F. Illuminati. Università degli Studi di Salerno

Transcription

1 CCCQS 2014 Evora 06/10/2014 Quantum Quantum frustration, frustration, entanglement, entanglement, and and frustration-driven frustration-driven quantum quantum hase hase transitions transitions F. Illuminati Università degli Studi di Salerno S. M. Giamaolo Universität Wien B. C. Hiesmayr Universität Wien

2 Overview 1) Frustrated systems 2) Toulouse criteria: classical 3) Universal measure of total frustration 4) Toulouse criteria: quantum 5) Frustration and entanglement 6) Valence bonds: frustration-driven transitions

3 Defining and characterizing frustration Many body systems: global H sum of local terms H = h Frustration: imossibility to satisfy simultaneously all local terms h? Sources of frustration: Classical World Nontrivial geometry of the underlying hysical sace, e.g.: Heisenberg antiferromagnet on the 2-d Kagomé lattice Cometing interactions on different length scales, e.g. sin chains with antiferromagnetic n.n. and n.n.n. Interactions. Sources of frustration: Quantum World Entanglement: Non-commutativity of the different local interaction terms

4 Classical Toulouse criteria for frustration [Formulation 1]: A classical Hamiltonian system is frustrated iff it is imossible to transform it in a fully ferromagnetic model only by means of local sin inversions [Formulation 2]: A classical Hamiltonian is frustrated iff there exists at least one closed loo for which : ( 1) N af = 1 where N af is the number of antiferromagnetic bonds. Dicotomic: only two ossible answer: yes or no Ferromagnetic Links Anti-ferromagnetic Links

5 Limitations of the Toulouse criteria in the quantum regime Entanglement: T.C. do not detect quantum frustration Classical Ising ferromagnet All local terms commute H= J [S 1 z S 2 z +S 2 z S 3 z ] Minimum of the local energy terms: each air of sins aligned. Global ground state: all sins aligned. No frustration. T.C. ok! Quantum XX Hamiltonian Local terms do not commute H= J [(S 1 z S 2 z +S 1 x S 2 x )+(S 2 z S 3 z +S 2 x S 3 x )] The ground state of each air in vacuum is a maximally entangled Bell state. But sin 2 cannot be maximally entangled symultaneously with sins 1 and 3. Monogamy of entanglement ---> Frustration. However, according to the T.C., there is no frustration!

6 Universal measure of total frustration Measure of frustration: the degree of incomatibility between the local vacuum ground sace and the dressed one, namely, the sace of the reduced local density matrices in the resence of the many-body interactions. Π f =1 Tr (ρ Π ) rojector onto the local ground sace (local GS in vacuum ) ρ rojection of the global GS on the local GS Frustration-free INES (INEquality Saturating): Quantum Frustration f =ε (d ) =0 Non-INES: quantum and geometric frustration (d f =ε ) (d ) >0 f >ε

7 Quantum Toulouse Criteria If the global ground sace has degeneracy > 1, the measure of local frustration can deend on the choice of the articular ground state Maximally Mixed Ground State: convex combination with equal weights of all degenerate ground states. The MMGS reserves the same symmetries of the Global Hamiltonian Quantum Touluse Criteria: A model is rototye if 1) there exists at least one local ground state common to all local terms; 2) all couling vectors are ferromagnetic. Conjectures: Quantum Toulouse criterion I - All rototye models are INES. Quantum Toulouse criterion II All models obtained from rototye models by local unitary oerations and artial transositions are INES. No rigorous roof yet. Suorted by vast numerical evidence.

8 Frustration and Entanglement Pure Ground state ε (d ) Biartite entanglement monotone on states with Schmidt rank > d. Vanishing on states with Schmidt rank < d. ε (1) Biartite GS Entanglement between the local subsystem and the rest of the system R. Distance from the set of bisearable ure states. Mixed Ground State Sum of the (convex-roof) biartite entanglement between and R and of the classical correlations established by a local measurement erformed on by an ancillary system A. (d ε ) (d =E ) (d ) R+C A

9 Frustration and Entanglement: generic Heisenberg models (sin ½) - I H= h x h =(i, j) =α i, j S x i S x y j +α i, j S y i S y z j +α i, j S i z S j z H reserve arity along the three sin directions x, y and z ρ = 1 4 +g zz 0 0 g xx g yy g zz g xx yy +g 0 g xx yy 1 +g 4 g zz g xx g yy g zz admits as eigenstates the maximally entangled Bell states If all h admit a common ground state with d>1 the system is frustration free Absence of quantum frustration

10 Frustration and Entanglement: generic Heisenberg models (sin ½) - II ρ = 1 4 +g zz 0 0 g xx g yy g zz g xx yy +g 0 g xx yy 1 +g 4 g zz g xx g yy g zz _ij has as eigenstates the Bell states, and d=1 (nondeg. antiferr. local GS) Local-term concurrence C_ij (1 C ij =max (0,1 2 ε ) ij ) max (0, 1 2 f ij ) j max (0,1 2 f ij ) 2 2 = j C ij τi =1 General relation between frustration and monogamy of entanglement!

11 VBS (dimerized GS): transition to QF (INES) δ=0 δ=π/4 δ=π/2

12 Frustration-driven transition to VBS: observable Behavior of the static structure factor aroaching the Majumdar- Ghosh oint J_2/J_1 = 1/2 S f (k)= 1 N i,j cos(ka i j ) S i S j φ=0.05 π φ=0.22π φ=0.45 π

13 Conclusions & Outlook Summary: 1) Universal measure of total frustration 2) General relation with GS entanglement 3) QuantumToulouse criteria 4) Relation between frustration and monogamy of entanglement in generic Heisenberg models 5) VBS: transition from geometric to quantum frustration Memos for future directions: 1) Scaling behavior, area laws, and dynamics. Existence of a frustration length? 2) Relations with genuine multiartite entanglement. 3) Frustration and globally ordered hases (e.g. toological order).

14 REFERENCES S. M. Giamaolo, G. Gualdi, A. Monras, and F. I., Phys. Rev. Lett. 107, (2011) U. Marzolino, S. M. Giamaolo, and F. I., Phys. Rev. A 88, (R) (2013) S. M. Giamaolo, B. C. Hiesmayr, and F. I., arxiv:1410.xxxx