The phase diagram of polar condensates
|
|
- Jesse Tate
- 5 years ago
- Views:
Transcription
1 The phase diagram of polar condensates Taking the square root of a vortex Austen Lamacraft [with Andrew James] arxiv: University of Virginia September 23, 2010 KITP, UCSB Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
2 Outline 1 Magnetism in Bose condensates 2 Order parameters and topology in polar condensates 3 Domain walls, disclinations, and the phase diagram in 2D 4 Generalized XY model: view from field theory Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
3 Outline 1 Magnetism in Bose condensates 2 Order parameters and topology in polar condensates 3 Domain walls, disclinations, and the phase diagram in 2D 4 Generalized XY model: view from field theory Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
4 Magnetic Bose condensates BEC = magnetism for bosons with spin! Condensate wavefunction φ(r) 0 is a spinor and must choose a direction in spin space. Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
5 Magnetic Bose condensates BEC = magnetism for bosons with spin! Condensate wavefunction φ(r) 0 is a spinor and must choose a direction in spin space. Example: spin-polarized Hydrogen [Siggia & Ruckenstein, 1980] ( ) ( φ1/2 φ = = e iψ e iϕ/2 ) cos θ/2 e iϕ/2 sin θ/2 φ 1/2 All states (of fixed norm) obtained by rotation of reference state Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
6 Magnetic Bose condensates BEC = magnetism for bosons with spin! Condensate wavefunction φ(r) 0 is a spinor and must choose a direction in spin space. Example: spin-polarized Hydrogen [Siggia & Ruckenstein, 1980] ( ) ( φ1/2 φ = = e iψ e iϕ/2 ) cos θ/2 e iϕ/2 sin θ/2 φ 1/2 All states (of fixed norm) obtained by rotation of reference state Does magnetism = BEC? or can ordering happen sequentially as temperature lowered? Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
7 Higher spin gives additional possibilities Consider the spin-1 state φ 1 0 φ = φ 0 = 1 φ 1 0 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
8 Higher spin gives additional possibilities Consider the spin-1 state φ 1 0 φ = φ 0 = 1 φ 1 0 Not hard to show that φ ( m S (1)) φ = 0 (S (1) the spin-1 matrices) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
9 Higher spin gives additional possibilities Consider the spin-1 state φ 1 0 φ = φ 0 = 1 φ 1 0 Not hard to show that φ ( m S (1)) φ = 0 Yet evidently there is still an axis (headless?) (S (1) the spin-1 matrices) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
10 Higher spin gives additional possibilities Consider the spin-1 state φ 1 0 φ = φ 0 = 1 φ 1 0 Not hard to show that φ ( m S (1)) φ = 0 Yet evidently there is still an axis (headless?) (S (1) the spin-1 matrices) Fixing the nature of the BEC requires some dynamical input (interactions) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
11 Contact interactions in a spin-1 gas Total spin 2 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
12 Contact interactions in a spin-1 gas Total spin 0 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
13 Contact interactions in a spin-1 gas H int = i<j = i<j δ(r i r j ) [g 0 P 0 + g 2 P 2 ] δ(r i r j ) [c 0 + c 2 S i S j ] c 0 = (g 0 + 2g 2 ) /3 c 2 = (g 2 g 0 ) /3 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
14 Mean field ground states H int = i<j = i<j δ(r i r j ) [g 0 P 0 + g 2 P 2 ] δ(r i r j ) [c 0 + c 2 S i S j ] Energy of state Ψ m1 m N (r 1,..., r N ) = φ m1 (r 1 ) φ mn (r N ) involves H int = c 2 2 (φ Sφ) 2 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
15 Mean field ground states H int = i<j = i<j δ(r i r j ) [g 0 P 0 + g 2 P 2 ] δ(r i r j ) [c 0 + c 2 S i S j ] Energy of state Ψ m1 m N (r 1,..., r N ) = φ m1 (r 1 ) φ mn (r N ) involves H int = c 2 2 (φ Sφ) 2 c 2 < 0 (e.g. 87 Rb) φ Sφ is maximized c 2 > 0 (e.g. 23 Na) φ Sφ is minimized Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
16 Spin-1 states: Cartesian representation ( φ = a + ib ) S (1) i jk = iɛ ijk φ S (1) φ = 2a b Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
17 Spin-1 states: Cartesian representation ( φ = a + ib ) S (1) i jk = iɛ ijk φ S (1) φ = 2a b c 2 < 0 (e.g. 87 Rb) φ Sφ is maximized a and b perpendicular c 2 > 0 (e.g. 23 Na) φ Sφ is minimized a and b parallel Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
18 Spin-1 states: Cartesian representation ( φ = a + ib ) S (1) i jk = iɛ ijk φ S (1) φ = 2a b c 2 < 0 (e.g. 87 Rb) φ Sφ is maximized a and b perpendicular Order parameter manifold is SO(3) c 2 > 0 (e.g. 23 Na) φ Sφ is minimized a and b parallel Order parameter written as φ = ne iθ Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
19 Spin-1 states: Cartesian representation ( φ = a + ib ) S (1) i jk = iɛ ijk φ S (1) φ = 2a b c 2 < 0 (e.g. 87 Rb) φ Sφ is maximized a and b perpendicular Order parameter manifold is SO(3) Ferromagnetic condensate c 2 > 0 (e.g. 23 Na) φ Sφ is minimized a and b parallel Order parameter written as φ = ne iθ Polar condensate Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
20 The Bose Ferromagnet Stamper-Kurn group, Berkeley Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
21 Outline 1 Magnetism in Bose condensates 2 Order parameters and topology in polar condensates 3 Domain walls, disclinations, and the phase diagram in 2D 4 Generalized XY model: view from field theory Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
22 Order parameter manifold for polar condensates Mean field ground state written φ = ne iθ There is an evident redundancy: (n, θ) and ( n, θ + π) are the same Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
23 Order parameter manifold for polar condensates Mean field ground state written φ = ne iθ There is an evident redundancy: (n, θ) and ( n, θ + π) are the same Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
24 Order parameter manifold for polar condensates Mean field ground state written φ = ne iθ There is an evident redundancy: (n, θ) and ( n, θ + π) are the same Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
25 Disclinations in nematic liquid crystals Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
26 Consequences for Kosterlitz Thouless transition Circulation quantum is halved v dl = θ = m m πn = h 2m n, n Z Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
27 Consequences for Kosterlitz Thouless transition Circulation quantum is halved v dl = θ = m m πn = h 2m n, n Z Consider free energy of a single half vortex / disclination E = n s 2m dr v 2 = πn s 2 4m ln ( ) L 2 S = k B ln ξ ( πns 2 F = U TS = 4m 2k BT ( ) L ξ ) ln ( ) L ξ Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
28 Consequences for Kosterlitz Thouless transition Circulation quantum is halved v dl = θ = m m πn = h 2m n, n Z Consider free energy of a single half vortex / disclination E = n s 2m dr v 2 = πn s 2 4m ln ( ) L 2 S = k B ln ξ ( πns 2 F = U TS = 4m 2k BT ( ) L ξ ) ln ( ) L ξ Vanishes for n s = 8 π mk B T 2 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
29 Consequences for Kosterlitz Thouless transition Circulation quantum is halved v dl = θ = m m πn = h 2m n, n Z Consider free energy of a single half vortex / disclination E = n s 2m dr v 2 = πn s 2 4m ln ( ) L 2 S = k B ln ξ ( πns 2 F = U TS = 4m 2k BT ( ) L ξ ) ln ( ) L ξ Vanishes for n s = 8 π mk B T 2 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
30 KT transition mediated by half-vortices / disclinations Jump is 4 bigger than usual! (Korshunov, 1985) n KT /2 = 4 n KT = 8 π mk B T c π 2 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
31 KT transition mediated by half-vortices / disclinations PRL 97, (2006) P H Y S I C A L R E V I E W L E T T E R S week ending 22 SEPTEMBER 2006 Topological Defects and the Superfluid Transition of the s 1 Spinor Condensate in Two Dimensions Subroto Mukerjee, 1,2 Cenke Xu, 1 and J. E. Moore 1,2 1 Department of Physics, University of California, Berkeley, California 94720, USA Γ T/π T/π T Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
32 What about the n degrees of freedom? Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
33 What about the n degrees of freedom? A simple model is H = t <ij> φ i φ j + c.c = 2t ij n i n j cos (θ i θ j ) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
34 What about the n degrees of freedom? A simple model is H = t <ij> φ i φ j + c.c = 2t ij n i n j cos (θ i θ j ) Taking the continuum limit.. H a 2 d t dr [( θ) 2 + ( n) 2] Identifying (n, θ) and ( n, θ + π) ties half vortices to disclinations Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
35 What about the n degrees of freedom? A simple model is H = t <ij> φ i φ j + c.c = 2t ij n i n j cos (θ i θ j ) Taking the continuum limit.. H a 2 d t dr [( θ) 2 + ( n) 2] Identifying (n, θ) and ( n, θ + π) ties half vortices to disclinations Beneath KT transition, half vortices are absent and one can treat the n degrees of freedom as Heisenberg spins Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
36 What about the n degrees of freedom? A simple model is H = t <ij> φ i φ j + c.c = 2t ij n i n j cos (θ i θ j ) Taking the continuum limit.. H a 2 d t dr [( θ) 2 + ( n) 2] Identifying (n, θ) and ( n, θ + π) ties half vortices to disclinations Beneath KT transition, half vortices are absent and one can treat the n degrees of freedom as Heisenberg spins Mermin Wagner theorem says they do not order at finite temperature. Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
37 Outline 1 Magnetism in Bose condensates 2 Order parameters and topology in polar condensates 3 Domain walls, disclinations, and the phase diagram in 2D 4 Generalized XY model: view from field theory Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
38 The Zeeman effect Within the spin-1 multiplet, the Zeeman energy is H Z,m = pm + qm 2 p B linear and q B 2 /A HF quadratic Zeeman effects Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
39 The Zeeman effect Within the spin-1 multiplet, the Zeeman energy is H Z,m = pm + qm 2 p B linear and q B 2 /A HF quadratic Zeeman effects Spin conservation makes p irrelevant, leading only to precession Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
40 The Zeeman effect Within the spin-1 multiplet, the Zeeman energy is H Z,m = pm + qm 2 p B linear and q B 2 /A HF quadratic Zeeman effects Spin conservation makes p irrelevant, leading only to precession At large q m = 0 state only occupied: ordering via usual KT transition Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
41 The Zeeman effect Within the spin-1 multiplet, the Zeeman energy is H Z,m = pm + qm 2 p B linear and q B 2 /A HF quadratic Zeeman effects Spin conservation makes p irrelevant, leading only to precession At large q m = 0 state only occupied: ordering via usual KT transition Basic problem: How does the phase diagram evolve with q from the 1 2KT transition mediated by half vortices to the usual KT transition? Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
42 Consequences of quadratic Zeeman effect H QZ = q i φ i S 2 z φ i = q i (1 n 2 z,i) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
43 Consequences of quadratic Zeeman effect H QZ = q i φ i S 2 z φ i = q i (1 n 2 z,i) q > 0 favors alignment of n in z-direction (easy axis) H = 2t ij n i n j cos (θ i θ j ) q i n 2 z,i Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
44 Consequences of quadratic Zeeman effect H QZ = q i φ i S 2 z φ i = q i (1 n 2 z,i) q > 0 favors alignment of n in z-direction (easy axis) H = 2t ij n i n j cos (θ i θ j ) q i n 2 z,i Large q fixes n: regular KT transition Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
45 Half vortices terminate soliton walls Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
46 Conjectured phase diagram Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
47 Monte Carlo simulation Include hopping of singlet pairs φ φ = cos(2θ) H = ij [2t n i n j cos (θ i θ j ) + u cos(2 [θ i θ j ])] q i n 2 z,i Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
48 Monte Carlo simulation Intermediate phase with singlet pair quasi-long range order D 2 N T/t O q (data at u = 1) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
49 Heat Capacity 10 5 q=1 Specific heat L= T q= L= T Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
50 Binder cummulant Φ = i φ i U 4 = ( Φ Φ ) 2 ( Φ Φ ) 2 q=1 Binder cumulant L= T Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
51 Ising scaling for the lower transition U 4 = f ( ) ( L = f L ξ T T c T c ν) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
52 Ising scaling for the lower transition Binder cumulant L= T (T-Tc)L Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
53 Ising scaling for the lower transition Binder cumulant L= T (T-Tc)L Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
54 What about finite magnetization? At finite magnetization have spin-flop transition: n flops into x y plane Analogous to antiferromagnet Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
55 What about finite magnetization? At finite magnetization have spin-flop transition: n flops into x y plane Analogous to antiferromagnet Zero temperature Heisenberg fixed point Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
56 Outline 1 Magnetism in Bose condensates 2 Order parameters and topology in polar condensates 3 Domain walls, disclinations, and the phase diagram in 2D 4 Generalized XY model: view from field theory Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
57 Generalized XY model H gen = ij ( ) cos(θ i θ j ) + (1 ) cos(2θ i 2θ j ) Korshunov (1985), Grinstein & Lee (1985) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
58 Field theoretic view: how can domain walls end? 1 1 Thanks to Paul Fendley for discussions Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
59 Field theoretic view: how can domain walls end? 1 Recall disorder operator µ in Ising model 1 Thanks to Paul Fendley for discussions Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
60 Field theoretic view: how can domain walls end? 1 Recall disorder operator µ in Ising model Half vortex insertion tied to disorder operator 1 Thanks to Paul Fendley for discussions Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
61 Phase diagram from µ cos(φ/2) perturbation Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
62 Phase diagram from µ cos(φ/2) perturbation 1 2KT occurs when cos (φ/2) becomes relevant if µ 0 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
63 Phase diagram from µ cos(φ/2) perturbation 1 2 KT occurs when cos (φ/2) becomes relevant if µ 0 Along Ising line: continuous transition until µ cos(φ/2) relevant Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
64 Scaling with µ cos(φ/2) perturbation H vortex = λ µ cos φ/2 Three couplings to keep track of 1 λ, half vortex fugacity 2 t I, energy operator for Ising 3 Stiffness K of superfluid Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
65 Scaling with µ cos(φ/2) perturbation H vortex = λ µ cos φ/2 Three couplings to keep track of 1 λ, half vortex fugacity 2 t I, energy operator for Ising 3 Stiffness K of superfluid ( dλ dl = πk 4 dt I dl = t I + λ2 2 dk dl = λ2 ) λ λt I Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
66 RG flow for K > 15/2π Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, / 32
Unusual criticality in a generalized 2D XY model
Unusual criticality in a generalized 2D XY model Yifei Shi, Austen Lamacraft, Paul Fendley 22 October 2011 1 Vortices and half vortices 2 Generalized XY model 3 Villain Model 4 Numerical simulation Vortices
More informationVortices and other topological defects in ultracold atomic gases
Vortices and other topological defects in ultracold atomic gases Michikazu Kobayashi (Kyoto Univ.) 1. Introduction of topological defects in ultracold atoms 2. Kosterlitz-Thouless transition in spinor
More informationPhase transitions beyond the Landau-Ginzburg theory
Phase transitions beyond the Landau-Ginzburg theory Yifei Shi 21 October 2014 1 Phase transitions and critical points 2 Laudau-Ginzburg theory 3 KT transition and vortices 4 Phase transitions beyond Laudau-Ginzburg
More informationMagnetism in ultracold gases
Magnetism in ultracold gases Austen Lamacraft Theoretical condensed matter and atomic physics April 10th, 2009 faculty.virginia.edu/austen/ Outline Magnetism in condensed matter Ultracold atomic physics
More informationCold Atomic Gases. California Condensed Matter Theory Meeting UC Riverside November 2, 2008
New Physics with Interacting Cold Atomic Gases California Condensed Matter Theory Meeting UC Riverside November 2, 2008 Ryan Barnett Caltech Collaborators: H.P. Buchler, E. Chen, E. Demler, J. Moore, S.
More informationCollective Effects. Equilibrium and Nonequilibrium Physics
Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 3, 3 March 2006 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/course/ Caltech
More informationMichikazu Kobayashi. Kyoto University
Topological Excitations and Dynamical Behavior in Bose-Einstein Condensates and Other Systems Michikazu Kobayashi Kyoto University Oct. 24th, 2013 in Okinawa International Workshop for Young Researchers
More informationObservation of topological phenomena in a programmable lattice of 1800 superconducting qubits
Observation of topological phenomena in a programmable lattice of 18 superconducting qubits Andrew D. King Qubits North America 218 Nature 56 456 46, 218 Interdisciplinary teamwork Theory Simulation QA
More informationLecture II: A. Entanglement in correlated systems B. Introduction to topological defects
Lecture II: A. Entanglement in correlated systems B. Introduction to topological defects Ann Arbor, August 10, 2010 Joel Moore University of California, Berkeley and Lawrence Berkeley National Laboratory
More informationNon-equilibrium phenomena in spinor Bose gases. Dan Stamper-Kurn University of California, Berkeley
Non-equilibrium phenomena in spinor Bose gases Dan Stamper-Kurn University of California, Berkeley outline Introductory material Interactions under rotational symmetry Energy scales Ground states Spin
More informationVortex lattice transitions in cyclic spinor condensates
Vortex lattice transitions in cyclic spinor condensates Ryan Barnett, 1 Subroto Mukerjee,,3 and Joel E. Moore,3 1 Department of Physics, California Institute of Technology, MC 114-36, Pasadena, California
More informationMany-body physics 2: Homework 8
Last update: 215.1.31 Many-body physics 2: Homework 8 1. (1 pts) Ideal quantum gases (a)foranidealquantumgas,showthatthegrandpartitionfunctionz G = Tre β(ĥ µ ˆN) is given by { [ ] 1 Z G = i=1 for bosons,
More information5 Topological defects and textures in ordered media
5 Topological defects and textures in ordered media In this chapter we consider how to classify topological defects and textures in ordered media. We give here only a very short account of the method following
More informationDeconfined Quantum Critical Points
Deconfined Quantum Critical Points Leon Balents T. Senthil, MIT A. Vishwanath, UCB S. Sachdev, Yale M.P.A. Fisher, UCSB Outline Introduction: what is a DQCP Disordered and VBS ground states and gauge theory
More informationKosterlitz-Thouless Transition
Heidelberg University Seminar-Lecture 5 SS 16 Menelaos Zikidis Kosterlitz-Thouless Transition 1 Motivation Back in the 70 s, the concept of a phase transition in condensed matter physics was associated
More informationSpinor Bose gases lecture outline
Spinor Bose gases lecture outline 1. Basic properties 2. Magnetic order of spinor Bose-Einstein condensates 3. Imaging spin textures 4. Spin-mixing dynamics 5. Magnetic excitations We re here Coupling
More informationSuperfluidity in bosonic systems
Superfluidity in bosonic systems Rico Pires PI Uni Heidelberg Outline Strongly coupled quantum fluids 2.1 Dilute Bose gases 2.2 Liquid Helium Wieman/Cornell A. Leitner, from wikimedia When are quantum
More informationThe XY model, the Bose Einstein Condensation and Superfluidity in 2d (I)
The XY model, the Bose Einstein Condensation and Superfluidity in 2d (I) B.V. COSTA UFMG BRAZIL LABORATORY FOR SIMULATION IN PHYSICS A Guide to Monte Carlo Simulations in Statistical Physics by Landau
More informationVortices and vortex states of Rashba spin-orbit coupled condensates
Vortices and vortex states of Rashba spin-orbit coupled condensates Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University March 5, 2014 P.N, T.Duric, Z.Tesanovic,
More informationGinzburg-Landau theory of supercondutivity
Ginzburg-Landau theory of supercondutivity Ginzburg-Landau theory of superconductivity Let us apply the above to superconductivity. Our starting point is the free energy functional Z F[Ψ] = d d x [F(Ψ)
More informationTopological Phase Transitions
Chapter 5 Topological Phase Transitions Previously, we have seen that the breaking of a continuous symmetry is accompanied by the appearance of massless Goldstone modes. Fluctuations of the latter lead
More informationDefects in topologically ordered states. Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014
Defects in topologically ordered states Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014 References Maissam Barkeshli & XLQ, PRX, 2, 031013 (2012) Maissam Barkeshli, Chaoming Jian, XLQ,
More information!onformali" Los# J.-W. Lee D. T. Son M. Stephanov D.B.K. arxiv: Phys.Rev.D80:125005,2009
!onformali" Los# J.-W. Lee D. T. Son M. Stephanov D.B.K arxiv:0905.4752 Phys.Rev.D80:125005,2009 Motivation: QCD at LARGE N c and N f Colors Flavors Motivation: QCD at LARGE N c and N f Colors Flavors
More informationXY model: particle-vortex duality. Abstract
XY model: particle-vortex duality Subir Sachdev Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA Dated: February 7, 2018) Abstract We consider the classical XY model in D
More informationNon-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates)
Non-magnetic states Two spins, i and j, in isolation, H ij = J ijsi S j = J ij [Si z Sj z + 1 2 (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j, E ij = 3J ij /4 2 The
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 7: Magnetic excitations - Phase transitions and the Landau mean-field theory. - Heisenberg and Ising models. - Magnetic excitations. External parameter, as for
More informationSpontaneous Symmetry Breaking in Bose-Einstein Condensates
The 10th US-Japan Joint Seminar Spontaneous Symmetry Breaking in Bose-Einstein Condensates Masahito UEDA Tokyo Institute of Technology, ERATO, JST collaborators Yuki Kawaguchi (Tokyo Institute of Technology)
More informationIntroduction to the Berezinskii-Kosterlitz-Thouless Transition
Introduction to the Berezinskii-Kosterlitz-Thouless Transition Douglas Packard May 9, 013 Abstract This essay examines the Berezinskii-Kosterlitz-Thousless transition in the two-dimensional XY model. It
More informationBerry-phase Approach to Electric Polarization and Charge Fractionalization. Dennis P. Clougherty Department of Physics University of Vermont
Berry-phase Approach to Electric Polarization and Charge Fractionalization Dennis P. Clougherty Department of Physics University of Vermont Outline Quick Review Berry phase in quantum systems adiabatic
More informationQuantum spin systems - models and computational methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction
More informationSuperfluid vortex with Mott insulating core
Superfluid vortex with Mott insulating core Congjun Wu, Han-dong Chen, Jiang-ping Hu, and Shou-cheng Zhang (cond-mat/0211457) Department of Physics, Stanford University Department of Applied Physics, Stanford
More informationEffects of spin-orbit coupling on the BKT transition and the vortexantivortex structure in 2D Fermi Gases
Effects of spin-orbit coupling on the BKT transition and the vortexantivortex structure in D Fermi Gases Carlos A. R. Sa de Melo Georgia Institute of Technology QMath13 Mathematical Results in Quantum
More informationSuperfluidity and Condensation
Christian Veit 4th of June, 2013 2 / 29 The discovery of superfluidity Early 1930 s: Peculiar things happen in 4 He below the λ-temperature T λ = 2.17 K 1938: Kapitza, Allen & Misener measure resistance
More informationQuantum Field Theory. Kerson Huang. Second, Revised, and Enlarged Edition WILEY- VCH. From Operators to Path Integrals
Kerson Huang Quantum Field Theory From Operators to Path Integrals Second, Revised, and Enlarged Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA I vh Contents Preface XIII 1 Introducing Quantum Fields
More informationHow spin, charge and superconducting orders intertwine in the cuprates
How spin, charge and superconducting orders intertwine in the cuprates Eduardo Fradkin University of Illinois at Urbana-Champaign Talk at the Kavli Institute for Theoretical Physics Program on Higher temperature
More informationTopological defects and its role in the phase transition of a dense defect system
Topological defects and its role in the phase transition of a dense defect system Suman Sinha * and Soumen Kumar Roy Depatrment of Physics, Jadavpur University Kolkata- 70003, India Abstract Monte Carlo
More informationbeen succeeded in 1997 Rb, 23 Na, 7 Li, 1 H, 85 Rb, 41 K, 4 He, 133 Cs, 174 Yb, 52 Cr, 40 Ca, 84 Sr, 164 Dy Laser cooling Trap of atoms 87
Non-Abelian Vortices and Their Non-equilibrium Michikazu Kobayashi a University of Tokyo November 18th, 2011 at Keio University 2 nd Workshop on Quarks and Hadrons under Extreme Conditions - Lattice QCD,
More informationXY model in 2D and 3D
XY model in 2D and 3D Gabriele Sicuro PhD school Galileo Galilei University of Pisa September 18, 2012 The XY model XY model in 2D and 3D; vortex loop expansion Part I The XY model, duality and loop expansion
More informationHIGH DENSITY NUCLEAR CONDENSATES. Paulo Bedaque, U. of Maryland (Aleksey Cherman & Michael Buchoff, Evan Berkowitz & Srimoyee Sen)
HIGH DENSITY NUCLEAR CONDENSATES Paulo Bedaque, U. of Maryland (Aleksey Cherman & Michael Buchoff, Evan Berkowitz & Srimoyee Sen) What am I talking about? (Gabadadze 10, Ashcroft 89) deuterium/helium at
More informationHidden Symmetry and Quantum Phases in Spin 3/2 Cold Atomic Systems
Hidden Symmetry and Quantum Phases in Spin / Cold Atomic Systems Congjun Wu Kavli Institute for Theoretical Physics, UCSB Ref: C. Wu, Mod. Phys. Lett. B 0, 707, (006); C. Wu, J. P. Hu, and S. C. Zhang,
More informationCritical Spin-liquid Phases in Spin-1/2 Triangular Antiferromagnets. In collaboration with: Olexei Motrunich & Jason Alicea
Critical Spin-liquid Phases in Spin-1/2 Triangular Antiferromagnets In collaboration with: Olexei Motrunich & Jason Alicea I. Background Outline Avoiding conventional symmetry-breaking in s=1/2 AF Topological
More informationStrongly correlated systems in atomic and condensed matter physics. Lecture notes for Physics 284 by Eugene Demler Harvard University
Strongly correlated systems in atomic and condensed matter physics Lecture notes for Physics 284 by Eugene Demler Harvard University January 29, 2018 2 Chapter 4 Spinor condensates 4.1 Two component mixtures
More informationQuantum disordering magnetic order in insulators, metals, and superconductors
Quantum disordering magnetic order in insulators, metals, and superconductors Perimeter Institute, Waterloo, May 29, 2010 Talk online: sachdev.physics.harvard.edu HARVARD Cenke Xu, Harvard arxiv:1004.5431
More informationTopology in Magnetism a phenomenological account. Wednesday: vortices Friday: skyrmions
Topology in Magnetism a phenomenological account Wednesday: vortices Friday: skyrmions Henrik Moodysson Rønnow Laboratory for Quantum Magnetism (LQM), Institute of Physics, EPFL Switzerland Many figures
More informationRenormalization group analysis for the 3D XY model. Francesca Pietracaprina
Renormalization group analysis for the 3D XY model Statistical Physics PhD Course Statistical physics of Cold atoms 5/04/2012 The XY model Duality transformation H = J i,j cos(ϑ i ϑ j ) Plan of the seminar:
More informationValence Bonds in Random Quantum Magnets
Valence Bonds in Random Quantum Magnets theory and application to YbMgGaO 4 Yukawa Institute, Kyoto, November 2017 Itamar Kimchi I.K., Adam Nahum, T. Senthil, arxiv:1710.06860 Valence Bonds in Random Quantum
More information6 Kosterlitz Thouless transition in 4 He thin films
6 Kosterlitz Thouless transition in 4 He thin films J.M. Kosterlitz and D.J. Thouless, Phys. 5, L124 (1972); ibid. 6, 1181 (1973) Topological phase transitions associated with: 1. dislocation-pair unbinding
More informationlattice that you cannot do with graphene! or... Antonio H. Castro Neto
Theoretical Aspects What you can do with cold atomsof on agraphene honeycomb lattice that you cannot do with graphene! or... Antonio H. Castro Neto 2 Outline 1. Graphene for beginners 2. Fermion-Fermion
More informationAn imbalanced Fermi gas in 1 + ɛ dimensions. Andrew J. A. James A. Lamacraft
An imbalanced Fermi gas in 1 + ɛ dimensions Andrew J. A. James A. Lamacraft 2009 Quantum Liquids Interactions and statistics (indistinguishability) Some examples: 4 He 3 He Electrons in a metal Ultracold
More information1 Superfluidity and Bose Einstein Condensate
Physics 223b Lecture 4 Caltech, 04/11/18 1 Superfluidity and Bose Einstein Condensate 1.6 Superfluid phase: topological defect Besides such smooth gapless excitations, superfluid can also support a very
More informationPhase transitions in Hubbard Model
Phase transitions in Hubbard Model Anti-ferromagnetic and superconducting order in the Hubbard model A functional renormalization group study T.Baier, E.Bick, C.Krahl, J.Mueller, S.Friederich Phase diagram
More informationDeconfined Quantum Critical Points
Deconfined Quantum Critical Points Outline: with T. Senthil, Bangalore A. Vishwanath, UCB S. Sachdev, Yale L. Balents, UCSB conventional quantum critical points Landau paradigm Seeking a new paradigm -
More informationNon-abelian statistics
Non-abelian statistics Paul Fendley Non-abelian statistics are just plain interesting. They probably occur in the ν = 5/2 FQHE, and people are constructing time-reversal-invariant models which realize
More informationQuantum critical transport, duality, and M-theory
Quantum critical transport, duality, and M-theory hep-th/0701036 Christopher Herzog (Washington) Pavel Kovtun (UCSB) Subir Sachdev (Harvard) Dam Thanh Son (Washington) Talks online at http://sachdev.physics.harvard.edu
More informationWORLD SCIENTIFIC (2014)
WORLD SCIENTIFIC (2014) LIST OF PROBLEMS Chapter 1: Magnetism of Free Electrons and Atoms 1. Orbital and spin moments of an electron: Using the theory of angular momentum, calculate the orbital
More informationPlease read the following instructions:
MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This
More informationThe Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs
The Gross-Pitaevskii Equation and the Hydrodynamic Expansion of BECs i ( ) t Φ (r, t) = 2 2 2m + V ext(r) + g Φ (r, t) 2 Φ (r, t) (Mewes et al., 1996) 26/11/2009 Stefano Carignano 1 Contents 1 Introduction
More informationGlobal phase diagrams of two-dimensional quantum antiferromagnets. Subir Sachdev Harvard University
Global phase diagrams of two-dimensional quantum antiferromagnets Cenke Xu Yang Qi Subir Sachdev Harvard University Outline 1. Review of experiments Phases of the S=1/2 antiferromagnet on the anisotropic
More informationHexatic and microemulsion phases in a 2D quantum Coulomb gas
Hexatic and microemulsion phases in a 2D quantum Coulomb gas Bryan K Clark (University of Illinois at Urbana Champaign) Michele Casula (Ecole Polytechnique, Paris) David M Ceperley (University of Illinois
More informationPhysics 127b: Statistical Mechanics. Landau Theory of Second Order Phase Transitions. Order Parameter
Physics 127b: Statistical Mechanics Landau Theory of Second Order Phase Transitions Order Parameter Second order phase transitions occur when a new state of reduced symmetry develops continuously from
More informationGround State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory
Ground State Patterns of Spin-1 Bose-Einstein condensation via Γ-convergence Theory Tien-Tsan Shieh joint work with I-Liang Chern and Chiu-Fen Chou National Center of Theoretical Science December 19, 2015
More informationField Theories in Condensed Matter Physics. Edited by. Sumathi Rao. Harish-Chandra Research Institute Allahabad. lop
Field Theories in Condensed Matter Physics Edited by Sumathi Rao Harish-Chandra Research Institute Allahabad lop Institute of Physics Publishing Bristol and Philadelphia Contents Preface xiii Introduction
More informationDrag force and superfluidity in the supersolid striped phase of a spin-orbit-coupled Bose gas
/ 6 Drag force and superfluidity in the supersolid striped phase of a spin-orbit-coupled Bose gas Giovanni Italo Martone with G. V. Shlyapnikov Worhshop on Exploring Nuclear Physics with Ultracold Atoms
More informationRenormalization of Tensor Network States. Partial order and finite temperature phase transition in the Potts model on irregular lattice
Renormalization of Tensor Network States Partial order and finite temperature phase transition in the Potts model on irregular lattice Tao Xiang Institute of Physics Chinese Academy of Sciences Background:
More informationDimerized & frustrated spin chains. Application to copper-germanate
Dimerized & frustrated spin chains Application to copper-germanate Outline CuGeO & basic microscopic models Excitation spectrum Confront theory to experiments Doping Spin-Peierls chains A typical S=1/2
More informationPhase diagram of spin 1/2 bosons on optical lattices
Phase diagram of spin 1/ bosons on optical lattices Frédéric Hébert Institut Non Linéaire de Nice Université de Nice Sophia Antipolis, CNRS Workshop Correlations, Fluctuations and Disorder Grenoble, December
More informationFundamentals and New Frontiers of Bose Einstein Condensation
Contents Preface v 1. Fundamentals of Bose Einstein Condensation 1 1.1 Indistinguishability of Identical Particles.......... 1 1.2 Ideal Bose Gas in a Uniform System............ 3 1.3 Off-Diagonal Long-Range
More informationSuperfluid Helium-3: From very low Temperatures to the Big Bang
Superfluid Helium-3: From very low Temperatures to the Big Bang Universität Frankfurt; May 30, 2007 Dieter Vollhardt Contents: The quantum liquids 3 He and 4 He Superfluid phases of 3 He Broken symmetries
More informationNoncollinear spins in QMC: spiral Spin Density Waves in the HEG
Noncollinear spins in QMC: spiral Spin Density Waves in the HEG Zoltán Radnai and Richard J. Needs Workshop at The Towler Institute July 2006 Overview What are noncollinear spin systems and why are they
More informationMagnetic phenomena in a spin-1 quantum gas. Dan Stamper-Kurn UC Berkeley, Physics Lawrence Berkeley National Laboratory, Materials Sciences
Magnetic phenomena in a spin-1 quantum gas Dan Stamper-Kurn UC Berkeley, Physics Lawrence Berkeley National Laboratory, Materials Sciences Spinor gases 37 electrons J = 1/2 Energy F=2 Optically trapped
More informationSuperfluidity of a 2D Bose gas (arxiv: v1)
Superfluidity of a 2D Bose gas (arxiv:1205.4536v1) Christof Weitenberg, Rémi Desbuquois, Lauriane Chomaz, Tarik Yefsah, Julian Leonard, Jérôme Beugnon, Jean Dalibard Trieste 18.07.2012 Phase transitions
More informationDesign and realization of exotic quantum phases in atomic gases
Design and realization of exotic quantum phases in atomic gases H.P. Büchler and P. Zoller Theoretische Physik, Universität Innsbruck, Austria Institut für Quantenoptik und Quanteninformation der Österreichischen
More informationSolitonic elliptical solutions in the classical XY model
Solitonic elliptical solutions in the classical XY model Rodrigo Ferrer, José Rogan, Sergio Davis, Gonzalo Gutiérrez Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago,
More informationDuality, Statistical Mechanics and Random Matrices. Bielefeld Lectures
Duality, Statistical Mechanics and Random Matrices Bielefeld Lectures Tom Spencer Institute for Advanced Study Princeton, NJ August 16, 2016 Overview Statistical mechanics motivated by Random Matrix theory
More informationIntertwined Orders in High Temperature Superconductors
Intertwined Orders in High Temperature Superconductors! Eduardo Fradkin University of Illinois at Urbana-Champaign! Talk at SCES@60 Institute for Condensed Matter Theory University of Illinois at Urbana-Champaign
More informationElectronic Liquid Crystal Phases in Strongly Correlated Systems
Electronic Liquid Crystal Phases in Strongly Correlated Systems Eduardo Fradkin University of Illinois at Urbana-Champaign Talk at the workshop Materials and the Imagination, Aspen Center of Physics, January
More informationNotes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model
Notes on Renormalization Group: Berezinskii-Kosterlitz-Thouless (BKT) transition and Sine-Gordon model Yi Zhou (Dated: December 4, 05) We shall discuss BKT transition based on +D sine-gordon model. I.
More informationPhysics 4022 Notes on Density Matrices
Physics 40 Notes on Density Matrices Definition: For a system in a definite normalized state ψ > the density matrix ρ is ρ = ψ >< ψ 1) From Eq 1 it is obvious that in the basis defined by ψ > and other
More informationLong-range order in (quasi-) 2D systems
PHYS598PTD A.J.Leggett 2016 Lecture 9 Long-range order in (quasi-) 2D systems 1 Long-range order in (quasi-) 2D systems Suppose we have a many-body system that is described by some order parameter Ψ. The
More informationFunctional renormalization for ultracold quantum gases
Functional renormalization for ultracold quantum gases Stefan Floerchinger (Heidelberg) S. Floerchinger and C. Wetterich, Phys. Rev. A 77, 053603 (2008); S. Floerchinger and C. Wetterich, arxiv:0805.2571.
More informationFrom laser cooling to BEC First experiments of superfluid hydrodynamics
From laser cooling to BEC First experiments of superfluid hydrodynamics Alice Sinatra Quantum Fluids course - Complement 1 2013-2014 Plan 1 COOLING AND TRAPPING 2 CONDENSATION 3 NON-LINEAR PHYSICS AND
More informationSpin Models and Gravity
Spin Models and Gravity Umut Gürsoy (CERN) 6th Regional Meeting in String Theory, Milos June 25, 2011 Spin Models and Gravity p.1 Spin systems Spin Models and Gravity p.2 Spin systems Many condensed matter
More informationSupersymmetry breaking and Nambu-Goldstone fermions in lattice models
YKIS2016@YITP (2016/6/15) Supersymmetry breaking and Nambu-Goldstone fermions in lattice models Hosho Katsura (Department of Physics, UTokyo) Collaborators: Yu Nakayama (IPMU Rikkyo) Noriaki Sannomiya
More informationSuperfluidity and Superconductivity
Superfluidity and Superconductivity These are related phenomena of flow without resistance, but in very different systems Superfluidity: flow of helium IV atoms in a liquid Superconductivity: flow of electron
More informationPhase transitions in Bi-layer quantum Hall systems
Phase transitions in Bi-layer quantum Hall systems Ming-Che Chang Department of Physics Taiwan Normal University Min-Fong Yang Departmant of Physics Tung-Hai University Landau levels Ferromagnetism near
More informationBerezinskii-Kosterlitz-Thouless Phase Transition and Superfluidity In Two Dimensions. University of Science and Technology of China
Lecture on Advances in Modern Physics Berezinskii-Kosterlitz-Thouless Phase Transition and Superfluidity In Two Dimensions Youjin Deng University of Science and Technology of China 2016-12-03 Popular
More informationGEOMETRICALLY FRUSTRATED MAGNETS. John Chalker Physics Department, Oxford University
GEOMETRICLLY FRUSTRTED MGNETS John Chalker Physics Department, Oxford University Outline How are geometrically frustrated magnets special? What they are not Evading long range order Degeneracy and fluctuations
More informationSpin liquid phases in strongly correlated lattice models
Spin liquid phases in strongly correlated lattice models Sandro Sorella Wenjun Hu, F. Becca SISSA, IOM DEMOCRITOS, Trieste Seiji Yunoki, Y. Otsuka Riken, Kobe, Japan (K-computer) Williamsburg, 14 June
More informationLuigi Paolasini
Luigi Paolasini paolasini@esrf.fr LECTURE 4: MAGNETIC INTERACTIONS - Dipole vs exchange magnetic interactions. - Direct and indirect exchange interactions. - Anisotropic exchange interactions. - Interplay
More informationFully symmetric and non-fractionalized Mott insulators at fractional site-filling
Fully symmetric and non-fractionalized Mott insulators at fractional site-filling Itamar Kimchi University of California, Berkeley EQPCM @ ISSP June 19, 2013 PRL 2013 (kagome), 1207.0498...[PNAS] (honeycomb)
More informationSuperfluid Helium-3: From very low Temperatures to the Big Bang
Superfluid Helium-3: From very low Temperatures to the Big Bang Dieter Vollhardt Yukawa Institute, Kyoto; November 27, 2007 Contents: The quantum liquids 3 He and 4 He Superfluid phases of 3 He Broken
More informationQuantum phase transitions and the Luttinger theorem.
Quantum phase transitions and the Luttinger theorem. Leon Balents (UCSB) Matthew Fisher (UCSB) Stephen Powell (Yale) Subir Sachdev (Yale) T. Senthil (MIT) Ashvin Vishwanath (Berkeley) Matthias Vojta (Karlsruhe)
More informationSuperconducting properties of carbon nanotubes
Superconducting properties of carbon nanotubes Reinhold Egger Institut für Theoretische Physik Heinrich-Heine Universität Düsseldorf A. De Martino, F. Siano Overview Superconductivity in ropes of nanotubes
More informationSupplementary Figure 3: Interaction effects in the proposed state preparation with Bloch oscillations. The numerical results are obtained by
Supplementary Figure : Bandstructure of the spin-dependent hexagonal lattice. The lattice depth used here is V 0 = E rec, E rec the single photon recoil energy. In a and b, we choose the spin dependence
More informationPhysics 505 Homework No. 8 Solutions S Spinor rotations. Somewhat based on a problem in Schwabl.
Physics 505 Homework No 8 s S8- Spinor rotations Somewhat based on a problem in Schwabl a) Suppose n is a unit vector We are interested in n σ Show that n σ) = I where I is the identity matrix We will
More informationLow dimensional quantum gases, rotation and vortices
Goal of these lectures Low dimensional quantum gases, rotation and vortices Discuss some aspect of the physics of quantum low dimensional systems Planar fluids Quantum wells and MOS structures High T c
More informationSummer School on Novel Quantum Phases and Non-Equilibrium Phenomena in Cold Atomic Gases. 27 August - 7 September, 2007
1859-5 Summer School on Novel Quantum Phases and Non-Equilibrium Phenomena in Cold Atomic Gases 27 August - 7 September, 2007 Dipolar BECs with spin degrees of freedom Yuki Kawaguchi Tokyo Institute of
More informationVortex States in a Non-Abelian Magnetic Field
Vortex States in a Non-Abelian Magnetic Field Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University SESAPS November 10, 2016 Acknowledgments Collin Broholm IQM
More informationPlease read the following instructions:
MIDTERM #1 PHYS 33 (MODERN PHYSICS II) DATE/TIME: February 16, 17 (8:3 a.m. - 9:45 a.m.) PLACE: RB 11 Only non-programmable calculators are allowed. Name: ID: Please read the following instructions: This
More informationThe Remarkable Superconducting Stripe Phase of the High Tc Superconductor La2-xBaxCuO4 near x=1/8
The Remarkable Superconducting Stripe Phase of the High Tc Superconductor La2-xBaxCuO4 near x=1/8 Eduardo Fradkin University of Illinois at Urbana-Champaign Seminar at the Department of Physics Harvard
More information