Quantum quenches in the thermodynamic limit
|
|
- Kenneth Neal
- 5 years ago
- Views:
Transcription
1 Quantum quenches in the thermodynamic limit Marcos Rigol Department of Physics The Pennsylvania State University Correlations, criticality, and coherence in quantum systems Evora, Portugal October 9, 204 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, 204 / 34
2 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
3 Quenches in one-dimensional superlattices Quantum dynamics in a D superlattice Trotzky et al. (Bloch s group), Nature Phys. 8, 325 (202). Initial state Unitary dynamics under the Bose-Hubbard Hamiltonian Experimental results ( ) vs exact t-dmrg calculations (lines) without free parameters local observables (top) vs nonlocal observables (bottom) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
4 Relaxation dynamics after turning off a superlattice Density profile Momentum profile τ=0 τ=0 n n k x/a 0 π π/2 0 π/2 π ka MR, A. Muramatsu, and M. Olshanii, PRA 74, (2006). MR, V. Dunjko, V. Yurovsky, and M. Olshanii, PRL 98, (2007). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
5 Unitary dynamics after a sudden quench If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a few-body observable O will evolve following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ/ ψ 0. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
6 Unitary dynamics after a sudden quench If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a few-body observable O will evolve following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ/ ψ 0. What is it that we call thermalization? O(τ) = O(E 0 ) = O(T ) = O(T, µ). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
7 Unitary dynamics after a sudden quench If the initial state is not an eigenstate of Ĥ ψ 0 α where Ĥ α = E α α and E 0 = ψ 0 Ĥ ψ 0, then a few-body observable O will evolve following O(τ) ψ(τ) Ô ψ(τ) where ψ(τ) = e iĥτ/ ψ 0. What is it that we call thermalization? O(τ) = O(E 0 ) = O(T ) = O(T, µ). One can rewrite O(τ) = α,α C α C αe i(e α Eα )τ/ O α α where ψ 0 = α C α α. Taking the infinite time average (diagonal ensemble ˆρ DE α C α 2 α α ) τ O(τ) = lim dτ Ψ(τ τ τ ) Ô Ψ(τ ) = C α 2 O αα Ô DE, 0 α which depends on the initial conditions through C α = α ψ 0. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
8 Description after relaxation (lattice models) Hard-core boson (spinless fermion) Hamiltonian L ) Ĥ = t (ˆb iˆb i+ + H.c. + V ˆn iˆn i+ t (ˆb ) iˆb i+2 + H.c i= Nonintegrable: t = V 0 Dynamics vs statistical ensembles n(k) V ˆn iˆn i+2 Integrable: t = V = 0 time average thermal GGE n(k) initial state time average thermal 0.2 -π -π/2 0 π/2 π ka MR, PRL 03, (2009), PRA 80, (2009), π -π/2 0 π/2 π ka MR, Dunjko, Yurovsky, and Olshanii, PRL 98, (2007),... Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
9 Eigenstate thermalization Eigenstate thermalization hypothesis [Deutsch, PRA (99); Srednicki, PRE 50, 888 (994).] b b in an The expectation value hα O αi of a few-body observable O eigenstate of the Hamiltonian αi, with energy Eα, of a many-body b at the mean energy Eα : system is equal to the thermal average of O b b ME (Eα ). hα O αi = hoi Nonintegrable Integrable (ρ GGE = P m λm Iˆm ) ρ(e) exact ρ(e) microcan. ρ(e) canonical - ρ(e) exact ρ(e) microcan. ρ(e) canonical ρ(e)[j ] n(kx=0) 2-0 e ZGGE ρ(e)[j ] n(kx=0) E[J] E[J] MR, Dunjko, and Olshanii, Nature 452, 854 (2008). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
10 Time fluctuations and their scaling with system size δn k0. t =V =0.24 L=2 L=24 δn k0. t =V =0.2 0 δn k0. 0 δn k0. t =V =0.03 t =V = τ Bounds Relative differences (struct. factor) (G) P. Reimann, PRL 0, (2008). k δn(τ) = N(k, τ) N diag(k) (G) Linden et al., PRE 79, 0603 (2009). k N (N) Cramer et al., PRL 00, (2008). diag(k) (N) Venuti&Zanardi, PRE 87, 0206 (203). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
11 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
12 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α O αα Oαα typical N states One needs: O typical α α Otypical αα Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
13 Time fluctuations Are they small because of dephasing? Ô(t) Ô(t) = α,α α α C α C αe i(e α Eα )t O α α α,α α α e i(e α Eα )t O α N α states N 2 states N states O typical α α Otypical α α Time average of Ô Ô = α C α 2 O αα α O αα Oαα typical N states One needs: O typical α α Otypical αα MR, PRA 80, (2009) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
14 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
15 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 October 9, 204 / 34
16 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 October 9, 204 / 34
17 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 October 9, 204 / 34
18 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 October 9, 204 / 34
19 Linked-Cluster Expansions Extensive observables Ô per lattice site (O) in the thermodynamic limit O = c L(c) W O (c) where L(c) is the number of embeddings of cluster c Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
20 Linked-Cluster Expansions Extensive observables Ô per lattice site (O) in the thermodynamic limit O = c L(c) W O (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 O(c) is the result for O in cluster c W O (c) = O(c) s c W O (s). O(c) = Tr ˆρ GC c = Zc GC Z GC c = Tr } {Ô ˆρ GC c, and the s sum runs over all subclusters of c. exp (Ĥc µ ˆN c)/k B T {exp } (Ĥc µ ˆN c)/k B T Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
21 Linked-Cluster Expansions In HTEs O(c) is expanded in powers of β and only a finite number of terms is retained. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
22 Linked-Cluster Expansions In HTEs O(c) is expanded in powers of β and only a finite number of terms is retained. In numerical linked cluster expansions (NLCEs) an exact diagonalization of the cluster is used to calculate O(c) at any temperature. (Spins models in the square, triangular, and kagome lattices) MR, T. Bryant, and R. R. P. Singh, PRL 97, (2006). MR, T. Bryant, and R. R. P. Singh, PRE 75, 068 (2007). (t-j model in the square lattice) MR, T. Bryant, and R. R. P. Singh, PRE 75, 069 (2007). PRL 98, (2007), PRB 76, (2007), PRB 83, 3443 (20), PRA 84, 0536 (20), PRB 84, 2244 (20), PRB 85, (202), PRA 86, (202), PRL 09, (202), CPC 84, 557 (203), PRB 88, 2527 (203),... Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
23 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
24 Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice Bond clusters c L(c) Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
25 Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) No. of bonds topological clusters Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
26 Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) iii) Find all subclusters of a given topological cluster No. of bonds topological clusters Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
27 Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) iii) Find all subclusters of a given topological cluster iv) Diagonalize the topological clusters and compute the observables No. of bonds topological clusters Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
28 Numerical Linked Cluster Expansions i) Find all clusters that can be embedded on the lattice ii) Group the ones with the same Hamiltonian (Topological cluster) iii) Find all subclusters of a given topological cluster iv) Diagonalize the topological clusters and compute the observables v) Compute the weight of each cluster and compute the direct sum of the weights E Heisenberg Model QMC bonds 3 bonds MR et al., PRE 75, 068 (2007). B. Tang et al., CPC 84, 557 (203). T Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
29 Numerical Linked-Cluster Expansions Square clusters c L(c) /2 No. of squares topological clusters Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
30 Numerical Linked-Cluster Expansions Square clusters c L(c) 2 / Heisenberg Model QMC bonds 3 bonds E T MR et al., PRE 75, 068 (2007). 4 sites 5 sites 4 squares 5 squares B. Tang et al., CPC 84, 557 (203). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
31 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
32 Resummation algorithms We can define partial sums n O n = S i, with S i = L(c i ) W O (c i ) c i i= where all clusters c i share a given characteristic (no. of bonds, sites, etc). Goal: Estimate O = lim n O n from a sequence {O n }, with n =,..., N. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
33 Resummation algorithms We can define partial sums n O n = S i, with S i = L(c i ) W O (c i ) c i i= where all clusters c i share a given characteristic (no. of bonds, sites, etc). Goal: Estimate O = lim n O n from a sequence {O n }, with n =,..., N. Wynn s algorithm: where ε (k ) n ε ( ) n = 0, ε (0) n = O n, ε (k) n = ε (k 2) n+ + = ε (k ) n+ ε (k ) n. Brezinski s algorithm [θ ( ) n = 0, θ (0) n = O n ]: θ (2k+) n where 2 θ (k) n = θ (2k ) n + θ n (2k) = θ (k) n+2 2θ(k) n+ + θ(k) n., θ (2k+2) n = θ (2k) ε (k ) n n+ + θ(2k) n+ θ(2k+) n+ 2 θ (2k+) n Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
34 Resummation results (Heisenberg model) E Energy (square lattice) QMC Wynn 6 Euler ED 6 C v C v (square lattice) QMC Wynn 6 Euler ED 6 ED T T MR, T. Bryant, and R. R. P. Singh, PRE 75, 068 (2007). B. Tang, E. Khatami, and MR, Comput. Phys. Commun. 84, 557 (203). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
35 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
36 Diagonal ensemble and NLCEs The initial state is in thermal equilibrium in contact with a reservoir ˆρ I a c = e (Ec a µ I N c a )/T I a c a c Zc I, where Zc I = e (Ec a µi N c a )/T I, a a c (Ea) c are the eigenstates (eigenvalues) of the initial Hamiltonian ĤI c in c. MR, PRL 2, 7060 (204). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
37 Diagonal ensemble and NLCEs The initial state is in thermal equilibrium in contact with a reservoir ˆρ I a c = e (Ec a µ I N c a )/T I a c a c Zc I, where Zc I = e (Ec a µi N c a )/T I, a a c (Ea) c are the eigenstates (eigenvalues) of the initial Hamiltonian ĤI c in c. At the time of the quench ĤI c Ĥc, the system is detached from the reservoir. Writing the eigenstates of ĤI c in terms of the eigenstates of Ĥc where ˆρ DE c lim τ τ τ 0 dτ ˆρ(τ) = α W c α α c α c a Wα c = e (Ec a µ I N c a )/T I α c a c 2, α c (ε c α) are the eigenstates (eigenvalues) of the final Hamiltonian Ĥc in c. Z I c MR, PRL 2, 7060 (204). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
38 Diagonal ensemble and NLCEs The initial state is in thermal equilibrium in contact with a reservoir ˆρ I a c = e (Ec a µ I N c a )/T I a c a c Zc I, where Zc I = e (Ec a µi N c a )/T I, a a c (Ea) c are the eigenstates (eigenvalues) of the initial Hamiltonian ĤI c in c. At the time of the quench ĤI c Ĥc, the system is detached from the reservoir. Writing the eigenstates of ĤI c in terms of the eigenstates of Ĥc where ˆρ DE c lim τ τ τ 0 dτ ˆρ(τ) = α W c α α c α c a Wα c = e (Ec a µ I N c a )/T I α c a c 2, α c (ε c α) are the eigenstates (eigenvalues) of the final Hamiltonian Ĥc in c. Using ˆρ DE c in the calculation of O(c), NLCEs allow one to compute observables in the DE in the thermodynamic limit. MR, PRL 2, 7060 (204). Z I c Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
39 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
40 Models and quenches Hard-core bosons in D lattices at half filling (µ I = 0) L ) Ĥ = t (ˆb iˆb i+ + H.c. + V ˆn iˆn i+ t (ˆb ) iˆb i+2 + H.c. i= + V ˆn iˆn i+2 Quench: T I, t I = 0.5, V I =.5, t I = V I = 0 t = V =.0, t = V Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
41 Models and quenches Hard-core bosons in D lattices at half filling (µ I = 0) L ) Ĥ = t (ˆb iˆb i+ + H.c. + V ˆn iˆn i+ t (ˆb ) iˆb i+2 + H.c. i= + V ˆn iˆn i+2 Quench: T I, t I = 0.5, V I =.5, t I = V I = 0 t = V =.0, t = V NLCE with maximally connected clusters (l = 8 sites) Energy: E DE = Tr[Ĥ ˆρDE ] Convergence: (O ens ) l = Oens l O ens 8 O8 ens (E DE ) Convergence of E DE with l (E DE ) l T I = T I = l l t =V =0 t =V =0. t =V =0.25 t =V = T I Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
42 Few-body experimental observables in the DE Momentum distribution ˆm k = e ik(j j ) ˆρ jj L jj (m k ) Initial t =V =0, DE t =V =0, GE t =V =0.5, DE t =V =0.5, GE 0 π/4 π/2 3π/4 π k Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
43 Few-body experimental observables in the DE Momentum distribution ˆm k = e ik(j j ) ˆρ jj L jj Differences between DE and GE k δ(m) l = (m k) DE l (m k ) GE 8 k (m k) GE 8 (m k ) (m) DE GE T I Initial t =V =0, DE t =V =0, GE t =V =0.5, DE t =V =0.5, GE 0 π/4 π/2 3π/4 π k δ(m) l t =V =0 t =V =0.025 t =V =0. t =V =0.5 T I =2 T I = l Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
44 Outline Introduction Quantum quenches Many-body quantum systems in thermal equilibrium 2 Numerical Linked Cluster Expansions Direct sums Resummations 3 Quantum quenches in the thermodynamic limit Diagonal ensemble and NLCEs Quenches in the t-v -t -V chain (thermalization) Quenches in XXZ chain from a Neel state (QA vs GGE) 4 Conclusions Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
45 Failure of the GGE based on local quantities XXZ (integrable) Hamiltonian ( ) Ĥ = J σi x σi+ x + σ y i σy i+ + σz i σi+ z i Quench starting from the Neel state to different values of Quench action solution differs from GGE based on local quantities: B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, MR, and J.-S. Caux, PRL 3, 7202 (204). B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács, PRL 3, 7203 (204). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
46 Failure of the GGE based on local quantities XXZ (integrable) Hamiltonian ( ) Ĥ = J σi x σi+ x + σ y i σy i+ + σz i σi+ z i Quench starting from the Neel state to different values of Quench action solution differs from GGE based on local quantities: B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, MR, and J.-S. Caux, PRL 3, 7202 (204). B. Pozsgay, M. Mestyán, M. A. Werner, M. Kormos, G. Zaránd, and G. Takács, PRL 3, 7203 (204). Can we use NLCEs for ground states (pure states)? Using the parity symmetry of the clusters: a e c = 2 ( ) a o c = 2 ( ) W c,e/o α = αc e/o a e/o c 2 Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
47 Results for spin-spin correlations <σ z σ2 z > <σ z σ2 z > <σ z σ2 z > DE GE QA Resum l <σ z σ2 z > = =4 =7 =0 MR, PRE 90, 0330(R) (204) (a) (b) (c) (d) <σ z σ3 z > <σ z σ3 > <σ z σ3 z > <σ z σ3 z > (e) (f) (g) (h) l Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
48 Quench action, GGE, and NLCE (a) σ z σz δ σ z σz 3 B. Wouters et al., PRL 3, 7202 (204) σ z σz 3 (b) σ z σz 3 sp σ z σz 3 GGE σ z σz 3 NLCE δ σ z σ z 3 = σ z σ z 3 GGE / σ z σ z 3 QA Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
49 Conclusions NLCEs provide a general framework to study the diagonal ensemble in lattice systems after a quantum quench in the thermodynamic limit. NLCE results suggest that few-body observables thermalize in nonintegrable systems while they do not thermalize in integrable systems. As one approaches the integrable point DE-NLCEs behave as NLCEs for equilibrium systems approaching a phase transition. This suggests that a transition to thermalization may occur as soon as one breaks integrability. The GGE based on known local conserved quantities does not describe observables after relaxation, while the QA does, as suggested by the NLCE results. New things to be learned about integrable systems! Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
50 Collaborators Michael Brockmann (ITP Amsterdam) Jean-Sébastien Caux (ITP Amsterdam) Jacopo De Nardis (ITP Amsterdam) Davide Fioretto (ITP Amsterdam) Bram Wouters (ITP Amsterdam) PRL 3, 7202 (204). Supported by: Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
51 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 DQMC of a 2D system with: U = 6t, V = 0.04t, T = 0.3t and 560 fermions Density Density fluctuations Spin correlations Pairing correlations S. Chiesa, C. N. Varney, MR, and R. T. Scalettar, PRL 06, (20). Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
52 Dispersion of the energy in the DE Dispersion of the energy E 2 = L ( Ĥ2 Ĥ 2 ) 8 E DE GE δ( E 2 ) l Deviations from the GE δ(o) l = ODE l T I = O GE 8 O8 GE 0 t =V =0 2 t =V = l 4 8 t =V =0.5, diff. init. state T The dispersion of the energy (and of the particle number) in the DE depends on the initial state independently of whether the system is integrable or not. Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
53 Few-body experimental observables in the DE 0.03 nn kinetic energy K = t i ˆb iˆb i+ Differences between DE and GE δ(k) l = KDE l K GE 8 K GE 8 δ(k) t =V =0 t =V =0.025 t =V =0. t =V = T δ(k) l t =V =0 t =V =0.025 t =V =0. t =V =0.5 T I =2 T I = l Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
54 NLCEs vs exact diagonalization (equilibrium, t = 0) (E GE ) l (K GE ) l V =0 V = T= T=5 δ(e ED ) L δ(k ED ) L L L ( E 2 GE ) l l l ED from L. Santos and MR, PRE 82, 0330 (200). (m GE ) l δ(m ED ) L L Marcos Rigol (Penn State) NLCEs for the diagonal ensemble October 9, / 34
From unitary dynamics to statistical mechanics in isolated quantum systems
From unitary dynamics to statistical mechanics in isolated quantum systems Marcos Rigol Department of Physics The Pennsylvania State University The Tony and Pat Houghton Conference on Non-Equilibrium Statistical
More informationFluctuation-dissipation theorem in isolated quantum systems out of equilibrium
San Jose State University From the SelectedWorks of Ehsan Khatami Fluctuation-dissipation theorem in isolated quantum systems out of equilibrium Ehsan Khatami, University of California, Davis Guido Pupillo,
More informationFabian Essler (Oxford)
Quantum Quenches in Integrable Models Fabian Essler (Oxford) Collaborators: J.S. Caux (Amsterdam), M. Fagotti (Oxford) earlier work: P. Calabrese (Pisa), S. Evangelisti (Oxford), D. Schuricht (Aachen)
More informationTime Evolving Block Decimation Algorithm
Time Evolving Block Decimation Algorithm Application to bosons on a lattice Jakub Zakrzewski Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Center, Jagiellonian University,
More informationA quantum dynamical simulator Classical digital meets quantum analog
A quantum dynamical simulator Classical digital meets quantum analog Ulrich Schollwöck LMU Munich Jens Eisert Freie Universität Berlin Mentions joint work with I. Bloch, S. Trotzky, I. McCulloch, A. Flesch,
More informationWhat is thermal equilibrium and how do we get there?
arxiv:1507.06479 and more What is thermal equilibrium and how do we get there? Hal Tasaki QMath 13, Oct. 9, 2016, Atlanta 40 C 20 C 30 C 30 C about the talk Foundation of equilibrium statistical mechanics
More informationUnder what conditions do quantum systems thermalize?
Under what conditions do quantum systems thermalize? 1 / 9 Under what conditions do quantum systems thermalize? New insights from quantum information theory Christian Gogolin, Arnau Riera, Markus Müller,
More informationNON-EQUILIBRIUM DYNAMICS IN
NON-EQUILIBRIUM DYNAMICS IN ISOLATED QUANTUM SYSTEMS Masud Haque Maynooth University Dept. Mathematical Physics Maynooth, Ireland Max-Planck Institute for Physics of Complex Systems (MPI-PKS) Dresden,
More informationDephasing, relaxation and thermalization in one-dimensional quantum systems
Dephasing, relaxation and thermalization in one-dimensional quantum systems Fachbereich Physik, TU Kaiserslautern 26.7.2012 Outline 1 Introduction 2 Dephasing, relaxation and thermalization 3 Particle
More informationSeparation of relaxation time scales in a quantum Newton s cradle
Separation of relaxation time scales in a quantum Newton s cradle Rianne van den Berg Universiteit van Amsterdam CRM: Beyond integrability Research team UvA Bram Wouters PhD student BNL Robert Konik PI
More informationDiagrammatic Green s Functions Approach to the Bose-Hubbard Model
Diagrammatic Green s Functions Approach to the Bose-Hubbard Model Matthias Ohliger Institut für Theoretische Physik Freie Universität Berlin 22nd of January 2008 Content OVERVIEW CONSIDERED SYSTEM BASIC
More informationquasi-particle pictures from continuous unitary transformations
quasi-particle pictures from continuous unitary transformations Kai Phillip Schmidt 24.02.2016 quasi-particle pictures from continuous unitary transformations overview Entanglement in Strongly Correlated
More informationSimulating Quantum Systems through Matrix Product States. Laura Foini SISSA Journal Club
Simulating Quantum Systems through Matrix Product States Laura Foini SISSA Journal Club 15-04-2010 Motivations Theoretical interest in Matrix Product States Wide spectrum of their numerical applications
More informationMany-Body Localization. Geoffrey Ji
Many-Body Localization Geoffrey Ji Outline Aside: Quantum thermalization; ETH Single-particle (Anderson) localization Many-body localization Some phenomenology (l-bit model) Numerics & Experiments Thermalization
More information(Dynamical) quantum typicality: What is it and what are its physical and computational implications?
(Dynamical) : What is it and what are its physical and computational implications? Jochen Gemmer University of Osnabrück, Kassel, May 13th, 214 Outline Thermal relaxation in closed quantum systems? Typicality
More informationCorrelated Electron Dynamics and Nonequilibrium Dynamical Mean-Field Theory
Correlated Electron Dynamics and Nonequilibrium Dynamical Mean-Field Theory Marcus Kollar Theoretische Physik III Electronic Correlations and Magnetism University of Augsburg Autumn School on Correlated
More informationNumerical Studies of Correlated Lattice Systems in One and Two Dimensions
Numerical Studies of Correlated Lattice Systems in One and Two Dimensions A Dissertation submitted to the Faculty of the Graduate School of Arts and Sciences of Georgetown University in partial fulfillment
More informationFermionic tensor networks
Fermionic tensor networks Philippe Corboz, Institute for Theoretical Physics, ETH Zurich Bosons vs Fermions P. Corboz and G. Vidal, Phys. Rev. B 80, 165129 (2009) : fermionic 2D MERA P. Corboz, R. Orus,
More informationTaming the non-equilibrium: Equilibration, thermalization and the predictions of quantum simulations
Taming the non-equilibrium: Equilibration, thermalization and the predictions of quantum simulations Jens Eisert Freie Universität Berlin v KITP, Santa Barbara, August 2012 Dynamics and thermodynamics
More informationTime-dependent DMRG:
The time-dependent DMRG and its applications Adrian Feiguin Time-dependent DMRG: ^ ^ ih Ψ( t) = 0 t t [ H ( t) E ] Ψ( )... In a truncated basis: t=3 τ t=4 τ t=5τ t=2 τ t= τ t=0 Hilbert space S.R.White
More informationThe nature of superfluidity in the cold atomic unitary Fermi gas
The nature of superfluidity in the cold atomic unitary Fermi gas Introduction Yoram Alhassid (Yale University) Finite-temperature auxiliary-field Monte Carlo (AFMC) method The trapped unitary Fermi gas
More informationarxiv: v1 [cond-mat.stat-mech] 8 Oct 2016
arxiv:1610.02495v1 [cond-mat.stat-mech] 8 Oct 2016 On Truncated Generalized Gibbs Ensembles in the Ising Field Theory F. H. L. Essler 1, G. Mussardo 2,3 and M. Panfil 4 1 The Rudolf Peierls Centre for
More informationarxiv: v4 [cond-mat.stat-mech] 25 Jun 2015
Thermalization of entanglement Liangsheng Zhang, 1 Hyungwon Kim, 1, 2 and David A. Huse 1 1 Physics Department, Princeton University, Princeton, NJ 08544 2 Department of Physics and Astronomy, Rutgers
More informationLinked-Cluster Expansions for Quantum Many-Body Systems
Linked-Cluster Expansions for Quantum Many-Body Systems Boulder Summer School 2010 Simon Trebst Lecture overview Why series expansions? Linked-cluster expansions From Taylor expansions to linked-cluster
More informationQuantum spin systems - models and computational methods
Summer School on Computational Statistical Physics August 4-11, 2010, NCCU, Taipei, Taiwan Quantum spin systems - models and computational methods Anders W. Sandvik, Boston University Lecture outline Introduction
More informationTypical quantum states at finite temperature
Typical quantum states at finite temperature How should one think about typical quantum states at finite temperature? Density Matrices versus pure states Why eigenstates are not typical Measuring the heat
More informationHigh-Temperature Criticality in Strongly Constrained Quantum Systems
High-Temperature Criticality in Strongly Constrained Quantum Systems Claudio Chamon Collaborators: Claudio Castelnovo - BU Christopher Mudry - PSI, Switzerland Pierre Pujol - ENS Lyon, France PRB 2006
More informationTuning the Quantum Phase Transition of Bosons in Optical Lattices
Tuning the Quantum Phase Transition of Bosons in Optical Lattices Axel Pelster 1. Introduction 2. Spinor Bose Gases 3. Periodically Modulated Interaction 4. Kagome Superlattice 1 1.1 Quantum Phase Transition
More informationTrapped ion spin-boson quantum simulators: Non-equilibrium physics and thermalization. Diego Porras
Trapped ion spin-boson quantum simulators: Non-equilibrium physics and thermalization Diego Porras Outline Spin-boson trapped ion quantum simulators o Rabi Lattice/Jahn-Teller models o Gauge symmetries
More informationLecture 4: Entropy. Chapter I. Basic Principles of Stat Mechanics. A.G. Petukhov, PHYS 743. September 7, 2017
Lecture 4: Entropy Chapter I. Basic Principles of Stat Mechanics A.G. Petukhov, PHYS 743 September 7, 2017 Chapter I. Basic Principles of Stat Mechanics A.G. Petukhov, Lecture PHYS4: 743 Entropy September
More informationQuantum Cluster Methods (CPT/CDMFT)
Quantum Cluster Methods (CPT/CDMFT) David Sénéchal Département de physique Université de Sherbrooke Sherbrooke (Québec) Canada Autumn School on Correlated Electrons Forschungszentrum Jülich, Sept. 24,
More informationMeron-Cluster and Nested Cluster Algorithms: Addressing the Sign Problem in Quantum Monte Carlo Simulations
Meron-Cluster and Nested Cluster Algorithms: Addressing the Sign Problem in Quantum Monte Carlo Simulations Uwe-Jens Wiese Bern University IPAM Workshop QS2009, January 26, 2009 Collaborators: B. B. Beard
More informationR. Citro. In collaboration with: A. Minguzzi (LPMMC, Grenoble, France) E. Orignac (ENS, Lyon, France), X. Deng & L. Santos (MP, Hannover, Germany)
Phase Diagram of interacting Bose gases in one-dimensional disordered optical lattices R. Citro In collaboration with: A. Minguzzi (LPMMC, Grenoble, France) E. Orignac (ENS, Lyon, France), X. Deng & L.
More informationDynamics of Quantum Many Body Systems Far From Thermal Equilibrium
Dynamics of Quantum Many Body Systems Far From Thermal Equilibrium Marco Schiro CNRS-IPhT Saclay Schedule: Friday Jan 22, 29 - Feb 05,12,19. 10h-12h Contacts: marco.schiro@cea.fr Lecture Notes: ipht.cea.fr
More informationSpin liquid phases in strongly correlated lattice models
Spin liquid phases in strongly correlated lattice models Sandro Sorella Wenjun Hu, F. Becca SISSA, IOM DEMOCRITOS, Trieste Seiji Yunoki, Y. Otsuka Riken, Kobe, Japan (K-computer) Williamsburg, 14 June
More informationGeneral quantum quenches in the transverse field Ising chain
General quantum quenches in the transverse field Ising chain Tatjana Puškarov and Dirk Schuricht Institute for Theoretical Physics Utrecht University Novi Sad, 23.12.2015 Non-equilibrium dynamics of isolated
More informationQuantum many-body systems and tensor networks: simulation methods and applications
Quantum many-body systems and tensor networks: simulation methods and applications Román Orús School of Physical Sciences, University of Queensland, Brisbane (Australia) Department of Physics and Astronomy,
More informationAuxiliary-field Monte Carlo methods in Fock space: sign problems and methods to circumvent them
Auxiliary-field Monte Carlo methods in Fock space: sign problems and methods to circumvent them Introduction Yoram Alhassid (Yale University) Finite-temperature auxiliary-field Monte Carlo methods in Fock
More informationMany-Body Fermion Density Matrix: Operator-Based Truncation Scheme
Many-Body Fermion Density Matrix: Operator-Based Truncation Scheme SIEW-ANN CHEONG and C. L. HENLEY, LASSP, Cornell U March 25, 2004 Support: NSF grants DMR-9981744, DMR-0079992 The Big Picture GOAL Ground
More informationSimulations of Quantum Dimer Models
Simulations of Quantum Dimer Models Didier Poilblanc Laboratoire de Physique Théorique CNRS & Université de Toulouse 1 A wide range of applications Disordered frustrated quantum magnets Correlated fermions
More informationClassical and quantum simulation of dissipative quantum many-body systems
0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 0 20 32 Classical and quantum simulation of dissipative quantum many-body systems
More informationObservation of topological phenomena in a programmable lattice of 1800 superconducting qubits
Observation of topological phenomena in a programmable lattice of 18 superconducting qubits Andrew D. King Qubits North America 218 Nature 56 456 46, 218 Interdisciplinary teamwork Theory Simulation QA
More informationAuxiliary-field quantum Monte Carlo methods for nuclei and cold atoms
Introduction Auxiliary-field quantum Monte Carlo methods for nuclei and cold atoms Yoram Alhassid (Yale University) Auxiliary-field Monte Carlo (AFMC) methods at finite temperature Sign problem and good-sign
More information(De)-localization in mean-field quantum glasses
QMATH13, Georgia Tech, Atlanta October 10, 2016 (De)-localization in mean-field quantum glasses Chris R. Laumann (Boston U) Chris Baldwin (UW/BU) Arijeet Pal (Oxford) Antonello Scardicchio (ICTP) CRL,
More informationAshvin Vishwanath UC Berkeley
TOPOLOGY + LOCALIZATION: QUANTUM COHERENCE IN HOT MATTER Ashvin Vishwanath UC Berkeley arxiv:1307.4092 (to appear in Nature Comm.) Thanks to David Huse for inspiring discussions Yasaman Bahri (Berkeley)
More informationGround-state properties, excitations, and response of the 2D Fermi gas
Ground-state properties, excitations, and response of the 2D Fermi gas Introduction: 2D FG and a condensed matter perspective Auxiliary-field quantum Monte Carlo calculations - exact* here Results on spin-balanced
More informationEfficient time evolution of one-dimensional quantum systems
Efficient time evolution of one-dimensional quantum systems Frank Pollmann Max-Planck-Institut für komplexer Systeme, Dresden, Germany Sep. 5, 2012 Hsinchu Problems we will address... Finding ground states
More informationQuantum simulation with string-bond states: Joining PEPS and Monte Carlo
Quantum simulation with string-bond states: Joining PEPS and Monte Carlo N. Schuch 1, A. Sfondrini 1,2, F. Mezzacapo 1, J. Cerrillo 1,3, M. Wolf 1,4, F. Verstraete 5, I. Cirac 1 1 Max-Planck-Institute
More informationClassical Monte Carlo Simulations
Classical Monte Carlo Simulations Hyejin Ju April 17, 2012 1 Introduction Why do we need numerics? One of the main goals of condensed matter is to compute expectation values O = 1 Z Tr{O e βĥ} (1) and
More informationFirst Problem Set for Physics 847 (Statistical Physics II)
First Problem Set for Physics 847 (Statistical Physics II) Important dates: Feb 0 0:30am-:8pm midterm exam, Mar 6 9:30am-:8am final exam Due date: Tuesday, Jan 3. Review 0 points Let us start by reviewing
More informationBose-Hubbard Model (BHM) at Finite Temperature
Bose-Hubbard Model (BHM) at Finite Temperature - a Layman s (Sebastian Schmidt) proposal - pick up Diploma work at FU-Berlin with PD Dr. Axel Pelster (Uni Duisburg-Essen) ~ Diagrammatic techniques, high-order,
More informationTRANSPORT OF QUANTUM INFORMATION IN SPIN CHAINS
TRANSPORT OF QUANTUM INFORMATION IN SPIN CHAINS Joachim Stolze Institut für Physik, Universität Dortmund, 44221 Dortmund Göttingen, 7.6.2005 Why quantum computing? Why quantum information transfer? Entangled
More informationLecture 6 Photons, electrons and other quanta. EECS Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku
Lecture 6 Photons, electrons and other quanta EECS 598-002 Winter 2006 Nanophotonics and Nano-scale Fabrication P.C.Ku From classical to quantum theory In the beginning of the 20 th century, experiments
More informationin-medium pair wave functions the Cooper pair wave function the superconducting order parameter anomalous averages of the field operators
(by A. A. Shanenko) in-medium wave functions in-medium pair-wave functions and spatial pair particle correlations momentum condensation and ODLRO (off-diagonal long range order) U(1) symmetry breaking
More informationRenormalization of Tensor- Network States Tao Xiang
Renormalization of Tensor- Network States Tao Xiang Institute of Physics/Institute of Theoretical Physics Chinese Academy of Sciences txiang@iphy.ac.cn Physical Background: characteristic energy scales
More information1 Fluctuations of the number of particles in a Bose-Einstein condensate
Exam of Quantum Fluids M1 ICFP 217-218 Alice Sinatra and Alexander Evrard The exam consists of two independant exercises. The duration is 3 hours. 1 Fluctuations of the number of particles in a Bose-Einstein
More informationMultifractality in simple systems. Eugene Bogomolny
Multifractality in simple systems Eugene Bogomolny Univ. Paris-Sud, Laboratoire de Physique Théorique et Modèles Statistiques, Orsay, France In collaboration with Yasar Atas Outlook 1 Multifractality Critical
More informationQuantum Quenches in Chern Insulators
Quantum Quenches in Chern Insulators Nigel Cooper Cavendish Laboratory, University of Cambridge CUA Seminar M.I.T., November 10th, 2015 Marcello Caio & Joe Bhaseen (KCL), Stefan Baur (Cambridge) M.D. Caio,
More informationLie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains
.. Lie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains Vladimir M. Stojanović Condensed Matter Theory Group HARVARD UNIVERSITY September 16, 2014 V. M. Stojanović (Harvard)
More informationAngular Momentum set II
Angular Momentum set II PH - QM II Sem, 7-8 Problem : Using the commutation relations for the angular momentum operators, prove the Jacobi identity Problem : [ˆL x, [ˆL y, ˆL z ]] + [ˆL y, [ˆL z, ˆL x
More informationTensor network simulation of QED on infinite lattices: learning from (1 + 1)d, and prospects for (2 + 1)d
Tensor network simulation of QED on infinite lattices: learning from (1 + 1)d, and prospects for (2 + 1)d Román Orús University of Mainz (Germany) K. Zapp, RO, Phys. Rev. D 95, 114508 (2017) Goal of this
More informationH ψ = E ψ. Introduction to Exact Diagonalization. Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden
H ψ = E ψ Introduction to Exact Diagonalization Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden http://www.pks.mpg.de/~aml laeuchli@comp-phys.org Simulations of
More informationSupersymmetry breaking and Nambu-Goldstone fermions in lattice models
YKIS2016@YITP (2016/6/15) Supersymmetry breaking and Nambu-Goldstone fermions in lattice models Hosho Katsura (Department of Physics, UTokyo) Collaborators: Yu Nakayama (IPMU Rikkyo) Noriaki Sannomiya
More informationMagnetism and Superconductivity on Depleted Lattices
Magnetism and Superconductivity on Depleted Lattices 1. Square Lattice Hubbard Hamiltonian: AF and Mott Transition 2. Quantum Monte Carlo 3. The 1/4 depleted (Lieb) lattice and Flat Bands 4. The 1/5 depleted
More informationarxiv: v3 [cond-mat.str-el] 15 Dec 2014
Quench action approach for releasing the Néel state into the spin-1/ XXZ chain arxiv:1408.5075v3 cond-mat.str-el] 15 Dec 014 1. Introduction M. Brockmann, B. Wouters, D. Fioretto, J. De Nardis, R. Vlijm,
More informationMachine learning quantum phases of matter
Machine learning quantum phases of matter Summer School on Emergent Phenomena in Quantum Materials Cornell, May 2017 Simon Trebst University of Cologne Machine learning quantum phases of matter Summer
More informationSecond Lecture: Quantum Monte Carlo Techniques
Second Lecture: Quantum Monte Carlo Techniques Andreas Läuchli, New states of quantum matter MPI für Physik komplexer Systeme - Dresden http://www.pks.mpg.de/~aml aml@pks.mpg.de Lecture Notes at http:www.pks.mpg.de/~aml/leshouches
More informationProperties of many-body localized phase
Properties of many-body localized phase Maksym Serbyn UC Berkeley IST Austria QMATH13 Atlanta, 2016 Motivation: MBL as a new universality class Classical systems: Ergodic ergodicity from chaos Quantum
More informationThermalization in Quantum Systems
Thermalization in Quantum Systems Jonas Larson Stockholm University and Universität zu Köln Dresden 18/4-2014 Motivation Long time evolution of closed quantum systems not fully understood. Cold atom system
More informationEQUATION OF STATE OF THE UNITARY GAS
EQUATION OF STATE OF THE UNITARY GAS DIAGRAMMATIC MONTE CARLO Kris Van Houcke (UMass Amherst & U of Ghent) Félix Werner (UMass Amherst) Evgeny Kozik Boris Svistunov & Nikolay Prokofev (UMass Amherst) (ETH
More informationCoupled Cluster Method for Quantum Spin Systems
Coupled Cluster Method for Quantum Spin Systems Sven E. Krüger Department of Electrical Engineering, IESK, Cognitive Systems Universität Magdeburg, PF 4120, 39016 Magdeburg, Germany sven.krueger@e-technik.uni-magdeburg.de
More informationManybody wave function and Second quantization
Chap 1 Manybody wave function and Second quantization Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan (Dated: March 7, 2013) 1 I. MANYBODY WAVE FUNCTION A. One particle
More informationDiagrammatic Monte Carlo methods for Fermions
Diagrammatic Monte Carlo methods for Fermions Philipp Werner Department of Physics, Columbia University PRL 97, 7645 (26) PRB 74, 15517 (26) PRB 75, 8518 (27) PRB 76, 235123 (27) PRL 99, 12645 (27) PRL
More informationCorrelated Phases of Bosons in the Flat Lowest Band of the Dice Lattice
Correlated Phases of Bosons in the Flat Lowest Band of the Dice Lattice Gunnar Möller & Nigel R Cooper Cavendish Laboratory, University of Cambridge Physical Review Letters 108, 043506 (2012) LPTHE / LPTMC
More informationEntanglement spectrum as a tool for onedimensional
Entanglement spectrum as a tool for onedimensional critical systems MPI-PKS, Dresden, November 2012 Joel Moore University of California, Berkeley, and Lawrence Berkeley National Laboratory Outline ballistic
More informationSqueezing and superposing many-body states of Bose gases in confining potentials
Squeezing and superposing many-body states of Bose gases in confining potentials K. B. Whaley Department of Chemistry, Kenneth S. Pitzer Center for Theoretical Chemistry, Berkeley Quantum Information and
More informationNon-equilibrium dynamics of isolated quantum systems
EPJ Web of Conferences 90, 08001 (2015) DOI: 10.1051/ epjconf/ 20159008001 C Owned by the authors, published by EDP Sciences, 2015 Non-equilibrium dynamics of isolated quantum systems Pasquale Calabrese
More informationMagnetism and Superconductivity in Decorated Lattices
Magnetism and Superconductivity in Decorated Lattices Mott Insulators and Antiferromagnetism- The Hubbard Hamiltonian Illustration: The Square Lattice Bipartite doesn t mean N A = N B : The Lieb Lattice
More informationMean-field theory for extended Bose-Hubbard model with hard-core bosons. Mathias May, Nicolas Gheeraert, Shai Chester, Sebastian Eggert, Axel Pelster
Mean-field theory for extended Bose-Hubbard model with hard-core bosons Mathias May, Nicolas Gheeraert, Shai Chester, Sebastian Eggert, Axel Pelster Outline Introduction Experiment Hamiltonian Periodic
More informationField-Driven Quantum Systems, From Transient to Steady State
Field-Driven Quantum Systems, From Transient to Steady State Herbert F. Fotso The Ames Laboratory Collaborators: Lex Kemper Karlis Mikelsons Jim Freericks Device Miniaturization Ultracold Atoms Optical
More informationarxiv: v2 [cond-mat.quant-gas] 17 Jun 2014
How to experimentally detect a GGE? - Universal Spectroscopic Signatures of the GGE in the Tons gas Garry Goldstein and Natan Andrei Department of Physics, Rutgers University and Piscataway, New Jersey
More informationThermal pure quantum state
Thermal pure quantum state Sho Sugiura ( 杉浦祥 ) Institute for Solid State Physics, Univ. Tokyo Collaborator: Akira Shimizu (Univ. Tokyo) SS and A.Shimizu, PRL 108, 240401 (2012) SS and A.Shimizu, PRL 111,
More informationAdvanced Computation for Complex Materials
Advanced Computation for Complex Materials Computational Progress is brainpower limited, not machine limited Algorithms Physics Major progress in algorithms Quantum Monte Carlo Density Matrix Renormalization
More informationReal-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach
Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach Frithjof B Anders Institut für theoretische Physik, Universität Bremen Concepts in Electron Correlation,
More informationMP 472 Quantum Information and Computation
MP 472 Quantum Information and Computation http://www.thphys.may.ie/staff/jvala/mp472.htm Outline Open quantum systems The density operator ensemble of quantum states general properties the reduced density
More informationQuantum quenches in 2D with chain array matrix product states
Quantum quenches in 2D with chain array matrix product states Andrew J. A. James University College London Robert M. Konik Brookhaven National Laboratory arxiv:1504.00237 Outline MPS for many body systems
More informationQuantum correlations and entanglement in far-from-equilibrium spin systems
Quantum correlations and entanglement in far-from-equilibrium spin systems Salvatore R. Manmana Institute for Theoretical Physics Georg-August-University Göttingen PRL 110, 075301 (2013), Far from equilibrium
More information7.4. Why we have two different types of materials: conductors and insulators?
Phys463.nb 55 7.3.5. Folding, Reduced Brillouin zone and extended Brillouin zone for free particles without lattices In the presence of a lattice, we can also unfold the extended Brillouin zone to get
More informationDiagrammatic Monte Carlo simulation of quantum impurity models
Diagrammatic Monte Carlo simulation of quantum impurity models Philipp Werner ETH Zurich IPAM, UCLA, Jan. 2009 Outline Continuous-time auxiliary field method (CT-AUX) Weak coupling expansion and auxiliary
More informationNon-perturbative effects in ABJM theory
October 9, 2015 Non-perturbative effects in ABJM theory 1 / 43 Outline 1. Non-perturbative effects 1.1 General aspects 1.2 Non-perturbative aspects of string theory 1.3 Non-perturbative effects in M-theory
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationGround State Projector QMC in the valence-bond basis
Quantum Monte Carlo Methods at Work for Novel Phases of Matter Trieste, Italy, Jan 23 - Feb 3, 2012 Ground State Projector QMC in the valence-bond basis Anders. Sandvik, Boston University Outline: The
More informationMomentum-space and Hybrid Real- Momentum Space DMRG applied to the Hubbard Model
Momentum-space and Hybrid Real- Momentum Space DMRG applied to the Hubbard Model Örs Legeza Reinhard M. Noack Collaborators Georg Ehlers Jeno Sólyom Gergely Barcza Steven R. White Collaborators Georg Ehlers
More informationMagnets, 1D quantum system, and quantum Phase transitions
134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider
More informationSUPPLEMENTARY INFORMATION
SUPPLEMENTARY INFORMATION doi:10.1038/nature10748 Contents 1 Quenches to U /J = 5.0 and 7.0 2 2 Quasiparticle model 3 3 Bose Hubbard parameters 6 4 Ramp of the lattice depth for the quench 7 WWW.NATURE.COM/NATURE
More informationStorage of Quantum Information in Topological Systems with Majorana Fermions
Storage of Quantum Information in Topological Systems with Majorana Fermions Leonardo Mazza Scuola Normale Superiore, Pisa Mainz September 26th, 2013 Leonardo Mazza (SNS) Storage of Information & Majorana
More informationComputational Approaches to Quantum Critical Phenomena ( ) ISSP. Fermion Simulations. July 31, Univ. Tokyo M. Imada.
Computational Approaches to Quantum Critical Phenomena (2006.7.17-8.11) ISSP Fermion Simulations July 31, 2006 ISSP, Kashiwa Univ. Tokyo M. Imada collaboration T. Kashima, Y. Noda, H. Morita, T. Mizusaki,
More informationReal-Space RG for dynamics of random spin chains and many-body localization
Low-dimensional quantum gases out of equilibrium, Minneapolis, May 2012 Real-Space RG for dynamics of random spin chains and many-body localization Ehud Altman, Weizmann Institute of Science See: Ronen
More informationMixing Quantum and Classical Mechanics: A Partially Miscible Solution
Mixing Quantum and Classical Mechanics: A Partially Miscible Solution R. Kapral S. Nielsen A. Sergi D. Mac Kernan G. Ciccotti quantum dynamics in a classical condensed phase environment how to simulate
More informationTypicality paradigm in Quantum Statistical Thermodynamics Barbara Fresch, Giorgio Moro Dipartimento Scienze Chimiche Università di Padova
Typicality paradigm in Quantum Statistical Thermodynamics Barbara Fresch, Giorgio Moro Dipartimento Scienze Chimiche Università di Padova Outline 1) The framework: microcanonical statistics versus the
More information