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 Systems The study of entanglement in strongly correlated systems has lived a series of important advances in recent years, in turn underpinning a better understanding of the quantum properties of matter. Concerning numerics, examples of these are e.g. new numerical methods based on tensor networks, as well as advances in quantum Monte Carlo, exact diagonalizations, and continuous unitary transformations... quasi-particle pictures from continuous unitary transformations 1
overview continuous unitary transformations (CUTs) spirit quasi-particle pictures perturbative CUTs (pcuts) transverse-field Ising model on the three-dimensional Swedenborgite lattice graph-based CUTs (gcuts) generalized notion of linked-cluster theorem quasi-particle decay momentum-space CUTs magnon and Higgs particles in an ordered quantum magnet quasi-particle pictures from continuous unitary transformations 2
numerical tools for quantum-lattice models exact diagonalization (ED) quantum Monte Carlo (QMC) tensor-network approaches DMRG, ipeps,...... quasi-particle pictures from continuous unitary transformations 3
numerical tools for quantum-lattice models exact diagonalization (ED) quantum Monte Carlo (QMC) tensor-network approaches DMRG, ipeps,...... continuous unitary transformations (CUTs) linked-cluster expansions (LCEs) non-perturbative LCEs quasi-particle pictures from continuous unitary transformations 4
continuous unitary transformations (CUTs) CUT flow equation effective model '94 Wegner '93/94 Glazek/Wilson infinite hierarchy of coupled equations truncation schemes necessary! see also works by Wegner, Stein, Toda, Mielke, Kehrein,... quasi-particle pictures from continuous unitary transformations 5
effective models and quasi-particles correlated system photobucket.com low energy quasi-particle pictures from continuous unitary transformations 6
quasi-particle conserving CUT CUT counting operator with quasi-particle generator quasi-particle conserving effective Hamiltonian with '97 Stein '98 Mielke '00 Knetter/Uhrig quasi-particle pictures from continuous unitary transformations 7
truncation schemes perturbative CUT high-order series expansions thermodynamic limit graph-based CUT combine graph theory with CUTs thermodynamic limit Knetter/Uhrig, EPJB (2000) Knetter/KPS/Uhrig, J. Phys. A (2003) Coester/KPS, PRE (2015) Yang/KPS, EPL (2011) Coester/Clever/Herbst/Capponi/KPS, EPL (2015) Coester/KPS (2016) self-similar CUT operator flow thermodynamic limit see also posters by David Schneider and Serkan Sahin! Heidbrink/Uhrig, PRL (2002) Fischer/Duffe/Uhrig, NJP (2010) Krull/Drescher/Uhrig, PRB (2013) Powalski/Uhrig/KPS, PRL (2015) quasi-particle pictures from continuous unitary transformations 8
truncation schemes perturbative CUT high-order series expansions thermodynamic limit Knetter/Uhrig, EPJB (2000) Knetter/KPS/Uhrig, J. Phys. A (2003) Coester/KPS, PRE (2015) graph-based CUT combine graph theory with CUTs thermodynamic limit Yang/KPS, EPL (2011) Coester/Clever/Herbst/Capponi/KPS, EPL (2015) Coester/KPS (2016) self-similar CUT operator flow thermodynamic limit see also posters by David Schneider and Serkan Sahin! Heidbrink/Uhrig, PRL (2002) Fischer/Duffe/Uhrig, NJP (2010) Krull/Drescher/Uhrig, PRB (2013) Powalski/Uhrig/KPS, PRL (2015) quasi-particle pictures from continuous unitary transformations 9
set up consider lattice of supersites (spin, dimer,...) rewrite exactly unperturbed part diagonal in supersites mit interaction couples supersites (colors) mit quasi-particle pictures from continuous unitary transformations 10
perturbative CUT pcut if spectrum of is equidistant in with then perturbative CUT yields model-independently quasi-particle pictures from continuous unitary transformations 11
perturbative CUT pcut model-independent effective Hamiltonian not normal-ordered (model-dependent part!) pcut as a linked-cluster expansion quasi-particle pictures from continuous unitary transformations 12
pcut as a linked-cluster expansion divide in finite clusters and embed in thermodynamic limit quasi-particle pictures from continuous unitary transformations 13
linked-cluster expansions (LCEs) quantum many-body problem lattice with sites and links focus on zero temperature quasi-particle pictures from continuous unitary transformations 14
linked-cluster expansions (LCEs) Goal calculate high-order series expansion (order 10-20) in use extrapolation schemes to access quantum critical regime determine properties of elementary excitations quasi-particle pictures from continuous unitary transformations 15
linked-cluster expansions (LCEs) ent e ct quasi-particle pictures from continuous unitary transformations 16
linked-cluster expansions (LCEs) i-particles, e.g. magnons, triplons, anyons,... momentum, dispersion one quasi-particle dispersion two quasi-particle interaction spectral densities topological order / anyons '96 Gelfand '00 Trebst et al. '01 Zheng et al. '01 Knetter/KPS/Uhrig '01 Knetter et al.. '01 KPS/Knetter/Uhrig '08- Dusuel/Vidal/KPS quasi-particle pictures from continuous unitary transformations 17
linked-cluster expansions (LCEs) Properties thermodynamic limit scales well with increasing dimensions perturbative, but extrapolations quasi-particle pictures from continuous unitary transformations 18
linked-cluster expansions LCEs Properties thermodynamic limit scales well with increasing dimensions perturbative, but extrapolations Challenges systems with various couplings long-range interactions disorder non-perturbative extensions white graphs Coester/KPS, PRE (2015) quasi-particle pictures from continuous unitary transformations 19
2 TFIM on the Swedenborgite lattice swede 1 si 1d chains 0 lattice quasi-particle pictures from continuous unitary transformations 20
decoupled chains on the Swedenborgite lattice (J 1 = 0) rated si-1d chain quant s quasi-particle pictures from continuous unitary transformations 21
decoupled chains on the Swedenborgite lattice (J 1 = 0) lar lattice quasi-particle pictures from continuous unitary transformations 22
TFIM on the Swedenborgite lattice swede 1 si 1d chains 0 lattice quasi-particle pictures from continuous unitary transformations 23
fully frustrated Ising models h=0: extensively many ground states J=0: fully polarized phase (trivial) generically "order by disorder" trivial phase QPT ordered phase (h=0) extensive degeneracy J J -J Moessner/Sondi/Chandra, PRL (2000) Moessner/Sondi, PRB (2001) J/h quasi-particle pictures from continuous unitary transformations 24
Ising model on the Swedenborgite lattice (h = 0) two classical spin liquids I and II hrandt/fritz, PRB (2014) Phase II sub-extensive degeneracy macrog mo Phase I extensive degeneracy quasi-particle pictures from continuous unitary transformations 25
kagome ice in a transverse field (J 2 = 0) 1 h=0: infinitely many ground states (kagome ice) J=0: fully polarized phase 1? kagome ice J /h 1 quasi-particle pictures from continuous unitary transformations 26
kagome ice in a transverse field (J 2 = 0) 1 h=0: infinitely many ground states (kagome ice) J=0: fully polarized phase 1 polarized phase kagome ice a, PRL (2000) Moessner/Sondi, PRB (2001) J /h 1 quasi-particle pictures from continuous unitary transformations 27
disorder by disorder (J 2 = 0) 1 polarized phase J /h 1 disorder by disorder scenario for kagome TFIM Powalski/Coester/Moessner/KPS, PRB (2013) disorder by disorder for fully-frustrated diamond chain can be deduced analytically Coester/Malitz/Fey/KPS, PRB (2013) quasi-particle pictures from continuous unitary transformations 28
TFIM on the Swedenborgite lattice limit o- g mo quasi-particle pictures from continuous unitary transformations 29
LCEs on the Swedenborgite lattice high-field expansion in and up to order 11 linktypes = "color" white-graph expansion ignore color for generating topologically distinct graphs, but do an optimal bookkeeping! Coester/KPS, PRE (2015) quasi-particle pictures from continuous unitary transformations 30
TFIM on the Swedenborgite lattice limit o- g mo chains quasi-particle pictures from continuous unitary transformations 31
k d d c u d TFIM on the Swedenborgite lattice limit chains quasi-particle pictures from continuous unitary transformations 32
TFIM on the Swedenborgite lattice limit chains quasi-particle pictures from continuous unitary transformations 33
TFIM on the Swedenborgite lattice limit ef classi s chains quasi-particle pictures from continuous unitary transformations 34
"! ( # " # ' & % " $ # TFIM on the Swedenborgite lattice limit ef classi s all configurations stay degenerate in, +* ) chains quasi-particle pictures from continuous unitary transformations 35
3. / 0 1-9 4 3 4 8 7 3 5 0 6 4 TFIM on the Swedenborgite lattice limit ef classi configurations s in / = <; : chains quasi-particle pictures from continuous unitary transformations 36
C? @ A B > I D C D H G C E A F D TFIM on the Swedenborgite lattice limit o- ef classi gent stripe structures nerate s @ M LKJ chains neracy on finite clusters and at least up quasi-particle pictures from continuous unitary transformations 37
S O P Q R N Y T S T X W Q V S U T TFIM on the Swedenborgite lattice white graphs efficient disorder by disorder ield limit ef classi id P ] \[ Z chains quasi-particle pictures from continuous unitary transformations 38
perturbative CUTs and LCEs Properties thermodynamic limit scales well with increasing dimensions perturbative, but extrapolations Challenges systems with various couplings long-range interactions disorder non-perturbative extensions white graphs Coester/KPS, PRE (2015) quasi-particle pictures from continuous unitary transformations 39
truncation schemes high-order series expansions thermodynamic limit Knetter/Uhrig, EPJB (2000) Knetter/KPS/Uhrig, J. Phys. A (2003) Coester/KPS, PRE (2015) graph-based CUT combine graph theory with CUTs thermodynamic limit Yang/KPS, EPL (2011) Coester/Clever/Herbst/Capponi/KPS, EPL (2015) Coester/KPS (2016) self-similar CUT operator flow thermodynamic limit Heidbrink/Uhrig, PRL (2002) Fischer/Duffe/Uhrig, NJP (2010) Krull/Drescher/Uhrig, PRB (2013) Powalski/Uhrig/KPS, PRL (2015) quasi-particle pictures from continuous unitary transformations 40
non-perturbative linked-cluster expansions (NLCEs) do graph expansion as before calculate non-perturbatively (e.g. ED) on graphs! divide in finite clusters and embed in thermodynamic limit quasi-particle pictures from continuous unitary transformations 41
non-perturbative linked-cluster expansions (NLCEs) see also poster by Dominik Ixert! ground-state ener '84 Irving/Hamer thermodynamic quantities for quantum spin models (ED) '06 Rigol/Bryant/Singh excitations and effective models '11 Yang/KPS (gcut) '15 Coester/Clever/Herbst/Capponi/KPS entanglement entropies '13 Kallin/Hyatt/Singh/Melko (ED,DMRG) '14 Stoudenmire/Gustainis/Johal/Wessel/Melko quantum quenches and many-body localization '14 Rigol in the thermodynamic limit (ED) '15 Tang/Iyer/Rigol quasi-particle pictures from continuous unitary transformations 42
graph-based CUTs Idea generate topologically distinct graphs solve flow-equation on each graph block-diagonalization unitary transformation block-diagonal in particle-number operator Yang/KPS EPL (2011) quasi-particle pictures from continuous unitary transformations 43
graph-based CUTs rate topologically distinct graphs unit ion it Y quasi-particle pictures from continuous unitary transformations 44
graph-based CUTs Idea generate topologically distinct graphs quation on each graph unit ion uti T for ground-state ener st identical gence cted for finite correlation length Yang/K quasi-particle pictures from continuous unitary transformations 45
graph-based continuous unitary transformations (gcuts) one-particle dispersion hopping matrix on each cluster embedding restores translation symmetry Fourier transformation yields quasi-particle pictures from continuous unitary transformations 46
graph-based continuous unitary transformations (gcuts) one-particle dispersion hopping matrix on each cluster embedding restores translation symmetry? really true non-perturbatively? quasi-particle pictures from continuous unitary transformations 47
graph-based continuous unitary transformations (gcuts) consider two clusters Coester/Clever/Herbst/Capponi/K (2015) quasi-particle pictures from continuous unitary transformations 48
graph-based continuous unitary transformations (gcuts) consider two clusters (2015) -targets quasi-particle pictures from continuous unitary transformations 49
graph-based continuous unitary transformations (gcuts) consider two clusters Coester/Clever/Herbst/Capponi/KPS (2015) non-per nts relation to intruder states in quantum chemistry Malrieu et al. (1985) breakdown of conventional NLCE quasi-particle pictures from continuous unitary transformations 50
graph-based continuous unitary transformations (gcuts) consider two clusters Coester/Clever/Herbst/Capponi/KPS (2015) need level crossing! generalized notion of cluster additivity needed! quasi-particle pictures from continuous unitary transformations 51
graph-based continuous unitary transformations (gcuts) 5 quasi-particle pictures from continuous unitary transformations 52
graph-based continuous unitary transformations (gcuts) fre one can quasi-particle pictures from continuous unitary transformations 53
graph-based continuous unitary transformations (gcuts) two-leg Heisenberg ladder λ=0 : product state of singlets, excitations local triplets finite : triplon quasi-particles, finite dispersion '03 KPS/Uhrig quasi-particle pictures from continuous unitary transformations 54
graph-based continuous unitary transformations (gcuts) two-leg Heisenberg ladder apply standard and optimized gcuts consider graphs up to 12 dimers Coester/Clever/Herbst/Capponi/KPS, EPL (2015) quasi-particle pictures from continuous unitary transformations 55
graph-based continuous unitary transformations (gcuts) two-leg Heisenberg ladder effect of anti-level crossing! Coester/Clever/Herbst/Capponi/KPS, EPL (2015) quasi-particle pictures from continuous unitary transformations 56
graph-based continuous unitary transformations (gcuts) two-leg Heisenberg ladder Coester/Clever/Herbst/Capponi/KPS, EPL (2015) quasi-particle pictures from continuous unitary transformations 57
graph-based continuous unitary transformations (gcuts) two-leg Heisenberg ladder comparison gcut/dmrg DMRG data from Schmidinger et al., PRB (2013) Coester/KPS (2016) quasi-particle pictures from continuous unitary transformations 58
summary - graph-based CUTs non-perturbative LCEs for excited states, fundamental challenge due to reduced symmetry of graphs (intruder states) generalized notion of cluster additivity challenge can be solved with graph-based CUTs (also possible for true quasi-particle decay!) quasi-particle pictures from continuous unitary transformations 59
truncation schemes high-order it Knetter/U A e gr it ang/k (2011) Coester/Clever/Herbst/Capponi/K (2015) operator it He (2002) (2010) Krul (2015) quasi-particle pictures from continuous unitary transformations 60
set up pcut/gcut good for gapped qua phases pace re truncat rators ( ield theory ) square latti beyond isenber quasi-particle pictures from continuous unitary transformations 61
momentum space CUTs Fractional spinons in the square lattice Heisenberg? Dalla Piazza et al., Nat. Phys. (2015) quasi-particle pictures from continuous unitary transformations 62
momentum space CUTs quantitative magnon description roton minimum from magnon-higgs scattering square lattice, spin 1/2 Dyson-Maleev representation CUT in momentum space truncation based on scaling dimensions alski/uhrig/kps, PRL (2015) quasi-particle pictures from continuous unitary transformations 63
momentum space CUTs longitudinal dynamic structure factor sonance alski/uhrig/kps, PRL (2015) quasi-particle pictures from continuous unitary transformations 64
kps ESCS, Benasque, 24.02.2015 summary continuous unitary transformations (CUTs) spirit quasi-particle pictures perturbative CUTs (pcuts) polarized phase transverse-field Ising model on the three-dimensional Swedenborgite lattice L) graph-based CUTs (gcuts) generalized notion of linked-cluster theorem quasi-particle decay momentum-space CUTs magnon and Higgs particles in an ordered quantum magnet quasi-particle pictures from continuous unitary transformations 65
quasi-particle pictures from continuous unitary transformations w n f a j e j g a { f q s ` w l ` } y q w ian g f p _ x a ` _ ^ e eder o ` n m l j l e ~ ~ w n m l j r g _ e i h f } f b l ` g a w w n m l j r g _ e i h g a { _ f z s ` _ x n m l j r g _ e i h _ f g a f e b a ` _ ^ a _ a ` _ ^ w v n g p o f h g ` _ x a _ w v n g p o f _ g t h s l f s s ` q e p o y n m l j r g _ e i h _ f g a f e b a ` _ ^ pcut r g _ e i h ` w b l ` w { r i (T l j r s a e w } f q Utr g _ e i h ` f p o ` y q eder _ p T t g kps ESCS, Benasque, 24.02.2015 collaborators wede gcut d) 66
kps ESCS, Benasque, 24.02.2015 extensions to be continued... quasi-particle pictures from continuous unitary transformations 67