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Ground state factorization versus frustration in spin systems

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									Ground state factorization versus
   frustration in spin systems

                       Gerardo Adesso
                School of Mathematical Sciences
                   University of Nottingham


joint work with S. M. Giampaolo and F. Illuminati (University of Salerno)


                        "Hamiltonian & Gaps", 7/9/2010                      1
Outline

•   Spin systems and frustration
•   What we want to do and why
•   Theory of ground state factorization
•   Factorized solutions to frustration-free models
•   Frustration vs factorization and order
•   Summary and outlook


                   "Hamiltonian & Gaps", 7/9/2010     2
Quantum spin systems
                                            • N spin-1/2 particles
                                              on regular lattices
                                                 – anisotropic
                                                   interactions of
                                                   arbitrary range
                                                 – arbitrary spatial
                                                   dimension
                                                 – translationally
                                                   invariant & PBC
                                                 – external field along z

   1X r x x      r y y        r z z
                                          X
                                             z
H=   Jx Si Sl + Jy Si S l + J z Si Sl ¡ h   Si
   2
       i;l                                                        i
                (
                r= i - l   )
                "Hamiltonian & Gaps", 7/9/2010                              3
Ground states

• No known exact analytical solution in general, except for
  a few simple subcases (Ising, XY,...) and now a wider class
  of models with nearest-neighbor interactions (see JE)
• Difficult to be determined even numerically, especially
  for high-dimensional lattices (2D, 3D, ...)
• Rich phenomenology: different magnetic orderings,
  critical points and quantum phase transitions
• Typically exhibit highly correlated quantum fluctuations,
  i.e., they are typically entangled



                     "Hamiltonian & Gaps", 7/9/2010             4
Frustration
                        • Occurs when the ground state of the
                          system cannot satisfy all the couplings
                        • Even richer phase diagram (high
             AF   ?       degeneracy), hence even harder to
   AF
                          find ground states
        AF              • In frustrated systems a magnetic order
                          does not freeze, which typically results
                          in even more correlations
• At the root of statistically fascinating phenomena and exotic
  phases such as spin liquids and glasses
• Frustrated systems may play a crucial role to model high-Tc
  superconductivity and certain biological processes


                        "Hamiltonian & Gaps", 7/9/2010               5
Relevant questions
• How to define natural signatures and measures of
  (classical and/or quantum) frustration?
• More generally, is it possible to tune an external field
  so that a many-body model admits as exact ground
  state a completely factorized (“classical-like”) state?
   – This would be an instance of mean field becoming exact

• If yes, under which conditions? Does this possibility
  depend on the presence of frustration? In turn, does
  the fulfillment or not of this condition define a regime
  of weak versus strong frustration?

                     "Hamiltonian & Gaps", 7/9/2010           6
Ground state factorization
               ground =    ⊗ ⊗ ⊗ ⊗                  …


• Answer to the 2° question: YES! There can exist special
  points in the phase diagram of a spin system such that the
  ground state is exactly a completely uncorrelated tensor
  product of single-spin states: factorized ground state
• The “factorization point” is obtained for specific, finite values
  of the external magnetic field (dubbed factorizing field) which
  depend on the Hamiltonian parameters
• First devised by Kurmann, Thomas and Muller (1982) for 1-d
  Heisenberg chains with nearest neighbor antiferromagnetic
  interactions


                       "Hamiltonian & Gaps", 7/9/2010                 7
Motivations
 • Many-body condensed matter perspective
    – To find exact particular solutions to non-exactly solvable models
    – To devise ansatz for perturbative analyses, DMRG, …
 • Quantum information and technology perspective
    – For several applications (e.g. quantum state transfer, dense coding,
      resource engineering for one-way quantum computation), both in
      the case of protocols relying on “natural” ground state
      entanglement for quantum communication (in which case
      factorization points should be avoided!), and for tasks which
      instead require a qubit register initialized in a product state
 • Statistical perspective
    – To investigate the occurrence of “phase transitions in
      entanglement” with no classical counterpart
    – For frustrated systems: to characterize the frustration-driven
      transition between order (signaled by a factorized ground state)
      and disorder (landmarked by correlations in the ground state), thus
      achieving a quantitative handle on the frustration degree

                         "Hamiltonian & Gaps", 7/9/2010                      8
History
• Direct method (product-state ansatz)
 Analitic brute-force method, guess a product state and verify that it is the ground state
     via the Schrödinger equation, becomes nontrivial for more complicate models …

    – Kurmann et al. (1982): 1d Heisenberg, nearest neighbors
    – Hoeger et al. (1985); Rossignoli et al. (2008): 1d Heisenberg, arbitrary
      interaction range
    – Dusuel & Vidal (2005): Fully connected Lipkin-Meshkov-Glick model
    – Giorgi (2009): Dimerized XY chains


• Numerical method (Monte Carlo simulations)
        Nightmarish for spatial dimensions bigger than two (never attempted !)

    – Roscilde et al. (2004, 2005): 1d & 2d Heisenberg, nearest neighbors


                               "Hamiltonian & Gaps", 7/9/2010                                9
Our method
• Quantum informational approach
   – Inspired by tools of entanglement theory
   – Fully analytic method
   – Requires no ansatz: the magnetic order, energy, and specific
     form of the factorized ground state are obtained as a result of
     the method
   – Encompasses previous findings and enables the identification of
     novel factorization points
   – Provides self-contained necessary and sufficient conditions for
     ground state factorization (in absence of frustration) in terms of
     the Hamiltonian parameters
   – Straightforwardly applied to cases with arbitrary range of the
     interactions and arbitrary spatial dimension (e.g. cubic
     Heisenberg lattices), and to systems with spatial anisotropy, etc.


 S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)
                                    "Hamiltonian & Gaps", 7/9/2010                                         10
The ingredients /1

                                    ½k
• Under translational invariance, the ground state is completely
  factorized iff the entanglement between any spin and the
  block of all the remaining ones vanishes, i.e., if the marginal
  (linear) entropy of a generic spin, say on site k, is zero
                                          £ x 2         y 2   z i2
                                                                   ¤
• We have: SL(½k ) = 4Det½k = 1 ¡ 4 hSk i + hSk i + hSk
  so this factorization condition would depend on the
  magnetizations, which are indeed the objects one cannot
  compute in general models


                       "Hamiltonian & Gaps", 7/9/2010                  11
The ingredients /2
                               Uk

                                         ½k
• A generic N-qubit state is factorized iff for any qubit k there exist a unique
  Hermitian, traceless, unitary operator Uk (which takes in general the form
  of a linear combination of the three Pauli matrices), whose action on qubit
  k leaves the global state unchanged (Giampaolo & Illuminati, 2007)
• We can define in general the “entanglement excitation energy” (EXE)
  associated to spin k as the increase in energy after perturbing the system,
  in its ground state, via this special local unitary Uk (Giampaolo et al., 2008)
                                     y
  In formula: ¢E (Uk ) = hªjUk H Uk jªi ¡ hªjH jªi

• One can prove that, under translational invariance and under the
  hypothesis [H,Sa]≠0 (a=x,y,z), the ground state is completely factorized iff
  the entanglement excitation energy vanishes for any generic spin k

                            "Hamiltonian & Gaps", 7/9/2010                          12
The ingredients /3


1.       A factorized ground state must have vanishing local entropy
2.       A factorized ground state must have vanishing EXE
3.       The ground state must minimize the energy
4.       The Hamiltonian model H does not give rise to frustration




           general theory of ground state factorization

     S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)
                                        "Hamiltonian & Gaps", 7/9/2010                                         13
Net interactions
• All the results (form of the state, factorizing field, conditions
  for ground state factorization) are only functions of the
  Hamiltonian coupling parameters and of lattice geometry
  factors, or more compactly, of the “net interactions”:
                  ¥                                        – Zr is the coordination number, i.e.
  J    A
      x ,y
              =   å             r
                        (- 1) Z r J ,     r
                                          x ,y
                                                             the number of spins at a distance r
                                                             from a given site
                  r=1
                   ¥                                       – the magnetic order is determined by
  J    F
      x ,y
              =   å     Z r J xr,y ,                                       F     A    F   A
                                                                ¹ = min fJx ; Jx ; Jy ; Jy g
                  r=1                                   8 F
                    ¥                                   > Jx
                                                        > F             )   Ferrom. order along x ;

                  å
       A ,F                                             <
  J   x ,y
              =          Z r J zr .                  ¹=
                                                          Jy
                                                           A
                                                        > Jx
                                                                        )
                                                                        )
                                                                            Ferrom. order along y ;
                                                                            Antiferrom. order along x ;
                                                        > A
                                                        :
                   r=1                                    Jy            )   Antiferrom. order along y .

  S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)
                                       "Hamiltonian & Gaps", 7/9/2010                                14
Results: Frustration-free




  S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)
                                    "Hamiltonian & Gaps", 7/9/2010     15
Heisenberg lattices
               1D nearest neighbor
                                                                    • The method is versatile
                                                                      and the result is totally
       Kurmann‘82                  AF (y)                             general: the complexity
                                                                      is the same for any
                                                                      spatial dimension, one
                                              AF (x)                  only needs to put the
                                                                      correct coordination
           F (x)                                                      numbers in the
                                                                      definition of the net
                                                                      interactions
                      F (y)                                           (e.g. for nearest
                                                                      neighbor models: Z=2 for
                                                                      chains, Z=4 for planes,
                                                                      Z=6 for cubic lattices)

  S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 100, 197201 (2008); Phys. Rev. B 79, 224434 (2009)
                                     "Hamiltonian & Gaps", 7/9/2010                                         16
Other applications

     • Long-range and infinite-range models
     • Models with spatial anisotropy

              a)                                   b)




              c)                                   d)



     • …

  S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. B 79, 224434 (2009)
                                    "Hamiltonian & Gaps", 7/9/2010     17
Frustrated systems
                                         AF

                                         AF     AF

• We consider a subclass of the original Hamiltonian, comprising models with
  anisotropic antiferromagnetic (along x) interactions up to a maximum range rmax
• Frustration arises from the interplay between the couplings at different ranges
• We focus on 1d systems (chains) of infinite length
• For simplicity, we consider the interaction anisotropies independent on the
  distance, but overall the couplings are rescaled by a range-dependent factor fr
• If all the fr’s beyond r=1 vanish, the system is not frustrated. Vice versa, if the
  fr’s are all equal, the system is fully frustrated.
            X                                                                  X
                               x x          y y          z z                        z
 H=                    fr (Jx Si Si+r + Jy Si Si+r + Jz Si Si+r ) ¡ h              Si
        i;r·rmax                                                               i

     S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                        "Hamiltonian & Gaps", 7/9/2010                  18
Short-range systems
• Simplest case: rmax = 2                          (nearest and next-nearest neighbors)

    – We set f1 = 1, f2 ≡ f
    – The parameter f ∊[0,1] plays the role of a “frustration degree”
        (a more general definition of frustration degree was given by Sen(De) et al., PRL 2008)



• Magnetic order of the ground state
 f<½                                                              standard antiferromagnet


 f≥½                                                              dimerized antiferromagnet


  S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                     "Hamiltonian & Gaps", 7/9/2010                               19
Factorized ground states
• We can determine in general the form of the candidate factorized state
  and the factorizing field
                   p                    p
                 1 J J          (1 ¡ f ) Jx Jy f < 1=2
                                 p
           hf =         x y =
                 2              f Jx Jy          f ¸ 1=2

• The nontrivial part is now in the verification steps
   – We find that for the candidate factorized state to be an eigenstate, a
      necessary condition is
       Jz = 0 (other possibilities lead to saturation instead of proper factorization)
   – By decomposing the Hamiltonian into triplet terms, we can derive a
      sufficient condition for the candidate state to be the ground state, by
      testing whether its projection on three spins is the ground state of
      the triplet Hamiltonian
   – For frustration-free, the factorized state was always the ground state

   S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                      "Hamiltonian & Gaps", 7/9/2010                     20
Factorization vs frustration
 • From the triplet decomposition we find, analytically, that if the frustration
   is weaker than a “critical” value, f £ f º 1 J x - J xJ y + J y ,
                                                          c
                                                                2       Jx + Jy
    then the ground state is factorized
                                                                 • The actual “compatibility
                                                                   threshold” (i.e. the maximum
                                                                   frustration degree that allows
                                                                   ground state factorization) can
                                                                   be determined numerically by
                                                                   considering decompositions
                                                                   into blocks of more than three
                                                                   spins.
                                                                 • Above this boundary the system
         ground state factorization                                admits a factorized eigenstate
                                                                   at h=hf, but this does not
                                                                   minimize energy and instead
                                                                   the ground state is entangled
   S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                      "Hamiltonian & Gaps", 7/9/2010                           21
Remarks
                                                                                    ( h = hf )
FRUSTRATION



                                                                      Factorized dimerized excited eigenstate

                                                                      Factorized antiferromagnetic excited eigenstate

                                                                      Factorized antiferromagnetic ground state



   • Frustration naturally induces correlations which tend to suppress ground
     state factorization: for strong enough frustration it is not energetically
     favourable for the system to arrange in a factorized state (although a
     factorized state can exist in the higher-energy spectrum)
   • At the factorizing field, we witness a first order quantum phase transition
     (level crossing) from a factorized to an entangled ground state when the
     frustration crosses the compatibility threshold
   • Qualitative agreement with the results on the scaling of correlations (Sen(De)
     et al., PRL 2008) and on tensor network representability (Eisert et al., PRL 2010)

              S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                                 "Hamiltonian & Gaps", 7/9/2010                                     22
Remarks
                                                                                    ( h = hf )
FRUSTRATION



                                                                      Factorized dimerized excited eigenstate

                                                                      Factorized antiferromagnetic excited eigenstate

                                                                      Factorized antiferromagnetic ground state



   • Reversing the perspective, we can define the regime of weak frustration as
     the one compatible with ground state factorization, and the regime of
     strong frustration as the one where no factorization points are allowed.
   • Ground state factorization implies a definite magnetic order, thus it is a
     precursor to a quantum phase transition, with critical field hc≥hf
   • The regime of strong frustration is thus characterized by the fact that a
     magnetic order does not freeze even at zero temperature (in layman’s
     words, the ground state remains always entangled), in accordance with
     other criteria to assess the frustration degree (Ramirez, Balents, ...)

              S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                                 "Hamiltonian & Gaps", 7/9/2010                                     23
Longer-range models
• The same general features emerge by investigating frustrated
  systems with interactions beyond next-nearest neighbors
    – Factorized eigenstates are only allowed for Jz=0
       (this limitation could be relaxed in more general non-translationally-invariant models
       where the anisotropies depend individually on the distance)
    – There is a compatibility threshold dividing the phase diagram into a
      region of weak frustration/order/ground state factorization and a
      region of strong frustration/disorder/ground state entanglement

 Compatibility thresholds: Maximum value of the                                rmax = 4
 frustration f as a function of the ratio Jy =Jx for
 which ground state factorization points exist in frus-
 trated antiferromagnets with rmax = 4. The black
 line stands for systems with f2 = f , f3 = f =2 and
 f4 = f =3, the red line for f2 = f , f3 = f =2 and p
 f4 = f =4, and the blue line for f2 = f , f3 = f = 2
 and f4 = f =6.



     S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                        "Hamiltonian & Gaps", 7/9/2010                          24
Infinite-range models
                                                             fr


• To verify ground state factorization in fully connected models (rmax=∞),
  one should decompose the Hamiltonian in terms involving n→∞ spins,
  i.e., basically solve the Hamiltonian itself!
• A workaround is possible if the frustration coefficients fr follow a
  decreasing functional law with r and vanish in the limit r→∞
• In this case one can impose a cutoff and deal with decompositions into
  blocks of a finite number n of neighboring spins which are most effectively
  coupled
• Then one takes the limit n→∞. Numerically, this means that ground state
  factorization occurs if, for n large enough, the difference D between the
  minimum eigenvalue m of the n-spin Hamiltonian component and the
  energy associated to the candidate factorized state, vanishes
  asymptotically. If r’ is the cutoff range (such that n=2r’+1), then
                         0       0                P
                    D(r ) = ¹(r ) + 1 (Jx + Jy ) r=1 (¡1)l fl
                                       4            l

  S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                     "Hamiltonian & Gaps", 7/9/2010             25
Results for infinite range
• Case fr=1/r2                    (weak frustration)
   – The difference D(r’)=0 for any cutoff r’                                 p
         Exact factorized ground state at h f = ( ¼ 2 = 1 2 )                    Jx Jy


• Case fr=1/r                     (medium frustration)
   – The difference D(r’) seems to converge
     to 0 (more numerics needed)
         Conjectured factorized         p
          ground state at h f = l n ( 2 ) J x J y

                                                               D
             _
• Case fr=1/√r                    (strong frustration)
   – The difference D(r’) converges
     to a finite value
         No factorized ground state !


    S. M. Giampaolo, GA, F. Illuminati, Phys. Rev. Lett. 104, 207202 (2010)
                                       "Hamiltonian & Gaps", 7/9/2010                     26
Summary
• We approached the problem of finding exact factorized ground state
  solutions to general cooperative spin models
• We devised a method to identify and fully characterize such solutions
  thanks to some tools borrowed from quantum information theory
• In frustration-free systems, necessary and sufficient conditions are derived
  and several novel factorized exact solutions are straightforwardly
  obtained for various translationally invariant models, with interactions of
  arbitrary range, and arbitrary lattice spatial dimension
• In frustrated systems, a universal behaviour emerged in which frustration
  and ground state factorization are competing phenomena, the former
  inducing correlations and disorder, and the latter relying on ordered,
  uncorrelated magnetic arrangements. Notably short-range as well as
  infinite-range (weakly) frustrated antiferromagnetic models have been
  shown to admit exact factorized solutions
• Ground state factorization is an effective tool to probe quantitatively
  frustrated quantum systems. The possibility vs impossibility of having a
  classical-like ground state at a given value of the magnetic field defines
  the regimes of weak vs strong frustration


                           "Hamiltonian & Gaps", 7/9/2010                        27
Discussion and outlook
• In this talk we only considered spin-1/2 systems, however due to a
  theorem by Kurmann et al. (1982), any spin-S (S> ½) Hamiltonian which is
  of the same form as a spin-1/2 Hamiltonian that admits a factorized
  ground state at h=hf, will also admit a factorized ground state at the same
  value of the field: greater scope of our results
• For a generic model (frustrated or not), the factorizing field (when it
  exists) is a precursor to the critical field associated to a quantum phase
  transition (where the external field is the order parameter), i.e. hf≤hc
• A fascinating perspective is the investigation of the ground state
  entanglement structure near a factorization point: it is conjectured that
  entanglement undergoes a global reshuffling and can change its typology
  (demonstrated in the XY and XXZ models, Amico et al. 2007): an
  “entanglement phase transition” with no classical counterpart, and
  signaled by a diverging range of pairwise entanglement
• More perspectives
    – Generalize the method: relax translational invariance, identify exactly
      dimerized solutions, etc.
    – Area laws for frustrated systems?
    – Define a measure of frustration, able to distinguish quantum from classical
    – ...
                             "Hamiltonian & Gaps", 7/9/2010                         28
Thank you
    ?




            "Hamiltonian & Gaps", 7/9/2010   29

								
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