Boson Pairing and Unusual Criticality GGI, May 2012 Austen - PowerPoint PPT Presentation

Pair condensates Interplay of strings and vortices Novel signatures Boson Pairing and Unusual Criticality GGI, May 2012 Austen Lamacraft Y. Shi, P. Fendley, AL, Phys. Rev. Lett. 107, 240601 (2011) A. J. A. James, AL, Phys. Rev. Lett. 106, 140402 (2011)

Pair condensates Interference of independent BECs Interplay of strings and vortices M. R. Andrews et al. (1997) Novel signatures

Pair condensates Interplay of strings and vortices Novel signatures Interference of independent BECs N � e in ∆ θ | n � L | N − n � R Definite phase | ∆ θ � = n =0 � 2 π d (∆ θ ) | ∆ θ � e − in ∆ θ Definite number | n � L | N − n � R = 2 π 0

Pair condensates Interplay of strings and vortices Novel signatures Pair condensates: Ising variables in an XY system Take a condensate of molecules and split it N / 2 � | Ψ � = | 2 n � L | N − 2 n � R n =0

Pair condensates Interplay of strings and vortices Novel signatures Pair condensates: Ising variables in an XY system Dissociate pairs

Pair condensates Interplay of strings and vortices Novel signatures Pair condensates: Ising variables in an XY system What is the resulting state? Superposition involves only even numbers of atoms N / 2 N N | 2 n � L | N − 2 n � R = 1 | n � L | N − n � R + 1 � � � ( − 1) n | n � L | N − n � R 2 2 n =0 n =0 n =0 = 1 2 ( | ∆ θ = 0 � + | ∆ θ = π � )

Pair condensates Interplay of strings and vortices Novel signatures Pair condensates: Ising variables in an XY system What are the consequences for the phase diagram?

Pair condensates Interplay of strings and vortices Novel signatures Outline Pair condensates Interplay of strings and vortices Novel signatures

Pair condensates Interplay of strings and vortices Novel signatures Outline Pair condensates Interplay of strings and vortices Novel signatures

Pair condensates Interplay of strings and vortices Novel signatures Vortices give a twist in 2D Quantized vortices: phase increases by 2 π × q (Integer q ) v = 1 m ∇ θ = 1 ˆ e θ m r � L � � Kinetic energy = mn d r v 2 = π n m ln 2 ξ

Pair condensates Interplay of strings and vortices Novel signatures The Kosterlitz–Thouless transition Consider free energy of a single integer vortex � L � 2 Entropy, S = k B ln ξ � L � � π n � F = E − TS = m − 2 k B T ln ξ

Pair condensates Interplay of strings and vortices Novel signatures The Kosterlitz–Thouless transition Consider free energy of a single integer vortex � L � 2 Entropy, S = k B ln ξ � L � � π n � F = E − TS = m − 2 k B T ln ξ Vanishes at k B T c = π n 2 m

Pair condensates Interplay of strings and vortices Novel signatures A simple model for pair condensates Hopping Pair hopping � �� � � �� � � � � H GXY = − ∆ cos( θ i − θ j ) + (1 − ∆) cos (2 θ i − 2 θ j ) � ij � Korshunov (1985), Lee & Grinstein (1985) • ∆ = 1 is usual XY; ∆ = 0 is π -periodic XY • ∆ > 0 → metastable min. → line tension along π -phase jump

Pair condensates Interplay of strings and vortices Novel signatures “Half” Kosterlitz–Thouless transition at ∆ = 0 � L � � E = mn d r v 2 = π n 4 m ln 2 ξ � π n � L � � F = U − TS = 4 m − 2 k B T ln ξ

Pair condensates Interplay of strings and vortices Novel signatures “Half” Kosterlitz–Thouless transition at ∆ = 0 � L � � E = mn d r v 2 = π n 4 m ln 2 ξ � π n � L � � F = U − TS = 4 m − 2 k B T ln ξ Vanishes at k B T c = π n 8 m

Pair condensates Interplay of strings and vortices Novel signatures “Half” Kosterlitz–Thouless transition at ∆ = 0 � L � � E = mn d r v 2 = π n 4 m ln 2 ξ � π n � L � � F = U − TS = 4 m − 2 k B T ln ξ Vanishes at k B T c = π n 8 m

Pair condensates Interplay of strings and vortices Novel signatures Schematic phase diagram � H GXY = − [∆ cos( θ i − θ j ) + (1 − ∆) cos (2 θ i − 2 θ j )] � ij �

Pair condensates Interplay of strings and vortices Novel signatures Schematic phase diagram � H GXY = − [∆ cos( θ i − θ j ) + (1 − ∆) cos (2 θ i − 2 θ j )] � ij � What is the nature of phase transition along dotted line?

Pair condensates Interplay of strings and vortices Novel signatures Almost tetratic phases of colloidal rectangles 1 1 Kun Zhao et al. , PRE (2007)

Pair condensates Interplay of strings and vortices Novel signatures Almost tetratic phases of colloidal rectangles 1 θ 2 • � cos(2 θ ) � � = 0 Nematic disordered by π -disclinations • � cos(4 θ ) � � = 0 Tetratic disordered by π 2 -disclinations θ 1 1 Kun Zhao et al. , PRE (2007)

Pair condensates Interplay of strings and vortices Novel signatures Outline Pair condensates Interplay of strings and vortices Novel signatures

Pair condensates Interplay of strings and vortices Novel signatures How things change on the Ising critical line Redo KT argument accounting for string Partition function with string connecting x , y defines � µ ( x ) µ ( y ) � µ ’s are disorder operators

Pair condensates Interplay of strings and vortices Novel signatures How things change on the Ising critical line Disorder operators dual to σ ( x ) of Ising model. At T c 1 � σ ( x ) σ ( y ) � = � µ ( x ) µ ( y ) � = | x − y | 1 / 4 1 Take one end of string to ∞ i.e. edge of system: � µ ( x ) � → L 1 / 8 Contributes + k B T ln L to free energy F = − k B T ln Z 8 � π n � � L � 4 m − 15 F = U − TS = 8 k B T ln ξ k B T c = 2 π n 15 m Dissociation at higher temperatures than for ‘free’ half vortices

Pair condensates Interplay of strings and vortices Novel signatures How things change on the Ising critical line Direct Ising transition to low temperature phase!

Pair condensates Interplay of strings and vortices Novel signatures RG flow Keep track of 1. Stiffness J (or J ∗ = J − 1 ) 2. Deviation from Ising critical line κ = K − K c 3. Half vortex fugacity z 1 � 15 � 8 − π z 1 − κ z 1 dz 1 dl = 4 J ∗ 2 dl = π 2 z 2 dJ ∗ 1 4 dl = κ − z 2 d κ 1 4

Pair condensates Interplay of strings and vortices Novel signatures RG flow Keep track of 1. Stiffness J (or J ∗ = J − 1 ) 2. Deviation from Ising critical line κ = K − K c 3. Half vortex fugacity z 1 J > 15 2 π 0.8 z 1 0.4 0 � 0.2 0 0.2 Κ

Pair condensates Interplay of strings and vortices Novel signatures RG flow Keep track of 1. Stiffness J (or J ∗ = J − 1 ) 2. Deviation from Ising critical line κ = K − K c 3. Half vortex fugacity z 1 J < 15 2 π 0.8 z 1 0.4 0 � 0.2 0 0.2 Κ

Pair condensates Interplay of strings and vortices Novel signatures Numerical simulation using worm algorithm    − J ∗ ij + K ∗ � � � n 2 ( − 1) n ij Z = exp  2 2 { n ij } � ij � � ij � ∇· n =0 Simulate worldlines of bosons, not spin variables

Pair condensates Interplay of strings and vortices Novel signatures Numerical simulation using worm algorithm � π d θ c � � Z = w ( θ a − θ b ) 2 π − π c � ab � w ( θ ) written in Villain form: w ( θ ) ≡ w V ( θ ) + e − K w V ( θ − π ) ∞ ∞ 2 ( θ +2 π p ) 2 ∝ � � e in θ e − J ∗ e − J 2 n 2 w V ( θ ) ≡ p = −∞ n = −∞ J ∗ = J − 1 , sinh K ∗ sinh K = 1

Pair condensates Interplay of strings and vortices Novel signatures Numerical simulation using worm algorithm    − J ∗ ij + K ∗ � � � n 2 ( − 1) n ij Z = exp  2 2 { n ij } � ij � � ij � ∇· n =0 J ∗ = J − 1 , sinh K ∗ sinh K = 1

Pair condensates Interplay of strings and vortices Novel signatures Numerical simulation using worm algorithm

Pair condensates Interplay of strings and vortices Novel signatures Use of sectors Calculation of superfluid stiffness Υ = ∂ 2 F ∂θ 2 2 Υ = T 2 � W 2 � W = ( W x , W y ), vector of windings halfKT_fitting 4 L=20 L=40 L=80 2 L=160 0 Winding_Variance -2 -4 -6 -8 -10 -12 -15 -10 -5 0 5 10 J_rescaled 2 Pollock & Ceperley (1987)

Pair condensates Interplay of strings and vortices Novel signatures Use of sectors Keeping track of parity of winding gives sectors of Ising model Z PP Z PA Z AP Z AA

Pair condensates Interplay of strings and vortices Novel signatures Use of sectors Critical ratios known from CFT 2 Z PP = 1 . 8963 . . . 1 √ Z AP = Z PA = 2 Z AA = 0 . 4821 . . . Use to locate phase transition from finite size scaling 2 P. Ginsparg, Les Houches lectues (1989)

Pair condensates Interplay of strings and vortices Novel signatures Use of sectors Z AP + Z PA = 0 . 372 . . . 2 Z PP Crossing_J=2.8 0.9 "SectorL=20J=2.8.dat" "SectorL=40J=2.8.dat" 0.8 "SectorL=80J=2.8.dat" "SectorL=160J=2.8.dat" 0.7 0.6 0.5 Zap/Zpp 0.4 0.3 0.2 0.1 0 0.878 0.88 0.882 0.884 0.886 0.888 0.89 0.892 0.894 0.896 0.898 0.9 K

Pair condensates Interplay of strings and vortices Novel signatures Quantum Transition 2D classical picture also applies to (1+1)D quantum transition Ejima et al. (2011)