Exact regimes of collapsed and extra two-string solutions in the two down-spin sector of the spin-1/2 massive XXZ spin chain

Exact regimes of collapsed and extra two-string

solutions in the two down-spin sector of the

spin-1/2 massive XXZ spin chain

Takashi Imoto

In collaboration with Jun Sato and Tetsuo Deguchi

(arXiv:1807.08885)

Talk plan

We analyze the Bethe ansatz equations of the anti-ferromagnetic massive XXZ spin chain in two down-spin sector.

▶ We show that an extra two string solution appears, which is not predicted by the string hypothesis.

▶ We obtain the condition of the collapse. The collapse means some of two-string solutinos predicted by the string-hypothesis become real solutions.

Spin-1/2 XXZ chain Hamiltonians

The Hamiltonian of the XXZ spin chain under the P.B.C is given by

H

=

∑

[

σ

σ

+

σ

σ

+ ∆

σ

σ

]

where

σ

(

a

=

X

,

Y

,

Z

)

are the Pauli matices defined on the

jth site

,

∆

denotes the anisotropic parameter, and

N is the site number.

▶

|∆| >

1: massive regime

▶

₋

1

≤ ∆ ≤

1: massless regime

Bethe ansatz equations(BAE)

1

In M down-spin sector with rapidities

λ

, λ

, · · · , λ

, the Bethe

ansatz equations(BAE) of the spin-1/2 XXZ Heisenberg spin chain

in the massive regime are given by

Arctangent function-part 1

A factor of RHS of the BAE(1) sin(λj− λk+iζ) sin(λj− λk−iζ) =tan(λj− λk)/tanh(ζ) +i tan(λj− λk)/tanh(ζ) −i = (−1)exp [ −2i tan−1 ( tan(λj− λk) tanh(ζ) )] . (2)

In the right hand side of (2) the differenceλj− λk may be greater thanπ/2. In order to make the phase factor continuous atλj− λk =π/2 Takahashi introduced the following function2_{for real variableλ,}

θn(λ; ζ) =2 tan−1 ( tanλ tanh(nζ/2) ) +2π [ 2λ + π 2π ] (3) where[x]is the greatest integer that is not larger thanx. We remark

−π/2<Re(tan−1(x))< π/2 for−∞ <x< ∞. This formulation is not analytically continuous atλ = π/2. The logarithmic form of the BAE is

θ1(λj;ζ) = 2π NIj+ 1 N M ∑ k,j,k=1 θ2(λj− λk;ζ). (4)





This formulation is not analytically continuous atλ = π/2.





Arctangent function-part 2

In order to derive complex solutions we analytically continue the functionsθn(λ) through the two branches of the logarithmic function.

θn(λ; ζ) = 1 i ( log(+) ( tanλ tanh(nζ/2) ) −log(s) ( tanλ tanh(nζ/2) )) . (5) whereα, β ∈ R,n∈ Z, log(s)_{(α +iβ) =i(θ}(s)_{(α +iβ) +2πn) +}1 2log(α 2_+β2_{) (6)} log(+)(α +iβ) =i(θ(+)_{(α +iβ) +} 2πn) +1 2log(α 2 +β2) (7) and θ(s)_{(α +iβ) =} tan−1(β/α) + πH(−α)sgn(β+) (forα ,0) sgn(β+)π/2 (forα =0) θ(+)_{(α +} iβ) =_tan −1_{(β/α) + πH(−α) +2πH(α)H(−β) (forα ,0)} sgn(β+)π/2+2πH(−β) (forα =0) Here the step functionH(x)and the sign functionssgn(x+),sgn(x−)are given by

H(x) =_1 (for x>0)

Arctangent function-part 3

In this talk, we use the following definition.

θn(λ; ζ) = 1 i ( log(+) ( tanλ tanh(nζ/2) ) −log(s) ( tanλ tanh(nζ/2) )) .

Therefore, the logarithmic form of the BAE is

θ1(λj;ζ) = 2π NJj+ 1 N M ∑ k,j,k=1 θ2(λj− λk;ζ). When we use this definition, the arctangent function 2tan[−1(α +iβ)is

2tan[−1(α +iβ) =1 i ( log(+)(1+i(α +iβ)) −log(s)(1−i(α +iβ)) ) =tan−1( α 1− β ) +πH(β −1)sgn(α+) +2πH(1− β)H(−α) +tan−1( α 1+β ) − πH(−β −1)sgn(α₋) + 1 2ilog (_α2_{+ (β −1)}2 α2_{+ (β +1)}2 ) . (8)





This formulation is analytically continuous atλ = π/2 forλ ∈ C.

The logarithmic form of BAE in the massive regime of

XXZ spin chain

The logarithmic form of Bethe ansatz equations are given by

2

tan

[

(

tan

λ

tanh

(

ζ/

2

)

)

=

2

π

N

J

+

1

N

∑

2

tan

[

(

tan

(

λ

− λ

)

tanh

ζ

)

(9)

J

≡

1

2

(

N

−

M

−

1

) (

mod1

)

for

i

=

1

,

2

, · · · ,

M

Massive regime in two down-spin sector

The logarithmic form of Bethe ansatz equations in two down-spin sector

2 [

tan

(

tan

λ

tanh

ζ/2

)

=

2

π

N

J

+

2

N

[

tan

(

tan(

λ

− λ

)

tanh

ζ

)

(10)

2 [

tan

(

tan

λ

tanh

ζ/2

)

=

2

π

N

J

+

2

N

[

tan

(

tan(

λ

− λ

)

tanh

ζ

)

(11)

We call

J

, J

the Bethe quantum numbers.

Self-conjugacy

{λ

1

, λ

, · · · , λ

} = {λ

, λ

, · · · , λ

}.

(12)

2-string solution

We assume the form of a two-string solution as

λ

=

x +

i

2

ζ + iδ, λ

=

x

−

i

2

ζ − iδ (x, δ ∈ R)

(13)

where

x: string center,

δ: string deviation. We remark δ > −ζ/2.

The Bethe ansatz equation(10) is equivalent to 2π NJ1= tan −1( a 1− b ) + tan−1 ( a 1 +b ) +π ( H(b− 1) + 2H(1 − b)H(−a) −H(δ) N ) | {z } real part +1 2ilog {(_a2_{+ (b− 1)}2 a2_{+ (b + 1)}2 )(_{(tanh(ζ + 2δ)/ tanh(ζ) + 1)}2 (tanh(ζ + 2δ)/ tanh(ζ) − 1)2 )1/N} | {z } imaginary part . (14) a := tanx(1− w 2_t2₎ t(1 + (tan2_x)w2_t2₎ b := (1 + tan2_x)w ( 1 + (tan2_x)w2_t2) (15)

wherew = tanh(ζ/2 + δ)/tanh (ζ/2), t = tanh (ζ/2) and H(y) = {

1 fory> 0 0 otherwise . We remark thatw depend onδ.

Counting function

We define the counting function as

The real part of the Bethe ansatz equation Z1(δ(w),x(w), ζ) := J1 N Z2(δ(w),x(w), ζ) := J2 N. (16)

We obtain the relation betweenZ1(w)andZ2(w): Z2(w)−Z1(w) =

1

NH(δ) −H(−a). (17)





If we find the Bethe quantum numbersJ1,J2which the solution of the BAE exists corrsponding to, we obtain the solutions of BAE corresponding to the Bethe quantum numbers numerically.





The imaginary part of the Bethe ansatz equation

( a2_{+ (b−1)}2 a2_{+ (b+1)}2 ) = ( (tanh(ζ +2δ)/tanh(ζ) +1)2 (tanh(ζ +2δ)/tanh(ζ) −1)2 )1/N . (18)





We regard the imaginary part(18) of the Bethe ansatz equation(10) as a con-straint onx andλ. With using this constraint, we analyze the real part.

The imaginary part of the Bethe ansatz equation(10) is equivalent

to the quadratic equation of

X

=

tan

_{x as follows.}

A (w)X2+B(w)X + C(w) = 0, (19) where A (w) = w2(1 +wt2)2 {( −(1 − w)(1 − wt2₎ 1 +w2_t2 )2}1/N − w2_{(1− wt}2₎2 {( (1 +w)(1 + wt2₎ 1 +w2_t2 )2}1/N . B(w) = {_{(1− w}2_t2₎2 t2 + 2w(1 + w)(1 + wt 2₎ }{(_{−(1 − w)(1 − wt}2₎ 1 +w2_t2 )2}1/N − { (1− w2t2)2 t2 − 2w(1 − w)(1 − wt 2₎ }{( (1 +w)(1 + wt2) 1 +w2_t2 )2}1/N . C(w) = (1 + w)2 {(_{−(1 − w)(1 − wt}2₎ 1 +w2_t2 )2}1/N − (1 − w)2 {(_{(1 +w)(1 + wt}2₎ 1 +w2_t2 )2}1/N .

wherew = tanh(ζ/2 + δ)/tanh (ζ/2), t = tanh (ζ/2). Solutions of eq(19) is given by

We consider the interval ofw satisfying(tan2_{x =)X(w)>0.}

Proposition

There doesn’t exist w satisfying X+(w)>0. X₋(w)>0if and only if we have A(w)<0.

Therefore,the domain of definition for the counting functionZ(w)is equivalent to the interval ofw satisfying A(w)<0.

Stable and unstable regime

▶ stable regime:A(0)>0 case (i.e.X(0)<0) ▶ unstable regime:A(0)≤0 case (i.e.X(0)≥0)

A(0) =0⇔tanh2(ζ/2) = 1

Monotonicity of the counting function

We definew1,w2,w3as

w1≡min{w|A(w)≥0}, w2≡1, w3≡max{w|A(w)≥0} (23) We partially prove the monotonicity of the counting functionZ(w)inw∈ [w1,w2] andw∈ [w2,w3].

▶ w∈ [w1,w2]in stable regime,Z(w)decreases monotonically ▶ w∈ [w2,w3]in stable regime,Z(w)increases monotonically ▶ w∈ [w2,w3]in unstable regime,Z(w)increases monotonically

▶ w∈ [w1,w2]in unstable regime, the monotonicity of the counting function Z(w)is conjecture but numerically checked.

Monotonicity of the counting function

We definew1,w2,w3as

w1≡min{w|A(w)≥0}, w2≡1, w3≡max{w|A(w)≥0} (24) We partially prove the monotonicity of the counting functionZ(w)inw∈ [w1,w2] andw∈ [w2,w3].

▶ w∈ [w1,w2]in stable regime,Z(w)decreases monotonically ▶ w∈ [w2,w3]in stable regime,Z(w)increases monotonically ▶ w∈ [w2,w3]in unstable regime,Z(w)increases monotonically

▶ w∈ [w1,w2]in unstable regime, the monotonicity of the counting function Z(w)is conjecture but numerically checked.

narrow pair/wide pair

If

w

<

1, we call the solution of BAE a narrow pair, because

δ

is

negative.

If

w

>

1, we call the solution of BAE a wide pair, because

δ

is

positive.

We remark

w

=

tanh

(

ζ/

2

+

δ)

/

tanh

(

ζ/

2

)

,

t

=

tanh

(

ζ/

2

)

Stable regime

Real part of the Bethe ansatz equation

NZ

(w) = J

(25)

where

t = tanh(ζ/2) and w = tanh(ζ/2 + δ)/ tanh(ζ/2)

If

w

< 1, we call the solution of BAE a narrow pair, because δ is negative.

If

w

> 1, we call the solution of BAE a wide pair, because δ is positive.





Given the Bethe quantum number

J

has the point of intersection

with the counting function

Z (w), we obtain the solutions of BAE

cor-responding to the Bethe quantum numbers numerically.

Unstable regime

Real part of the Bethe ansatz equation

NZ

(w) = J

(26)

where

t = tanh(ζ/2) and w = tanh(ζ/2 + δ)/ tanh(ζ/2)

If

w

< 1, We call the solution of BAE a narrow pair, because δ is negative.

If

w

> 1, We call the solution of BAE a wide pair, because δ is positive.





Given the Bethe quantum number

J

has the point of intersection

with the counting function

Z (w), we obtain the solutions of BAE

cor-responding to the Bethe quantum numbers numerically.

Bethe quantum numbers

N 4 ≤ J1< N 2 (tanx> 0) − N 2 < J1≤ − N 4 (tanx< 0) narrow pair(unstable regime)

N 4 ≤ J1< N πtan−1 ( √ N− (1 + t2₎ 1− (N − 1)t2 ) (tanx> 0) −N 2 < J1≤ − N 4 (tanx< 0) singular solution (J1, J2) = ( N 4 − 1 2, N 4 + 1 2) and (− N 4 − 1 2, − N 4 + 1

2) for site numberN = 4n n∈ Z (J1, J2) = ( N 4, N 4) and (− N 4, − N

Collapse and extra two string solution

The conditions that the collapse ofm two-string solutions occurs in the chain of N sites are given by

Z₀(ζ,N)<N− (1+2m)

2N form=1,2, · · · . (28) In the limit of sendingζto 0(the XXX limit)

lim ζ→0Z (ζ,N) 0 = 1 πtan−1 ( √ N−1 ) <N− (1+2m) 2N . (29)

It coincides with that of the XXX chain.4

On the other hand, an extra pair of two-strings appears. If the following condition is satisfied, the number of two string-solutions is by two larger than the number due to the string solution. We call it the extra two string solution. The condition that the extra two string solution appear is given by

N−1 2N <Z

(ζ,N)

0 . (30)

The following is a summary of the above

Conjecture

If N and

ζ satisfy

tanh

(

ζ/2) <

1

− (N − 1) tan

(

_{π(1 + 2m)/2N}

)

(N

− 1) − tan

(

_{π(1 + 2m)/2N}

)

(31)

the collapse of m two-string solutions occurs.

Conjecture

If N and

ζ satisfy

tanh

(

ζ/2) >

1

− (N − 1) tan

(

π(1 + 2m)/2N

)

(N

− 1) − tan

(

_{π(1 + 2m)/2N}

) ,

(32)

an extra pair of two-string solutions appears.





If we prove the monotonicity of counting function on [w

, w

] , these

conjectures become theorems. We have checked it numerically.

Figure of collapse and extra two-string solutions-part 1

∆ =

cosh

ζ

▶

tanh

(

ζ/

2

)

≥

:stable regime

Figure of collapse and extra two-string solutions-part 2

∆ =

cosh

ζ

▶

m1: 1 collapsed regime

▶

m2: 2 collapsed regime

Numerical values for extra two-string solution

Numerical two-string solutions of the BAE for

N

=

12 in the two

down-spin sector, at

ζ =

0

.

6. No.1 and No.6 are extra complex

solutions, which are not predicted by the string hypothesis.

Numerical values for singular solution

Numerical two-string solutions of the BAE for

N

=

12 in the two

down-spin sector, at

ζ =

0

.

6. No.9 is the singular solution. It has

another set of quantum number

(

5

/

2

,

7

/

2

)

Conclusion

▶

We have obtained the condition of the collapse. (i.e. Some of

two-string solutions predicted by the string-hypothesis

become real solutions.)

▶

We have shown that the extra two string solution appears. (i.e.

The number of two-string solutions by two larger than that of

the string hypothesis.)

Rigolous proof of complex solutions approaching

complete strings exponentially with respect to

w

We have obtained next two fact about the complete solutions.

Fact

If

N is large enough, w

becomes close to 1 exponentially fast with

respect to

N.

Fact

If

N is large enough, w

becomes close to 1 exponentially fast with

respect to

N.

Therefore

Fact

Numerical solutions of two-srings

We check this result with the numerical solutions. Parameters

(∆

,

N

)

are

∆ =

1

.

001 and

N

=

1000

,

2000

,

3000

,

6000





When

N is large, the string deviation is small.

Check point of monotonicity of counting function