Anisotropic Long Range Spin Systems Nicol` o Defenu Scuola - PowerPoint PPT Presentation

Anisotropic Long Range Spin Systems Nicol` o Defenu Scuola Internazionale Superiore di Studi Avanzati, Trieste (Italy). ndefenu@sissa.it July 26, 2016 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 1 / 21

Overview Long Range Interactions 1 Spin systems LR Spin Systems 2 Traditional results Controversy Effective Dimension The Anisotropic case 3 Dimensional Analysis Three regimes Effective dimension Critical exponents Nicol` o Defenu (SISSA) Functional RG July 26, 2016 2 / 21

Long range interacting systems. 1 r d + σ Nicol` o Defenu (SISSA) Functional RG July 26, 2016 3 / 21

Spin Systems. Why spin systems spin systems are the testbed of statistical mechanics. Various Monte Carlo (MC) and perturbative results available. Diverse interesting physical problems in a single formalism. Issues: Phase diagram for diverse interaction shapes. Description of different symmetry groups. Description of high order critical points. Nicol` o Defenu (SISSA) Functional RG July 26, 2016 4 / 21

Spin Systems Lattice Hamiltonian 1 H = − J � | i − j | d + σ S i S j 2 ij Nicol` o Defenu (SISSA) Functional RG July 26, 2016 5 / 21

Spin Systems Lattice Hamiltonian 1 H = − J � | i − j | d + σ S i S j 2 ij Mean Field Propagator � G ( q ) − 1 = J ( q ) = d d x J ( i − j ) e iq · ( i − j ) Nicol` o Defenu (SISSA) Functional RG July 26, 2016 5 / 21

Spin Systems Lattice Hamiltonian 1 H = − J � | i − j | d + σ S i S j 2 ij Mean Field Propagator � G ( q ) − 1 = J ( q ) = d d x J ( i − j ) e iq · ( i − j ) Leading momentum term q → 0 G − 1 ( q ) ∝ q σ lim if σ ≤ 2 q → 0 G − 1 ( q ) ∝ q 2 lim σ > 2 if Nicol` o Defenu (SISSA) Functional RG July 26, 2016 5 / 21

Long range interactions in d dimensions Traditional Results Three regimes: 0 < σ < d / 2 Mean field exponents ( η = 2 − σ and ν = σ − 1 ). d / 2 < σ < 2 Long range exponents ( η ≡ η ( σ ) and ν ≡ ν ( σ )). σ > 2 Short range exponents ( η = η SR and ν = ν SR ). a a M.E. Fisher et al. PRL 29,14 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 6 / 21

Long range interactions in d dimensions Traditional Results Three regimes: 0 < σ < d / 2 Mean field exponents ( η = 2 − σ and ν = σ − 1 ). d / 2 < σ < 2 Long range exponents ( η ≡ η ( σ ) and ν ≡ ν ( σ )). σ > 2 Short range exponents ( η = η SR and ν = ν SR ). a a M.E. Fisher et al. PRL 29,14 Peculiar Long Range Behavior Using ǫ -expansion technique with ǫ = 2 σ − d or 1 / N expansion is possible to calculate the critical exponent η . η = 2 − σ + O ( ǫ 3 ) Exact at any order in ǫ . η = 2 − σ for all σ < 2. Discontinuity in σ = 2. Nicol` o Defenu (SISSA) Functional RG July 26, 2016 6 / 21

4.0 3.5 d l e i F n a Short Range e M 3.0 d Long Range 2.5 2.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 σ Removal of the discontinuity Sak’s Results The anomalous dimension cannot be less than η SR , σ < σ ∗ η = 2 − σ σ > σ ∗ η = η SR where σ ∗ = 2 − η SR . No discontinuity is present. Nicol` o Defenu (SISSA) Functional RG July 26, 2016 7 / 21

Removal of the discontinuity Sak’s Results The anomalous dimension cannot be less than η SR , σ < σ ∗ η = 2 − σ σ > σ ∗ η = η SR where σ ∗ = 2 − η SR . No discontinuity is present. System regimes 4.0 3.5 d l e i F n a Short Range e M 3.0 d Long Range 2.5 2.0 1.0 1.2 1.4 1.6 1.8 2.0 2.2 σ Nicol` o Defenu (SISSA) Functional RG July 26, 2016 7 / 21

Monte Carlo Results: Controversy Luijte and Blote a results (2002) seemed to confirm Sak results, but new, more complete, results (2013) b question on Sak validity a E. Luijte & H.W. Blote PRL 89, 025703 b M. Picco, arXiv:1207.1018 VS Figure: MC 2013 Figure: MC 2002 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 8 / 21

Effective Dimension Ginzburg-Landau Free Energy � − Z k ψ ∆ ψ + µψ 2 + g ψ 4 � d d SR x � Φ SR = + · · · � � 2 ψ − Z 2 , k ψ ∆ ψ + µψ 2 + g ψ 4 � σ d d LR x Φ LR = − Z k ψ ∆ + · · · Nicol` o Defenu (SISSA) Functional RG July 26, 2016 9 / 21

Effective Dimension Ginzburg-Landau Free Energy � − Z k ψ ∆ ψ + µψ 2 + g ψ 4 � d d SR x � Φ SR = + · · · � � 2 ψ − Z 2 , k ψ ∆ ψ + µψ 2 + g ψ 4 � σ d d LR x Φ LR = − Z k ψ ∆ + · · · Effective dimension results Z k = Z 2 , k = 1 → d SR = 2 d LR σ Z 2 , k = 1 → d LR = (2 − η SR ) d LR σ Nicol` o Defenu (SISSA) Functional RG July 26, 2016 9 / 21

Qualitative Description I Approximation Level: No anomalous dimension σ : Exact N → ∞ , Correct σ ranges, σ ∗ = 2 d SR = 2 d LR Nicol` o Defenu (SISSA) Functional RG July 26, 2016 10 / 21

Qualitative Description I Approximation Level: No anomalous dimension σ : Exact N → ∞ , Correct σ ranges, σ ∗ = 2 d SR = 2 d LR II Approximation Level: Pure Long range case : Exact N → ∞ , Correct σ ranges, σ ∗ = 2 − η SR d SR = (2 − η SR ) d LR σ Nicol` o Defenu (SISSA) Functional RG July 26, 2016 10 / 21

Qualitative Description I Approximation Level: No anomalous dimension σ : Exact N → ∞ , Correct σ ranges, σ ∗ = 2 d SR = 2 d LR II Approximation Level: Pure Long range case : Exact N → ∞ , Correct σ ranges, σ ∗ = 2 − η SR d SR = (2 − η SR ) d LR σ III Approximation Level: Mixed theory space Competition between Short and Long range fixed points: ✟✟ ❍❍ d SR ✟ ❍ Fixed Points Solutions and Stability | ν LR ( d, σ ) − 2 σ ν SR ( D eff ) | η 2 1 0.01 1 0.008 0 η SR θ − 1 0.004 − 2 0 d 1 d 0 . 8 σ ∗ 0 . 9 σ ∗ σ ∗ 2 σ ∗ 2 σ ∗ 2 2 2 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 10 / 21

Quantitative Results Short Range Corrections Short Range corrections spoil dimensional equivalence. Small every- where but at σ ≃ σ ∗ . Correlation Length Exponent 1.6 1.0 1.5 0.8 1.4 0.6 1 1 1.3 1.10 Ν LR Ν LR 1.05 0.4 1.00 1.2 0.95 0.90 0.2 1.1 0.85 0.80 0.75 1.0 0.0 1.0 1.2 1.4 1.6 1.8 2.0 1.5 1.6 1.7 1.8 1.9 2.0 1.0 1.2 1.4 1.6 1.8 2.0 Σ Σ Nicol` o Defenu (SISSA) Functional RG July 26, 2016 11 / 21

Anisotropic O ( N ) models. Lattice Hamiltonian J � S i S j J ⊥ S i S j � � H = − δ ( r ⊥ , ij ) − δ ( r � , ij ) . r d 1 + σ r d 2 + τ 2 2 i � = j � , ij i � = j ⊥ , ij Nicol` o Defenu (SISSA) Functional RG July 26, 2016 12 / 21

Anisotropic O ( N ) models. Lattice Hamiltonian J � S i S j J ⊥ S i S j � � H = − δ ( r ⊥ , ij ) − δ ( r � , ij ) . r d 1 + σ r d 2 + τ 2 2 i � = j � , ij i � = j ⊥ , ij Mean Field Propagator q → 0 G ( q ) − 1 = lim q → 0 J ( q ) = Z � q σ � + Z ⊥ q τ ⊥ + µ + O ( q 2 ) lim Nicol` o Defenu (SISSA) Functional RG July 26, 2016 12 / 21

Anisotropic O ( N ) models. Lattice Hamiltonian J � S i S j J ⊥ S i S j � � H = − δ ( r ⊥ , ij ) − δ ( r � , ij ) . r d 1 + σ r d 2 + τ 2 2 i � = j � , ij i � = j ⊥ , ij Mean Field Propagator q → 0 G ( q ) − 1 = lim q → 0 J ( q ) = Z � q σ � + Z ⊥ q τ ⊥ + µ + O ( q 2 ) lim Effective field theory − Z � 2 φ ( x )∆ σ/ 2 φ ( x ) − Z � � � � d d x 2 φ ( x )∆ τ/ 2 φ ( x ) + ... + U ( φ ( x )) Nicol` o Defenu (SISSA) Functional RG July 26, 2016 12 / 21

Physical Realizations. Quantum Lattice Hamiltonian σ ( z ) σ ( z ) H = − J i j σ ( x ) � � | i − j | d + σ − h , i 2 i � = j i Nicol` o Defenu (SISSA) Functional RG July 26, 2016 13 / 21

Physical Realizations. Quantum Lattice Hamiltonian σ ( z ) σ ( z ) H = − J i j σ ( x ) � � | i − j | d + σ − h , i 2 i � = j i Mapping between Quantum LR and Anisotropic LR σ i → S i d 1 = d d 2 = z σ = σ τ = 2 . The Quantum LR Ising is obtained for z = 1 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 13 / 21

Critical Behavior Asymptotic propagators − 1 G ( q 1 , q 1 ) ≈ q − σ + η σ 1 ) ≈ q − τ + η τ G (1 , q 2 q − θ G ( q 1 q θ , 1) 1 2 2 Correlation Lengths ξ � ≈ | T − T c | − ν 1 ξ ⊥ ≈ | T − T c | − ν 2 , Nicol` o Defenu (SISSA) Functional RG July 26, 2016 14 / 21

Critical Behavior Asymptotic propagators − 1 G ( q 1 , q 1 ) ≈ q − σ + η σ 1 ) ≈ q − τ + η τ G (1 , q 2 q − θ G ( q 1 q θ , 1) 1 2 2 Correlation Lengths ξ � ≈ | T − T c | − ν 1 ξ ⊥ ≈ | T − T c | − ν 2 , Anisotropy index σ − η σ = ν 2 = θ. τ − η τ ν 1 Nicol` o Defenu (SISSA) Functional RG July 26, 2016 14 / 21

Critical Behavior Asymptotic propagators − 1 G ( q 1 , q 1 ) ≈ q − σ + η σ 1 ) ≈ q − τ + η τ G (1 , q 2 q − θ G ( q 1 q θ , 1) 1 2 2 Correlation Lengths ξ � ≈ | T − T c | − ν 1 ξ ⊥ ≈ | T − T c | − ν 2 , Anisotropy index σ − η σ = ν 2 = θ. τ − η τ ν 1 Mean field Results ν 1 = σ − 1 , ν 2 = τ − 1 . η σ = η τ = 0 , Nicol` o Defenu (SISSA) Functional RG July 26, 2016 14 / 21