|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| The '''AKLT model''' is an extension of the one-dimensional [[quantum mechanics|quantum]] [[Heisenberg model (quantum)|Heisenberg spin model]]. The proposal and exact solution of this model by [[Ian Affleck|Affleck]], [[Elliott H. Lieb|Lieb]], Kennedy and Tasaki<ref name="Affleck:1987" /> provided crucial insight into the physics of the spin-1 Heisenberg chain.<ref> | | <br><br>Cardsharp® is a great chef's knife The blade is sharp rated 'very good' by CATRA long and skinny adequate to reduce bread and also very useful for quickly slicing of extremely thin slices of greens such as tomatoes, cucumber, garlic etc. Great for camping and expedition use Cardsharp® is an essential out of doors companion/survival tool. It is important to keep hydrated in the course of a picnic with all of the frisbee throwing or soccer tossing. Provide water, sodas, fruit beverages, beer, wine, coffee, tea or everything else you like to drink. Usually be safe if you are ingesting alcohol. Have a [http://wiki.Apsinfotech.com/index.php?title=Best_Military_Knife_Brand selected driver] and adhere to [http://Wiki.paconun.eu/index.php/The_Gentleman_s_Pocket_Knife park guidelines].<br>Total, this is not the nicest Spyderco I've witnessed-visually talking, of course. The blade is as sharp as you would count on from a Spyderco, and it will certainly complete the job, what ever that occupation might be. The knife itself isn't as appealing as other folks I've noticed from Spyderco, but that is a matter of opinion, and probably not an situation for you. I might suggest this knife to anyone who would like a relaxed carry to use for generating slim slices or deep cuts. Folders are truly successful as effectively as expense successful cases for your businesses. That is why firm delivers you least expensive pocket folder printing answers [http://Wiki.Tes.Mi.it/mediawiki/index.php/Red_Pocket_Knife_Brands worldwide].<br>If you are an outdoorsman, there are a number of equipment you will want to have with you in your outside package. The trick is to uncover the right equipment that will enable you to travel light while not lacking out on any essential items of products that will make your tenting, looking, or climbing experience far more enjoyable. Study on to discover out more [http://www.thebestpocketknifereviews.com/best-pocket-knife-brand-good-knives-in-the-world/ Best Folding Knife Ever Made]! go through more I haven't experienced ample time to truly comment the sturdiness, but this is a well designed blade that feels excellent even though in use. I adore the vertical belt loop as properly. I regret getting rid of my old tactical, but this makes that lose a minor considerably less agonizing.<br>Im sorry, he stated. I didnt recognise you without the He twirled his fingers spherical his head to [http://wiki.myriadtalk.com/index.php?title=What_Is_The_Legal_Length_Of_A_Pocket_Knife_In_Texas signify] the veil. She was hunting straight at him, and he had no idea what that appear was intended to imply. Which is absurd, he additional speedily. I can see your confront so I dont know who you are. Whats your identify? He lay even now, striving not to consider about the darkness. All his daily life hed slept with a lamp burning, but there was no lamp in this area and he hadnt sent for one. Was it?<br>Later, Bassos mother determined she should have given her also significantly (comprehensible, in the situation). Also, the girl might have had a weak heart or some similar problem. It was unfortunate, of course, but just 1 of individuals things. Bassos mother paid out for a coffin and a plot in the public cemetery. It was, she felt, the least she could do. As the door closed, Aelius leaned forward across his desk. The boy was viewing him, to see what would come about subsequent. He wasnt afraid, he wasnt smug. That on your own was sufficient to verify that he was who he stated he was. What variety of lewd suggestion? Aelius requested.<br>Difficult, but beautiful, figured green linen Micarta surrounds this hybrid beast with a bead blasted four.five" blade and Karambit type finger ring. The Micarta sheath has a customized created "S" shaped serpentine clip for idea up, or suggestion down have. You can also get rid of the clip, and location it on the other side for boot, belt, remaining or proper handed undercover transport. This is the huge [http://www.thebestpocketknifereviews.com/best-pocket-knife-brand-good-knives-in-the-world/ The Best Pocket Knife 2013] brother to my smaller "Neckolace" designed knife which CRKT is at present making. Completely handmade. $450. Nowadays , do not get oneselfCheck price tag prior to the Price tag Up. Buck Elk Manage Folding Hunter Knife with B&C Medallion You will not regret if verify price. |
| F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983), Phys.
| |
| Lett. A 93, 464 (1983); I. Affleck and F. D. M. Haldane, Phys.
| |
| Rev. B 36, 5291 (1987); I. Affleck, J. Phys.: Condens.
| |
| Matter. 1, 3047 (1989).
| |
| </ref> It has also served as a useful testbed for such concepts as valence bond solid order, [[symmetry protected topological order]]<ref>
| |
| Zheng-Cheng Gu and [[Xiao-Gang Wen]]
| |
| [http://arxiv.org/abs/0903.1069 Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order] | |
| Phys. Rev. B80, 155131 (2009).
| |
| </ref><ref name="Pollmann:2012" /><ref>Xie Chen, Zheng-Cheng Gu, [[Xiao-Gang Wen]],
| |
| [http://arxiv.org/abs/1008.3745 Classification of Gapped Symmetric Phases in 1D Spin Systems] ''Phys. Rev. B'' 83, 035107 (2011); | |
| Xie Chen, Zheng-Xin Liu, [[Xiao-Gang Wen]],
| |
| [http://arxiv.org/abs/1106.4752 2D symmetry protected topological orders and their protected gapless edge excitations] ''Phys. Rev. B'' 84, 235141 (2011)</ref> and matrix product state wavefunctions.
| |
| | |
| == Background ==
| |
| | |
| A major motivation for the AKLT model was the [[Majumdar-Ghosh Model|Majumdar-Ghosh chain]]. Because two out of every set of three neighboring spins in a Majumdar-Ghosh ground state are paired into a singlet, or valence bond, the three spins together can never be found to be in a spin 3/2 state. In fact, the Majumdar-Ghosh Hamiltonian is nothing but the sum of all projectors of three neighboring spins onto a 3/2 state.
| |
| | |
| The main insight of the AKLT paper was that this construction could be generalized to obtain exactly solvable models for spin sizes other than 1/2. Just as one end of a valence bond is a spin 1/2, the ends of two valence bonds can be combined into a spin 1, three into a spin 3/2, etc.
| |
| | |
| == Definition ==
| |
| | |
| Affleck et al. were interested in constructing a one-dimensional state with a valence bond between every pair of sites. Because this leads to two spin 1/2s for every site, the result must be the wavefunction of a spin 1 system.
| |
| | |
| For every adjacent pair of the spin 1s, two of the four constituent spin 1/2s are stuck in a total spin zero state. Therefore each pair of spin 1s is forbidden from being in a combined spin 2 state. By writing this condition as a sum of projectors, AKLT arrived at the following Hamiltonian
| |
| | |
| <math> \hat H = \sum_j \vec{S}_j \cdot \vec{S}_{j+1} + \frac{1}{3} (\vec{S}_j \cdot \vec{S}_{j+1})^2 </math> | |
| | |
| This Hamiltonian is similar to the spin 1, one-dimensional [[quantum mechanics|quantum]] [[Heisenberg model (quantum)|Heisenberg spin model]]
| |
| but has an additional spin interaction term.
| |
| | |
| == Ground State ==
| |
| | |
| By construction, the ground state of the AKLT Hamiltonian is the valence bond solid with a single valence bond connecting every neighboring pair of sites.
| |
| Pictorially, this may be represented as
| |
| | |
| [[File:AKLT GroundState.png]]
| |
| | |
| Here the solid points represent spin 1/2s which are put into singlet states. The lines connecting the spin 1/2s are the valence bonds indicating the pattern of singlets. The ovals are projection operators which "tie" together two spin 1/2s into a single spin 1, projecting out the spin 0 or singlet subspace and keeping only the spin 1 or triplet subspace. The symbols +, 0 and - label the standard spin 1 basis states (eigenstates of the <math>S^z</math> operator).<ref name="Schollwoeck:2011" />
| |
| | |
| ===Spin 1/2 Edge States===
| |
| | |
| For the case of spins arranged in a ring (periodic boundary conditions) the AKLT construction yields a unique ground state. But for the case of an open chain, the first and
| |
| last spin 1 have only a single neighbor, leaving one of their constituent spin 1/2s unpaired. As a result, the ends of the chain behave like free spin 1/2 moments even though
| |
| the system consists of spin 1s only.
| |
| | |
| The spin 1/2 edge states of the AKLT chain can be observed in a few different ways. For short chains, the edge states mix into a singlet or a triplet giving either a unique ground state or a three-fold multiplet of ground states. For longer chains, the edge states decouple exponentially quickly as a function of chain length leading to a ground state manifold that is four-fold degenerate.<ref name="Kennedy:1990" /> By using a numerical method such as [[DMRG]] to measure the local magnetization along the chain, it is also possible to see the edge states directly and to show that they can be removed by placing actual spin 1/2s at the ends.<ref name="White:1993" /> It has even proved possible to detect the spin 1/2 edge states in measurements of a quasi-1D magnetic compound containing a small amount of impurities whose role is to break the chains into finite segments.<ref name="Hagiwara:1990" />
| |
| | |
| ===Matrix Product State Representation===
| |
| | |
| The simplicity of the AKLT ground state allows it to be represented in compact form as a [[matrix product state]].
| |
| This is a wavefunction of the form
| |
| | |
| <math>|\Psi\rangle = \sum_{\{s\}} \text{Tr}[A^{s_1} A^{s_2} \ldots A^{s_N}] |s_1 s_2 \ldots s_N\rangle</math>.
| |
| | |
| Here the As are a set of 3 matrices labeled by <math>s_j</math> and the trace comes from assuming periodic boundary conditions.
| |
| | |
| The AKLT ground state wavefunction corresponds to the choice:<ref name="Schollwoeck:2011" />
| |
| | |
| <math>A^{+} = \sqrt{\frac{2}{3}}\ \sigma^{+} </math>
| |
| | |
| <math>A^{0} = \frac{-1}{\sqrt{3}}\ \sigma^{z}</math>
| |
| | |
| <math>A^{-} = -\sqrt{\frac{2}{3}}\ \sigma^{-}</math>
| |
| | |
| where the <math>\sigma\text{'s}</math> are [[Pauli matrices]].
| |
| | |
| == Generalizations and Extensions ==
| |
| | |
| The AKLT model has been solved on lattices of higher dimension,<ref name="Affleck:1987"/><ref name="Wei:2011"/> even in [[quasicrystals]] {{citation needed|date=March 2013}}.
| |
| The model has also been constructed for higher Lie algebras including [[SU(n)]],<ref name="Greiter:2007a"/><ref name="Greiter:2007b"/> [[SO(n)]],<ref name="Tu:2008"/> [[Sp(n)]] <ref name="Schuricht:2008"/> and extended to the [[quantum groups]] SUq(n).<ref name="SantosParaan2012"/>
| |
| | |
| == References ==
| |
| {{Reflist|refs=
| |
| <ref name="Pollmann:2012">
| |
| {{cite journal
| |
| |last1=Pollmann |first1=F.
| |
| |last2=Berg |first2=E.
| |
| |last3=Turner |first3=Ari M.
| |
| |last4=Oshikawa |first4=Masaki
| |
| |year=2012
| |
| |title=Symmetry protection of topological phases in one-dimensional quantum spin systems
| |
| |journal=Phys. Rev. B
| |
| |volume=85 |issue=7 |pages=075125
| |
| |doi=10.1103/PhysRevB.85.075125
| |
| |bibcode = 2012PhRvB..85g5125P |arxiv = 0909.4059 }}</ref>
| |
| <ref name="Affleck:1987"> | |
| {{cite journal
| |
| |last1=Affleck |first1=Ian
| |
| |last2=Kennedy |first2=Tom
| |
| |last3=Lieb |first3=Elliott H.
| |
| |last4=Tasaki |first4=Hal
| |
| |year=1987
| |
| |title=Rigorous results on valence-bond ground states in antiferromagnets
| |
| |journal=[[Physical Review Letters]]
| |
| |volume=59 |issue=7 |pages=799–802
| |
| |bibcode=1987PhRvL..59..799A
| |
| |doi=10.1103/PhysRevLett.59.799
| |
| |pmid=10035874
| |
| }}</ref>
| |
| <ref name="Schollwoeck:2011">
| |
| {{cite journal
| |
| |last1=Schollwöck |first1=Ulrich
| |
| |year=2011
| |
| |title=The density-matrix renormalization group in the age of matrix product states
| |
| |journal=[[Annals of Physics]]
| |
| |volume=326 |pages=96–192
| |
| |arxiv=1008.3477
| |
| |bibcode=2011AnPhy.326...96S
| |
| |doi=10.1016/j.aop.2010.09.012
| |
| }}</ref>
| |
| <ref name="Kennedy:1990">
| |
| {{cite journal
| |
| |last1=Kennedy |first1=Tom
| |
| |year=1990
| |
| |title=Exact diagonalisations of open spin-1 chains
| |
| |journal=J. Phys. Condens. Matter
| |
| |volume=2 |issue=26 |pages=5737
| |
| |doi=10.1088/0953-8984/2/26/010
| |
| |bibcode = 1990JPCM....2.5737K }}</ref>
| |
| <ref name="White:1993">
| |
| {{cite journal
| |
| |last1=White |first1=Steven
| |
| |last2=Huse |first2=David
| |
| |year=1993
| |
| |title=Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S=1 Heisenberg chain
| |
| |journal=Phys. Rev. B
| |
| |volume=48 |issue=6 |pages=3844–3852
| |
| |doi=10.1103/PhysRevB.48.3844
| |
| |bibcode = 1993PhRvB..48.3844W }}</ref>
| |
| <ref name="Hagiwara:1990">
| |
| {{cite journal
| |
| |last1=Hagiwara |first1=M.
| |
| |last2=Katsumata |first2=K.
| |
| |last3=Affleck |first3=Ian
| |
| |last4=Halperin |first4=B.I.
| |
| |last5=Renard |first5=J.P.
| |
| |year=1990
| |
| |title=Observation of S=1/2 degrees of freedom in an S=1 linear-chain Heisenberg antiferromagnet
| |
| |journal=Phys. Rev. Lett.
| |
| |volume=65 |issue=25 |pages=3181–3184
| |
| |doi=10.1103/PhysRevLett.65.3181
| |
| |bibcode = 1990PhRvL..65.3181H }}</ref>
| |
| <ref name="Wei:2011">
| |
| {{cite journal
| |
| |last1=Wei |first1=T.-C.
| |
| |last2=Affleck |first2=I.
| |
| |last3=Raussendorf |first3=R.
| |
| |year=2011
| |
| |title=Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource
| |
| |journal=Phys. Rev. Lett.
| |
| |volume=106 |issue=7 |pages=070501
| |
| |doi=10.1103/PhysRevLett.106.070501
| |
| |arxiv = 1009.2840 |bibcode = 2011PhRvL.106g0501W }}</ref>
| |
| | |
| <ref name="SantosParaan2012">
| |
| {{cite journal
| |
| |last1=Santos|first1=R. A.
| |
| |last2=Paraan|first2=F. N. C.
| |
| |last3=Korepin|first3=V. E.
| |
| |last4=Klümper|first4=A.
| |
| |title=Entanglement spectra of the q-deformed Affleck-Kennedy-Lieb-Tasaki model and matrix product states
| |
| |journal=EPL (Europhysics Letters)
| |
| |volume=98
| |
| |issue=3
| |
| |year=2012
| |
| |pages=37005
| |
| |issn=0295-5075
| |
| |doi=10.1209/0295-5075/98/37005|arxiv = 1112.0517 |bibcode = 2012EL.....9837005S }}</ref>
| |
| | |
| <ref name="Greiter:2007a">
| |
| {{cite journal
| |
| |last1=Greiter |first1=Martin
| |
| |last2=Rachel |first2=Stephan
| |
| |last3=Schuricht |first3=Dirk
| |
| |year=2007
| |
| |title=Exact results for SU(3) spin chains: Trimer states, valence bond solids, and their parent Hamiltonians
| |
| |journal=Phys. Rev. B
| |
| |volume=75 |issue=6 |pages=060401(R)
| |
| |doi=10.1103/PhysRevB.75.060401
| |
| |arxiv = cond-mat/0701354 |bibcode = 2007PhRvB..75f0401G }}</ref>
| |
| | |
| <ref name="Greiter:2007b">
| |
| {{cite journal
| |
| |last1=Greiter |first1=Martin
| |
| |last2=Rachel |first2=Stephan
| |
| |year=2007
| |
| |title=Valence bond solids for SU(n) spin chains: Exact models, spinon confinement, and the Haldane gap
| |
| |journal=Phys. Rev. B
| |
| |volume=75 |issue=18 |pages=184441
| |
| |doi=10.1103/PhysRevB.75.184441
| |
| |arxiv = cond-mat/0702443 |bibcode = 2007PhRvB..75r4441G }}</ref>
| |
| | |
| <ref name="Tu:2008">
| |
| {{cite journal
| |
| |last1=Tu |first1=Hong-Hao
| |
| |last2=Zhang |first2=Guang-Ming
| |
| |last3=Xiang |first3=Tao
| |
| |year=2008
| |
| |title=Class of exactly solvable SO(n) symmetric spin chains with matrix product ground states
| |
| |journal=Phys. Rev. B
| |
| |volume=78 |issue=9 |pages=094404
| |
| |doi=10.1103/PhysRevB.78.094404
| |
| |arxiv = 0806.1839 |bibcode = 2008PhRvB..78i4404T }}</ref>
| |
| | |
| <ref name="Schuricht:2008">
| |
| {{cite journal
| |
| |last1=Schuricht |first1=Dirk
| |
| |last2=Rachel |first2=Stephan
| |
| |year=2008
| |
| |title=Valence bond solid states with symplectic symmetry
| |
| |journal=Phys. Rev. B
| |
| |volume=78 |issue=1 |pages=014430
| |
| |doi=10.1103/PhysRevB.78.014430
| |
| |arxiv = 0805.3918 |bibcode = 2008PhRvB..78a4430S }}</ref>
| |
| | |
| }}
| |
| | |
| [[Category:Spin models]]
| |
| [[Category:Statistical mechanics]]
| |
| [[Category:Quantum magnetism]]
| |
| [[Category:Lattice models]]
| |
Cardsharp® is a great chef's knife The blade is sharp rated 'very good' by CATRA long and skinny adequate to reduce bread and also very useful for quickly slicing of extremely thin slices of greens such as tomatoes, cucumber, garlic etc. Great for camping and expedition use Cardsharp® is an essential out of doors companion/survival tool. It is important to keep hydrated in the course of a picnic with all of the frisbee throwing or soccer tossing. Provide water, sodas, fruit beverages, beer, wine, coffee, tea or everything else you like to drink. Usually be safe if you are ingesting alcohol. Have a selected driver and adhere to park guidelines.
Total, this is not the nicest Spyderco I've witnessed-visually talking, of course. The blade is as sharp as you would count on from a Spyderco, and it will certainly complete the job, what ever that occupation might be. The knife itself isn't as appealing as other folks I've noticed from Spyderco, but that is a matter of opinion, and probably not an situation for you. I might suggest this knife to anyone who would like a relaxed carry to use for generating slim slices or deep cuts. Folders are truly successful as effectively as expense successful cases for your businesses. That is why firm delivers you least expensive pocket folder printing answers worldwide.
If you are an outdoorsman, there are a number of equipment you will want to have with you in your outside package. The trick is to uncover the right equipment that will enable you to travel light while not lacking out on any essential items of products that will make your tenting, looking, or climbing experience far more enjoyable. Study on to discover out more Best Folding Knife Ever Made! go through more I haven't experienced ample time to truly comment the sturdiness, but this is a well designed blade that feels excellent even though in use. I adore the vertical belt loop as properly. I regret getting rid of my old tactical, but this makes that lose a minor considerably less agonizing.
Im sorry, he stated. I didnt recognise you without the He twirled his fingers spherical his head to signify the veil. She was hunting straight at him, and he had no idea what that appear was intended to imply. Which is absurd, he additional speedily. I can see your confront so I dont know who you are. Whats your identify? He lay even now, striving not to consider about the darkness. All his daily life hed slept with a lamp burning, but there was no lamp in this area and he hadnt sent for one. Was it?
Later, Bassos mother determined she should have given her also significantly (comprehensible, in the situation). Also, the girl might have had a weak heart or some similar problem. It was unfortunate, of course, but just 1 of individuals things. Bassos mother paid out for a coffin and a plot in the public cemetery. It was, she felt, the least she could do. As the door closed, Aelius leaned forward across his desk. The boy was viewing him, to see what would come about subsequent. He wasnt afraid, he wasnt smug. That on your own was sufficient to verify that he was who he stated he was. What variety of lewd suggestion? Aelius requested.
Difficult, but beautiful, figured green linen Micarta surrounds this hybrid beast with a bead blasted four.five" blade and Karambit type finger ring. The Micarta sheath has a customized created "S" shaped serpentine clip for idea up, or suggestion down have. You can also get rid of the clip, and location it on the other side for boot, belt, remaining or proper handed undercover transport. This is the huge The Best Pocket Knife 2013 brother to my smaller "Neckolace" designed knife which CRKT is at present making. Completely handmade. $450. Nowadays , do not get oneselfCheck price tag prior to the Price tag Up. Buck Elk Manage Folding Hunter Knife with B&C Medallion You will not regret if verify price.