Phase structure and real-time dynamics of the massive Thirring model - PowerPoint PPT Presentation

Phase structure and real-time dynamics of the massive Thirring model in 1+1 dimensions using tensor-network methods Phys. Rev. D 100 (2019) 094504 C.-J. David Lin National Chiao-Tung University, Taiwan with Mari Carmen Banuls (MPQ Munich), Krzysztof Cichy (Adam Mickiewicz Univ.), Hao-Ti Hung (National Taiwan Univ.), Ying-Jer Kao (National Taiwan Univ.), Yu-Ping Lin (Univ. of Colorado, Boulder), David T.-L. Tan (National Chaio-Tung Univ.) TNSAA 2019 Taipei 06/12/2019

LGT in the early days

Beginning of MC simulations for LGT

Success for simple quantities The BMW collaboration, science 322 (2008)

Success for less simple quantities G. A. Cowan (LHCb collaboration), arXiv:1708.08628.

Motivation for HEP Things that are challenging for Euclidean MC simulations ……. See talks by Kuhn and Nakamura Further examples: light-cone physics, inelastic scattering,…

Motivation for HEP Topology freezing Bazavov et al ., Phys. Rev. D 98 (2018) 074512

Feasibility (toy-model) studies for HEP

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 Picture from: K. Huang and J. Polonyi, 1991 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 t n a v e l e r r i s s a m

Beyond the SM, composite Higgs? Fermion favours ~1000 TeV ? Need large anomalous dim to suppress FCNC ? Searched up here ~2 TeV Higgs boson ~125 GeV The Higgs boson is light 12

The “conformal windows” Figure credit: F. Sannino 13


 
 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 XXZ ¯ X S z H sim = + λ n � S target ν ( g ) JW-trans of the total fermion number, n =0 Bosonise to topological index in the SG theory.


 
 
 
 
 
 
 Simulation details for the phase structure Matrix product operator for the Hamiltonian (bulk) 
 • 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 B C W [ n ] = 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 Simulation parameters • Twenty values of , ranging from -0.9 to 1.0 ∆ ( g ) Fourteen values of , ranging from 0 to 0.4 m 0 a ˜ Bond dimension D = 50 , 100 , 200 , 300 , 400 , 500 , 600 System size N = 400 , 600 , 800 , 1000

Convergence different convergence properties observed

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 ∼ − Scaling observed at for , and for all values of at ∆ ( g ) < m 0 a 6 = 0 ˜ m 0 a = 0 d ∆ ( g ) = 0 ˜ ⇠ � 0 . 7 In the critical phase, c = 1

Density-density correlators 1 S z ( n ) S z ( n + x ) � 1 JW trans C zz ( x ) = h ¯ ψψ ( x 0 + x ) ¯ X X X ψψ ( x 0 ) i conn � � � � � � ! S z ( n ) S z ( n + 1) N x N 0 n n n try fitting to zz ( x ) = β x α and C pow C pow − exp ( x ) = Bx η A x zz 1.1 pow fit α -1 C zz ma=0.005 fitted values of A pow-exp fit η C zz ma=0.02 ma=0.08 1.05 -1.5 ma=0.3 1 -2 the parameter α and η fo rs: N = 1000, ˜ m 0 a = 0 . 02. 0.95 -2.5 0.9 -3 0.85 -3.5 -4 0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 ∆ (g) ∆ (g) Evidence for a critical phase

Soliton (string) correlators 1 JW trans C string ( x ) = h ψ † ( x 0 + x ) ψ ( x 0 ) i X S + ( n ) S z ( n + 1) · · · S z ( n + x � 1) S − ( n + x ) � � � � � � ! N x n try fitting to string ( x ) = β x α + C ( x ) = Bx η A x + C and C pow − exp C pow string 0.7 pow fit α C string 0.00 ma=0.005 the parameter α and η fo pow-exp fit η fitted values of C C string ma=0.02 -0.20 0.6 rs: N = 1000, ˜ m 0 a = 0 . 02. ma=0.08 ma=0.3 -0.40 0.5 -0.60 -0.80 0.4 the string order -1.00 0.3 -1.20 -1.40 0.2 -1.60 0.1 -1.80 -2.00 0 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 ∆ (g) ∆ (g) Similar behaviour in A. Evidence for a critical phase

Chiral condensate � � � = 1 � � � h ¯ X � � χ = a ˆ ψψ i ( � 1) n S z � � N n � � � � n Extrapolated to infinite D and N Massive phase e s a h p e v i s s a M Evidence for criticality from other quantities Chiral condensate is not an order parameter

Phase structure of the Thirring model quench and real-time dynamic