Spin chains: Thermodynamics and criticality M.A. Rodr guez - PowerPoint PPT Presentation

Spin chains: Thermodynamics and criticality M.A. Rodr´ ıguez Universidad Complutense de Madrid, Spain Joint work with F. Finkel, A. Gonz´ alez-L´ opez and I. Le´ on Merino IbortFest Madrid. March 9, 2018 M.A. Rodr´ ıguez (UCM) Criticality in spin chains 1 / 69

Outline 1 Introduction 2 The models 3 Partition function 4 Associated vertex models 5 Thermodynamics 6 The su ( m | n ) chains 7 Conclusions M.A. Rodr´ ıguez (UCM) Criticality in spin chains 2 / 69

Introduction Spin chains of Haldane–Shastry type have been extensively studied as the prototypical examples of one-dimensional lattice models with long-range interactions, due to their remarkable physical and mathematical properties. Applications: conformal field theory fractional statistics and anyons, quantum chaos vs. integrability quantum information theory quantum simulation of long-range magnetism. M.A. Rodr´ ıguez (UCM) Criticality in spin chains 3 / 69

HS spin chain Connection with the (dynamical) spin Sutherland model Polychronakos’s freezing trick → chain’s partition function Other models: ◮ Calogero → Polychronakos–Frahm (PF) spin chain ◮ Inozemtsev → Frahm–Inozemtsev (FI) spin chain spin = su ( m ) spin Supersymmetric models: su ( m | n ), sites are either su ( m ) bosons or su ( n ) fermions M.A. Rodr´ ıguez (UCM) Criticality in spin chains 4 / 69

Thermodynamics of spin chains of HS type Haldane Transfer matrix method, used by Frahm and Inozemtsev (magnetization in an external constant magnetic field) Spin 1 / 2 chains of HS type in a constant magnetic field (Enciso, Finkel, Gonz´ alez-L´ opez) Supersymmetric case, su (1 | 1) HS chain (with a chemical potential term): equivalence to a free, translationally invariant fermion system (Carrasco, Finkel, Gonz´ alez-L´ opez, Rodr´ ıguez, Tempesta) It cannot be applied to the su (1 | 1) PF and FI chains nor to chains of HS type with m > 1 or n > 1 M.A. Rodr´ ıguez (UCM) Criticality in spin chains 5 / 69

Conformal field theory Connection between su (2) HS chain and the level-1 su (2) Wess–Zumino–Novikov–Witten conformal field theory (CFT) su ( n ) HS chain (with no magnetic field or chemical potential term) is critical (gapless), with central charge c = n − 1. Extended to the su ( m | n ), m � 1, PF chain with central charge c = m − 1 + n / 2 su (1 | 1) HS chain with a chemical potential: critical with central charge c = 1 (for a certain range of values of the chemical potential M.A. Rodr´ ıguez (UCM) Criticality in spin chains 6 / 69

Thermodynamics and critical behavior of su ( m | n ) spin chains of HS type with a general chemical potential term chains’ partition functions (connection with vertex models) transfer matrix free energy per site in the thermodynamic limit thermodynamics and criticality of supersymmetric chains of HS type with 1 � m , n � 2 ◮ Low-temperature behavior of the free energy per site ◮ Values of the chemical potentials for which these chains are critical, central charge. ◮ Phase transitions at zero temperature M.A. Rodr´ ıguez (UCM) Criticality in spin chains 7 / 69

Spin chains The Hamiltonian � J ij ( ✶ − P ( m | n ) H 0 = ) ij 1 � i < j � N Haldane, Shastry, Polychronakos, Frahm, Inozemtsiev, . . . Spin states, su ( M ): V = ⊗ N i =1 R M , dim V = M N | s 1 , . . . , s N � , s i ∈ { 1 , . . . , M } , Coupling constants: J ij > 0 Exchange operators: P ( m | n ) ij M.A. Rodr´ ıguez (UCM) Criticality in spin chains 8 / 69

• Polychronakos-Frahm (PF): J J ij = ( ξ i − ξ j ) 2 , ξ i ≡ zeros of Hermite polynomials • Haldane-Shastry (HS): J ξ i = i π J ij = 2 sin 2 ( ξ i − ξ j ) , N • Frahm-Inozemtsiev (FI): J e 2 ξ i ≡ zeros of Laguerre polynomials J ij = 2 sinh 2 ( ξ i − ξ j ) , M.A. Rodr´ ıguez (UCM) Criticality in spin chains 9 / 69

Exchange operators Bosonic model ( s i ∈ { 1 , . . . , m } ) Polychronakos P ij | s 1 , . . . , s i , . . . , s j , . . . , s N � = | s 1 , . . . , s j , . . . , s i , . . . , s N � Supersymmetric model ( s i ∈ { 1 , . . . , m + n } ) Basu-Mallick, Bondyopadhaya, Hikami, Sen, Gonz´ alez-L´ opez, Finkel, Enciso, Barba, . . . s i ∈ B = { 1 , . . . , m } ≡ bosons s i ∈ F = { m + 1 , . . . , m + n } ≡ fermions P ij | s 1 , . . . , s i , . . . , s j , . . . , s N � = ǫ i , i +1 ,..., j | s 1 , . . . , s j , . . . , s i , . . . , s N �  1 , s i , s j bosons   ( − 1) p ,  { s i , s j } ≡ { fermion , boson } ,  ǫ i , i +1 ,..., j = p = number of fermions in positions i + 1 , . . . , j − 1     − 1 , s i , s j fermions M.A. Rodr´ ıguez (UCM) Criticality in spin chains 10 / 69

Supersymmetric su (1 | 1) , N = 3   0 0 0 0 0 0 0 0 − 1 − 1 0 2 0 0 0 0     − 1 − 1 0 2 0 0 0 0     H HS = 2 − 1 − 1 0 2 0 0 0 0     0 0 0 0 4 − 1 1 0 3     0 0 0 0 − 1 4 − 1 0     0 0 0 0 1 − 1 4 0   0 0 0 0 0 0 0 6 1 = 2 + 4 q 2 + 2 q 4 , Z (1 | 1) q = e − β , E = { 0 2 , 2 4 , 4 2 } , β = 3 k B T M.A. Rodr´ ıguez (UCM) Criticality in spin chains 11 / 69

The Calogero-Sutherland model Rational case, scalar model a − 1 � x i + a 2 � � H sc = − ∂ 2 x 2 i + 2 a ( x i − x j ) 2 i < j i i N � E 0 = aN ( a ( N − 1) + 1) E = E 0 + 2 a n i , i =1 n = ( n 1 , . . . , n N ) ∈ Z N + M.A. Rodr´ ıguez (UCM) Criticality in spin chains 12 / 69

The spin Calogero-Sutherland model Rational case, spin dynamical model a − P ( m | n ) � x i + a 2 � � ∂ 2 x 2 H 0 = − i + 2 a ( x i − x j ) 2 i i i < j N � E = E 0 + 2 a n i i =1 �� � ψ s n ( x ) = ρ ( x )Λ ( m | n ) x n i i | s 1 , . . . , s N � i j � Λ ( m | n ) = Λ ( m | n ) ρ ( x ) = e − a j x 2 K ij P ( m | n ) � | x i − x j | a , 2 ij i < j M.A. Rodr´ ıguez (UCM) Criticality in spin chains 13 / 69

Chemical potential In the non-supersymmetric case we can add a magnetic field (a chemical potencial in the supersymmetric case) to study the behavior of the system regarding the number of particles of different types: m + n − 1 � H µ = − µ α N α α =1 N α number operator of α ∈ { 1 , . . . , n + m } type particles. N α | s 1 · · · s N � = N α ( s ) | s 1 · · · s N � , N � N α ( s ) ≡ δ s i ,α i =1 is the number of spins of type α in the state | s 1 · · · s N � . M.A. Rodr´ ıguez (UCM) Criticality in spin chains 14 / 69

The Hamiltonian is: H = H 0 + 2 a J H µ The operators N α commute with the exchange operators and the energy spectrum of the total Hamiltonian H is: n i − 2 a � � E s n = E 0 + 2 a µ s i J i i M.A. Rodr´ ıguez (UCM) Criticality in spin chains 15 / 69

Symmetries of the Hamiltonian m + n − 1 � � H ( m | n ) = J ij (1 − P ( m | n ) ) − µ α N α ≡ H 0 + H 1 , ij i < j α =1 • H ( m | n ) , is related to H ( n | m ) by a duality relation. U : Σ ( m | n ) → Σ ( n | m ) , � i i π ( s i ) | s ′ 1 · · · s ′ U | s 1 · · · s N � = ( − 1) N � s ′ π ( s i ) = 0 , s i ∈ B , π ( s i ) = 1 , s i ∈ F , i = m + n + 1 − s i U − 1 P ( n | m ) U = − P ( m | n ) U − 1 N α U = N m + n +1 − α , , ij ij U − 1 H ( n | m ) U = E 0 − H ( m | n ) � � � µ α →− µ m + n +1 − α , E 0 ≡ 2 J ij . � i < j M.A. Rodr´ ıguez (UCM) Criticality in spin chains 16 / 69

Thus the spectra of H ( n | m ) and H ( m | n ) are related by E ( n | m ) ( µ 1 , . . . , µ m + n ) = E 0 − E ( m | n ) ( − µ m + n , . . . , − µ 1 ) . k k • Changes in the labeling of the bosonic or fermionic degrees of freedom: T αβ : Σ ( m | n ) → Σ ( m | n ) , α � = β ∈ { 1 , . . . , m + n } replacing all the s k ’s equal to α by β , and vice versa. If π ( α ) = π ( β ), T αβ commutes with P ( m | n ) , and with H 0 . ij T − 1 T − 1 αβ N α T αβ = N β , αβ N β T αβ = N α T − 1 αβ N γ T αβ = N γ ( γ � = α, β ) , m + n � T − 1 αβ H T αβ = H 0 − µ α N β − µ β N α − µ γ N γ . γ =1 γ � = α,β M.A. Rodr´ ıguez (UCM) Criticality in spin chains 17 / 69

E ( m | n ) ( . . . , µ α , . . . , µ β , . . . ) = E ( m | n ) ( . . . , µ β , . . . , µ α , . . . ) k k ( π ( α ) = π ( β )) the spectrum of H is invariant under permutations of the bosonic or fermionic chemical potentials among themselves. E ( n | m ) ( µ 1 , . . . , µ m + n ) = E 0 − E ( m | n ) • ( − µ α 1 , . . . , − µ α m + n ) , k k ( α 1 , . . . , α m + n ) = permutation of (1 , . . . , m + n ) with { α 1 , . . . , α m } = { n + 1 , . . . , n + m } , { α m +1 , . . . , α m + n } = { 1 , . . . , n } . M.A. Rodr´ ıguez (UCM) Criticality in spin chains 18 / 69

Hamiltonians a − 1 � x i + a 2 � � H sc = − ∂ 2 x 2 i + 2 a ( x i − x j ) 2 i < j i i a − P ( m | n ) � x i + a 2 � � ∂ 2 x 2 H 0 = − i + 2 a ( x i − x j ) 2 i i i < j m + n − 1 J � � ( ξ i − ξ j ) 2 ( ✶ − P ( m | n ) H 0 = ) , H µ = − µ α N α ij i < j α =1 H = H 0 + 2 a J H µ , H = H 0 + H µ H = H sc + 2 a � J H � ξ i → x i M.A. Rodr´ ıguez (UCM) Criticality in spin chains 19 / 69