Bell-paired states inducing volume law for Entanglement Entropy in - PowerPoint PPT Presentation

Bell-paired states inducing volume law for Entanglement Entropy in fermionic lattices Simone Paganelli in collaboration with: Giacomo Gori (SISSA) Auditya Sharma (Tel Aviv University) Pasquale Sodano (IIP , Natal) Andrea Trombettoni (SISSA) G. Gori et al. PRB 91 , 245138 (2015)

Outlook • Entanglement entropy in many-body systems: area law • Free fermion’s entanglement entropy • Bell-paired states, nontrivial Fermi surface and violation of the area-law • Examples • Conclusions

Relevance of the EE and its scaling Black holes, holographic principle Analogy with Bekenstein-Hawking area law: a black hole carries an entropy that is proportional to its horizon area A. Characterization of quantum phases Universal behavior near critical points. Indicator of phases that cannot be described by local order parameters (informations about central charge, topological charge, Kondo length …) (X. G. Wen, Phys. Rev. B 40 7387 (1989)), (E. Witten, Adv. Theory Math. 2 253 (1998)), (E. Nussinov et al. , Ann. Phys. 3 22 977 (2009)) Efficiency of simulations Efficiency of 1D algorithms such as DMRG is based on the Area law behaviour (U. Schollwöck, Rev. Mod. Phys. 77 259 (2005)) Nonequilibrium dynamics, quenches Propagation of excitation after a quench S A ( t ) ∝ ∂ ( A )

Entanglement Entropy (EE) • Measure of the entanglement of a pure state ρ ¯ A A • Bipartition into two subsystems: and ρ A = Tr ¯ A ρ S A = − Tr A ( ρ A ln ρ A ) In QM positive entropies may arise in a subsystem even at T=0. Entanglement produces a loss of information if one observes only a subsystem S A = 0 ρ = ρ A ρ ¯ A ρ 6 = ρ A ρ ¯ S A 6 = 0 A

How does the EE scale with the size L of the subsystem? • Scaling of the entropy. Thermal/classical states: extensive entropy (volume law). • Typical ground states: area law. The EE grows as the boundary of the subsystem S A ∼ L d − 1 • 1D gapped systems , short range, finite interaction strengths: area law proved by Hastings (M. B. Hastings, Phys. Rev. B 69 104431 (2007)) • Gapless systems: CFT, logarithmic divergence. EE proportional to the central charge (C. F. Holzhey et al., Nucl. Phys. B 424 443 (1994)), (G. J. Vidal et al., Phys. Rev. Lett. 424 227902 (2003)) Free fermions S A ∼ L d − 1 log L Logarithmic deviation to the area law

Entanglement entropy for free fermions c † X H = − I t IJ c J I,J EE in terms of correlation matrix L X S A = − [(1 − C γ ) ln (1 − C γ ) + C γ ln C γ ] γ =1 C ij = h Ψ | c † eigenvalues of the correlation matrix C γ i c j | Ψ i . i,j: index of the subsystem’s sites translational invariance: plane wave solutions 1 dk Z e ikJ ψ k ( J ) = 2 π e ik ( I − J ) C IJ = √ N S E<E F Independent of the values of the energy

Non-local power law hopping • 1D free fermions with PBC, half filling • Long range hopping: can it determine a violation of the area law beyond the logarithmic correction? ( I = J, 0 t I,J = t I 6 = J, | I − J | α p | I − J | p = min( | I − J | , N S − | I − J | )

Non-local power law hopping • 1D free fermions with PBC, half filling • Long range hopping: can it determine a violation of the area law beyond the logarithmic correction? ( I = J, 0 t I,J = t I 6 = J, | I − J | α p | I − J | p = min( | I − J | , N S − | I − J | ) EE does not depend on α Still Area law!

Non-local power law hopping • 1D free fermions with PBC, half filling • Long range hopping: can it determine a violation of the area law beyond the logarithmic correction? ( I = J, 0 t I,J = t I 6 = J, A | I − J | α p | I − J | p = min( | I − J | , N S − | I − J | ) EE does not depend on α Still Area law! A • Solid line: nearest neighbors • dashed line: α = 2 • Inset: nearest and next-nearest neighbors model with different hopping’s signs

Non-local power law hopping • 1D free fermions with PBC, half filling • Long range hopping: can it determine a violation of the area law beyond the logarithmic correction? ( I = J, 0 t I,J = t I 6 = J, A | I − J | α p | I − J | p = min( | I − J | , N S − | I − J | ) EE does not depend on α Still Area law! A EE depends on the topology of the • Solid line: nearest neighbors • dashed line: α = 2 Fermi surface • Inset: nearest and next-nearest neighbors model with different hopping’s signs Z k d Z k c dk dk 2 π e ik ( I − J ) − 2 π e ik ( I − J ) C IJ = k a k b

Let’s try all the possible filling configurations 7 • • • • • • ◦ ◦ ◦ ◦ ◦◦ • • ◦ • • • ◦ • ◦ ◦ ◦◦ • • • • • ◦ • ◦ ◦ ◦ ◦◦ • • ◦ • • • ◦ ◦ • ◦ ◦◦ • • • • • ◦ ◦ • ◦ ◦ ◦◦ • • ◦ • • ◦ • • ◦ ◦ ◦◦ 6 • • • • • ◦ ◦ ◦ • ◦ ◦◦ • • ◦ • • ◦ • ◦ • ◦ ◦◦ • • • • ◦ • • ◦ ◦ ◦ ◦◦ • • ◦ • • ◦ • ◦ ◦ • ◦◦ • • • • ◦ • ◦ • ◦ ◦ ◦◦ • • ◦ • • ◦ ◦ • • ◦ ◦◦ • • • • ◦ • ◦ ◦ • ◦ ◦◦ • • ◦ • • ◦ ◦ • ◦ • ◦◦ 5 • • • • ◦ ◦ • • ◦ ◦ ◦◦ • • ◦ • ◦ • • ◦ • ◦ ◦◦ • • • • ◦ ◦ • ◦ • ◦ ◦◦ • • ◦ • ◦ • • ◦ ◦ • ◦◦ • • • • ◦ ◦ • ◦ ◦ • ◦◦ • • ◦ • ◦ • ◦ • • ◦ ◦◦ 4 • • • • ◦ ◦ ◦ • • ◦ ◦◦ • • ◦ • ◦ • ◦ • ◦ • ◦◦ S/ ln 2 • • • ◦ • • • ◦ ◦ ◦ ◦◦ • • ◦ • ◦ ◦ • • ◦ • ◦◦ • • • ◦ • • ◦ • ◦ ◦ ◦◦ • • ◦ ◦ • • • ◦ • ◦ ◦◦ • • • ◦ • • ◦ ◦ • ◦ ◦◦ • • ◦ ◦ • • ◦ • ◦ • ◦◦ 3 • • • ◦ • ◦ • • ◦ ◦ ◦◦ • • ◦ ◦ • • ◦ ◦ • • ◦◦ • • • ◦ • ◦ • ◦ • ◦ ◦◦ • • ◦ ◦ • ◦ • • ◦ • ◦◦ • • • ◦ • ◦ • ◦ ◦ • ◦◦ • ◦ • • • • ◦ • ◦ ◦ ◦◦ 2 • • • ◦ • ◦ ◦ • • ◦ ◦◦ • ◦ • • • • ◦ ◦ • ◦ ◦◦ • • • ◦ • ◦ ◦ • ◦ • ◦◦ • ◦ • • • ◦ • ◦ • ◦ ◦◦ • • • ◦ ◦ • • • ◦ ◦ ◦◦ • ◦ • • • ◦ • ◦ ◦ • ◦◦ • • • ◦ ◦ • • ◦ • ◦ ◦◦ • ◦ • • ◦ • • ◦ • ◦ ◦◦ 1 • • • ◦ ◦ • • ◦ ◦ • ◦◦ • ◦ • • ◦ • ◦ • ◦ • ◦◦ • • • ◦ ◦ • ◦ • • ◦ ◦◦ • ◦ • • ◦ ◦ • • ◦ • ◦◦ • • • ◦ ◦ • ◦ • ◦ • ◦◦ • ◦ • ◦ • • ◦ • ◦ • ◦◦ • • • ◦ ◦ ◦ • • • ◦ ◦◦ • ◦ • ◦ • ◦ • ◦ • ◦ • ◦ 0 0 2 4 6 8 10 12 Half filling L N S = 12 • Maximal EE for alternating filling of the wave vectors • Periodicity in k-space: piecewise linear behavior

States with maximal EE V = A ⊕ ¯ • Single particle vector space A • Basis in the complementary subsystems α j i 2 ¯ | α j i 2 A , | ¯ A 1 1 1 p p p | β 1 i = 2( | α 1 i + | ¯ α 1 i ) , | β 2 i = 2( | α 2 i + | ¯ α 2 i ) , . . . , | β | A | i = 2( | α | A | i + | ¯ α | A | i ) • single particle occupied states (not necessarily wave vectors) constructed in this way maximize the EE for half filling • Also all the Rényi entropies • Bell-paired states

alternate occupation of the ( N S / 2 e 2 π in k J/ ( N S / 2) 1 p for J  N S / 2 h J | α k i = momentum space for J > N S / 2 0 ( for J  N S / 2 0 h J | ¯ α k i = 1 N S / 2 e 2 π in k J/ ( N S / 2) p ± for J > N S / 2 , | C IJ | A. contiguous occupation of k B. “zigzag” state C. two filled and two empty momenta alternates D. Bell-paired state with random connection between the complementary subsystems

Examples of models violating the area law A model can be constructed for the GS to have the structure of the Bell- paired states. c † X H = − I t IJ c J I,J ⇢ for | I − J | p = N S t t I,J = 2 otherwise, 0

Long-Range with a Magnetic flux t I,J = t · e i φ d I,J φ = 2 π Φ | I − J | α N S p α = 0 . 1 α = 0 . 4 N=100 Φ = 0 . 1 Φ = 0 . 001 Φ = 0 . 1 Φ = 0 . 3 fitting function: S=a L

Fermi surface with an accumulation point ✓ 1 ◆ ε k = − t · sin , k α Fermi surface: {± 1 π , ± 1 1 π 2 α ± π 3 α . . . } α d box = Set with box counting dimension: Fit function: α + 1 S = a + bL β L=1…128. Short range: S=a+b ln L

Random Long-Range hopping t I,J = t · η I,J • Breaking of translational symmetry | I − J | α p • Logarithm behavior for α � 1 • Linear behavior for α ⌧ 1 η I,J = ± 1 Fit function: S = a + bL β

Conclusions • Violation of the area law has been investigated for free-fermions • Long range hopping is not a sufficient condition. A more complex structure of the spectrum and of the Fermi surface is needed and/or a breaking of the translational invariance. • Explicit construction of the states maximizing the EE in terms of Bell pairs. • Explicit examples: magnetic phase; FS with an accumulation point • Preliminary results for a disordered lattice