Classifying Two-dimensional Superfluids: Why There Is More .
Classifying two-dimensional superfluids:why there is more to cuprate superconductivitythan the condensation of charge -2e Cooper pairscond-mat/0408329, cond-mat/0409470, and to appearLeon Balents (UCSB)Lorenz Bartosch (Yale)Anton Burkov (UCSB)Subir Sachdev (Yale)Krishnendu Sengupta (Toronto)Talk online:Sachdev
I.OutlineBose-Einstein condensation and superfluidityII. Vortices in the superfluidMagnus forces, duality, and point vortices asdual “electric” chargesIII. The superfluid-Mott insulator quantum phase transitionIV. Vortices in superfluids near the superfluid-insulatorquantum phase transitionThe Hofstadter Hamiltonian and vortex flavorsV. The cuprate superconductorsThe “quantum order” of the superconducting state:evidence for vortex flavors
I. Bose-Einstein condensation and superfluidity
Superfluidity/superconductivityoccur in: liquid 4He metals Hg, Al, Pb, Nb,Nb3Sn . liquid 3He neutron stars cuprates La2-xSrxCuO4,YBa2Cu3O6 y . M3C60 ultracold trapped atoms MgB2
The Bose-Einstein condensate:A macroscopic number of bosons occupy thelowest energy quantum stateSuch a condensate also forms in systems of fermions, wherethe bosons are Cooper pairs of fermions:Pair wavefunction in cuprates:kykx(Ψ k x2 k y2) ( S 0 )
Velocity distribution function of ultracold 87Rb atomsM. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)
Superflow:The wavefunction of the condensateiθ ( r )Ψ ΨeSuperfluid velocityvs m θ(for non-Galilean invariant superfluids,the co-efficient of θ is modified)
II. Vortices in the superfluidMagnus forces, duality, and pointvortices as dual “electric” charges
Excitations of the superfluid: Vortices
Observation of quantized vortices in rotating 4HeE.J. Yarmchuk, M.J.V. Gordon, andR.E. Packard,Observation of Stationary VortexArrays in Rotating Superfluid Helium,Phys. Rev. Lett. 43, 214 (1979).
Observation of quantized vortices in rotating ultracold NaJ. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle,Observation of Vortex Lattices in Bose-Einstein Condensates,Science 292, 476 (2001).
Quantized fluxoids in YBa2Cu3O6 yJ. C. Wynn, D. A. Bonn, B.W. Gardner, Yu-Ju Lin, Ruixing Liang, W. N. Hardy,J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).
Excitations of the superfluid: 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)
III. The superfluid-Mott insulatorquantum phase transition
Apply a periodic potential (standing laser beams)to trapped ultracold bosons (87Rb)
Momentum distribution function of bosonsBragg reflections of condensate at reciprocal lattice vectorsM. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Momentum distribution function of bosonsBragg reflections of condensate at reciprocal lattice vectorsM. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
Superfluid-insulator quantum phase transition at T 0V0 0ErV0 13ErV0 3ErV0 7ErV0 10ErV0 14ErV0 16ErV0 20Er
Superfluid-insulator quantum phase transition at T 0M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
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 1/2Ψ 0Weak interactions: superfluidityC. 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)
Bosons at filling fraction f 1/2Ψ 0Weak interactions: superfluidityC. 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)
Bosons at filling fraction f 1/2Ψ 0Weak interactions: superfluidityC. 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)
Bosons at filling fraction f 1/2Ψ 0Weak interactions: superfluidityC. 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)
Bosons at filling fraction f 1/2Ψ 0Weak interactions: superfluidityC. 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)
Bosons at filling fraction f 1/2Ψ 0Strong interactions: insulatorC. 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)
Bosons at filling fraction f 1/2Ψ 0Strong interactions: insulatorC. 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)
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
Insulating phases of bosons at filling fraction f 1/2 Charge densitywave (CDW) order12( Valence bondsolid (VBS) order)Valence bondsolid (VBS) orderCan define a common CDW/VBS order using a generalized "density" ρ ( r ) ρQ eiQ .rAll insulators have Ψ 0 and ρQ 0 for certain QC. 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)Q
IV. Vortices in superfluids near thesuperfluid-insulator quantum phase transitionThe Hofstadter Hamiltonianand vortex flavors
Upon approaching the insulator, the phase ofthe condensate becomes “uncertain”.Vortices cost less energy and vortex-antivortex pairs proliferate.The quantum mechanics of vortices plays acentral role in the superfluid-insulatorquantum phase transition.
A3A2A4A1A1 A2 A3 A4 2π fwhere f is the boson filling fraction.
Bosons at filling fraction f 1 At f 1, the “magnetic” flux per unit cell is 2π,and the vortex does not pick up any phase fromthe boson density. The effective dual “magnetic” field acting on thevortex is zero, and the corresponding componentof the Magnus force vanishes.
Bosons at rational filling fraction f p/qQuantum mechanics of the vortex “particle” in aperiodic potential with f flux quanta per unit cellSpace group symmetries of Hofstadter Hamiltonian:Tx , Ty : Translations by a lattice spacing in the x, y directionsR : Rotation by 90 degrees.Magnetic space group:TxTy e 2π if TyTx ;R 1Ty R Tx ; R 1Tx R Ty 1 ; R 4 1The low energy vortex states must form arepresentation of this algebra
Vortices in a superfluid near a Mott insulator at filling f p/qHofstadter spectrum of the quantum vortex “particle”with field operator ϕAt filling f p / q, there are q speciesof vortices, ϕ (with 1 q ),associated with q degenerate minima inthe vortex spectrum. These vortices realizethe smallest, q -dimensional, representation ofthe magnetic algebra.Tx : ϕ ϕ 1;1R :ϕ qTy : ϕ eq2π i f2π i mfϕe mm 1ϕ
Vortices in a superfluid near a Mott insulator at filling f p/qi The excitations of the superfluid are described by thequantum mechanics of q flavors of low energy vorticesmoving in zero dual "magnetic" field.i The Mott insulator is a Bose-Einstein condensate ofvortices, with ϕ 0i Any set of values of ϕbreaks the space groupsymmetry, and the orientation of the vortex condensate,ϕ , in flavor space determines the CDW/VBS order.
Mott insulators obtained by condensing vorticesSpatial structure of insulators for q 2 (f 1/2) 12( )
Field theory with projective symmetrySpatial structure of insulators for q 4 (f 1/4 or 3/4)a b unit cells;q , q , ab ,abqall integers
V. The cuprate superconductorsThe “quantum order” of thesuperconducting state:evidence for vortex flavors
Superconductivity ofholes, of density δ,moving on the squarelattice of Cu sites.LaOCu
Experiments on the cuprate superconductors show: Tendency to produce “density” wave order nearwavevectors (2π/a)(1/4,0) and (2π/a)(0,1/4). Proximity to a Mott insulator at hole density δ 1/8with long-range “density” wave order at wavevectors(2π/a)(1/4,0) and (2π/a)(0,1/4). Vortex/anti-vortex fluctuations for a widetemperature range in the normal state
The cuprate superconductor Ca2-xNaxCuO2Cl2Multiple order parameters: superfluidity and density wave.Phases: Superconductors, Mott insulators, and/or supersolidsT. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano,H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).
Distinct experimental charcteristics of underdoped cuprates at T TcMeasurements of Nernst effect are well explained by a modelof a liquid of vortices and anti-vorticesN. P. Ong, Y. Wang, S. Ono, Y.Ando, and S. Uchida, Annalender Physik 13, 9 (2004).Y. Wang, S. Ono, Y. Onose, G.Gu, Y. Ando, Y. Tokura, S.Uchida, and N. P. Ong, Science299, 86 (2003).
Distinct experimental charcteristics of underdoped cuprates at T TcSTM measurements observe “density” modulations with aperiod of 4 lattice spacingsLDOS of Bi2Sr2CaCu2O8 δ at 100 K.M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004).
Pinned vortices in the superfluidAny pinned vortex breaks the space group symmetry,and so has a preferred orientation in flavor space. Thisnecessarily leads to modulations in the local density ofstates over the spatial region where the vortex executesits quantum zero point motion.In the cuprates, assuming boson density density of Cooper pairs we haveρ MI 7 /16, and q 16 (both models in part B yield this value of q). Somodulation must have period a b with 16 / a, 16 / b, and ab /16 all integers.
Vortex-induced LDOS of Bi2Sr2CaCu2O8 δ integratedfrom 1meV to 12meV at 4KVortices have haloswith LDOSmodulations at aperiod 4 latticespacings7 pAb0 pA100ÅJ. Hoffman, E. W. Hudson, K. M. Lang,V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida,and J. C. Davis, Science 295, 466 (2002).Prediction of VBS ordernear vortices: K. Parkand S. Sachdev, Phys.Rev. B 64, 184510(2001).
Measuring the inertial mass of a vortex
Measuring the inertial mass of a vortexPreliminary estimates for the BSCCO experiment:Inertial vortex mass mv 10meVortex magnetoplasmon frequency ν p 1 THz 4 meVFuture experiments can directly detect vortex zero point motionby looking for resonant absorption at this frequency.Vortex oscillations can also modify the electronic density of states.
SuperfluidsSuperfluids nearnear MottMott insulatorsinsulatorsThe Mott insulator has average Cooper pair density, f p/qper site, while the density of the superfluid is close (but neednot be identical) to this value meininmultiplemultiple(usually(usually q)q)“flavors”“flavors” onononthethevortexvortexflavorflavorspace.space. tuteaa“quantum“quantumorder”order” ion.
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.
orthographic drawings To draw nets for three-dimensional figures. . .And Why To make a foundation drawing, as in Example 3 You will study both two-dimensional and three-dimensional figures in geometry. A drawing on a piece of paper is a two-dimensional object. It has length and width. Your textbook is a three-dimensional object.
instabilities that occurred in two dimensional and three dimensional simulations are performed by Van Berkel et al. (2002) in a thermocline based water storage tank. In two-dimensional simulations the entrainment velocity was 40% higher than that found in the corresponding three dimensional simulations.
The Matlab Hilbert transform operates on one-dimensional data. For the work described here, it is necessary to adapt the HT to two-dimensional data. I do this by analogy to the two-dimensional Fourier Transform (strictly, the "discrete Fourier Transform"). Basic equations (e.g. Lim, 1990) show that the two-dimensional discrete Fourier
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6. Why Table 1 A 5-Why Analysis Question Table Build the Why Tree One Cause Level at a Time Many people start into a 5-Why analysis by using the 5-Why Table. With each Why question they put in an answer and then ask the next Why question. This question-and-answer tic-tac-toe
In Unit 6 the children are introduced to three-dimensional shapes and their properties, and through the use of “math nets” they discover the two-dimensional shapes that comprise each three-dimensional shape. The children will learn to identify three-dimensional shapes (cone, cube, cylinder, sphere, pyramid, rectangular prism) in the .
The nanocomposites have different phases as zero-dimensional (core shell), one-dimensional (nanowires and nanotubes), two- dimensional (lamellar) and three-dimensional (metal matrix composites) [8]. Based on their structural characteristics these nanocomposites are classified as nano-layered composites, nano-
An Introduction to Conditional Random Fields Charles Sutton1 and Andrew McCallum2 1 EdinburghEH8 9AB, UK, csutton@inf.ed.ac.uk 2 Amherst, MA01003, USA, mccallum@cs.umass.edu Abstract Often we wish to predict a large number of variables that depend on each other as well as on other observed variables. Structured predic- tion methods are essentially a combination of classi cation and graph-ical .
