Quantum Theory Of Vortices In - Harvard University
Quantum theory of vortices ind-wave superconductorsPhysical Review B 71, 144508 and 144509 (2005),Annals of Physics 321, 1528 (2006),Physical Review B 73, 134511 (2006),cond-mat/0606001.Leon Balents (UCSB)Lorenz Bartosch (Harvard)Anton Burkov (Harvard)Predrag Nikolic (Harvard)Subir Sachdev (Harvard)Krishnendu Sengupta (HRI, India)Talk online at http://sachdev.physics.harvard.edu
BCS theory of vortices in d-wave superconductors periodic potential strong Coulomb interactionsVortices in BCS superconductors near asuperconductor-Mott insulator transition at finite doping
The cuprate superconductor Ca2-xNaxCuO2Cl2T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J.C. Davis, Nature 430, 1001 (2004). Closely related modulations in superconductingBi2Sr2CaCu2O8 δ observed first by C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik,cond-mat/0201546 and Physical Review B 67, 014533 (2003).
The cuprate superconductor Ca2-xNaxCuO2Cl2Evidence that holes can form an insulating state with period 4modulation in the densityT. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J.C. Davis, Nature 430, 1001 (2004). Closely related modulations in superconductingBi2Sr2CaCu2O8 δ observed first by C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik,cond-mat/0201546 and Physical Review B 67, 014533 (2003).
STM around vortices induced by a magnetic field in the superconducting stateJ. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan,H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).3.0Local density of states (LDOS)RegularQPSRVortexDifferential Conductance (nS)2.52.01.51Å spatial resolutionimage of integratedLDOS ofBi2Sr2CaCu2O8 δ( 1meV to 12 meV)at B 5 Tesla.1.00.50.0-120-80-4004080Sample Bias (mV)120I. Maggio-Aprile et al. Phys. Rev. Lett. 75, 2754 (1995).S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).
Vortex-induced LDOS of Bi2Sr2CaCu2O8 δ integratedfrom 1meV to 12meV at 4KVortices havehalos withLDOSmodulations at aperiod 4 latticespacings7 pAb0 pA100ÅJ. Hoffman et al., Science 295, 466 (2002).G. Levy et al., Phys. Rev. Lett. 95, 257005 (2005).Prediction of periodic LDOSmodulations near vortices:K. Park and S. Sachdev, Phys.Rev. B 64, 184510 (2001).
Questions on the cuprate superconductors What is the quantum theory of the ground state as itevolves from the superconductor to the modulatedinsulator ? What happens to the vortices near such a quantumtransition ?
Outline The superfluid-insulator transition of bosons The quantum mechanics of vortices near the superfluidinsulator transitionDual theory of superfluid-insulator transition as theproliferation of vortex-anti-vortex pairs Influence of nodal quasiparticles on vortex dynamics ina d-wave superconductor
I. The superfluid-insulator transition ofbosons
Bosons at filling fraction f 1Weak interactions:superfluidityStrong interactions:Mott insulator whichpreserves all latticesymmetriesM. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Bosons at filling fraction f 1Ψ 0Weak interactions: superfluidity
Bosons at filling fraction f 1Ψ 0Weak interactions: superfluidity
Bosons at filling fraction f 1Ψ 0Weak interactions: superfluidity
Bosons at filling fraction f 1Ψ 0Weak interactions: superfluidity
Bosons at filling fraction f 1Ψ 0Strong interactions: insulator
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Weak interactions: superfluidity
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Weak interactions: superfluidity
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Weak interactions: superfluidity
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Weak interactions: superfluidity
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Weak interactions: superfluidity
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Strong interactions: insulator
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Strong interactions: insulator
Bosons at filling fraction f 1/2or S 1/2 XXZ modelΨ 0Strong interactions: insulatorInsulator has “density wave” order
Bosons at filling fraction f 1/2or S 1/2 XXZ model?InsulatorSuperfluidCharge densitywave (CDW) orderInteractions between bosons
Bosons at filling fraction f 1/2or S 1/2 XXZ model?InsulatorSuperfluidCharge densitywave (CDW) orderInteractions between bosons
Bosons at filling fraction f 1/2or S 1/2 XXZ model 12(?InsulatorSuperfluidValence bondsolid (VBS) orderInteractions between bosonsN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). )
Bosons at filling fraction f 1/2or S 1/2 XXZ model 12(?InsulatorSuperfluidValence bondsolid (VBS) orderInteractions between bosonsN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). )
Bosons at filling fraction f 1/2or S 1/2 XXZ model 12(?InsulatorSuperfluidValence bondsolid (VBS) orderInteractions between bosonsN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). )
Bosons at filling fraction f 1/2or S 1/2 XXZ model 12(?InsulatorSuperfluidValence bondsolid (VBS) orderInteractions between bosonsN. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). )
The superfluid-insulator quantum phase transitionKey difficulty: Multiple order parameters (BoseEinstein condensate, charge density wave, valencebond-solid order ) not related by symmetry, butclearly physically connected. Standard methods onlypredict strong first order transitions (for genericparameters).
The superfluid-insulator quantum phase transitionKey difficulty: Multiple order parameters (BoseEinstein condensate, charge density wave, valencebond-solid order ) not related by symmetry, butclearly physically connected. Standard methods onlypredict strong first order transitions (for genericparameters).Key theoretical tool: Quantum theory of vortices
Outline The superfluid-insulator transition of bosons The quantum mechanics of vortices near the superfluidinsulator transitionDual theory of superfluid-insulator transition as theproliferation of vortex-anti-vortex pairs Influence of nodal quasiparticles on vortex dynamics ina d-wave superconductor
II. The quantum mechanics of vortices near asuperfluid-insulator transitionDual theory of the superfluid-insulator transitionas the proliferation of vortex-anti-vortex-pairs
Excitations of the superfluid: Vortices and anti-vorticesCentral question:In two dimensions, we can view the vortices aspoint particle excitations of the superfluid. Whatis the quantum mechanics of these “particles” ?
In ordinary fluids, vortices experience the Magnus ForceFMFM ( mass density of air ) i ( velocity of ball ) i ( circulation )
Dual picture:The vortex is a quantum particle with dual “electric”charge n, moving in a dual “magnetic” field ofstrength h (number density of Bose particles)C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60,1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989)
Bosons on the square lattice at filling fraction f p/q
Bosons on the square lattice at filling fraction f p/q
Vortex theory of the superfluid-insulator transitionAs a superfluid approaches an insulating state, thedecrease in the strength of the condensate willlower the energy cost of creating vortex-antivortex pairs.
Vortex theory of the superfluid-insulator transitionProliferation of vortex-anti-vortex pairs.
Vortex theory of the superfluid-insulator transitionProliferation of vortex-anti-vortex pairs.
Vortex theory of the superfluid-insulator transitionProliferation of vortex-anti-vortex pairs.
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001)S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001)S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
Vortex-induced LDOS of Bi2Sr2CaCu2O8 δ integratedfrom 1meV to 12meV at 4KVortices havehalos withLDOSmodulations at aperiod 4 latticespacings7 pAb0 pA100ÅJ. Hoffman et al., Science 295, 466 (2002).G. Levy et al., Phys. Rev. Lett. 95, 257005 (2005).Prediction of periodic LDOSmodulations near vortices:K. Park and S. Sachdev, Phys.Rev. B 64, 184510 (2001).
Outline The superfluid-insulator transition of bosons The quantum mechanics of vortices near the superfluidinsulator transitionDual theory of superfluid-insulator transition as theproliferation of vortex-anti-vortex pairs Influence of nodal quasiparticles on vortex dynamics ina d-wave superconductor
III. Influence of nodal quasiparticles onvortex dynamics in a d-wavesuperconductorP. Nikolic
A finite effective massmv ΛvF2where Λ Δ is a high energy cutoff
sub-Ohmic damping withvΔ C1 v Universal function of vF 2F
Bardeen-Stephen viscous drag withvΔ C2 v Universal function of vF 2F
Bardeen-Stephen viscous drag withvΔ C2 v Universal function of vF 2FEffect of nodal quasiparticles on vortexdynamics is relatively innocuous.
Influence of the quantum oscillating vortex on the LDOSResonant feature near thevortex oscillation frequencyP. Nikolic, S. Sachdev, and L. Bartosch, cond-mat/0606001
Influence of the quantum oscillating vortex on the LDOS3.0RegularQPSRVortexDifferential Conductance (nS)2.52.01.51.00.5Resonant feature near thevortex oscillation frequency0.0-120-80-4004080120Sample Bias (mV)I. Maggio-Aprile et al. Phys. Rev. Lett. 75, 2754 (1995)S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).P. Nikolic, S. Sachdev, and L. Bartosch, cond-mat/0606001
ConclusionsConclusions ments. ssofofaavortexvortex scatteringlight-scatteringexperimentsexperiments nsistentconsistentwithwithLDOSLDOSspectrumspectrum Landau-Ginzburg-Wilsontheory.theory.
The cuprate superconductor Ca 2-x Na x CuO 2 Cl 2 Evidence that holes can form an insulating state with period 4 modulation in the density T. Hanaguri, C. Lupien, Y. K
ANALYSIS OF THE STABILITY OF MULTIPLE HELICAL VORTICES USING COMPLEX-STEP LINEARIZATION V. G. Kleine1,2, A. Hanifi1 & D. S. Henningson1 1FLOW, Dept. of Engineering Mechanics, KTH Royal Institute of Technology, Sweden 2Division of Aeronautical and Aerospace Engineering, Instituto Tecnológico de Aeronáutica, Brazil Abstract The system of vortices created by the hub and tip vortices of rotors .
I. Bose-Einstein condensation and superfluidity II. Vortices in the superfluid Magnus forces, duality, and point vortices as dual “electric” charges III. The superfluid-Mott insulator quantum phase transition IV. Vortices in superfluids near the superfluid-insulator quantum phase transition The Hofstadter Hamiltonian and vortex flavors V.
Chapter 2 - Quantum Theory At the end of this chapter – the class will: Have basic concepts of quantum physical phenomena and a rudimentary working knowledge of quantum physics Have some familiarity with quantum mechanics and its application to atomic theory Quantization of energy; energy levels Quantum states, quantum number Implication on band theory
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