintanilla, J. and Hooley, C. (2007). Quantitative determination of the Hubbard model phase diagram from optical lattice experiments by two-parameter scaling. Physical Review Letters: Moving Physics Forward [Online] 99. Available at: http://dx.doi.org/10.1103/PhysRevLett.99.240403.
We propose an experiment to obtain the phase diagram of the fermionic Hubbard model, for any dimensionality, using cold atoms in optical lattices. It is based on measuring the total energy for a sequence of trap profiles. It combines finite-size scaling with an additional “finite-curvature scaling” necessary to reach the homogeneous limit. We illustrate its viability in the 1D case, simulating experimental data in the Bethe-ansatz local-density approximation. Including experimental errors, the filling corresponding to the Mott transition can be determined with better than 3% accuracy.
• Quintanilla, J. and Campo, V. (2007). Electron in a tangled chain: Multifractality at the small-world critical point. Physical Review B: Condensed Matter and Materials Physics [Online] 75:1441204. Available at: http://dx.doi.org/10.1103/PhysRevB.75.144204.
We study a simple model of conducting polymers in which a single electron propagates through a randomly tangled chain. The model has the geometry of a small-world network, with a small density p of crossings of the chain acting as shortcuts for the electron. We use numerical diagonalization and simple analytical arguments to discuss the density of states, inverse participation ratios, and wave functions. We suggest that there is a critical point at p=0 and demonstrate finite-size scaling of the energy and wave functions at the lower band edge. The wave functions are multifractal. The critical exponent of the correlation length is consistent with criticality due to the small-world effect, as distinct from the previously discussed, dimensionality-driven Anderson transition.
• Quintanilla, J. and Schofield, A. (2006). Pomeranchuk and topological Fermi surface instabilities from central interactions. Physical Review B: Condensed Matter and Materials Physics 74.
We address at the mean field level the emergence of a Pomeranchuk instability in a uniform Fermi liquid with central particle-particle
interactions. We find that Pomeranchuk instabilities with all symmetries
except l=1 can take place if the interaction is repulsive and has
a finite range r(0) of the order of the interparticle distance. We
demonstrate this by solving the mean field equations analytically
for an explicit model interaction, as well as numerical results for
more realistic potentials. We find in addition to the Pomeranchuk
instability other, subtler phase transitions in which the Fermi surface
changes topology without rotational symmetry breaking. We argue that
such interaction-driven topological transitions may be as generic
to such systems as the Pomeranchuk instability.
• Hooley, C. and Quintanilla, J. (2006). Finite-curvature scaling in optical lattice systems. Physica B: Condensed Matter [Online] 378-80:1035-1036. Available at: http://dx.doi.org/10.1016/j.physb.2006.01.393.
We address the problem posed by the inhomogeneous trapping fields when using ultracold fermions to simulate strongly correlated electrons.
As a starting point, we calculate the density of states for a single
atom. Using semiclassical arguments, we show that this can be made
to evolve smoothly towards the desired limit by varying the curvature
of the field profile. Implications for mutually interacting atoms
in such potentials are briefly discussed. (c) 2006 Elsevier B.V.
• Hooley, C. and Quintanilla, J. (2004). Single-atom density of states of an optical lattice. Physical Review Letters: Moving Physics Forward [Online] 93:80404. Available at: http://dx.doi.org/10.1103/PhysRevLett.93.080404.
We consider a single atom in an optical lattice, subject to a harmonic trapping potential. The problem is treated in the tight-binding approximation,
with an extra parameter kappa denoting the strength of the harmonic
trap. It is shown that the kappa--&gt;0 limit of this problem is singular,
in the sense that the density of states for a very shallow trap (kappa--&gt;0)
is qualitatively different from that of a translationally invariant
lattice (kappa=0). The physics of this difference is discussed, and
densities of states and wave functions are exhibited and explained.
• Quintanilla, J. and Gyorffy, B. (2003). Cooper pairing with finite angular momentum: BCS versus Bose limits. Journal of Physics A: Mathematical and General 36:9379-9390.
We revisit the old problem of exotic superconductivity as Cooper pairing with finite angular momentum emerging from a central potential.
Based on some general considerations, we suggest that the phenomenon
is associated with interactions that keep electrons at some particular,
finite distance r(0), and occurs at a range of intermediate densities
n similar to 1/r(0)(3). We discuss the ground state and the critical
temperature in the framework of a standard functional-integral theory
of the BCS to Bose crossover. We find that, due to the lower energy
of two-body bound states with l = 0, the rotational symmetry of the
ground state is always restored on approaching the Bose limit. Moreover
in that limit the critical temperature is always higher for pairs
with l = 0. The breaking of the rotational symmetry of the continuum
by the superfluid state thus seems to be a property of weakly-attractive,
non-monotonic interaction potentials, at intermediate densities.
• Quintanilla, J., Capelle, K. and Oliveira, L. (2003). Comment on Anomalous proximity effect in underdoped YBa2Cu3O6+x Josephson junctions’’. Physical Review Letters: Moving Physics Forward 90.
• Quintanilla, J., Gyorffy, B., Annett, J. and Wallington, J. (2002). Cooper pairing with finite angular momentum via a central attraction: From the BCS to the Bose limits. Physical Review B: Condensed Matter and Materials Physics [Online] 66:214526. Available at: http://dx.doi.org/10.1103/PhysRevB.66.214526.
In the context of a simple model featuring an explicit, central interaction potential, and using a standard functional-integral technique, we
study superconductivity with angular momentum quantum number l=2
as an emergent property of the many-body system. Our interaction
potential is attractive at a finite distance r(0), and the breaking
of the rotational symmetry is the result of an interplay between
r(0) and the interparticle distance r(s). This interplay is generic
to interactions of this type and is responsible for the existence
of d-wave pairing for a range of densities. However, we find that
l=2 pairing takes place only in the BCS limit. In contrast, as the
Bose-Einstein (BE) limit is approached the internal energy of the
preformed pairs'' becomes the dominant contribution and there
is a quantum phase transition in which the s-wave symmetry is restored.
We also find that the limiting value of the critical temperature
is k(B)T(c)--&gt;3.315 h(2)/2m(*) [n/2(2l+1)](2/3), which coincides
with the usual result only for l=0; for l&gt;0, it differs in the degeneracy
factor 1/(2l+1), which lowers T-c. Our results thus place constraints
on exotic pairing in the BE limit, while at the same time indicating
a particularly interesting route to pairing with l&gt;0 in a BCS superconductor.
• Quintanilla, J. and Gyorffy, B. (2002). On the nature of the superconducting gap in the cuprates. Journal of Physics: Condensed Matter [Online] 14:6591-6600. Available at: http://dx.doi.org/10.1088/0953-8984/14/25/325.
Recent experiments indicate that the excitation spectrum of the cuprates is characterized, in the superconducting state, by two energy scales: the 'coherence energy' Δc and the 'pseudogap' Δp. Here we consider a simple generalization of the BCS model that yields exotic pairing and can describe, phenomenologically, the generic trends in the critical temperature T c of cuprate superconductors. We use the model to predict the gap in the single-particle spectrum arising from the superconductivity and we find evidence that it corresponds to the lower of the two energy scales, Δc, seen in the experiments. This provides further support to the view that the origin of the pseudogap is not superconducting fluctuations.
• Quintanilla, J. and Gyorffy, B. (2000). Finite range model interaction potential for d-wave superconductors:T-c versus doping in the cuprates. Physica B: Condensed Matter 284:421-422.
We study a simple, BCS like, model which describes unconventional superconductivity on the basis of an electron-electron attraction
corresponding to the delta-shell potential: U(r(12)) = - g delta(r(12)
- r(0)). It predicts a T-c versus doping behavior similar to that
characteristic of the high T-c cuprates. (C) 2000 Elsevier Science

### Internet publication

• Quintanilla, J. (2016). SPS Research into Superconductivity Highlighted in Physical Review Letters [Webpage]. Available at: https://www.kent.ac.uk/physical-sciences/news/front-page/new-form-superconductivity.html.
Research conducted by Dr Jorge Quintanilla as part of an international collaboration has been chosen as an Editors' Suggestion on the Physical Review Letters website. The full article can be accessed here. Below is a summary from Dr Quintanilla about what the research was and how it is useful.

### Thesis

• Irons, H. (2016). Experimental Implications of the Entanglement Transition in Clustered Quantum Materials.
Clustered quantum materials provide a new platform for the experimental study of many- body entanglement. Here we address a simple model featuring N interacting spins in a transverse field. The field can induce an entanglement transition (ET). We calculate the magnetisation, low-energy gap and neutron-scattering cross-section and find that the ET has distinct signatures, detectable at temperatures as high as 10% of the interaction strength. Unlike a quantum critical point, the signatures of the ET are stronger for smaller clusters.
• Tomasello, B. (2014). A Quantum Mechanical Study of the Dynamical Properties of Spin-Ice Materials.
The spin-ice materials Ho2Ti2O7 (HTO) and Dy2Ti2O7 (DTO) are part of a large family of compounds called magnetic pyrochlore oxides. Typically, the magnetism of these systems arise from the rare-earth ions RE3+ which sits at the vertices of a lattice of corner-sharing tetrahedra and couple with the degrees of freedom of the crystal leading to a wide spectrum of exotic phenomena. In spin-ices the magnetic moments of the individual RE3+ ions are large enough to let their mutual dipolar interactions be the leading factors for the thermodynamics. Moreover, the strength and the symmetries of their local crystalline environment are such that each ion behaves like a magnetic dipole with only two allowed configurations: it points either parallel or opposite to the ?111? axis joining the vertex where it sits to the centre of the tetrahedron. As a result, the ground state of the system is macroscopically degenerate because the ferromagnetic interactions between the 4 Ising-like spins in each tetrahedron cannot be satisfied simultaneously, and in turn the manifold of possible configurations minimising the energy (2 spins inward - 2 spins outward) increases with the size of the system. This exotic ground state is such that the Ising configurations of the dipoles map to the disordered vector-displacements of the protons in the water molecules of conventional ice; hence the name spin-ice. Violations of the (2in-2out) ice rules take the systems out of the ground state into more energetic configurations. More precisely, the flipping of a spin between two adjacent ground-state tetrahedra creates a local excitation (1in-3out in a tetrahedron, 3in-1out in the neighbouring one) that is made of two fractionalised opposite magnetic fluxes. Once the two fluxes are created they can separate and freely hop across the lattice as their motion does not involve ahy higher order violation of the ice-rules. The low temperature properties of spin-ice is found to depend heavily, if not exclusively, on the density and mobility of such flux-defects which effectively behave as magnetic monopoles mutually interacting via a Coulomb potential. Beside the success of many experiments which exploit the physics of the monopoles in their exclusively classical formulation, there has been an increasing curiosity about the microscopic mechanisms which dictate their propagation across the lattice. At present, the dynamics of the monopoles are still puzzling showing different responses under different probes and non-identical behaviours between the two compounds HTO and DTO. As the monopoles themselves consist of packed magnetic fluxes originating from the magnetic moments of the RE3+ ions, investigating the microscopical mechanisms underlying their motion requires revisiting the foundations of the classicality which emerges form the quantum substrate of the interactions of the magnetic ions. This is the subject of the work presented in this thesis. With particular focus on the interplay between the local crystal-symmetries and the mutual interactions between the RE3+ ions, the present study gives an accurate de- scription of the microscopic mechanisms which occur in the pyrochlore substrate in the presence of a monopole. The results suggest that the motion of the monopoles is achieved thanks to the spin-tunnelling of the RE3+ ions which account for the flipping of the Ising spins, necessary for the propagation of a monopole. A major improvement we add to the standard theory of spin ice, is the role of the exchange interactions that are overcome by the dipolar ones in the ice-state but posses a dynamical resilience which manifests in the presence of a monopole. Furthermore, the present study brings to light the articulated statistical structure of the kinematic spin-constraints which are expected to dictate the diffusion of the free monopoles and their response under dif- ferent probes also in conditions out of equilibrium. From a more general perspective, the mathematical and physical models developed during this work promise to be of in- terest also in other magnetic systems. Primarily, in the other pyrochlore oxides whose microscopical structure is akin the spin-ice one, secondarily, in other RE3+ compounds where the interplay between quantum and classical physics leads to the manifestation of unusual dynamical effects.

### Visual media

• Higgins, C., Hinman, N., de Lanerolle, A., Marshall, J., Moore, K., Newport, R., Park, A., Osborne, M., Quintanilla, J. and Wheatley, J. (2013). ’Copper, Aluminium, Green and Red’ - Artists And Scientists Discuss Carl Andre’s Work. [Youtube]. Turner Contemporary Art Gallery. Available at: https://www.youtube.com/watch?v=lJlvUmIFgFM.
In February 2013 Turner Contemporary held an event, led by Practical Philosopher Ayisha de Lanerolle, for artists and scientists to discuss Carl Andre's declaration: 'The periodic table of elements is for me what the colour spectrum is for a painter. . . Copper is more profoundly different from aluminium than green is from red'. Multi-award winning animation studio Cognitive Media have created a short animated film 'Copper, Aluminium, Green and Red' representing the ideas that emerged from this discussion.

### Forthcoming

• Ghosh, S., Annett, J. and Quintanilla, J. (2019). Time-reversal symmetry breaking in superconductors through loop super-current order. Cornell University - Pre-Print Server [Online]. Available at: https://arxiv.org/abs/1803.02618.
We propose a superconducting instability where microscopic loop super-currents form spontaneously within a unit cell at the critical temperature, Tc. Such currents break time-reversal symmetry (TRS) without needing an unconventional pairing mechanism. Using Ginzburg-Landau theory, we describe in detail how they emerge in a toy model. We discuss the crystal symmetry requirements more generally and show that they are met by the Re6X (X=Zr, Hf, Ti) family of TRS-breaking, but otherwise seemingly conventional, superconductors. We estimate an upper bound for the resulting internal fields, which is consistent with recent muon-spin relaxation experiments.
• Shang, T., Ghosh, S., Chang, L., Baines, C., Lee, M., Zhao, J., Verezhak, J., Gawryluk, J., Pomjakushina, E., Shi, M., Medarde, M., Mesot, J., Quintanilla, J. and Shiroka, T. (2019). Time-reversal symmetry breaking and unconventional superconductivity in Zr3Ir: A new type of noncentrosymmetric superconductor. arXiv.org [Online]. Available at: https://arxiv.org/abs/1901.01414.
We report the discovery of a new type of unconventional noncentrosymmetric superconductor, Zr3Ir with Tc=2.2 K, investigated mostly via muon-spin rotation/relaxation (μSR) techniques. Its bulk superconductivity was characterized using magnetic susceptibility, electrical resistivity, and heat capacity measurements. The low-temperature superfluid density, determined via transverse-field μSR and electronic specific heat, suggests a fully-gapped superconducting state. The spontaneous magnetic fields, revealed by zero-field μSR below Tc, indicate the breaking of time-reversal symmetry in Zr3Ir and, hence, the unconventional nature of its superconductivity. Our results suggest that Zr3Ir belongs to a new class of materials, suitable for studying the interplay of gauge-, spatial-inversion-, and time-reversal symmetries.
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