Modern methods of statistical physics František Slanina Institute of Physics, Academy of Sciences of the Czech Republic, Prague slanina@fzu.cz www.fzu.cz/ slanina Ising model Renormalisation group Modern methods of statistical physics p.1/20
Ising model spins: S i {+1, 1} H = J <ij> S i S j h i S i < ij >: neighbours on a lattice (graph) Modern methods of statistical physics p.2/20
Ising model spins: S i {+1, 1} H = J <ij> S i S j h i S i < ij >: neighbours on a lattice (graph) Linear chain: Modern methods of statistical physics p.2/20
Ising model spins: S i {+1, 1} H = J <ij> S i S j h i S i < ij >: neighbours on a lattice (graph) Linear chain: Square lattice Triangular lattice Modern methods of statistical physics p.2/20
Ising model spins: S i {+1, 1} H = J <ij> S i S j h i S i < ij >: neighbours on a lattice (graph) Linear chain: Square lattice Triangular lattice Duality Honeycomb lattice Modern methods of statistical physics p.2/20
Mean-field solution 1. Naive (Bragg-Wiliams) k neighbours, H eff = J neighbours of 0 S 0S j, Modern methods of statistical physics p.3/20
Mean-field solution 1. Naive (Bragg-Wiliams) k neighbours, H eff = J neighbours of 0 S 0S j, neighbours of 0 S j k S m = tanh βjkm Modern methods of statistical physics p.3/20
Mean-field solution 1. Naive (Bragg-Wiliams) k neighbours, H eff = J neighbours of 0 S 0S j, neighbours of 0 S j k S m = tanh βjkm 1 0.5 0-0.5 βjk = 0.7 βjk = 1.5-1 -1.5-1 -0.5 0 0.5 1 1.5 Modern methods of statistical physics p.3/20
Mean-field solution 1. Naive (Bragg-Wiliams) k neighbours, H eff = J neighbours of 0 S 0S j, neighbours of 0 S j k S m = tanh βjkm 1 0.5 0-0.5 βjk = 0.7 βjk = 1.5-1 -1.5-1 -0.5 0 0.5 1 1.5 tanh x = x 1 3 x3 +... m = 0 or m = 3 βjk 1 (βjk) 3 Modern methods of statistical physics p.3/20
Mean-field solution 1. Naive (Bragg-Wiliams) k neighbours, H eff = J neighbours of 0 S 0S j, neighbours of 0 S j k S m = tanh βjkm 1 0.5 0-0.5 βjk = 0.7 βjk = 1.5-1 -1.5-1 -0.5 0 0.5 1 1.5 tanh x = x 1 3 x3 +... m = 0 or m = 3 βjk 1 (βjk) 3 Critical temperature: β c = 1 Jk, m (T c T ) 1/2 crit. exp.: β = 1 2 Modern methods of statistical physics p.3/20
Mean-field solution Sophisticated: Modern methods of statistical physics p.4/20
Mean-field solution Sophisticated: Fully connected graph Self-consistent equations for local order parameter. (magnetisation) Modern methods of statistical physics p.4/20
Mean-field solution Sophisticated: Fully connected graph Self-consistent equations for local order parameter. (magnetisation) Bethe-Peierls (=Bethe lattice) Recurrent equations for locally conditioned partition function. Modern methods of statistical physics p.4/20
Mean-field solution 2. Fully connected H = Jk N Partition function (ij) S i S j h i S i = 1 2 Jk N ( i S i ) 2 h i S i + Jk 2 }{{} we drop Z = {S i } e βh[{s i}] = {S i } e 1 2 β Jk N (P i S i) 2 +βh P i S i Trick: Hubbard-Stratonovich transform e 1 2 A2 = dx 2π e 1 2 x2 +xa Z = N 2π e N [ ( 1 2 x2 ln 2 cosh(x )] βjk+βh) dx Saddle-point method: e Nφ(x) e Nφ(x ) φ(x) has minimum at x = x : φ (x ) = 0, φ (x ) > 0 Modern methods of statistical physics p.5/20
Mean-field solution 2. Fully connected H = Jk N Partition function (ij) S i S j h i S i = 1 2 Jk N ( i S i ) 2 h i S i + Jk 2 }{{} we drop Z = {S i } e βh[{s i}] = {S i } e 1 2 β Jk N (P i S i) 2 +βh P i S i Trick: Hubbard-Stratonovich transform e 1 2 A2 = dx 2π e 1 2 x2 +xa Z = N 2π e N [ ( 1 2 x2 ln 2 cosh(x )] βjk+βh) dx Saddle-point method: e Nφ(x) e Nφ(x ) φ(x) has minimum at x = x : φ (x ) = 0, φ (x ) > 0 Modern methods of statistical physics p.5/20
Mean-field solution 2. Fully connected H = Jk N Partition function (ij) S i S j h i S i = 1 2 Jk N ( i S i ) 2 h i S i + Jk 2 }{{} we drop Z = {S i } e βh[{s i}] = {S i } e 1 2 β Jk N (P i S i) 2 +βh P i S i Trick: Hubbard-Stratonovich transform e 1 2 A2 = dx 2π e 1 2 x2 +xa Z = N 2π e N [ ( 1 2 x2 ln 2 cosh(x )] βjk+βh) dx Saddle-point method: e Nφ(x) e Nφ(x ) φ(x) has minimum at x = x : φ (x ) = 0, φ (x ) > 0 Modern methods of statistical physics p.5/20
Mean-field solution 2. Fully connected H = Jk N Partition function (ij) S i S j h i S i = 1 2 Jk N ( i S i ) 2 h i S i + Jk 2 }{{} we drop Z = {S i } e βh[{s i}] = {S i } e 1 2 β Jk N (P i S i) 2 +βh P i S i Trick: Hubbard-Stratonovich transform e 1 2 A2 = dx 2π e 1 2 x2 +xa Z = N 2π e N [ ( 1 2 x2 ln 2 cosh(x )] βjk+βh) dx Saddle-point method: e Nφ(x) e Nφ(x ) φ(x) has minimum at x = x : φ (x ) = 0, φ (x ) > 0 Modern methods of statistical physics p.5/20
Mean-field solution 2. Fully connected H = Jk N Partition function (ij) S i S j h i S i = 1 2 Jk N ( i S i ) 2 h i S i + Jk 2 }{{} we drop Z = {S i } e βh[{s i}] = {S i } e 1 2 β Jk N (P i S i) 2 +βh P i S i Trick: Hubbard-Stratonovich transform e 1 2 A2 = dx 2π e 1 2 x2 +xa Z = N 2π e N [ ( 1 2 x2 ln 2 cosh(x )] βjk+βh) dx Saddle-point method: e Nφ(x) e Nφ(x ) φ(x) has minimum at x = x : φ (x ) = 0, φ (x ) > 0 Modern methods of statistical physics p.5/20
φ(x) = 1 2 x2 ln ( 2 cosh(x βjk + βh) ) redefine: m = x βjk. φ (x) = 0 m = tanh(βjkm + βh) (as before) Modern methods of statistical physics p.6/20
φ(x) = 1 2 x2 ln ( 2 cosh(x βjk + βh) ) redefine: m = x βjk. φ (x) = 0 m = tanh(βjkm + βh) (as before) susceptibility: χ = lim h 0 dm dh χ (T T c ) 1 γ = 1 Modern methods of statistical physics p.6/20
φ(x) = 1 2 x2 ln ( 2 cosh(x βjk + βh) ) redefine: m = x βjk. φ (x) = 0 m = tanh(βjkm + βh) (as before) susceptibility: χ = lim h 0 dm dh χ (T T c ) 1 γ = 1 free energy: substitute minimum into φ(x) φ(x) + ln 2 (T c T ) 2 specific heat C (T c T ) 0 α = 0 Modern methods of statistical physics p.6/20
φ(x) = 1 2 x2 ln ( 2 cosh(x βjk + βh) ) redefine: m = x βjk. φ (x) = 0 m = tanh(βjkm + βh) (as before) susceptibility: χ = lim h 0 dm dh χ (T T c ) 1 γ = 1 free energy: substitute minimum into φ(x) φ(x) + ln 2 (T c T ) 2 specific heat C (T c T ) 0 α = 0 Just at critical point: m = tanh(m + βh) h 1 3 δ = 3 Modern methods of statistical physics p.6/20
φ(x) = 1 2 x2 ln ( 2 cosh(x βjk + βh) ) redefine: m = x βjk. φ (x) = 0 m = tanh(βjkm + βh) (as before) susceptibility: χ = lim h 0 dm dh χ (T T c ) 1 γ = 1 free energy: substitute minimum into φ(x) φ(x) + ln 2 (T c T ) 2 specific heat C (T c T ) 0 α = 0 Just at critical point: m = tanh(m + βh) h 1 3 δ = 3 Summary of critical exponents β = 1 2 γ = 1 δ = 3 α = 0 Modern methods of statistical physics p.6/20
Z (S 0 1) } Bethe-lattice approximation 1 { 2 S 3 0 } Z (S 0 2) Z (S 0 3) Z(S 0 l) = all spins in l-th branch e βh[{s} only l-th branch] Modern methods of statistical physics p.7/20
Z (S 0 1) } Bethe-lattice approximation Iterative cutting of k branches { 2 S 3 0 } Z (S 0 2) Z (S 0 3) 1 Z(S 0 1) = Z(S 0 2) = Z(S 0 3) =... = Z(S 0 k) = Z(S 0 ) Z = S 0 Z k (S 0 ) Z(S 0 l) = all spins in l-th branch e βh[{s} only l-th branch] Modern methods of statistical physics p.7/20
Z (S 0 1) } Bethe-lattice approximation Iterative cutting of k branches { 2 S 3 0 } Z (S 0 2) Z (S 0 3) Z(S 0 l) = 1 all spins in l-th branch e βh[{s} only l-th branch] Z(S 0 1) = Z(S 0 2) = Z(S 0 3) =... = Z(S 0 k) = Z(S 0 ) Z n (S 0 ) = S 1 Z = S 0 Z k (S 0 ) e βj S 0S 1 [Z n 1 (S 1 )] k 1 Modern methods of statistical physics p.7/20
Fixed point for quantity x = Z( 1)/Z(+1) Modern methods of statistical physics p.8/20
Fixed point for quantity x = Z( 1)/Z(+1) x = e βj + e βj x k 1 e βj + e βj x k 1 Modern methods of statistical physics p.8/20
Fixed point for quantity x = Z( 1)/Z(+1) x = e βj + e βj x k 1 e βj + e βj x k 1 x n+1 1 0.8 0.6 0.4 0.2 e βj = 0.9 e βj = 0.6 0 0 0.2 0.4 0.6 0.8 1 x n Modern methods of statistical physics p.8/20
Fixed point for quantity x = Z( 1)/Z(+1) x = e βj + e βj x k 1 e βj + e βj x k 1 x n+1 1 0.8 0.6 0.4 0.2 e βj = 0.9 e βj = 0.6 0 0 0.2 0.4 0.6 0.8 1 x n Magnetisation m = 1 xk 1+x k Modern methods of statistical physics p.8/20
Fixed point for quantity x = Z( 1)/Z(+1) x = e βj + e βj x k 1 e βj + e βj x k 1 x n+1 1 0.8 0.6 0.4 0.2 e βj = 0.9 e βj = 0.6 0 0 0.2 0.4 0.6 0.8 1 Magnetisation m = 1 xk 1+x k Critical point tanh β c J = 1 1 k x n Modern methods of statistical physics p.8/20
One-dimensional Ising model H = J N 1 i=1 S i S i+1 JS N S 1 h N i=1 S i Modern methods of statistical physics p.9/20
One-dimensional Ising model H = J N 1 i=1 S i S i+1 JS N S 1 h N i=1 S i Z = {S i } e βh = {S i } eβ(js 1S 2 +h(s 1 +S 2 )/2) e β(js 3S 3 +h(s 2 +S 3 )/2)... e β(js NS 1 +h(s N +S 1 )/2) Modern methods of statistical physics p.9/20
One-dimensional Ising model H = J N 1 i=1 S i S i+1 JS N S 1 h N i=1 S i Z = {S i } e βh = {S i } eβ(js 1S 2 +h(s 1 +S 2 )/2) e β(js 3S 3 +h(s 2 +S 3 )/2)... e β(js NS 1 +h(s N +S 1 )/2) ( ) V (S, S ) = e β(jss +h(s+s )/2), or V = e β(j+h) e βj e βj e β(j h) Modern methods of statistical physics p.9/20
One-dimensional Ising model H = J N 1 i=1 S i S i+1 JS N S 1 h N i=1 S i Z = {S i } e βh = {S i } eβ(js 1S 2 +h(s 1 +S 2 )/2) e β(js 3S 3 +h(s 2 +S 3 )/2)... e β(js NS 1 +h(s N +S 1 )/2) ( ) V (S, S ) = e β(jss +h(s+s )/2), or V = e β(j+h) e βj e βj e β(j h) Z = TrV N = λ N 1 + λ N 2 λ N 1 (λ 1 > λ 2 ) Modern methods of statistical physics p.9/20
One-dimensional Ising model H = J N 1 i=1 S i S i+1 JS N S 1 h N i=1 S i Z = {S i } e βh = {S i } eβ(js 1S 2 +h(s 1 +S 2 )/2) e β(js 3S 3 +h(s 2 +S 3 )/2)... e β(js NS 1 +h(s N +S 1 )/2) ( ) V (S, S ) = e β(jss +h(s+s )/2), or V = e β(j+h) e βj e βj e β(j h) Z = TrV N = λ N 1 + λ N 2 λ N 1 (λ 1 > λ 2 ) Free energy density: f = β 1 ln λ 1 Modern methods of statistical physics p.9/20
One-dimensional Ising model H = J N 1 i=1 S i S i+1 JS N S 1 h N i=1 S i Z = {S i } e βh = {S i } eβ(js 1S 2 +h(s 1 +S 2 )/2) e β(js 3S 3 +h(s 2 +S 3 )/2)... e β(js NS 1 +h(s N +S 1 )/2) ( ) V (S, S ) = e β(jss +h(s+s )/2), or V = e β(j+h) e βj e βj e β(j h) Z = TrV N = λ N 1 + λ N 2 λ N 1 (λ 1 > λ 2 ) Free energy density: f = β 1 ln λ 1 Eigenvalues of V : λ 1,2 = e βj cosh βh ± e 2βJ sinh 2 βh + e 2βJ Modern methods of statistical physics p.9/20
Magnetisation: m = f h = e βj sinh βh e 2βJ sinh 2 βh + e 2βJ Modern methods of statistical physics p.10/20
Magnetisation: m = f h = e βj sinh βh e 2βJ sinh 2 βh + e 2βJ... no phase transition! Modern methods of statistical physics p.10/20
Magnetisation: m = f h = e βj sinh βh e 2βJ sinh 2 βh + e 2βJ Susceptibility:... no phase transition! χ = m h h=0 = βe 2βJ Modern methods of statistical physics p.10/20
Magnetisation: m = f h = e βj sinh βh e 2βJ sinh 2 βh + e 2βJ Susceptibility:... no phase transition! Correlation function (h = 0): χ = m h h=0 = βe 2βJ S i S j = 1 Z TrσV j i σv N (j i) = ( ) j i λ2 λ 1 ξ = (ln tanh βj) 1 1 2 e2βj diverges for β (or T 0) Modern methods of statistical physics p.10/20
Series expansions 1. High-temperature Z = e βh = {S i } {S i } <ij> e βjs is j Modern methods of statistical physics p.11/20
Series expansions 1. High-temperature Z = e βh = e βjsisj = (cosh βj) L {S i } {S i } <ij> {S i } <ij> (1 + S i S j tanh βj) Modern methods of statistical physics p.11/20
Series expansions 1. High-temperature Z = e βh = e βjsisj = (cosh βj) L {S i } {S i } <ij> {S i } <ij> (1 + S i S j tanh βj) Modern methods of statistical physics p.11/20
Series expansions 1. High-temperature Z = e βh = e βjsisj = (cosh βj) L {S i } {S i } <ij> {S i } <ij> (1 + S i S j tanh βj) Modern methods of statistical physics p.11/20
Series expansions 1. High-temperature Z = e βh = e βjsisj = (cosh βj) L {S i } {S i } <ij> {S i } <ij> (1 + S i S j tanh βj) 2. Low-temperature Z = 2e βjl e 2βJ l contours Modern methods of statistical physics p.11/20
Series expansions 1. High-temperature Z = e βh = e βjsisj = (cosh βj) L {S i } {S i } <ij> {S i } <ij> (1 + S i S j tanh βj) 2. Low-temperature Z = 2e βjl e 2βJ l contours Modern methods of statistical physics p.11/20
Series expansions 1. High-temperature Z = e βh = e βjsisj = (cosh βj) L {S i } {S i } <ij> {S i } <ij> (1 + S i S j tanh βj) 2. Low-temperature Z = 2e βjl e 2βJ l contours Modern methods of statistical physics p.11/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l Modern methods of statistical physics p.12/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l }{{} Z(βJ) Z(β J ) = 2N 1 (cosh βj) L e β J L Modern methods of statistical physics p.12/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l }{{} Z(βJ) Z(β J ) = 2N 1 (cosh βj) L e β J L Critical point: β J = βj β c J = 0.44068679... Modern methods of statistical physics p.12/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l }{{} Z(βJ) Z(β J ) = 2N 1 (cosh βj) L e β J L Critical point: β J = βj β c J = 0.44068679... Compare: mean-field β c J = 0.25, Bethe-Peierls β c J = 0.34657... Modern methods of statistical physics p.12/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l }{{} Z(βJ) Z(β J ) = 2N 1 (cosh βj) L e β J L Critical point: β J = βj β c J = 0.44068679... Compare: mean-field β c J = 0.25, Bethe-Peierls β c J = 0.34657... 2. Extraction of exponents: φ(x) = l=0 a l x l A(x x c ) γ Modern methods of statistical physics p.12/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l }{{} Z(βJ) Z(β J ) = 2N 1 (cosh βj) L e β J L Critical point: β J = βj β c J = 0.44068679... Compare: mean-field β c J = 0.25, Bethe-Peierls β c J = 0.34657... 2. Extraction of exponents: φ(x) = l=0 a l x l A(x x c ) γ r l (l + 1) a l+1 a l l a l a l 1 1 x c Modern methods of statistical physics p.12/20
Analysis of the series expansions 1. Duality: define tanh β J = e 2βJ low-t Z(βJ) = 2 N (cosh βj) L contours (tanh βj)l high-t Z(βJ) = 2e βjl contours (tanh β J ) l }{{} Z(βJ) Z(β J ) = 2N 1 (cosh βj) L e β J L Critical point: β J = βj β c J = 0.44068679... Compare: mean-field β c J = 0.25, Bethe-Peierls β c J = 0.34657... 2. Extraction of exponents: φ(x) = l=0 a l x l A(x x c ) γ a l r l (l + 1) a l+1 l 1 a l a l 1 x c ( ) al s l l x c 1 + 1 γ a l 1 Modern methods of statistical physics p.12/20
Renormalisation group Modern methods of statistical physics p.13/20
Renormalisation group Old spins S i New spins S I Projector P ({ S I }, {S i }) Modern methods of statistical physics p.13/20
Renormalisation group Old spins S i New spins S I Projector P ({ S I }, {S i }) example: 1D Ising S 1 S 2 S 3 S4 S5 S6 S7 S S S S 1 2 3 4 Modern methods of statistical physics p.13/20
Renormalisation group Old spins S i New spins S I Projector P ({ S I }, {S i }) example: 1D Ising S 1 S 2 S 3 S4 S5 S6 S7 S S2 S3 S4 1 P = I δ( S I S 2I 1 ) Modern methods of statistical physics p.13/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng Modern methods of statistical physics p.14/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng H({S i }) = K 1 H 1 ({S i }) + K 2 H 2 ({S i }) +... H({ S I }) = K 1 H 1 ({ S I }) + K 2 H 2 ({ S I }) +... Modern methods of statistical physics p.14/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng H({S i }) = K 1 H 1 ({S i }) + K 2 H 2 ({S i }) +... H({ S I }) = K 1 H 1 ({ S I }) + K 2 H 2 ({ S I }) +... dynamical process (K 1, K 2,...) ( K 1, K 2,...) K i = R i (K 1, K 2,...) Modern methods of statistical physics p.14/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng H({S i }) = K 1 H 1 ({S i }) + K 2 H 2 ({S i }) +... H({ S I }) = K 1 H 1 ({ S I }) + K 2 H 2 ({ S I }) +... dynamical process (K 1, K 2,...) ( K 1, K 2,...) Schematically: K i = R i (K 1, K 2,...) one parameter: T c T Modern methods of statistical physics p.14/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng H({S i }) = K 1 H 1 ({S i }) + K 2 H 2 ({S i }) +... H({ S I }) = K 1 H 1 ({ S I }) + K 2 H 2 ({ S I }) +... dynamical process (K 1, K 2,...) ( K 1, K 2,...) Schematically: stable K i = R i (K 1, K 2,...) one parameter: T c T K 2 K Modern methods of statistical physics p.14/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng H({S i }) = K 1 H 1 ({S i }) + K 2 H 2 ({S i }) +... H({ S I }) = K 1 H 1 ({ S I }) + K 2 H 2 ({ S I }) +... dynamical process (K 1, K 2,...) ( K 1, K 2,...) Schematically: stable K i = R i (K 1, K 2,...) one parameter: unstable T c T K 2 K 2 K K Modern methods of statistical physics p.14/20
Old measure New measure µ({s i }) = e H({S i}) µ({ S I }) = {S i } P ({ S I }, {S i })µ({s i }) = e H({ S I }) Ng H({S i }) = K 1 H 1 ({S i }) + K 2 H 2 ({S i }) +... H({ S I }) = K 1 H 1 ({ S I }) + K 2 H 2 ({ S I }) +... dynamical process (K 1, K 2,...) ( K 1, K 2,...) Schematically: stable K i = R i (K 1, K 2,...) one parameter: unstable mixed T c T K 2 K 2 K 2 K K Modern methods of statistical physics p.14/20 K
Fixed points: K = R(K ) Modern methods of statistical physics p.15/20
Fixed points: K = R(K ) linearisation: δ K i = j R i K j δk j Modern methods of statistical physics p.15/20
Fixed points: K = R(K ) linearisation: δ K i = j R i K j δk j diagonalisation: Ū α = λ α U α Modern methods of statistical physics p.15/20
Fixed points: K = R(K ) linearisation: δ K i = j R i K j δk j diagonalisation: Ū α = λ α U α semigroup property λ α = b y α λ α > 1 repulsive y α > 0 U α relevant λ α < 1 attractive y α < 0 U α irrelevant Modern methods of statistical physics p.15/20
Fixed points: K = R(K ) linearisation: δ K i = j R i K j δk j diagonalisation: Ū α = λ α U α semigroup property λ α = b y α λ α > 1 repulsive y α > 0 U α relevant λ α < 1 attractive y α < 0 U α irrelevant Scaling: f(t c T, h) = b d f((t c T ) b y T, h y h ) Modern methods of statistical physics p.15/20
Fixed points: K = R(K ) linearisation: δ K i = j R i K j δk j diagonalisation: Ū α = λ α U α semigroup property λ α = b y α λ α > 1 repulsive y α > 0 U α relevant λ α < 1 attractive y α < 0 U α irrelevant Scaling: f(t c T, h) = b d f((t c T ) b y T, h y h ) α = 2 d y T β = d y h δ = y T y h d y h Modern methods of statistical physics p.15/20
Renormalisation group solution of 1D Ising model Modern methods of statistical physics p.16/20
Renormalisation group solution of 1D Ising model J = 1 4 ln cosh(2j + h) + 1 4 ln cosh(2j h) 1 2 ln cosh h h = h + 1 2 ln cosh(2j + h) 1 2 ln cosh(2j h) Modern methods of statistical physics p.16/20
Renormalisation group solution of 1D Ising model J = 1 4 ln cosh(2j + h) + 1 4 ln cosh(2j h) 1 2 ln cosh h h = h + 1 2 ln cosh(2j + h) 1 2 ln cosh(2j h) 5 4 h 3 2 1 0 0 0.5 1 1.5 2 2.5 3 J Modern methods of statistical physics p.16/20
Scaling relations f(t, h) = b d f(tb y T, hb y h ) ξ(t, h) = bξ(tb y T, hb y h ) C(r, t, h) = b 2yh 2d C(r/b, tb y T, hb y h ) Modern methods of statistical physics p.17/20
Scaling relations f(t, h) = b d f(tb y T, hb y h ) ξ(t, h) = bξ(tb y T, hb y h ) C(r, t, h) = b 2yh 2d C(r/b, tb y T, hb y h ) α = 2 d y T ν = 1 y T νd = 2 α Modern methods of statistical physics p.17/20
Scaling relations f(t, h) = b d f(tb y T, hb y h ) ξ(t, h) = bξ(tb y T, hb y h ) C(r, t, h) = b 2yh 2d C(r/b, tb y T, hb y h ) α = 2 d y T ν = 1 y T β = d y h δ = y T y h d y h γ = 2y h d y T νd = 2 α γ = β(δ 1) Modern methods of statistical physics p.17/20
Scaling relations f(t, h) = b d f(tb y T, hb y h ) ξ(t, h) = bξ(tb y T, hb y h ) C(r, t, h) = b 2yh 2d C(r/b, tb y T, hb y h ) α = 2 d y T ν = 1 y T β = d y h δ = y T y h d y h γ = 2y h d y T νd = 2 α γ = β(δ 1) α + 2β + γ = 2 Modern methods of statistical physics p.17/20
Scaling relations f(t, h) = b d f(tb y T, hb y h ) ξ(t, h) = bξ(tb y T, hb y h ) C(r, t, h) = b 2yh 2d C(r/b, tb y T, hb y h ) α = 2 d y T ν = 1 y T β = d y h δ = y T y h d y h γ = 2y h d y T η = 2y h + d + 2 νd = 2 α γ = β(δ 1) α + 2β + γ = 2 γ = ν(2 η) Modern methods of statistical physics p.17/20
Overview of Ising critical exponents exponent mean-field d = 2 Onsager d = 3 high-t expansion α discontinuity 0 0.119 ± 0.006 β 1/2 1/8 0.326 ± 0.004 γ 1 7/4 1.239 ± 0.003 δ 3 15 5 η 1/2 1 0.024 ± 0.007 ν 0 1/4 0.627 ± 0.002 Modern methods of statistical physics p.18/20
Liquid-gas transition Phase diagram of water Modern methods of statistical physics p.19/20
Liquid-gas transition V(r) 7 6 5 4 3 2 1 0-1 0 0.5 1 1.5 r 2 2.5 3 Lennard-Jones potential ˆ` r0 12 ` r0 6 V (r) = 4 r r Phase diagram of water Modern methods of statistical physics p.19/20
Liquid-gas transition V(r) 7 6 5 4 3 2 1 0-1 0 0.5 1 1.5 r 2 2.5 3 Lennard-Jones potential ˆ` r0 12 ` r0 6 V (r) = 4 r r V(r) 7 6 5 4 3 2 1 0-1 Phase diagram of water 0 0.5 1 1.5 r 2 2.5 3 Modern methods of statistical physics p.19/20
Lattice gas model Molecules on a lattice. Occupation numbers: n i {0, 1} H = <ij> ɛ n i n j Modern methods of statistical physics p.20/20
Lattice gas model Molecules on a lattice. Occupation numbers: n i {0, 1} H = <ij> ɛ n i n j Grand-canonical partition function Z = {n i } e βɛ P <ij> n in j βµ P i n i Modern methods of statistical physics p.20/20
Lattice gas model Molecules on a lattice. Occupation numbers: n i {0, 1} H = <ij> ɛ n i n j Grand-canonical partition function Z = {n i } e βɛ P <ij> n in j βµ P i n i Mapping on Ising model in magnetic field: n i = 1 2 (S i + 1), J = 1 4 ɛ, h = 1 2 µ + 1 4 ɛ k magnetisation density: M = 2ρ 1 Modern methods of statistical physics p.20/20