The Bethe Ansatz Heisenberg Model and Generalizations

Home , Bethe ansatz

F. Alexander Wolf

University of Augsburg

June 22 2011

1 / 33 Contents

1 Introduction

2 Ferromagnetic 1D Heisenberg model

3 Antiferromagnetic 1D Heisenberg model

4 Generalizations

5 Summary and References

2 / 33 Although eigenvalues and eigenstates of a ﬁnite system may be obtained from brute force numerical diagonalization Two important advantages of the Bethe ansatz all eigenstates characterized by set of quantum numbers → distinction according to speciﬁc physical properties in many cases: possibility to take thermodynamic limit, no system size restrictions

One shortcoming structure of obtained eigenstates in practice often to complicated to obtain correlation functions

Introduction Bethe ansatz Hans Bethe (1931): particular parametrization of eigenstates of the 1D Heisenberg model Bethe, ZS. f. Phys. (1931) Today: generalized to whole class of 1D quantum many-body systems

3 / 33 Introduction Bethe ansatz Hans Bethe (1931): particular parametrization of eigenstates of the 1D Heisenberg model Bethe, ZS. f. Phys. (1931) Today: generalized to whole class of 1D quantum many-body systems

Although eigenvalues and eigenstates of a ﬁnite system may be obtained from brute force numerical diagonalization Two important advantages of the Bethe ansatz all eigenstates characterized by set of quantum numbers → distinction according to speciﬁc physical properties in many cases: possibility to take thermodynamic limit, no system size restrictions

One shortcoming structure of obtained eigenstates in practice often to complicated to obtain correlation functions

3 / 33 Contents

1 Introduction

2 Ferromagnetic 1D Heisenberg model

3 Antiferromagnetic 1D Heisenberg model

4 Generalizations

5 Summary and References

4 / 33 Basic remarks: eigenstates

reference basis: {|σ1 . . . σN i} Bethe ansatz is basis tansformation z PN z z rotational symmetry z-axis ST ≡ n=1 Sn conserved: [H, ST ] = 0 z ⇒ block diagonalization by sorting basis according to hST i = N/2 − r where r = number of down spins

Ferromagnetic 1D Heisenberg model

Goal obtain exact eigenvalues and eigenstates with their physical properties

N X H = −J Sn · Sn+1 n=1 N X 1 + − − + = −J S S + S S + Sz Sz 2 n n+1 n n+1 n n+1 n=1

5 / 33 Ferromagnetic 1D Heisenberg model

Goal obtain exact eigenvalues and eigenstates with their physical properties

N X H = −J Sn · Sn+1 n=1 N X 1 + − − + = −J S S + S S + Sz Sz 2 n n+1 n n+1 n n+1 n=1

Basic remarks: eigenstates

5 / 33 block r = 1: one-particle excitations

with energy E = J(1 − cos k) + E0 magnons |ki N one-particle excitations correspond to elementary particles “magnons” with one particle states |ki

Note: not the lowest excitations!

Intuitive states

Lowest energy states intuitively obtained

block r = 0: groundstate

|Fi ≡ | ↑ ... ↑i

with energy E0 = −JN/4

6 / 33 Intuitive states

Lowest energy states intuitively obtained

block r = 0: groundstate

|Fi ≡ | ↑ ... ↑i

with energy E0 = −JN/4

block r = 1: one-particle excitations

with energy E = J(1 − cos k) + E0 magnons |ki N one-particle excitations correspond to elementary particles “magnons” with one particle states |ki

Note: not the lowest excitations!

6 / 33 ! N block r = 2: dim= = N(N − 1)/2 2

X − − |ψi = a(n1, n2)|n1, n2i where |n1, n2i ≡ Sn1 Sn2 |Fi 1≤n1

2[E − E0]a(n1, n2) = J[4a(n1, n2)−a(n1 −1, n2)

− a(n1 +1, n2) − a(n1, n2 −1) − a(n1, n2 +1)] for n2 > n1 +1

2[E − E0]a(n1, n2) = J[2a(n1, n2) − a(n1 −1, n2) − a(n1, n2 +1)]

for n2 = n1 +1

Systematic proceeding to obtain eigenstates block r = 1: dim= N N X |ψi = a(n)|ni n=1 H|ψi = E|ψi ⇔

2[E − E0]a(n) = J[2a(n) − a(n − 1) − a(n + 1)]

7 / 33 2[E − E0]a(n1, n2) = J[4a(n1, n2)−a(n1 −1, n2)

− a(n1 +1, n2) − a(n1, n2 −1) − a(n1, n2 +1)] for n2 > n1 +1

2[E − E0]a(n1, n2) = J[2a(n1, n2) − a(n1 −1, n2) − a(n1, n2 +1)]

for n2 = n1 +1

Systematic proceeding to obtain eigenstates block r = 1: dim= N N X |ψi = a(n)|ni n=1 H|ψi = E|ψi ⇔

2[E − E0]a(n) = J[2a(n) − a(n − 1) − a(n + 1)] ! N block r = 2: dim= = N(N − 1)/2 2

X − − |ψi = a(n1, n2)|n1, n2i where |n1, n2i ≡ Sn1 Sn2 |Fi 1≤n1

7 / 33 Systematic proceeding to obtain eigenstates block r = 1: dim= N N X |ψi = a(n)|ni n=1 H|ψi = E|ψi ⇔

2[E − E0]a(n) = J[2a(n) − a(n − 1) − a(n + 1)] ! N block r = 2: dim= = N(N − 1)/2 2

X − − |ψi = a(n1, n2)|n1, n2i where |n1, n2i ≡ Sn1 Sn2 |Fi 1≤n1

2[E − E0]a(n1, n2) = J[4a(n1, n2)−a(n1 −1, n2)

− a(n1 +1, n2) − a(n1, n2 −1) − a(n1, n2 +1)] for n2 > n1 +1

2[E − E0]a(n1, n2) = J[2a(n1, n2) − a(n1 −1, n2) − a(n1, n2 +1)]

for n2 = n1 +1

7 / 33 Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons

To summarize rewrite:

i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n , n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 Translational invariance:

Nk1 = 2πλ1 + θ, Nk2 = 2πλ2 − θ where λi ∈ {0, 1,..., N − 1}

with λi the integer (Bethe) quantum numbers

Two magnon excitations – eigenstates

Solution by parametrization

i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1, n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0

8 / 33 To summarize rewrite:

i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n , n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 Translational invariance:

Nk1 = 2πλ1 + θ, Nk2 = 2πλ2 − θ where λi ∈ {0, 1,..., N − 1}

with λi the integer (Bethe) quantum numbers

Two magnon excitations – eigenstates

Solution by parametrization

i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1, n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0

Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons

8 / 33 Translational invariance:

Nk1 = 2πλ1 + θ, Nk2 = 2πλ2 − θ where λi ∈ {0, 1,..., N − 1}

with λi the integer (Bethe) quantum numbers

Two magnon excitations – eigenstates

Solution by parametrization

i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1, n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0

Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons

To summarize rewrite:

i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n , n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2

8 / 33 Two magnon excitations – eigenstates

Solution by parametrization

i(k1n1+k2n2) 0 i(k1n2+k2n1) a(n1, n2) = Ae + A e where A ei(k1+k2) + 1 − 2eik1 ≡ eiθ = − A0 ei(k1+k2) + 1 − 2eik2 with energy E = J(1 − cos k1) + J(1 − cos k2) + E0

Note: only for A = A0 interpretation as direct product of two one-particle states, i.e. of two non-interacting magnons

To summarize rewrite:

i(k n +k n + 1 θ) i(k n +k n − 1 θ) θ k1 k2 a(n , n ) = e 1 1 2 2 2 + e 1 2 2 1 2 where 2 cot = cot − cot 1 2 2 2 2 Translational invariance:

Nk1 = 2πλ1 + θ, Nk2 = 2πλ2 − θ where λi ∈ {0, 1,..., N − 1} with λi the integer (Bethe) quantum numbers

8 / 33 Figure for N = 32 Karbach and Müller, Computers in

Physics (1997)

+ = 3N 2

1 2

+ = N

1 2

N(N − 1)/2 solutions:

class 1 (red): λ1 = 0

+ = N 2

1 2

⇒ k1 = 0, k2 = 2πλ2/N, θ = 0

class 2 (white): λ − λ ≥ 2

2 1

⇒ real solutions k , k

1 2

class 3 (blue): λ2 − λ1 < 2

⇒ complex solutions 0

0 N 1

k k 2 k1 ≡ 2 + iv, k2 ≡ 2 − iv

Two magnon excitations – eigenstates

Rewrite constraints

θ k k 2 cot = cot 1 − cot 2 2 2 2 Nk1 = 2πλ1 + θ

Nk2 = 2πλ2 − θ

9 / 33 Two magnon excitations – eigenstates

Rewrite constraints Figure for N = 32 Karbach and Müller, Computers in Physics (1997) θ k k 2 cot = cot 1 − cot 2 2 2 2

Nk1 = 2πλ1 + θ

+ = 3N 2

1 2

Nk2 = 2πλ2 − θ

+ = N

1 2

N(N − 1)/2 solutions:

class 1 (red): λ1 = 0

+ = N 2

1 2

⇒ k1 = 0, k2 = 2πλ2/N, θ = 0

class 2 (white): λ − λ ≥ 2

2 1

⇒ real solutions k , k

1 2

class 3 (blue): λ2 − λ1 < 2

⇒ complex solutions 0

0 N 1

k k 2 k1 ≡ 2 + iv, k2 ≡ 2 − iv

9 / 33 Two magnon excitations – dispersion

Nk1 = 2πλ1 + θ Nk2 = 2πλ2 − θ

⇒ k = k1 + k2 = 2π(λ1 + λ2)/N

Figure for N = 32 Karbach and Müller, Computers in Physics (1997) 4

C2 λ λ C3 , 1= 2 3 λ λ C3 , 1= 2-1 )/J 0 2 (E - E

0 0 1 2 3

k 10 / 33 Figure for N = 32 Karbach and Müller, Computers in Physics (1997) 4

C2 free magnons 3 )/J 0 2 (E - E

0 0 1 2 3 k

Two magnon excitations – physical properties classiﬁcation class 1 + 2: almost free scattering states, i.e. for N → ∞ degenerate with direct product of two non-interacting magnons class 3: bound states

11 / 33 Two magnon excitations – physical properties classiﬁcation class 1 + 2: almost free scattering states, i.e. for N → ∞ degenerate with direct product of two non-interacting magnons class 3: bound states

Figure for N = 32 Karbach and Müller, Computers in Physics (1997) 4

C2 free magnons 3 )/J 0 2 (E - E

0 0 1 2 3 k 11 / 33 Figure for N = 128 Karbach and Müller, Computers in Physics (1997)

0.2 )| 2 ,n 1 0.1 3 |a(n

2 k 0 1 -0.25 (n 0 2-n 0.25 1)/N 0

Two magnon excitations – class 3: bound states

J dispersion in thermodynamic limit (N → ∞): E = 2 (1 − cos k) + E0

12 / 33 Two magnon excitations – class 3: bound states

J dispersion in thermodynamic limit (N → ∞): E = 2 (1 − cos k) + E0

Figure for N = 128 Karbach and Müller, Computers in Physics (1997)

0.2 )| 2 ,n 1 0.1 3 |a(n

2 k 0 1 -0.25 (n 0 2-n 0.25 1)/N 0

12 / 33 Pr energy: E = J j=1(1 − cos kj ) + E0

quantum numbers: λi ∈ {0, 1,..., N − 1} determined via

θ k P ij ki j Nki = 2πλi + j6=i θij and 2 cot 2 = cot 2 − cot 2 for i, j = 1,..., r

Solution becomes tedious for N, r 1, but to analyze speciﬁc physical properties, it is sufﬁcient to study particular solutions

Bound states J bound states (class 3) in all subspaces r with dispersion E = r (1 − cos k) + E0 → lowest energy excitations → pure many-body feature

Eigenstates – states with r > 2

P |ψi = a(n1,..., nr )|n1,..., nr i 1≤n1<...

P Pr i P where a(n ,..., nr ) = exp i k n + θ 1 P∈Sr j=1 Pj j 2 i

13 / 33 quantum numbers: λi ∈ {0, 1,..., N − 1} determined via

θ k P ij ki j Nki = 2πλi + j6=i θij and 2 cot 2 = cot 2 − cot 2 for i, j = 1,..., r

Solution becomes tedious for N, r 1, but to analyze speciﬁc physical properties, it is sufﬁcient to study particular solutions

Bound states J bound states (class 3) in all subspaces r with dispersion E = r (1 − cos k) + E0 → lowest energy excitations → pure many-body feature

Eigenstates – states with r > 2

P |ψi = a(n1,..., nr )|n1,..., nr i 1≤n1<...

P Pr i P where a(n ,..., nr ) = exp i k n + θ 1 P∈Sr j=1 Pj j 2 i

Pr energy: E = J j=1(1 − cos kj ) + E0

13 / 33 Solution becomes tedious for N, r 1, but to analyze speciﬁc physical properties, it is sufﬁcient to study particular solutions

Bound states J bound states (class 3) in all subspaces r with dispersion E = r (1 − cos k) + E0 → lowest energy excitations → pure many-body feature

Eigenstates – states with r > 2

P |ψi = a(n1,..., nr )|n1,..., nr i 1≤n1<...

P Pr i P where a(n ,..., nr ) = exp i k n + θ 1 P∈Sr j=1 Pj j 2 i

Pr energy: E = J j=1(1 − cos kj ) + E0 quantum numbers: λi ∈ {0, 1,..., N − 1} determined via

θ k P ij ki j Nki = 2πλi + j6=i θij and 2 cot 2 = cot 2 − cot 2 for i, j = 1,..., r

13 / 33 Bound states J bound states (class 3) in all subspaces r with dispersion E = r (1 − cos k) + E0 → lowest energy excitations → pure many-body feature

Eigenstates – states with r > 2

P |ψi = a(n1,..., nr )|n1,..., nr i 1≤n1<...

P Pr i P where a(n ,..., nr ) = exp i k n + θ 1 P∈Sr j=1 Pj j 2 i

Pr energy: E = J j=1(1 − cos kj ) + E0 quantum numbers: λi ∈ {0, 1,..., N − 1} determined via

θ k P ij ki j Nki = 2πλi + j6=i θij and 2 cot 2 = cot 2 − cot 2 for i, j = 1,..., r

Solution becomes tedious for N, r 1, but to analyze speciﬁc physical properties, it is sufﬁcient to study particular solutions

13 / 33 Eigenstates – states with r > 2

P |ψi = a(n1,..., nr )|n1,..., nr i 1≤n1<...

P Pr i P where a(n ,..., nr ) = exp i k n + θ 1 P∈Sr j=1 Pj j 2 i

Pr energy: E = J j=1(1 − cos kj ) + E0 quantum numbers: λi ∈ {0, 1,..., N − 1} determined via

θ k P ij ki j Nki = 2πλi + j6=i θij and 2 cot 2 = cot 2 − cot 2 for i, j = 1,..., r

Solution becomes tedious for N, r 1, but to analyze speciﬁc physical properties, it is sufﬁcient to study particular solutions

Bound states J bound states (class 3) in all subspaces r with dispersion E = r (1 − cos k) + E0 → lowest energy excitations → pure many-body feature

13 / 33 Contents

1 Introduction

2 Ferromagnetic 1D Heisenberg model

3 Antiferromagnetic 1D Heisenberg model

4 Generalizations

5 Summary and References

14 / 33 Antiferromagnetic 1D Heisenberg model

N X H = J Sn · Sn+1 n=1 N X 1 + − − + = J S S + S S + Sz Sz 2 n n+1 n n+1 n n+1 n=1

Spectrum Eigenvalues inversed as compared to ferromagnetic Heisenberg model, e.g. |Fi ≡ | ↑ ... ↑i state with highest energy

Goals ground-state |Ai magnetic ﬁeld excitations

15 / 33 Ground-state

Classical candidate (no eigenstate): Néel state

|N1i ≡ |↑↓↑ · · · ↓i, |N2i ≡ |↓↑↓ · · · ↑i

Intuitive requirements for true ground-state |Ai: → full rotational invariance → zero magnetization, i.e. r = N/2

Starting from ferromagnetic case: Construction via excitation of N/2 (interacting) magnons from |Fi X |Ai = a(n1,..., nr )|n1,..., nr i with r = N/2

1≤n1<...

16 / 33 quantum numbers {λi } quantum numbers {Ii }

parametrization {ki }, {θij } parametrization {zi } obtained as

ki ≡ π − φ(zi ) where φ(z) ≡ 2 arctan z

θ k ij ki j sgn 2 cot 2 = cot 2 − cot 2 θij = π [<(zi − zj )] − φ (zi − zj )/2 P P Nki = 2πλi + j6=i θij Nφ(zi ) = 2πIi + j6=i φ (zi − zj )/2

such that N 1 N 3 N 1 |Ai ⇔ {I } = − + , − + ,..., − i A 4 2 4 2 4 2

Ground-state

ﬁnite N study reveals

|Ai ⇔ {λi }A = {1, 3, 5,..., N − 1}

17 / 33 Ground-state

ﬁnite N study reveals

|Ai ⇔ {λi }A = {1, 3, 5,..., N − 1}

quantum numbers {λi } quantum numbers {Ii }

parametrization {ki }, {θij } parametrization {zi } obtained as

ki ≡ π − φ(zi ) where φ(z) ≡ 2 arctan z

θ k ij ki j sgn 2 cot 2 = cot 2 − cot 2 θij = π [<(zi − zj )] − φ (zi − zj )/2 P P Nki = 2πλi + j6=i θij Nφ(zi ) = 2πIi + j6=i φ (zi − zj )/2 such that N 1 N 3 N 1 |Ai ⇔ {I } = − + , − + ,..., − i A 4 2 4 2 4 2

17 / 33 3 2 (a) (b) π i / i z 0 1 k

-3 0 -0.2 0 0.2 -0.2 0 0.2

ZN=Ii/N ZN=Ii/N

Karbach, Hu, and Müller, Computers in Physics (1998)

Ground-state

N 1 N 3 N 1 |Ai ⇔ {I } = − + , − + ,..., − i A 4 2 4 2 4 2 obtain zi and with that wave numbers ki by ﬁxed point iteration P Nφ(zi ) = 2πIi + j6=i φ (zi − zj )/2 (n+1) π 1 P (n) (n) ⇒ zi = tan N Ii + 2N j6=i 2 arctan (zi − zj )/2

18 / 33 Ground-state

3 2 (a) (b) π i / i z 0 1 k

-3 0 -0.2 0 0.2 -0.2 0 0.2

ZN=Ii/N ZN=Ii/N

Karbach, Hu, and Müller, Computers in Physics (1998) 18 / 33 For N → ∞ deﬁne continuous variable I ≡ Ii /N

N 1 r − 1 X 1 4X2 Z 1/4 Z ∞ (E − E )/(JN) = ε(z ) = ε(z ) = dI ε(z ) = dz σ ε(z ) F N i N Ii I 0 I i=1 N 1 −1/4 −∞ Ii =− 4 + 2

where dI 1 σ ≡ = from Nφ(z )=2πI +P 2 arctan (z −z )/2 0 dz 4 cosh(πz/4) i i j6=i i j

such that energy (E − EF )/(JN) = ln 2

Energy in the thermodynamic limit

Pr 2 (E − EF )/J = i=1 ε(zi ) where ε(zi ) = −2/(1 + zi ) Pr (remember (E − EF )/J = i=1(1 − cos ki ))) N 1 N 3 N 1 where the sum is over Ii ∈ − 4 + 2 , − 4 + 2 ,..., 4 − 2

19 / 33 N − 1 1 4X2 Z 1/4 Z ∞ = ε(z ) = dI ε(z ) = dz σ ε(z ) N Ii I 0 I N 1 −1/4 −∞ Ii =− 4 + 2

where dI 1 σ ≡ = from Nφ(z )=2πI +P 2 arctan (z −z )/2 0 dz 4 cosh(πz/4) i i j6=i i j

such that energy (E − EF )/(JN) = ln 2

Energy in the thermodynamic limit

Pr 2 (E − EF )/J = i=1 ε(zi ) where ε(zi ) = −2/(1 + zi ) Pr (remember (E − EF )/J = i=1(1 − cos ki ))) N 1 N 3 N 1 where the sum is over Ii ∈ − 4 + 2 , − 4 + 2 ,..., 4 − 2

For N → ∞ deﬁne continuous variable I ≡ Ii /N

r 1 X (E − E )/(JN) = ε(z ) F N i i=1

19 / 33 Z 1/4 Z ∞ = dI ε(zI ) = dz σ0ε(zI ) −1/4 −∞

where dI 1 σ ≡ = from Nφ(z )=2πI +P 2 arctan (z −z )/2 0 dz 4 cosh(πz/4) i i j6=i i j

such that energy (E − EF )/(JN) = ln 2

Energy in the thermodynamic limit

Pr 2 (E − EF )/J = i=1 ε(zi ) where ε(zi ) = −2/(1 + zi ) Pr (remember (E − EF )/J = i=1(1 − cos ki ))) N 1 N 3 N 1 where the sum is over Ii ∈ − 4 + 2 , − 4 + 2 ,..., 4 − 2

For N → ∞ deﬁne continuous variable I ≡ Ii /N

N 1 r − 1 X 1 4X2 (E − E )/(JN) = ε(z ) = ε(z ) F N i N Ii i=1 N 1 Ii =− 4 + 2

19 / 33 Z ∞ = dz σ0ε(zI ) −∞

where dI 1 σ ≡ = from Nφ(z )=2πI +P 2 arctan (z −z )/2 0 dz 4 cosh(πz/4) i i j6=i i j

such that energy (E − EF )/(JN) = ln 2

Energy in the thermodynamic limit

Pr 2 (E − EF )/J = i=1 ε(zi ) where ε(zi ) = −2/(1 + zi ) Pr (remember (E − EF )/J = i=1(1 − cos ki ))) N 1 N 3 N 1 where the sum is over Ii ∈ − 4 + 2 , − 4 + 2 ,..., 4 − 2

For N → ∞ deﬁne continuous variable I ≡ Ii /N

N 1 r − 1 X 1 4X2 Z 1/4 (E − E )/(JN) = ε(z ) = ε(z ) = dI ε(z ) F N i N Ii I i=1 N 1 −1/4 Ii =− 4 + 2

19 / 33 such that energy (E − EF )/(JN) = ln 2

Energy in the thermodynamic limit

Pr 2 (E − EF )/J = i=1 ε(zi ) where ε(zi ) = −2/(1 + zi ) Pr (remember (E − EF )/J = i=1(1 − cos ki ))) N 1 N 3 N 1 where the sum is over Ii ∈ − 4 + 2 , − 4 + 2 ,..., 4 − 2

For N → ∞ deﬁne continuous variable I ≡ Ii /N

N 1 r − 1 X 1 4X2 Z 1/4 Z ∞ (E − E )/(JN) = ε(z ) = ε(z ) = dI ε(z ) = dz σ ε(z ) F N i N Ii I 0 I i=1 N 1 −1/4 −∞ Ii =− 4 + 2 where dI 1 σ ≡ = from Nφ(z )=2πI +P 2 arctan (z −z )/2 0 dz 4 cosh(πz/4) i i j6=i i j

19 / 33 Energy in the thermodynamic limit

Pr 2 (E − EF )/J = i=1 ε(zi ) where ε(zi ) = −2/(1 + zi ) Pr (remember (E − EF )/J = i=1(1 − cos ki ))) N 1 N 3 N 1 where the sum is over Ii ∈ − 4 + 2 , − 4 + 2 ,..., 4 − 2

For N → ∞ deﬁne continuous variable I ≡ Ii /N

N 1 r − 1 X 1 4X2 Z 1/4 Z ∞ (E − E )/(JN) = ε(z ) = ε(z ) = dI ε(z ) = dz σ ε(z ) F N i N Ii I 0 I i=1 N 1 −1/4 −∞ Ii =− 4 + 2 where dI 1 σ ≡ = from Nφ(z )=2πI +P 2 arctan (z −z )/2 0 dz 4 cosh(πz/4) i i j6=i i j such that energy (E − EF )/(JN) = ln 2

19 / 33 Magnetic ﬁeld

N N X X z H = J Sn · Sn+1 − h Sn n=1 n=1

If ﬁeld h strong enough |Fi ≡ | ↑ ... ↑i will become ground-state

groundstate |Ai for very small h |Fi “overtakes” all other states with increasing h

saturation ﬁeld hS = 2J (=energy difference between state |Fi and |k = 0i)

20 / 33 Magnetization

0 0.5 (a) (b) 0.4 -0.2 0.3 )/JN z F m -0.4 0.2 (E - E

0.1 -0.6

0 0 0.25 0.5 0 1 2 z mz=ST/N h/J

Karbach, Hu, and Müller, Computers in Physics (1998) susceptibility inﬁnite slope at the saturation ﬁeld is pure quantum feature

21 / 33 Fundamental excitations are pairs of spinons magnon picture: remove one magnon from |Ai (N/2 → N/2 − 1 quantum numbers) spinon picture: representation as array (gaps are spinons)

Note: Spinons spin-1/2 particles, Magnons spin-1 particles

Two-spinon excitations ground-state N 1 N 3 N 1 |Ai ⇔ {I } = − + , − + ,..., − i A 4 2 4 2 4 2

-N/4+1/2 0 N/4-1/2

(0,0) E 0

(1,1) E q

(1,0) E q

(0,0) E q

Karbach, Hu, and Müller, Computers in Physics (1998)

22 / 33 Two-spinon excitations ground-state N 1 N 3 N 1 |Ai ⇔ {I } = − + , − + ,..., − i A 4 2 4 2 4 2

-N/4+1/2 0 N/4-1/2

(0,0) E 0

(1,1) E q

(1,0) E q

(0,0) E q

Karbach, Hu, and Müller, Computers in Physics (1998)

Fundamental excitations are pairs of spinons magnon picture: remove one magnon from |Ai (N/2 → N/2 − 1 quantum numbers) spinon picture: representation as array (gaps are spinons)

Note: Spinons spin-1/2 particles, Magnons spin-1 particles 22 / 33 Two-spinon excitations: dispersion

¯ ¯ Sum of two spinon wave numbers q = k1 + k2

in contrast to N/2 − 1 wave numbers ki in magnon picture

2 A (E-E )/J 1

0 0 0.5 1 q/π Karbach, Hu, and Müller, Computers in Physics (1998)

π q dispersion boundaries : L(q) = 2 J| sin q|, U (q) = πJ| sin 2 |

23 / 33 Contents

1 Introduction

2 Ferromagnetic 1D Heisenberg model

3 Antiferromagnetic 1D Heisenberg model

4 Generalizations

5 Summary and References

24 / 33 Hubbard model

X † X H = −t (ciscis + h.c.) + U ni↑ni↓ is i

Kondo model

X † † H = k ckscks + J ψ(r = 0)s σss0 ψ(r = 0)s0 · σ0 ks Z s-wave + low energy † † −→ H = −i dx ψ(x)s ∂x ψ(x)s + ψ(x = 0)s σss0 ψ(x = 0)s0 · σ0

Examples for models

Heisenberg model X 1 + − − + H = ±J S S + S S + Sz Sz 2 i i+1 i i+1 i i+1 i

25 / 33 Kondo model

X † † H = k ckscks + J ψ(r = 0)s σss0 ψ(r = 0)s0 · σ0 ks Z s-wave + low energy † † −→ H = −i dx ψ(x)s ∂x ψ(x)s + ψ(x = 0)s σss0 ψ(x = 0)s0 · σ0

Examples for models

Heisenberg model X 1 + − − + H = ±J S S + S S + Sz Sz 2 i i+1 i i+1 i i+1 i

Hubbard model

X † X H = −t (ciscis + h.c.) + U ni↑ni↓ is i

25 / 33 Z s-wave + low energy † † −→ H = −i dx ψ(x)s ∂x ψ(x)s + ψ(x = 0)s σss0 ψ(x = 0)s0 · σ0

Examples for models

Heisenberg model X 1 + − − + H = ±J S S + S S + Sz Sz 2 i i+1 i i+1 i i+1 i

Hubbard model

X † X H = −t (ciscis + h.c.) + U ni↑ni↓ is i

Kondo model

X † † H = k ckscks + J ψ(r = 0)s σss0 ψ(r = 0)s0 · σ0 ks

25 / 33 Examples for models

Heisenberg model X 1 + − − + H = ±J S S + S S + Sz Sz 2 i i+1 i i+1 i i+1 i

Hubbard model

X † X H = −t (ciscis + h.c.) + U ni↑ni↓ is i

Kondo model

X † † H = k ckscks + J ψ(r = 0)s σss0 ψ(r = 0)s0 · σ0 ks Z s-wave + low energy † † −→ H = −i dx ψ(x)s ∂x ψ(x)s + ψ(x = 0)s σss0 ψ(x = 0)s0 · σ0

25 / 33 Thus H|ψi = E|ψi −→ ha = Ea with X X h = −t ∆j + U δnj nl j j

Hubbard model

First steps of systematic solution allow to elucidate fundamental principles.

Hilbert space of N particles spanned by

X Y † |ψi = as1,...,sN (n1,..., nN ) cni si |vaci n1,...,nN i

26 / 33 Hubbard model

First steps of systematic solution allow to elucidate fundamental principles.

Hilbert space of N particles spanned by

X Y † |ψi = as1,...,sN (n1,..., nN ) cni si |vaci n1,...,nN i

Thus H|ψi = E|ψi −→ ha = Ea with X X h = −t ∆j + U δnj nl j j

26 / 33 Two particle case

h = −t(∆1 + ∆2) + Uδn1,n2

Consider n1 = n2 = n as third boundary for the system System consists of two regions A ∩ B ≡ [−L, n] ∩ [n, L]. Clearly, in both regions the Hamiltonian is of non-interacting form! In these subsets the solutions are given by plane waves again!

ansatz:

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

Hubbard model Take large lattice L → ∞ One particle case

h = −t∆ solved by plane waves

27 / 33 Consider n1 = n2 = n as third boundary for the system System consists of two regions A ∩ B ≡ [−L, n] ∩ [n, L]. Clearly, in both regions the Hamiltonian is of non-interacting form! In these subsets the solutions are given by plane waves again!

ansatz:

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

Hubbard model Take large lattice L → ∞ One particle case

h = −t∆ solved by plane waves Two particle case

h = −t(∆1 + ∆2) + Uδn1,n2

27 / 33 ansatz:

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

Hubbard model Take large lattice L → ∞ One particle case

h = −t∆ solved by plane waves Two particle case

h = −t(∆1 + ∆2) + Uδn1,n2

27 / 33 Hubbard model Take large lattice L → ∞ One particle case

h = −t∆ solved by plane waves Two particle case

h = −t(∆1 + ∆2) + Uδn1,n2

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

27 / 33 Note: remember the Heisenberg model

block r = 1: N X |ψi = a(n)|ni n=1 H|ψi = E|ψi ⇔

2[E − E0]a(n) = J [2a(n) − a(n − 1) − a(n + 1)] | {z } = ∆a(n)

block r = 2: X − − |ψi = a(n1, n2)|n1, n2i where |n1, n2i ≡ Sn1 Sn2 |Fi 1≤n1

for n2 > n1 +1 : 2[E − E0]a(n1, n2) =

= J [4a(n1, n2)−a(n1 −1, n2) − a(n1 +1, n2) − a(n1, n2 −1) − a(n1, n2 +1)] | {z } = (∆1 + ∆2)a(n1, n2)

for n2 = n1+1 : 2[E−E0]a(n1, n2) = J[2a(n1, n2)−a(n1−1, n2)−a(n1, n2+1)]

28 / 33 Need to relate the amplitudes As1,s2 and Bs1,s2 in both regions:

0 0 s1,s2 Bs ,s = Ss ,s A 0 , 0 1 2 1 2 s1 s2

Two-particle S-matrix Describes scattering processes in the basis of free particles!

To be obtained by use of symmetries and the Schroedinger equation at n1 = n2.

Summarize this viewpoint Hubbard, Heisenberg and Kondo model subject to local interaction. In the “free” regions, plain waves constitute solutions. Amplitudes of “free” regions related by two-particle S-matrix.

S-matrix and generalization to N particles (back to Hubbard model) We had

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

29 / 33 Two-particle S-matrix Describes scattering processes in the basis of free particles!

To be obtained by use of symmetries and the Schroedinger equation at n1 = n2.

S-matrix and generalization to N particles (back to Hubbard model) We had

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

Need to relate the amplitudes As1,s2 and Bs1,s2 in both regions:

0 0 s1,s2 Bs ,s = Ss ,s A 0 , 0 1 2 1 2 s1 s2

29 / 33 Summarize this viewpoint Hubbard, Heisenberg and Kondo model subject to local interaction. In the “free” regions, plain waves constitute solutions. Amplitudes of “free” regions related by two-particle S-matrix.

S-matrix and generalization to N particles (back to Hubbard model) We had

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

Need to relate the amplitudes As1,s2 and Bs1,s2 in both regions:

0 0 s1,s2 Bs ,s = Ss ,s A 0 , 0 1 2 1 2 s1 s2

Two-particle S-matrix Describes scattering processes in the basis of free particles!

To be obtained by use of symmetries and the Schroedinger equation at n1 = n2.

29 / 33 S-matrix and generalization to N particles (back to Hubbard model) We had

ik1n1+ik2n2 as1,s2 (n1, n2) = Ae (As1,s2 Θ(n1 − n2) + Bs1,s2 Θ(n2 − n1)) | {z } | {z } wavefunction in subset A wavefunction in subset B

Need to relate the amplitudes As1,s2 and Bs1,s2 in both regions:

0 0 s1,s2 Bs ,s = Ss ,s A 0 , 0 1 2 1 2 s1 s2

Two-particle S-matrix Describes scattering processes in the basis of free particles!

To be obtained by use of symmetries and the Schroedinger equation at n1 = n2.

29 / 33 Expand in plane waves over all permutations PR of regions ≡ Bethe ansatz:

P j kj nj X as1,...sN = Ae As1,...,sN (PR )Θ(nPR ) PR

N = 3 particles: → relate the amplitudes of two different regions R1 and R2

ij jk kl A(PR1 ) = S S S A(PR2 ) Usually there are several ways to relate different regions. The consistency of the ansatz requires uniqueness for different paths, i.e.

S23S13S12 = S12S13S23 → Yang-Baxter equation for three particles

Generalization to N particles, Yang Baxter Equation

Generalization to N particles N + 1 regions are obtained, in all of which solutions are given by plane waves and the interaction of which is described by the two-particle S-matrix

30 / 33 N = 3 particles: → relate the amplitudes of two different regions R1 and R2

ij jk kl A(PR1 ) = S S S A(PR2 ) Usually there are several ways to relate different regions. The consistency of the ansatz requires uniqueness for different paths, i.e.

S23S13S12 = S12S13S23 → Yang-Baxter equation for three particles

Generalization to N particles, Yang Baxter Equation

Generalization to N particles N + 1 regions are obtained, in all of which solutions are given by plane waves and the interaction of which is described by the two-particle S-matrix

Expand in plane waves over all permutations PR of regions ≡ Bethe ansatz:

P j kj nj X as1,...sN = Ae As1,...,sN (PR )Θ(nPR ) PR

30 / 33 Generalization to N particles, Yang Baxter Equation

Generalization to N particles N + 1 regions are obtained, in all of which solutions are given by plane waves and the interaction of which is described by the two-particle S-matrix

Expand in plane waves over all permutations PR of regions ≡ Bethe ansatz:

P j kj nj X as1,...sN = Ae As1,...,sN (PR )Θ(nPR ) PR

N = 3 particles: → relate the amplitudes of two different regions R1 and R2

ij jk kl A(PR1 ) = S S S A(PR2 ) Usually there are several ways to relate different regions. The consistency of the ansatz requires uniqueness for different paths, i.e.

S23S13S12 = S12S13S23 → Yang-Baxter equation for three particles

30 / 33 Contents

1 Introduction

2 Ferromagnetic 1D Heisenberg model

3 Antiferromagnetic 1D Heisenberg model

4 Generalizations

5 Summary and References

31 / 33 How may the eigenstates fail to have the Bethe form?

Implicit assumption was made: set of wave numbers {ki } is the same for all regions, i.e. momenta are conserved in interactions. Much stronger than energy or momentum conservation - except for a fermionic two-body interaction in 1D (where it is equivalent to momentum conservation). Feature of integrable models, i.e. of a additional dynamical symmetry expressed by an inﬁnite number of commuting conserved charges. Consequences: S-matrix factorizes in two-particle S-matrices,... But, no problem: all this guaranteed by successful check of the Yang-Baxter equation.

Summary of Section 4

Yang-Baxter equation If the S-matrix derived from the Hamiltonian satisﬁes the Yang-Baxter equation, the Bethe ansatz for the wave functions is consistent and the model is integrable.

32 / 33 Implicit assumption was made: set of wave numbers {ki } is the same for all regions, i.e. momenta are conserved in interactions. Much stronger than energy or momentum conservation - except for a fermionic two-body interaction in 1D (where it is equivalent to momentum conservation). Feature of integrable models, i.e. of a additional dynamical symmetry expressed by an inﬁnite number of commuting conserved charges. Consequences: S-matrix factorizes in two-particle S-matrices,... But, no problem: all this guaranteed by successful check of the Yang-Baxter equation.

Summary of Section 4

Yang-Baxter equation If the S-matrix derived from the Hamiltonian satisﬁes the Yang-Baxter equation, the Bethe ansatz for the wave functions is consistent and the model is integrable.

How may the eigenstates fail to have the Bethe form?

32 / 33 Much stronger than energy or momentum conservation - except for a fermionic two-body interaction in 1D (where it is equivalent to momentum conservation). Feature of integrable models, i.e. of a additional dynamical symmetry expressed by an inﬁnite number of commuting conserved charges. Consequences: S-matrix factorizes in two-particle S-matrices,... But, no problem: all this guaranteed by successful check of the Yang-Baxter equation.

Summary of Section 4

Yang-Baxter equation If the S-matrix derived from the Hamiltonian satisﬁes the Yang-Baxter equation, the Bethe ansatz for the wave functions is consistent and the model is integrable.

How may the eigenstates fail to have the Bethe form?

Implicit assumption was made: set of wave numbers {ki } is the same for all regions, i.e. momenta are conserved in interactions.

32 / 33 Feature of integrable models, i.e. of a additional dynamical symmetry expressed by an inﬁnite number of commuting conserved charges. Consequences: S-matrix factorizes in two-particle S-matrices,... But, no problem: all this guaranteed by successful check of the Yang-Baxter equation.

Summary of Section 4

Yang-Baxter equation If the S-matrix derived from the Hamiltonian satisﬁes the Yang-Baxter equation, the Bethe ansatz for the wave functions is consistent and the model is integrable.

How may the eigenstates fail to have the Bethe form?

32 / 33 But, no problem: all this guaranteed by successful check of the Yang-Baxter equation.

Summary of Section 4

Yang-Baxter equation If the S-matrix derived from the Hamiltonian satisﬁes the Yang-Baxter equation, the Bethe ansatz for the wave functions is consistent and the model is integrable.

How may the eigenstates fail to have the Bethe form?

32 / 33 Summary of Section 4

Yang-Baxter equation If the S-matrix derived from the Hamiltonian satisﬁes the Yang-Baxter equation, the Bethe ansatz for the wave functions is consistent and the model is integrable.

How may the eigenstates fail to have the Bethe form?

32 / 33 Exact ground-state of the antiferromagnetic case in the thermodynamic limit. Magentic ﬁeld and two spinon excitations.

References Section 1: Bethe, ZS. f. Phys. (1931) Section 2: Karbach and Müller, Computers in Physics (1997), arXiv: cond-mat/9809162 Section 3: Karbach, Hu, and Müller, Computers in Physics (1998), arXiv: cond-mat/9809163 Section 4: N. Andrei, “Integrable models in condensed matter physics”, ICTP lecture notes (1994), arXiv: cond-mat/9408101 Thank you for your attention!

Summary of Section 2 and 3 + References

Ferromagnetic and antiferromagnetic Heisenberg model Exact eigenstates and eigenenergies for the ferromagnetic case. Lowest excitations are bound states.

33 / 33 References Section 1: Bethe, ZS. f. Phys. (1931) Section 2: Karbach and Müller, Computers in Physics (1997), arXiv: cond-mat/9809162 Section 3: Karbach, Hu, and Müller, Computers in Physics (1998), arXiv: cond-mat/9809163 Section 4: N. Andrei, “Integrable models in condensed matter physics”, ICTP lecture notes (1994), arXiv: cond-mat/9408101 Thank you for your attention!

Summary of Section 2 and 3 + References

Ferromagnetic and antiferromagnetic Heisenberg model Exact eigenstates and eigenenergies for the ferromagnetic case. Lowest excitations are bound states. Exact ground-state of the antiferromagnetic case in the thermodynamic limit. Magentic ﬁeld and two spinon excitations.

33 / 33 Summary of Section 2 and 3 + References

33 / 33