Outline Introduction Spin systems Spin Wave Theory Quantum Monte Carlo Results Hard-core bosons on a triangular lattice with long range interaction with finite temperature l Maik 1 , 2 , Philipp Hauke 2 , Omjyoti Dutta 2 , Jakub Micha� Zakrzewski 1 , 3 and Maciej Lewenstein 2 , 4 1 Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagiello´ nski 2 ICFO – Institut de Ci` encies Fot` oniques, Mediterranean Technology Park 3 Mark Kac Complex Systems Research Center, Uniwersytet Jagiello´ nski 4 ICREA – Instituci` o Catalana de Recerca i Estudis Avan¸ cats September 17, 2012 Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Spin Wave Theory Quantum Monte Carlo Results Introduction 1 Quantum Simulators Trapped Ions Spin systems 2 1D system 2D system Spin Wave Theory 3 Quantum Monte Carlo 4 Results 5 Wigner Crystals Temperature Dependance Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Quantum Simulators Spin Wave Theory Trapped Ions Quantum Monte Carlo Results Quantum Simulators Why do we need to have quantum simulators? Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Quantum Simulators Spin Wave Theory Trapped Ions Quantum Monte Carlo Results Quantum Simulators Why do we need to have quantum simulators? Simulating quantum mechanical systems is very difficult. Number of parameters that describe a quantum state grow exponentially with the number of particles. (2 n for n spin 1/2 particles.) A way to solve this is to create a highly controlable system that efficiently simulates our system. Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Quantum Simulators Spin Wave Theory Trapped Ions Quantum Monte Carlo Results Trapped Ions Concept Effective Quantum Spin Systems with Trapped Ions D. Porras and J. Cirac, Phys. Rev. Lett. 92 , 207901 (2004) Proof-of-principle experiments Simulating a quantum magnet with trapped ions A. Friedenauer et al. , Nat. Phys. 4 , 757 (2008) Quantum simulation of frustrated Ising spins with trapped ions K. Kim et al. , Nature 465 , 590 (2010) Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems 1D system Spin Wave Theory 2D system Quantum Monte Carlo Results 1D Spin Chain Complete devil’s staircase and crystal-superfluid transitions in a dipolar XXZ spin chain: a trapped ion quantum simulation P. Hauke et al. , New Journal of Physics 12 , 113037 (2010) 1 � j + S y i S y � | i − j | 3 [cos θ ( S z i S z j ) + sin θ ( S x i S x S z H = J j )] − µ i i , j i Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems 1D system Spin Wave Theory 2D system Quantum Monte Carlo Results Magnetization Magnetic lobes of 1D spin chain Solved using Density Method Renormalization Group (DMRG) 60 site spin chain Long ranged interactions T = 0 Open Boundary Conditions. Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems 1D system Spin Wave Theory 2D system Quantum Monte Carlo Results Devil’s staircase θ = 0 Corresponds to the Ising model Creates a generalized Wigner crystal Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems 1D system Spin Wave Theory 2D system Quantum Monte Carlo Results The 2D model: Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems 1D system Spin Wave Theory 2D system Quantum Monte Carlo Results The 2D model: 6x6 triangular lattice with periodic boundary conditions. Long ranged spin-spin interactions (both hopping and dipolar) Ultra-frustrated Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems 1D system Spin Wave Theory 2D system Quantum Monte Carlo Results Frustration Prevents simultaneous NN model has 6 interactions minimization of interaction energies LR model has 36 interactions Creates degeneracies and a multitude of meta stable states Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Spin Wave Theory Quantum Monte Carlo Results Holstein-Primakoff bosons We start with our XXZ Spin Hamiltonian 1 � | i − j | 3 [cos θ ( S z i S z j ) + sin θ ( S x i S x j + S y i S y � S z H = J j )] − µ i i , j i Now we will use Holstein-Primakoff transformations in order to redefine our spins √ √ S − = ( 2 S − n ) a , S + = a † ( 2 S − n ), S z = n − S where n = a † a and [ a , a † ] = 1 and S is the total spin and the spins continue to obey their commutation relationships [ S α , S β ] = ıǫ αβγ S γ Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Spin Wave Theory Quantum Monte Carlo Results Approximation Let’s take a look at the square root term. √ √ 1 − n � 1 / 2 � 2 S − n = 2 S 2 S Now let’s expand the using Taylor series expansion ∞ √ ( − 1) n (2 n )! 2 − x 2 (1 − 2 n )( n !) 2 (4 n ) x n = 1 + x � 1 − x = 8 + ... n =0 So √ √ n 2 � 1 − n � 2 S − n = 2 S 4 S − 32 S 2 − ... Now we choose our spin to be S = 1 2 , then S − = a , S + = a † , S z = n − 1 2 Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Spin Wave Theory Quantum Monte Carlo Results Let’s now apply the transformations to the Hamiltonian → a † S − → a i , S + i , S z i → n − 1 i i 2 The new Hamiltonian now becomes: 1 � � n i n j − n i 2 − n j 2 + 1 �� � = cos θ H J | i − j | 3 4 i , j � sin θ �� 1 � � a † i a j + a † + J j a i | i − j | 3 2 i , j � n i − 1 � � − µ 2 i Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Spin Wave Theory Quantum Monte Carlo Results The simulation All simulations were run using the worm algorithm of the open source ALPS (Algorithms and Libraries for Physics Simulations) project. This algorithm, first created by N. Prokof’ev, works by sampling world lines in the path integral representation of the partition function in the grand canonical ensemble. Calulations are run in low but finite temperature. We are restricted to only studying negative θ due to the sign problem. The sign problem occurs when the hopping term is negative because negative probabilities arise in the partion function. Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Wigner Crystals Spin Wave Theory Temperature Dependance Quantum Monte Carlo Results Finite Temperature Devil’s Staircase Short ranged interactions 1.0 0.9 0.8 Ρ 0.7 0.6 Wigner crystal 0.5 0 1 2 3 4 5 Μ � J Longed ranged interactions θ = 0 1.0 T = 0 . 1 0.9 2/3 filling has largest plataeu. 0.8 Ρ 0.7 0.6 0.5 0 1 2 3 4 5 Μ � J Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Wigner Crystals Spin Wave Theory Temperature Dependance Quantum Monte Carlo Results Density and Superfluidity Short Ranged Long Ranged Dipole Long Ranged All Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Wigner Crystals Spin Wave Theory Temperature Dependance Quantum Monte Carlo Results Supersolids In order to properly investigate the existance of a supersolid we look at the two values: Structure factor 2 � �� � N � � � n i e ı Qr i / N 2 S ( Q ) = � � � � � � i =1 where the wave vector is Q = (4 π/ 3 , 0) Superfluid fraction ρ s = � W 2 � 4 β where W is the winding number fluctuation of world lines and β is the inverse temperature. Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Wigner Crystals Spin Wave Theory Temperature Dependance Quantum Monte Carlo Results 1 0.2 0.2 0.15 S � Q � , Ρ s 0.15 0.1 0.8 S � Q � 0.1 0.05 0.05 0 0.6 � 0.3 � 0.2 � 0.1 0 0 Ρ s � 0.3 � 0.2 � 0.1 0 Θ 0.2 0.4 Θ 0.15 S � Q � , Ρ s 0.1 0.2 0.05 0 0 � 0.3 � 0.2 � 0.1 0 � 0.3 � 0.2 � 0.1 0 Θ Θ Superfluid fraction and structure factor graphs taken at µ/ J = 0 for multiple system sizes (L = 6, 9 and 12). Lines get thicker and darker with system size increase. Maik et al. Hard-core bosons on a triangular lattice
Outline Introduction Spin systems Wigner Crystals Spin Wave Theory Temperature Dependance Quantum Monte Carlo Results 0.12 0.3 0.09 S � Q � , Ρ s 0.06 0.03 0.2 0 S � Q � , Ρ s 0 1 2 3 4 Μ � J 0.12 0.1 0.09 S � Q � , Ρ s 0.06 0.03 0 0 0 1 2 3 4 0 1 2 3 4 Μ � J Μ � J Superfluid fraction and structure factor graphs taken at θ = − 0 . 15 for multiple system sizes (L = 6, 9 and 12). Lines get thicker and darker with system size increase. Maik et al. Hard-core bosons on a triangular lattice
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