Quantum quenches in the thermodynamic limit Marcos Rigol Department of Physics The Pennsylvania State University Quantum dynamics in systems with many coupled degrees of freedom: challenges for theory Center for Free-Electron Laser Science, Hamburg March 26, 2014 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 1 / 30
Outline Introduction 1 Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches Quantum quenches in the thermodynamic limit 2 Diagonal ensemble and NLCEs Quantum quenches in one-dimension Conclusions 3 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 2 / 30
Outline Introduction 1 Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches Quantum quenches in the thermodynamic limit 2 Diagonal ensemble and NLCEs Quantum quenches in one-dimension Conclusions 3 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 3 / 30
Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 4 / 30
Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Exact diagonalization Exponential problem ⇒ Small systems ⇒ Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 4 / 30
Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Exact diagonalization Exponential problem ⇒ Small systems ⇒ Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! High temperature expansions Exponential problem ⇒ High temperatures Thermodynamic limit ⇒ Extrapolations to low T Can be used for any model! Can fail (at low T ) even when correlations are short ranged! Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 4 / 30
Finite temperature properties of lattice models Computational techniques for arbitrary dimensions Quantum Monte Carlo simulations Polynomial time ⇒ Large systems ⇒ Finite size scaling Sign problem ⇒ Limited classes of models Exact diagonalization Exponential problem ⇒ Small systems ⇒ Finite size effects No systematic extrapolation to larger system sizes Can be used for any model! High temperature expansions Exponential problem ⇒ High temperatures Thermodynamic limit ⇒ Extrapolations to low T Can be used for any model! Can fail (at low T ) even when correlations are short ranged! DMFT, DCA, DMRG, . . . Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 4 / 30
Linked-Cluster Expansions Extensive observables ˆ O per lattice site ( O ) in the thermodynamic limit � O = L ( c ) × W O ( c ) c where L ( c ) is the number of embeddings of cluster c Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 5 / 30
Linked-Cluster Expansions Extensive observables ˆ O per lattice site ( O ) in the thermodynamic limit � O = L ( c ) × W O ( c ) c where L ( c ) is the number of embeddings of cluster c and W O ( c ) is the weight of observable O in cluster c � W O ( c ) = O ( c ) − W O ( s ) . s ⊂ c O ( c ) is the result for O in cluster c � � ˆ ρ GC O ( c ) = Tr O ˆ , c 1 exp − ( ˆ H c − µ ˆ N c ) /k B T ρ GC ˆ = c Z GC c � N c ) /k B T � exp − ( ˆ H c − µ ˆ Z GC = Tr c and the s sum runs over all subclusters of c . Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 5 / 30
Linked-Cluster Expansions In HTEs O ( c ) is expanded in powers of β and only a finite number of terms are retained. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 6 / 30
Linked-Cluster Expansions In HTEs O ( c ) is expanded in powers of β and only a finite number of terms are retained. In NLCEs an exact diagonalization of the cluster is used to calculate O ( c ) at any temperature. MR, T. Bryant, and R. R. P . Singh, PRL 97 , 187202 (2006). MR, T. Bryant, and R. R. P . Singh, PRE 75 , 061118 (2007). MR, T. Bryant, and R. R. P . Singh, PRE 75 , 061119 (2007). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 6 / 30
Linked-Cluster Expansions In HTEs O ( c ) is expanded in powers of β and only a finite number of terms are retained. In NLCEs an exact diagonalization of the cluster is used to calculate O ( c ) at any temperature. MR, T. Bryant, and R. R. P . Singh, PRL 97 , 187202 (2006). MR, T. Bryant, and R. R. P . Singh, PRE 75 , 061118 (2007). MR, T. Bryant, and R. R. P . Singh, PRE 75 , 061119 (2007). 2D Hubbard-like models (square and honeycomb), spin models (kagome, checkerboard, pyroclore – experiments) MR and R. R. P . Singh, PRL 98 , 207204 (2007). MR and R. R. P . Singh, PRB 76 , 184403 (2007). E. Khatami and MR, PRB 83 , 134431 (2011). E. Khatami and MR, PRA 84 , 053611 (2011). E. Khatami, R. R. P . Singh, and MR, PRB 84 , 224411 (2011). E. Khatami, J. S. Helton, and MR, PRB 85 , 064401 (2012). E. Khatami and MR, PRA 86 , 023633 (2012). B. Tang, T. Paiva, E. Khatami, and MR, PRL 109 , 205301 (2012). B. Tang, T. Paiva, E. Khatami, and MR, PRB 88 , 125127 (2013). . . . Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 6 / 30
Outline Introduction 1 Computational techniques for quantum many-body problems Numerical Linked Cluster Expansions Quantum quenches Quantum quenches in the thermodynamic limit 2 Diagonal ensemble and NLCEs Quantum quenches in one-dimension Conclusions 3 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 7 / 30
Numerical Linked Cluster Expansions Bond clusters i) Find all clusters that can be c L(c) embedded on the lattice 1 1 2 2 3 2 4 4 5 4 6 2 7 4 8 4 9 8 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 8 / 30
Numerical Linked Cluster Expansions i) Find all clusters that can be No. of bonds topological clusters embedded on the lattice 0 1 1 1 ii) Group the ones with the 2 1 same Hamiltonian (Topo- 3 2 logical cluster) 4 4 5 6 6 14 7 28 8 68 9 156 10 399 11 1012 12 2732 13 7385 14 20665 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 8 / 30
Numerical Linked Cluster Expansions i) Find all clusters that can be No. of bonds topological clusters embedded on the lattice 0 1 1 1 ii) Group the ones with the 2 1 same Hamiltonian (Topo- 3 2 logical cluster) 4 4 5 6 iii) Find all subclusters of a 6 14 given topological cluster 7 28 8 68 9 156 10 399 11 1012 12 2732 13 7385 14 20665 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 8 / 30
Numerical Linked Cluster Expansions i) Find all clusters that can be No. of bonds topological clusters embedded on the lattice 0 1 1 1 ii) Group the ones with the 2 1 same Hamiltonian (Topo- 3 2 logical cluster) 4 4 5 6 iii) Find all subclusters of a 6 14 given topological cluster 7 28 8 68 iv) Diagonalize the topological 9 156 clusters and compute the 10 399 observables 11 1012 12 2732 13 7385 14 20665 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 8 / 30
Numerical Linked Cluster Expansions i) Find all clusters that can be Heisenberg Model embedded on the lattice 0 ii) Group the ones with the QMC 100 × 100 same Hamiltonian (Topo- 12 bonds -0.2 logical cluster) 13 bonds E -0.4 iii) Find all subclusters of a given topological cluster -0.6 iv) Diagonalize the topological clusters and compute the -0.8 observables 0.1 1 10 T v) Perform the subgraph MR et al. , PRE 75 , 061118 (2007). substraction to compute the B. Tang et al. , CPC 184 , 557 (2013). weight of each cluster Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 8 / 30
Numerical Linked-Cluster Expansions Site clusters No. of sites topological clusters c L(c) 1 1 1 1 2 1 3 1 2 2 4 3 3 2 5 4 6 10 4 4 7 19 8 51 5 4 9 112 6 2 10 300 11 746 7 1 12 2042 13 5450 8 4 14 15197 15 42192 9 8 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 9 / 30
Numerical Linked-Cluster Expansions Square clusters c L(c) 1 1 No. of squares topological clusters 2 1/2 0 1 1 1 3 1 2 1 3 2 4 5 5 11 4 1 5 2 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 10 / 30
Numerical Linked-Cluster Expansions Square clusters Heisenberg Model c L(c) 0 0 0 1 1 QMC 100 × 100 12 bonds -0.2 -0.2 -0.2 2 1/2 13 bonds E E E -0.4 -0.4 -0.4 3 1 14 sites 15 sites -0.6 -0.6 -0.6 4 squares 5 squares 4 1 -0.8 -0.8 -0.8 0.1 0.1 0.1 1 1 1 10 10 10 T T T 5 2 MR et al. , PRE 75 , 061118 (2007). B. Tang et al. , CPC 184 , 557 (2013). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble March 26, 2014 10 / 30
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