1. Block Entropy of the XYZ Model: Exact Results and Numerical Study
Cristian Degli Esposti Boschi CNISM and Dipartimento di Fisica, Università di Bologna Stefano Evangelisti Tesi di laurea in Fisica, Università di Bologna, Elisa Ercolessi, Francesco Ravanini, Fabio Ortolani, Dipartimento di Fisica and Sezione INFN, Bologna
Italian Quantum Information Science Conference,
5-8 nov 2009, Scuola Normale Superiore, Pisa

2. Given a bipartite quantum system (A+B), the von Neumann entropy
quantifies the entanglement of A with B if the whole system is in a pure state
in particular the ground state of the Hamiltonian on (A+B)
𝑆=− tr 𝐴 ρ 𝐴 log ρ 𝐴
ρ 𝐴 = tr 𝐵 ρ
ρ= ∣Ψ Ψ∣
ρ 0 = ∣ Ψ 0 Ψ 0 ∣
𝐻 𝐴+𝐵 ∣ Ψ 0 = 𝐸 0 ∣ Ψ 0

3. Critical points of 1D quantum systems are typically described by (1+1) conformal field theories.
Calabrese and Cardy [J. Stat. Mech. P06002 (2004)] computed exactly the block entropy in the conformal case. (Note added on the arXiv: corrections are necessary with block composed of disjoint blocks: Bari group, and others...).
As far as the noncritical (or massive or gapped) case is concerned, only two problems are addressed: Free bosonic theory and integrable models with a corner transfer matrix. General prediction for a single interval (one internal end)
𝑆= 𝑐 6 ln ξ 𝑎

4. The model studied here Spin-1/2 particles on a chain of L sites
Particular cases:
Only one nonvanishing coupling, say
Classical-like 1D Ising model
One vanishing coupling, say
XY model. Critical when
Solved by means of Jordan-Wigner + Bogolioubov transformations and asymptotics of Toeplitz determinants. Review in the following paper...
𝐻 𝑋𝑌𝑍 =− 𝑗 𝐽 𝑥 σ 𝑗 𝑥 σ 𝑗+1 𝑥 + 𝐽 𝑦 σ 𝑗 𝑦 σ 𝑗+1 𝑦 + 𝐽 𝑧 σ 𝑗 𝑧 σ 𝑗+1 𝑧
𝐽 𝑥
𝐽 𝑧
𝐽 𝑥 = 𝐽 𝑦

5. Its & Korepin, “The Fisher-Hartwig formula and entanglement entropy”, J. Stat. Phys., Online first (29/9/2009)

6. Three nonvanishing couplings:
At least two of them have the same sign, say
and . This common sign can always be taken positive or negative by means of a -rotation of the (conventionally) odd spins about z-axis.
The physical behaviour is dictated by the sign of the other coupling :
Ferromagnetism
Antiferromagnetism
(favourable for entanglement due to the tendency to form singlets)
𝐽 𝑥
𝐽 𝑦
𝐽 𝑧
𝐽 𝑧 >0
𝐽 𝑧 <0

7. Integrability and connection with 2D models in classical statistical mechanics
Back to the XXZ model, it is exactly solved by means of the Bethe ansatz.
It is also integrable: essentially this means that it admits an infinite number of (independent) conserved quantities. There is only a small number of “evident” symmetries:
Arbitrary rotation about z-axis [U(1)], -rotation about x- or y-axis [Z2], left-right reflection, possibly translational.

8. In the fully anisotropic XYZ model, the internal spin symmetry is reduced to Z2 X Z2. Nonetheless it is still integrable.
It is now convenient to exploit the dimensional crossover rule, that establishes a correspondence between D-dimensional quantum systems and classical systems in D+ spatial dimensions. Frequently one has =1, the additional dimension being represented by time.
For some 1D quantum lattice problems at zero temperature, the Hamiltonian is related to the trasfer matrix T of an equivalent 2D classical model in the canonical ensemble, and the two commute.

9. The parameters in the quantum model are related to the Boltzmann weights of classical variables configurations (“arrows”) at the sites (“vertex”) of a suitable 2D lattice.
XYZ corresponds to a 8-vertex model and XXZ reduces to 6-vertex model; both are integrable.

10. Corner transfer matrices
Remarkable properties [Peschel, Kaulke and Legeza, Ann. Physik (Leipzig) 8, 133 (1999)]:
(1) The block density matrix (the same used in the DMRG) is formally written as the partition function of the classical model with a cut
With a clever choice of the lattice orientation and of the boundary conditions A=B=C=D
ρ block =𝐴𝐵𝐶𝐷
Corner transfer matrices
Row-to-row transfer matrix T

11. and K has integer eigenvalues.
(2) Formally the block density matrix is written as a thermal state of a harmonic oscillator
and K has integer eigenvalues.
Hence, every quantity related to can be computed exactly; the only thing to know is the parameter . In particular
ρ block ∝exp −ε𝐾
ρ block
𝑆= 𝑗=1 ∞ 𝑗ε 1+exp 𝑗ε +ln 1+exp −𝑗ε
Strictly valid in the thermodynamic limit in noncritical cases. At criticality
ε~ 1 ln 𝐿

12. Baxter's parametrisation of the XYZ model with Jacobian functions
𝐽 𝑥 : 𝐽 𝑦 : 𝐽 𝑧 =1:Γ:Δ
Γ= 1+𝑘 sn 2 𝑖λ 1−𝑘 sn 2 𝑖λ
Δ= −cn 𝑖λ dn 𝑖λ 1−𝑘 sn 2 𝑖λ
𝑘′= 1− 𝑘 2
0<𝑘<1
0<λ<𝐼 𝑘′
Principal
regime
Complete elliptic
integral of the first kind
∣ 𝐽 𝑦 ∣≤ 𝐽 𝑥 ≤− 𝐽 𝑧
∣ 𝐽 𝑦 ∣≤ 𝐽 𝑥 ≤− 𝐽 𝑧

13. Once a common energy scale is fixed by, say, the other two couplings can be varied by tuning k and . However the entropy depends only on
𝐽 𝑥
ε=π λ 𝐼 𝑘 Δ
Details and reproduction of known limiting cases in Ercolessi, Evangelisti and Ravanini, “Exact entanglement entropy of the XYZ model and its sine-Gordon limit”, arXiv:
0.6 1 2 4 6 Γ

14. Numerical checks (DMRG)

15. Early attemps (i. e. forgetting about symmetries) with XXZ model
𝐽 𝑥 = 𝐽 𝑦 =1
Even-odd
effects at not sufficiently
large sizes
Saturation ???
Expected at 1.195

16. Analogous to even-odd effect due to
open boundary conditions
in critical systems
(vanishes algebraically)
Laflorencie et al.,
PRL 96, (2006)

17. Symmetry breaking in the gapped phase
XXZ: When the ground state is twofold degenerate in the thermodyamic limit and the Z2 up-down symmetry is spontaneously broken. There are two ground states with opposite value of the staggered (antiferro) magnetisation
The excitations above each of these two states are gapped with
Δ≤−1
𝑀 𝑠 = 𝑗 −1 𝑗 σ 𝑗 𝑧 𝐿
an order
parameter
𝐺∝exp − π −Δ−1 
(BKT
transition)

18. When L increases the two states are numerically
At finite size the ground state is unique but the energy difference between the two states and vanishes exponentially
∣ Ψ +
∣ Ψ −
δ𝐸∝ exp −𝐿 𝐿 𝐿 𝑝
with 𝐿 0 ∝ 𝐺 −1
When L increases the two states are numerically
degenerate and the DMRG will pick up a linear combination
∣Ψ 𝐿 =α ∣ Ψ + 𝐿 +β ∣ Ψ − 𝐿
This state (with real coefficients) has an average staggered magnetisation
from which we can read off , once Ms is known by a symmetry breaking construction.
𝑀 𝑠,DMRG = 2 ∣α∣ 2 −1 𝑀 𝑠

19. Entropy of the linear combination
ρ block = tr env ∣Ψ Ψ∣ = ∣α∣ 2 ρ + + ∣β∣ 2 ρ − +2αβμ
μ= tr env ∣ Ψ + Ψ − ∣ + ∣ Ψ − Ψ + ∣ 2
ρ ± = tr env ∣ Ψ ± Ψ ± ∣
 are connected by the up-down symmetry and give the same entropy. Moreover the trace over the environment (subsystem B) is expected to have a major contribution from the states obtained by restricting to B. So if
∣ Ψ ±
μ≃0
𝑆 α ≃𝑆 ρ + + 𝑆 deg α
𝑆 deg α =− ∣α∣ 2 log ∣α∣ 2 − 1− ∣α∣ 2 log 1− ∣α∣ 2

20. Symmetry breaking fields
− ℎ 𝑠 𝑗 −1 𝑗 σ 𝑗 𝑧
Staggered magnetic field along z:
− 𝐽 𝑧 𝑠 left σ 1 𝑧
− 𝐽 𝑧 σ 𝐿 𝑧 𝑠 right
Pinning fixed spins

21. For the XXZ case a small staggered field (to set to zero at the end of calculations) lifts the quasi-degeneracy at finite size as soon as that is approximately
𝑀 𝑠 ℎ 𝑠 𝐿>δ𝐸
𝐿>− ln 𝑀 𝑠 ℎ 𝑠 𝐺
XYZ: The fully anisotropic case is critical only when the two largest couplings have equal magnitude and disordered when We can adopt the same symmetry breaking mechanisms to select only one of the two ground states.
This was an implicit choice in the corner transferm matrix construction!
∣Δ∣ <1

22. Characteristic
size depends
logarithmically
on hs

23. Moving towards the critical
point the saturation from above (stag. fields) occurs
at a larger value and on longer length scales

24. Fully anisotropic
case (XYZ)
Different
choices of
parameters
but same 
DMRG
caveats

25. DMRG caveats
The influence of pinning fixed spins may be so strong that not only the ground-state degeneracy is lifted, but also the choice of block states retained during the DMRG is strongly biased: only a few states (in fact too few) that minimise the interaction with the boundary are retained and the representation in the truncated Hilbert space in the next steps is poor.
Practical solution: Even if the model is not strictly Z2 (up-down) invariant we consider at each step the symmetrised density matrix
ρ block + 𝑃 + ρ block 𝑃
𝑃 𝑗 = σ 𝑗 𝑥

26. Successful tests (pinning fixed spins)

27. Still even-odd effects before
a characteristic length scale

28. XYZ:

29. Accurate quantitative check: Scaling with L

30. Further developments:
The XYZ model with impurities is still integrable.
Is it still related to the 8-vertex classical 2D model, so that one can borrow these results?
Ambitious goal: Reproduce numerically the scaling limit that leads to the quantum sine-Gordon field theory [Luther, PRB 14, 2153 (1976)] so to check quantitatively the exact formula for the block entropy of the sine-Gordon model (Ercolessi, Evangelisti, Ravanini, 2009).
𝐴 𝑆𝐺 = d 2 𝑥 ∂ϕ 2 +𝑔cos βϕ 𝑥
𝑆= 𝑐 6 ln ξ 𝑔,β 𝑎 +𝑈 β

31. Thanks for your attention more about our activities ...
Supported by MIUR, PRIN n. 2007JHLPEZ

33. Definitions of Jacobian elliptic functions
Elliptic integral of
the first kind
𝑢= 0 φ d 𝑥 1− 𝑘 2 sin𝑥 −1 2
dn𝑢= 1− 𝑘 2 sinφ
sn𝑢=sinφ
cn𝑢=cosφ
Delta amplitude