Ising Model on Hyperbolic Lattices: toward Transverse Field Ising Model under Hyperbolic Deformation

Transcription

1 Ising Model on Hyperbolic Lattices: toward Transverse Field Ising Model under Hyperbolic Deformation T. Nishino, T. Iharagi (Kobe Universty) A. Gendiar (Slovak Academy of Sciences) H. Ueda (Osaka University) b a W c e d Hyperbolic Lattice

2 We are grateful.

3 !?

4 I m happy to meet you.

5 Ising Model on Hyperbolic Lattices: toward Transverse Field Ising Model under Hyperbolic Deformation T. Nishino, T. Iharagi (Kobe Universty) A. Gendiar (Slovak Academy of Sciences) H. Ueda (Osaka University) b a W c e d

6 There are various research field in hyperbolic space.

7 Before that, we start from the Square Lattice. Schroedinger s Dog

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9 Tutorials: from 2d Classical to 1d Quantum (Common Sense?!) Square Lattice (Classical) Anisotropic Ising Model H Ising {σ} = J 1 Σ i,j σ i,j σ i,j+1 + J 2 Σ i,j σ i,j σ i+1,j { } Indices: (i, j) = (horizontal, vertical) Ising Variable: σ i,j = ±1 Vertical Interaction: J 1 Σ i,j σ i,j σ i,j+1 Horizontal Interaction: J 2 Σ i,j σ i,j σ i+1,j Partition Function (K 1 = βj 1 and K 2 = βj 2 ) Z = {σ} exp [ ] βh Ising {σ} = ] exp [K 1 Σ i,j σ i,j σ i,j+1 + K 2 Σ i,j σ i,j σ i+1,j {σ} We express Z by means of Transfer Matrix.

10 ChessBoard Break up of the Square Lattice Consider the M 2N (horizonta vertical) Lattice Impose Periodic Boundary Condition to the vertical direction Z =Tr(T e T o ) N Transfer Matrices of the 2D Ising Model are defined as follows. T e = W 0 W 2 W 4 W M 2 T o = W 1 W 3 W 5 W M 1 Local Plaquette Weight W i = W (σ iσ i+1 σ i σ i+1 ) = W (σ i,j+1 σ i+1,j+1 σ i,j σ i+1,j ) W ( σ aσ b σ ) [ a σ b = exp K1 σ aσ a + K 1 σ bσ b + K 2 σ aσ b ] + K 2 σ a σ b (W can be interpreted as a vertex weight of 16vertex model.)

11 Local Weight W = e 2K 1 +2K e 2K 1 +2K 2 1 e 2K 1 2K 2 e 2K 1 2K e 2K 1 2K 2 e 2K 1 2K 2 1 e 2K 1 +2K e 2K 1 +2K 2 Critical Point Critical Point (From Duality) sinh 2K 1 sinh 2K 2 =1(From Duality) Anisotropic Case: sinh 2K 1 e 2K 1 /2 K 1 >> 1 >> K 2 sinh 2K 2 2K 2 W e 2K 1 Critical Point: K 2 e 2K K 2 e 2K 2 e 2K 2 0 e 2K 2 1 2K 2 0 e 2K 2 e 2K K 2 e 2K 2 0 e 2K 2 e 2K 2 1+2K 2 We ll see that this weight is equivalent to local (imaginary) time evolution in the transverse field Ising model. Transverse Field Ising Model Jump to Quantum 1D >>>>>

12 Transverse Field (Quantum) Ising Model (TFI) Ĥ TFI = J i ˆσ z i ˆσ z i+1 Γ i ˆσ x i = i ĥ i ĥ i = J ˆσ z i ˆσ z i+1 Γ 2 (ˆσ x i +ˆσ x i+1) Partition Function Z = Tre βĥtfi =Tr ( e βĥtfi /N ) N Tr ( e βĥe /N e βĥo /N ) N =Tr(Te T o ) N TrotterSuzuki Decomposition Transfer Matrices T e = e (β/n )(ĥ0 +ĥ2 +ĥ4 + ) = i=even T o = e (β/n )(ĥ1 +ĥ3 +ĥ5 + ) = i=odd (β/n ) ĥi e (β/n ) ĥi e

13 Locat Imaginary Time Evolution when N is very large (β/n ) ĥi e comparison W e 2K 1 1 β N ĥi = aws the relations 1+ βj N βγ 2N βγ βγ 2N 1 βj βγ 2N 0 βγ N 0 2N βγ 2N 2N 0 1 βj N βγ 0 βγ 2N 2N 1+2K 2 e 2K 2 e 2K 2 0 e 2K 2 1 2K 2 0 e 2K 2 e 2K K 2 e 2K 2 0 e 2K 2 e 2K 2 1+2K 2 1+ βj N with Classical 2D Ising Model with Anisotropy draws the correspondence between parameters. 2K 2 = βj N e 2K 1 = βγ 2N Critical Condition K 2 e 2K 1 1 draws the Quantum Criticality J = Γ.

14 Remark: a 1D Quantum Model can be (?!) derived as the anisotropic limit of a (or several) classical 2D Model.... if one allows negative or complex weights. ** Symmetry in 2D Classical models is NOT always high. Thus a quantum (?) model derived as the anisotropic limit of a certain 2D classical model might not be time reversal. Some times one gets nonhermite Hamiltonian.

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16 Classical Ising Model on Hyperbolic Lattices. b a W c e d

17 Typical Example: Ising Model on the (5,4) Lattice H = J Σ σi σj ( i and j are nearest neighbors ) ** Translationally Invariant ** We consider bulk properties. b a W c e d (... Boundary field shows different behavior...) Analytic Solution: not known yet. MPS formalism: can be applied Corner Transfer Matrix formalism (Baxter): efficient! CTMRG =?= DMRG =?= TNS =?= PEPS =?= TPVA

18 Recursive Structure of the (5,4) lattice in general, (q,p=even) lattice has the same property. P 1 2 W 4 3 2' C P 1' 3' 4' Half `row transfer matrix P (after taking configuration sum for spins inside, leaving those on the boundary) P 1 C W P a e b W d c 2' 3' 4' C P Corner transfer matrix C

19 (See P 1 C W P 2' 3' 4' C P DMRG? It is my field of research! P 1 2 W 4 3 2' C P 1' 3' 4' P C W P P C P C W P

20 The MeanField like 2nd order transition is observed < i > Spontaneous Magnetization T 0.40 < i +1 i > Internal Energy T This behavior is similar to what is observed in tensor network treatment of 2D TFI by Sandvik.

21 Transition point is NOT Critical, although the transition is Second order. Fig. 2. ξ m=50 m= k B T/J Correlation Length Correlation Length ξ with respect to the temperature ColumntoColumn Transfer Matrix P*P Upper or Lower half of the system can be identified as Wave Function of the (?) corresponding 1d quantum system. S m=50 m= k B T/J Entanglement Entropy Fig. 3. Entanglement entropy of the MPS in Eq. (2.12).

22 Generalizations: NNN Ising model (J1J2 Ising) H = J σ i σ j + J 2 ij =NN ik =NNN σ i σ k Paramagnetic k B T / J Ferromagnetic 2 nd order 1 st order Tricritical (??) Point! 0.5 T 0 (! c ) ! = J 2 / J 1

23 Generalizations: qstate Clock Models W (θ 1 θ 2 θ 3 θ 4 θ 5 )= { ( )} 5 J cos θi θ exp i+1 2 k B T i=1 H = J ij cos ( θ i θ j ), FIG. 4: The absolute value of the internal energy E (N). The open circles denote the jump in the case N = 3. (N) T Sch N=7 0.4 N=8 0.3 N=9 W N= W W W N=13! 1 0.1! 2 N=20 W! 5 W W N=30!! W W N 2 W W FIG. 6: The Schottky peak position T (N) Sch versus 1/N 2. C (N) (N) / C max C (N) T (N) C (T Sch ) C (T 0 (N) ) (N) T / T 0 FIG. 5: The rescaled specific heat C (N) /C (N) max versus the rescaled temperature T/T (N) 0. The inset shows a typical ex q=3: First IV. Order CONCLUSIONS Transition q>4: MeanField 2nd Order We have studied the Nstate clock models on the pentagonal lattice, which is a typical example of the hyperbolic lattices. The phase transition at the center of the system is observed by use of the CTMRG method. From the Further critical exponent Extensions: β = 1 2 for the spontaneous magnetization and the jump in the specific heat, we conclude thatdimer, the phase transition hard polygon, for N = 2 and Nvector, 4 is meanfield like, provided that the ferromagnetic boundary conditions Potts, are imposed. percolation, The Hausdorff... dimension, which is infinite for the hyperbolic lattices, is essential in the observed critical behavior. We conjecture that the phase as you like. transition deep inside the system is also present for systems with free boundary conditions. In the case when N = 3, where the system is equivalent to the 3state Potts model, we observed the firstorder phase transition. Since the qstate Potts model tends

24 Remark: Phase transitions on the Hyperbolic Lattice are meanfield like in most of the cases. Upper half, or Lower half of the system can be identified as a bra or ket states of some (??) quantum system.... how does the quantum system look like???

25 Corresponding Quantum 1D systems??? Missing Link: It is NOT easy to introduce anisotropy to 2D Hyperbolic Lattice.... return to Intuition, starting from continuous space... Assume the presence of continuous field in the 1+1 hyperbolic space.... if one discretize the space, one might obtain nonuniform Hamiltonian... (Hyperbolic Deformation) Tim e! " H = Σj cosh( jλ ) h j,j+1 Is the model well defined? Groundstate of such a Hamiltonian unifrom??? Poincare Disk

26 ) ] [ ] H c (λ) = J j cosh [ jλ ] σ z j σ z j+1 Γ j cosh [( j 1 2) λ ] σ x j [( ) ] n be interpreted as the one p λ=1, L=32, m= Γ=0.96 < Γσ x j > Γ= j Fig. 2. Onsite transverse interaction Γσ x j.

27 < σ z 0 > m=16 λ=1 λ=1/2 λ=1/4 λ=1/8 λ=1/ Γ jump is plotted in Fig. 5. The eighth power of the jump is proportional to λ. This dependence is consistent with the Ising universality. The length scale ξ 1/λ introduces an effective deviation to the transverse field of the amount Γ ξ ν = 1 ξ λ (3.1) from the criticality of the uniform TFI model, where Spontaneous magnetization ν = 1 is the critical exponent for the correlation length. jumps at the transition point. The observed jump in the spontaneous magnetization is therefore proportional to ( Γ) 1/8 λ 1/ st order Transition (?????) m=16 (< σ z 0 >) hidden effect of critical behavior λ

28 Enta 0.1 Entanglement Entropy Γ λ=1 λ=1/2 0.5 λ=1/4 Fig. 7. Entanglement λ=1/8 entropy with respect to λ. λ=1/ m=16. Bipartite entanglement entropy measured at the center of 16 system. 0.2 The fitting to the difference ERDF (B)(A) OPgives R&D, Projec lence lence for for Quantum Quantum Γ the esti Technologi cosh 0.1 λ in the small λ limit. CE This CE QUTE value QUTE SAV. is about the SAV. j 0 of log The 0.9 result 0.95 suggests that 1.1 differen entropy [(A)+(B)]/2 [(A)+(B)]/2 at at Γ = 1 is expressed as Appendix: Γ Duality Relati The undeformed TFI model 1 ordered:(a) log 1 λ Appendix: Duality R lement entropy with respect entanglement to λ. entropy written between by the new set of Paul Quantum ordered and disord states is a, constant, if the (3.2) the transformed correlation The undeformed Hamiltonian lengths of by the duality disordered:(b) relation. 1) TFI them h 2 Let (A)+(B) states are the same. 0.8 original We conjecture Hamiltonian, (B)(A) Pauli by operators the duality that wide range of λ. This dependence coincides with the τj x, where the relation. τ y j, and differ pa τ 1) j z. difference 0.6 σj x, σy j, and σz j can be expresse hat the leading (B)(A) termgives (1/2) of the the log entanglement estimate 2 captures changed. the boundary Pauli For operators finite condition, λ, τthe j x entropy, τ y j tran but, σj x ressed as (c/6) log W, where W is system 0.4 size, σy j, and σj xσz j can be ex λ limit. This value detail is about is not theclarified half Eq. yet. (A 2) has the form where la = τj 1τ z j z heisresult the central suggests charge. that 24) difference This is because of mation function is shifted 1/λ is σj x = by ( 0.2 j 2 τ1 j 4. Conclusion and rtional y between to theordered length scale and ξ, disordered and the TFI crossover Discussion model at Γ = 1 in Fig. 3 sug σ y m=16 j = 0 if gsthe to the correlation class where lengths c =1/2. We of have theintroduced both essential for the bulk part 0.01 the hyperbolic 0.1 deformation 1 σ y of thl j = t Entanglement Entropy Entanglement Entropy means means such such as the as Trotter the Trotte dec relation σj z λ σz j+1 = τ j x. Substitut H c (λ) in Eq. (1.3), we obtain t Acknowledgement Acknowledgement This This work work was was partly partly suppo JSPS JSPS Fellows, Fellows, and and GrantinA cosh (C) (C) No. No A. G. A. ack G ERDF OP R&D, Project Q H c (λ) = J j the index byλ 1/4 in advance an. We conjecture that 1Dthe TFIdifference model. It is shown that inner part of the gr es the boundary condition, but the j 1 σj z = τ x state is uniform, as as was observed other systems l u, j

29 Missing Link: Classical (isotropic) Ising on the Hyperbolic Lattice shows 2nd order transition. Quantum Ising under Hyperbolic Deformation shows 1st order transition. One should construct the anisotropic limit. ξ m=50 m=5 Classical k B T/J Fig. 2. Correlation Length ξ with respect to the temperature S m=50 m=5 Classical k B T/J Fig. 3. Entanglement entropy of the MPS in Eq. (2.12).

30 Summary: Phase transitions on the Hyperbolic Lattice are meanfield like in most of the Classical models. Hyperbolic Deformation applied to 1D quantum models can be considered as the anisotropic limit of the classical model living in the hyperbolic space. Unlike the square lattice, it is not straight forward to find out the quantumclassical correspondence by use of the Trotter decomposition.

31 Extensions: Spherical Deformation Well, tomorrow morning I fly back to Japan, and do a Lecture in Kobe. After that I come to Madrid again during the next week (!) and talk about another deformation applied to 1D quantum system, the Spherical deformation, in a workshop at La Coruna. Ĥ S = t N 1 l=1 2 sin lπ N ( ) ĉ lĉl+1 +ĉ l+1ĉl Boundary effect on the bond energy disappears completely!

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33 Thank you! Phase transitions on the Hyperbolic Lattice are meanfield like in most of the Classical models. Hyperbolic Deformation applied to 1D quantum models can be considered as the anisotropic limit of the classical model living in the hyperbolic space. Unlike the square lattice, it is not straight forward to find out the quantumclassical correspondence by use of the Trotter decomposition.