The phase diagram of polar condensates

The phase diagram of polar condensates Taking the square root of a vortex Austen Lamacraft [with Andrew James] arxiv:1009.0043 University of Virginia September 23, 2010 KITP, UCSB Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 1 / 32

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, 2010 2 / 32

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, 2010 4 / 32

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, 2010 4 / 32

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, 2010 4 / 32

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, 2010 5 / 32

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, 2010 5 / 32

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, 2010 5 / 32

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, 2010 5 / 32

Contact interactions in a spin-1 gas Total spin 2 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 6 / 32

Contact interactions in a spin-1 gas Total spin 0 Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 6 / 32

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, 2010 6 / 32

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, 2010 7 / 32

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, 2010 7 / 32

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, 2010 8 / 32

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, 2010 8 / 32

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, 2010 8 / 32

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, 2010 8 / 32

The Bose Ferromagnet Stamper-Kurn group, Berkeley Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 9 / 32

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, 2010 11 / 32

Disclinations in nematic liquid crystals Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 12 / 32

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, 2010 13 / 32

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, 2010 13 / 32

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, 2010 13 / 32

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, 2010 14 / 32

KT transition mediated by half-vortices / disclinations PRL 97, 120406 (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 0.35 0.3 0.25 Γ 0.2 0.15 8T/π 40 40 16 16 8 8 0.1 0.05 2T/π 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 T Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 14 / 32

What about the n degrees of freedom? Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 15 / 32

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, 2010 15 / 32

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, 2010 15 / 32

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, 2010 15 / 32

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, 2010 15 / 32

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, 2010 17 / 32

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, 2010 17 / 32

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, 2010 17 / 32

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, 2010 18 / 32

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, 2010 18 / 32

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, 2010 18 / 32

Half vortices terminate soliton walls Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 19 / 32

Conjectured phase diagram Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 20 / 32

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, 2010 21 / 32

Monte Carlo simulation Intermediate phase with singlet pair quasi-long range order 3 2.5 D 2 N T/t 1.5 1 O 0.5 0 0 2 4 6 8 10 q (data at u = 1) Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 21 / 32

Heat Capacity 10 5 q=1 Specific heat 8 6 4 4 3 2 1 L=8 16 24 32 0 0 1 2 3 4 5 T q=1 2 3 4 5 L=16 2 0 0 1 2 3 4 T Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 22 / 32

Binder cummulant Φ = i φ i U 4 = ( Φ Φ ) 2 ( Φ Φ ) 2 q=1 Binder cumulant 2.5 2 1.5 1.3 1.2 1.1 1 1.5 1.6 1.7 1.8 L=8 16 24 32 1 0 1 2 3 4 T Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 23 / 32

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, 2010 24 / 32

Ising scaling for the lower transition 1.4 1.4 Binder cumulant 1.3 1.2 1.3 1.2 L=8 16 24 32 40 1.1 1.1 1 1.66 1.67 1.68 1.69 T 1-0.5 0 0.5 (T-Tc)L Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 24 / 32

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, 2010 25 / 32

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, 2010 25 / 32

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, 2010 27 / 32

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, 2010 28 / 32

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, 2010 28 / 32

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, 2010 28 / 32

Phase diagram from µ cos(φ/2) perturbation Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 29 / 32

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, 2010 29 / 32

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, 2010 29 / 32

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, 2010 30 / 32

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 = 2 1 8 πk 4 dt I dl = t I + λ2 2 dk dl = λ2 ) λ + 1 2 λt I Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 30 / 32

RG flow for K > 15/2π Austen Lamacraft (University of Virginia) The phase diagram of polar condensates September 23rd, 2010 31 / 32