Investigation of the 1+1 dimensional Thirring model using the method - PowerPoint PPT Presentation

Investigation of the 1+1 dimensional Thirring model using the method of matrix product states C.-J. David Lin National Chiao-Tung University, Taiwan In collaboration with Mari Carmen Banuls (MPQ Munich), Krzysztof Cichy (Adam Mickiewicz Univ.), Ying-Jer Kao (National Taiwan Univ.), Yu-Ping Lin (Univ. of Colorado, Boulder), David T.-L. Tan (National Chaio-Tung Univ.) Lattice 2018 East Lansing, MI 26/07/2018

Outline • Preliminaries • Lattice simulations, the MPS and DMRG • Phase structure of the Thirring model • Remarks and outlook

Preliminaries

Motivation • Tensor network for lattice field theory • Topological phase transitions • Real-time dynamics (long-term goal)

The 1+1 dimensional Thirring model and its duality to the sine-Gordon model � ¯ � ψψ − g � � 2 � ψ , ¯ ψ i γ µ ∂ µ ψ − m 0 ¯ ¯ d 2 x � � ψ = ψγ µ ψ S Th 2 strong-weak duality g ↔ κ � 1 � � 2 ∂ µ φ ( x ) ∂ µ φ ( x ) + α 0 d 2 x S SG [ φ ] = κ 2 cos ( κφ ( x )) � 1 � → 1 � φ → φ / κ , and κ 2 = t 2 ∂ µ φ ( x ) ∂ µ φ ( x ) + α 0 cos ( φ ( x )) d 2 x − − − − − − − − − − − − t Works in the zero-charge sector

Dualities and phase structure Thirring sine-Gordon XY 4 π 2 T g − π K − π t The K-T phase transition at T ∼ K π / 2 in the XY model. g ⇠ � π / 2, Coleman’s instability point The phase boundary at t ∼ 8 π in the sine-Gordon theory. The cosine term becomes relevant or irrelevant. Thirring sine-Gordon 1 ¯ � µ 2 ⇡ ✏ µ ν @ ν � Λ ¯ ⇡ cos �

RG flows of the Thirring model dµ = − 64 π m 2 β g ≡ µ dg Λ 2 , dµ = − 2( g + π 2 ) 256 π 3 β m ≡ µdm ( g + π ) 2 Λ 2 m 3 . m − g + π Massless Thirring model is a conformal field theory mass relevant m mass irrelevant g = � π 2 , Coleman’s instability point g

Lattice simulations, the MPS and DMRG


 
 Operator formalism and the Hamiltonian Operator formaliam of the Thirring model Hamiltonian 
 • C.R. Hagen, 1967 " # ◆ − 1 � ¯ � ¯ ψψ + g � 2 � g ✓ 1 + 2 g Z � 2 � i ¯ ψγ 1 ∂ 1 ψ + m 0 ¯ ψγ 0 ψ ψγ 1 ψ H Th = dx 4 4 π Staggering, J-W transformation ( ): j ± iS y S ± • j = S x j J. Kogut and L. Susskind, 1975; A. Luther, 1976 N − 2 N − 1 N − 1  � 1 ✓ n + 1 ◆ ✓ n + 1 ◆ ✓ n +1 + 1 ◆ � ¯ X X ( � 1) n X S + n +1 + S + S z S z S z � � H XXZ = ν ( g ) n S − n +1 S − + a ˜ m 0 + ∆ ( g ) n 2 2 2 2 n n n 2 γ m 0 = m 0 ν ( g ) , ∆ ( g ) = cos ( γ ) , with γ = π � g ν ( g ) = π sin( γ ) , ˜ 2 projected to a sector of total spin ! 2 N − 1 H (penalty) ¯ = ¯ X S z H XXZ + λ n � S target XXZ JW-trans of the total fermion number, n =0 Bosonise to topological index in the SG theory.


 
 
 
 
 
 Simulation details Matrix product operator for the Hamiltonian 
 • 2 S − 2 λ S z ∆ S z β n S z + α 1 2 × 2 2 S + − 1 1 2 × 2 − 1 0 1 0 0 0 0 0 S − B C S + 0 0 0 0 0 W [ n ] = B C B C S z 0 0 0 1 0 B C B C S z 0 0 0 0 0 @ A 0 0 0 0 0 1 2 × 2 ! 4 + S 2 1 + ∆ β n = ∆ + ( − 1) n ˜ target m 0 a − 2 λ S target , α = λ N 4 Choices of parameters • Twenty values of , ranging from -0.9 to 1.0 ∆ ( g ) (run 1) m 0 a = 0 . 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 ˜ (run 2) m 0 a = 0 . 005 , 0 . 01 , 0 . 02 , 0 . 03 , 0 . 04 , 0 . 06 , 0 . 08 , 0 . 13 , 0 . 16 ˜ Bond dimension D = 50 , 100 , 200 , 300 , 400 , 500 , 600 System size N = 400 , 600 , 800 , 1000

6 Convergence of DMRG Start from random tensors at D=50, then go up in D • ⇠ ∆ ( g ) > DMRG converges fast at and • ⇠ � 0 . 7 m 0 a 6 = 0 ˜

Results for the phase structure

Entanglement entropy Calabrese-Cardy scaling and the central charge  N ⌘� S N ( n ) = c ⇣ π n 6 ln π sin + k , N 1 . 4 1 . 6 1 . 3 1 . 2 1 . 4 S N ( n ) S N ( n ) 1 . 1 1 . 2 1 . 0 D = 100 D = 100 1 . 0 0 . 9 D = 200 D = 200 0 . 8 D = 400 D = 400 0 . 8 ∆ ( g ) = − 0 . 88 , ˜ m 0 a = 0 . 0 ∆ ( g ) = 0 . 0 , ˜ m 0 a = 0 . 0 D = 600 D = 600 0 . 7 0 200 400 600 800 1000 0 200 400 600 800 1000 site n site n ∼ − Calabrese-Cardy scaling observed at all values of for m 0 a = 0 d ∆ ( g ) = 0 ˜

Entanglement entropy Calabrese-Cardy scaling and the central charge  N ⌘� S N ( n ) = c ⇣ π n 6 ln π sin + k , N 0 . 56 1 . 6 0 . 54 0 . 52 1 . 4 0 . 50 S N ( n ) S N ( n ) 1 . 2 0 . 48 1 . 0 0 . 46 D = 100 D = 100 D = 200 D = 200 0 . 8 0 . 44 D = 400 D = 400 ∆ ( g ) = − 0 . 88 , ˜ m 0 a = 0 . 2 ∆ ( g ) = 0 . 0 , ˜ m 0 a = 0 . 2 D = 600 D = 600 0 . 6 0 . 42 0 200 400 600 800 1000 0 200 400 600 800 1000 site n site n ⇠ � Calabrese-Cardy scaling observed at for m 0 a 6 = 0 ˜ ∆ ( g ) < ⇠ � 0 . 7

Entanglement entropy Calabrese-Cardy scaling and the central charge  N ⌘� S N ( n ) = c ⇣ π n 6 ln π sin + k , N 1 . 6 1 . 4 S N ( n ) 1 . 2 1 . 0 D = 100 D = 200 0 . 8 D = 400 ∆ ( g ) = − 0 . 88 , ˜ m 0 a = 0 . 2 D = 600 0 . 6 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 � N π sin( π n � 6 ln N ) Central charge is unity in the critical phase

Soliton correlators S. Mandelstam, 1975 ↵ ( x ) ↵ ( y ) = ⌥ i | 2 ⇡ ( x � y ) | − 1 | cµ ( x � y ) | − � 2 g 2 / (2 ⇡ ) 3 † Z y ⇢ � � ( ⇠ ) ⌥ 1 α = ± d ⇠ ˙ � 2 ⇡ i � − 1 2 i � [ � ( y ) � � ( x )] + O ( x � y ) 2 ⇥ : exp : x (35) Vertex operators Soliton operators connecting vortex and anti-vortex Power-law in the critical phase Power-law Exponential-law in the gapped phase Jordan-Wigner m e i ⇡ P n − 1 j = m +1 S z S + j S − transformation n

Soliton correlators + ( r ) ψ + (0) i , ¯ G ( r ) = h ψ † G ( r ) = G ( r ) /G (0) 0 0 m 0 a = 0 . 0 , D = 600 ˜ m 0 a = 0 . 2 , D = 600 ˜ − 2 − 2 − 4 G G ln ¯ ln ¯ − 4 ∆ ( g ) =-0.86 ∆ ( g ) =-0.86 ∆ ( g ) =-0.78 ∆ ( g ) =-0.78 − 6 ∆ ( g ) =-0.74 ∆ ( g ) =-0.74 − 6 ∆ ( g ) =-0.72 ∆ ( g ) =-0.72 − 8 ∆ ( g ) =-0.68 ∆ ( g ) =-0.68 ∆ ( g ) =-0.62 ∆ ( g ) =-0.62 − 8 − 10 0 1 2 3 4 5 0 1 2 3 4 5 ln ( r/a ) ln ( r/a ) Evidence for BKT phase transition

Chiral condensate � � � = 1 � � � h ¯ X � � χ = a ˆ ψψ i ( � 1) n S z � � N n � � � � n m 0 a = 0 . 0 ˜ Extrapolated to infinite D and N m 0 a = 0 . 1 ˜ 0 . 4 m 0 a = 0 . 2 ˜ m 0 a = 0 . 3 ˜ 0 . 3 m 0 a = 0 . 4 ˜ χ ˆ 0 . 2 0 . 1 0 . 0 − 1 . 00 − 0 . 75 − 0 . 50 − 0 . 25 0 . 00 0 . 25 0 . 50 0 . 75 1 . 00 ∆ ( g ) Zero-mass results reproduced using uMPS

Chiral condensate � � � = 1 � � � h ¯ X � � χ = a ˆ ψψ i ( � 1) n S z � � N n � � � � n Extrapolated to infinite D and N chiral symmetry is not spontaneously broken Curvature at small mass in the gapped phase

Mass gap M − 1 � H e ff [ M ] = Π M − 1 . . . Π 0 H Π 0 . . . Π M − 1 = H − E k | Ψ k ⟩⟨ Ψ k | . k =0 ( Π m ) k H k e ff e ff + � M − 1 H k e ff [M]= m =0 E m × | Ψ m ⟩ k e ff m 0 a = 0 . 0 ˜ m 0 a = 0 . 1 ˜ 0 . 8 m 0 a = 0 . 2 ˜ m 0 a = 0 . 3 ˜ 0 . 6 m 0 a = 0 . 4 ˜ E 1 − E 0 0 . 4 0 . 2 0 . 0 − 0 . 8 − 0 . 6 − 0 . 4 − 0 . 2 0 . 0 0 . 2 ∆ ( g )

Phase structure of the Thirring model dµ = − 64 π m 2 β g ≡ µ dg Λ 2 , dµ = − 2( g + π 2 ) 256 π 3 β m ≡ µdm ( g + π ) 2 Λ 2 m 3 . m − g + π Massless Thirring model is a conformal field theory am 0 gapped critical g 0 = 0, continuum limit g 0 = g c , Coleman’s instability point g 0


 Conclusion and outlook Evidence for BKT phase transition found using MPS 
 • Chiral symmetry is not spontaneously broken Current work for more detailed probe of the phase structure: 
 • More simulations at small fermion mass Eigenvalue spectrum of the transfer matrix Future projects: • Chemical potential Real-time evolution with a quench