The phase diagram of polar condensates
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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
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