Elementary functions in Thermodynamic Bethe Ansatz

0.3 MB·English

by  Junji Suzuki

Elementary functions
in
Thermodynamic Bethe Ansatz
5 ∗
1
0
2 J. Suzuki
†
r
p
A Department of Physics, Faculty of Science
Shizuoka University,
2
2
Ohya 836, Shizuoka,

] Japan
h
p
-
h
t January 2015
a
m
[

3
v Abstract
3
7 Some years ago, Fendley found an explicit solution to the thermo-
7
dynamic Bethe ansatz (TBA) equation for a = 2 supersymmetric
0 N
0 theory in 2D with a specific F-term. Motivated by this, we seek for
. explicitsolutionsforothersuper-potentialcasesutilizingtheideafrom
1
0 the ODE/IM correspondence. We find that the TBA equations, cor-
5
responding to a wider class of super-potentials, admit solutions in
1
: terms of elementary functions such as modified Bessel functions and
v
confluent hyper-geometric series.
i
X
r
a
∗Basedontalksgivenat“InfiniteAnalysis2014”(Tokyo,Feb. 2014)andat“Integrable
lattice models and quantum field theories” (Bad-Honnef, June 2014)
†e-mail: [email protected]
1
1 Introduction
The Thermodynamic Bethe Ansatz (TBA) is one of the most efficient tools
in the field of integrable systems [1]. Once input data such as factorized
S- matrices [2, 3], special patterns of Bethe ansatz roots (string hypothesis)
[4, 5], or the fusion relations [6, 7] are given, it provides finitely or infinitely
many coupledintegrals equationsasoutput. These equations makethequan-
titative analyses possible in integrable 1+1D quantum field theories of finite
size [2] or 1D quantum systems at finite temperatures [1]. The numerical
analysis provides physical quantities such as the specific heat or the mag-
netic susceptibility for the whole range of temperature [9], or flow of the
g function by the change in the system size [8]. On the other hand, some
limited information, such as the central charge, is available analytically from
the TBA equations. This is due to the fact that the nonlinearity of the TBA
equations defies explicit solutions in most of cases.
Someyearsago, Fendley[10]obtainedarareexample: anexplicit solution
in the massless limit of an integrable = 2 supersymmetric theory in 2D.
N
There is a deep structure behind the model which connects the solution
to the (massive) TBA equation and the solution to the Painlev´e III (PIII)
equation. The proof in [10] replies on the heavy machinery on the solution
to the PIII equation which has been developed in [11, 12, 13]. Especially
it utilizes the Tracy-Widom representation to the PIII solution, valid for a
massive theory in general, while the explicit solution in terms of elementary
function is possible only in the massless case. Then one may wonder if any
simpler derivation is possible for the result in [10], as the massless theory
possesses a larger symmetry and thus offers a simpler structure.
Inthiscommunication, wewillarguethattheOrdinaryDifferentialEquation/
Integrable Model (ODE/IM) 1 correspondence provides a much simpler ex-
planation of the solution. This program was actually suggested in [10]. We
will make it concrete. The Stokes multipliers τ, associated with a simple
ODE with an irregular singularity at infinity, turns out to provide the solu-
tionin[10]. This maysound oddasthereseems tobeno relationbetween the
original problem and the ODE thus there is no reason to consider a specific
ODE. There is, however, a relation. We take the super-potential correspond-
ing to Fendley’s solution. From the potential, we construct a function, which
solves an ODE. At this stage, the Stokes multiplier is a trivial constant. We
then “deform” the ODE by a weak gauge field (or small angular momen-
tum). Remarkably, the first nontrivial response of the Stokes multiplier to
the gauge field reproduces the solution in [10].
1For review see [16]
2
This immediately leads to a generalization. There exists a list of relevant
super-potentials [17, 18, 19] for the Landau-Ginzburg description of super-
conformal theories. The corresponding TBA equations for perturbed cases
are partially derived in [14, 15]. Thus, starting from one of the available po-
tentials, we can construct an ODE and evaluate the first non trivial response
to the weak gauge field. The resultant Stokes multipliers are then trans-
formed automatically into Y functions. We will show that these Y functions
solve TBA equations in perturbed = 2 minimal theories with the SU(2)
k
N
and with the SU(3) chiral rings, which generalize Fendley’s solution for the
1
SU(2) .
1
This paper is organized as follows. Section 2 is devoted to a short review
on the ODE/IM correspondence. Fendley’s solution is re-derived from the
Stokes multiplier associated to a special ODE in section 3. In section 4, we
apply the working hypothesis obtained in the previous section to the TBA
equations for perturbed = 2 minimal theories in 2D with the SU(2)
k
N
and the SU(n) chiral rings. We demonstrate the applications of the exact
1
solutionsinsection5. Section6isdevotedtoasummaryandfutureproblems.
2 The ODE/IM correspondence
We summarize results from the ODE/IM correspondence which are relevant
in the following discussions. For details, see [16].
We consider a simple ODE of nth order in the complex plane x C,
∈
dn
( )n−1 +(xnα E) ψ(x,E) = 0, (1)
− dxn −
(cid:16) (cid:17)
where α R .
≥−1
∈
Since it has the irregular singularity at , we conveniently divide the
∞
complex plane into sectors. Let be a sector in the complex plane,
j
S
2jπ π
= x argx < .
j
S − n(α+1) n(α+1)
n (cid:12) (cid:12) (cid:12) o
(cid:12)
(cid:12) (cid:12)
The sector thus includ(cid:12)es the positive real axis.
0
S
Let φ(x,E) be a solution to (1) which decays exponentially as x tends to
inside ,
0
∞ S
dpφ(x,E) x(1−n+2p)α xα+1
( 1)p 2 exp , x (2)
dxp ∼ − √ni(n−1)/2 −α+1 ∈ S0
(cid:16) (cid:17)
for p Z .
≥0
∈
3
The crucial observation in [21] is the “discrete rotational symmetry” of
(1): the invariance under the simultaneous transformations
x q−1x, E EΩn,
→ →
q = en(2απ+i1), Ω = q−α. (3)
We then introduce
φ = q(n−1)j/2φ(q−jx,ΩnjE). (4)
j
Thanks to the discrete rotational symmetry, any φ (j Z) is a solution to
j
∈
(1). The set (φ , ,φ ) forms the Fundamental System of Solutions
j j+n−1
···
(m)
(FSS) in . To see this, we introduce the Wronskian matrix Φ (x,E)
Sj j1,···,jm
(m) (m)
and the Wronskian W (x,E) = detΦ (x,E),
j1,···,jm j1,···,jm
φ φ
j1 ··· jm
. .
Φ(m) (x,E) = .. ..
j1,···,jm  
(m−1) (m−1)
φ φ
 j1 ··· jm 
 
where m n. Especially, when suffixes j are consecutive integers, e.g.,
≤ { }
(m) (m)
j = j+k 1 we write simply Φ (x,E) and their determinants W (x,E)
k − j j
(we drop the x dependency when m = n). By using the asymptotic form
(n)
(2), one can check W (E) = 1, hence the set (φ , ,φ ) is linearly
j j ··· j+n−1
independent.
We are interested in the relation among the FSS in different sectors. Let
(n)
us start from the relation between and . Two Wronskian matrices Φ
S0 S1 0
(n)
and Φ are simply connected by
1
Φ(n) = Φ(n) (n)(E) (5)
0 1 M
where
(1)
τ (E) 1 0 0
1 ···
(2)
τ (E) 0 1 0
(n)(E) =  − 1. ··· . (6)
M .. 1
 
( 1)n−1τ(n)(E) 0 0 0
 − 1 ··· 
 
(a)
The entries τ (E) are called the Stokes multipliers. The discrete rotational
1
symmetry then results in
Φ(n) = Φ(n) (n)(EΩnj). (7)
j j+1M1
4
The first observation of the ODE/IM correspondence is that this linear re-
lation, evaluated at the origin, can be identified with Baxter’s TQ relation
[22] for n = 2. That is, let
T (E) = τ(1)(EΩ−2), Q−(E) = φ(0,E), Q+(E) = φ′(0,E). (8)
1 1
Then one presents τ = φ /φ +φ /φ equivalently as
1 0 1 2 1
Q∓(EΩ−2) Q∓(EΩ2)
T1(E) = q∓21 Q∓(E) +q±21 Q∓(E) . (9)
This is known as the Dressed Vacuum Form (DVF) in integrable models.
Suppose that wave functions are given in advance. Then, thanks to the
normalization of φ, the Stokes multipliers are represented by wave functions,
e.g.,
(1) (n)
τ (E) = W (E). (10)
1 0,2,···,n
Let us introduce more generally
τ(a)(E) = W(n) (E). (11)
m 0,···,a−1,a+m,···,n+m−1
Some of them appear in the connection problem between and [35].
0 m
S S
The identity among Wronskians implies
τ(a)(E)τ(a)(EΩn) = τ(a+1)(E)τ(a−1)(EΩn)+τ(a) (E)τ(a) (EΩn) (12)
m m m m m+1 m−1
where 1 a n 1, m Z , τ(0) = τ(n) = 1 and τ(a) = 1.
≤ ≤ − ∈ ≥1 m m 0
After a suitable shift of the parameters and a change of variables (E
→
v,τ T), one arrives at the SU(n) T system [29],
→
T(a)(v +i)T(a)(v i) = T(a+1)(v)T(a−1)(v)+T(a) (v)T(a) (v). (13)
m m − m m m+1 m−1
(a) (0) (n)
The conditions T (v) = T (v) = T (v) = 1 are again imposed. By
0 m m
employing the further transformation [28, 29],
(a) (a)
T (v)T (v)
Y(a)(v) = m−1 m+1 , (14)
m (a+1) (a−1)
T (v)T (v)
m m
one obtains the SU(n) Y system,
(a) (a)
(1+Y (v))(1+Y (v))
Y(a)(v +i)Y(a)(v i) = m−1 m+1 . (15)
m m − (1+(Y(a+1)(v))−1)(1+(Y(a−1)(v))−1)
m m
The Y system for ADE scattering models was originally introduced in [30].
5
(a)
It is well known under the assumption of the analytic properties on Y
m
that the above algebraic equations can be transformed into the TBA equa-
tions. This also manifests the ODE/IM correspondence.
We can also formulate the problem on the positive real axis. To sim-
plify notations, let us concentrate on the case n = 2 (the radial Schro¨dinger
problem),
d2 ℓ(ℓ+1)
+(x2α E)+ ψ(x,E,ℓ) = 0. (16)
−dx2 − x2
(cid:16) (cid:17)
This is also regarded as the introduction of a gauge field when rewriting it
as
d ℓ d ℓ
( )( + )+(x2α E) ψ(x,E,ℓ) = 0. (17)
− dx − x dx x −
(cid:16) (cid:17)
While, in the absence of the gauge field, the Q function is directly related
to the value of the wave function at the origin, as in (8), it is no longer the
case with the presence of the gauge field. It is however shown in [23, 21] that
the Q function appears naturally if one considers the connection problem of
the FSS near the origin and the FSS at large x. Denote two solutions near
the origin,
1
χ±(x,E,ℓ) x±(ℓ+12)+12
∼ √2ℓ+1
and set more generally, analogously to (4),
χ±(x,E,ℓ) = q2jχ±(q−jx,Ω2jE,ℓ).
j
The x 0 behavior implies
→
χ±(x,E,ℓ) = q∓j(ℓ+21)χ±(x,E,ℓ).
j
The “radial” connection relation is given by 2 ,
φ(x,E,ℓ) = D−(E,ℓ)χ−(x,E,ℓ)+D+(E,ℓ)χ+(x,E,ℓ), (18)
or equivalently,
φ (x,E,ℓ) = D−(EΩ2j,ℓ)χ−(x,E,ℓ)+D+(EΩ2j,ℓ)χ+(x,E,ℓ).
j j j
The connection relations among φ (x,E,ℓ) assume the same form, e.g, (5).
j
One then derives the DVF in the radial problem as
D∓(EΩ−2,ℓ) D∓(EΩ2,ℓ)
T1(E,ℓ) = q∓(12+ℓ) D∓(E,ℓ) +q±(21+ℓ) D∓(E,ℓ) . (19)
2We change the sign of D+ from [21]
6
By comparing with (9), one concludes that D± generalizes Q± for non-zero
ℓ case. They recover (9) by putting ℓ = 0. In terms of D±, the Wronskian
representation of the generalized Stokes multipliers (10) is given by
Tj(E,ℓ) =q−(j+1)(ℓ+21)D+(EΩj+1,ℓ)D−(EΩ−(j+1),ℓ)
q(j+1)(ℓ+12)D−(EΩj+1,ℓ)D+(EΩ−(j+1),ℓ). (20)
−
This is to be identified with the quantum Wronskian relation [24], except
for a difference in normalization as discussed in [21].
3 Revisiting Fendley’s solution
In [14, 15], a class of integrable =2 supersymmetric theories in 2D, de-
N
scribed by Landau-Ginzburg actions, has been analyzed. For models with
spontaneously broken Z symmetry a set of TBA equations has been pro-
n
posed. Especially in the latter paper, direct relations of the solution to TBA
equations and solutions to PIII or to affine Toda equations are argued.
When the super-potential is given by W(X) = X3 X, the explicit TBA
3 −
equations read,
∞ dθ′ 1
A(θ,µ) = 2u(θ,µ) ln(1+B(θ′,µ)2),
− 2π cosh(θ θ′)
Z−∞ −
∞ dθ′ 1
B(θ,µ) = e−A(θ′,µ). (21)
− 2π cosh(θ θ′)
Z−∞ −
In the above, u(θ,µ) = µcoshθ and µ corresponds to a physical mass. It
reduces to eθ/2 in the massless limit [31].
Fendley found the following explicit solution [10] for the massless case,
d
e−A(θ) = 2π (Ai(z))2,
− dz
d
B(θ) = 2π Ai(zeiπ3)Ai(ze−iπ3) (22)
dz
where z = (3eθ/4)2/3.
We will re-derive the solution from the ODE side, starting from,
d2
+(x E) ψ(x,E) = 0 (23)
−dx2 −
(cid:16) (cid:17)
which is the case (n,α) = (2, 1) in (1). It follows from (3) that
2
q = e23πi, Ω = e−π3i.
7
It is well known that the Airy function solves this equation. Respecting the
leading asymptotic from (2), the desired solution of (23) is given by
2π
φ(x,E) = Ai(x E). (24)
i −
r
This immediately gives Q± 3 in (8)
Q−(E) = 2πAi( E),
i − (25)
Q+(E) = q 2π d Ai( E).
 − i dE −
q
We remark that the ODE (23) is not totally independent of the original
problem.
Although in the = 2 symmetric theory, the argument of the super-
N
potential W(X) is a super-field X, we allow for a usual variable in W(x).
Then the solution φ(x,E) has a well known integral representation,
3 −1
φ(x,E) = e(x−E)2W(z(x−E) 2)dz. (26)
ZC
The contour must be chosen so as to reproduce the asymptotic behavior (2)
of φ. Respecting the “discrete rotational symmetry” and the change of the
integration contour, it is easy to show
φ(x,E)+e23πiφ(e23πix,e23πiE)+e−23πiφ(e−23πix,e−23πiE) = 0. (27)
This is equivalent to the three terms relation for the Airy function and it
leads to the conclusion τ = T = 1. We can easily check this by using the
1 1
DVF(9) and (24). By choosing the upper index in (9) we have
Ai(Ee2πi) Ai(Ee−2πi) Ai(Ee2πi) Ai(Ee−2πi)
T1(E) = e−π3i 3 +eπ3i 3 = e23πi 3 e−23πi 3 .
Ai(E) Ai(E) − Ai(E) − Ai(E)
Thus T = 1 thanks to (27).
1
Now T system is trivially represented as
T2 = 1, T = 0. (28)
1 2
It simply gives a trivial solution of TBA, Y = 0, which is far from Fendley’s
1
solution.
3 Actually, the role played by the Airy function in = 2 SUSY theory, especially
N
its relation to Q± has been firstly noted in [20], independently from [10], exactly in the
context of the ODE/IM correspondence.
8
We then “deform” the ODE by the nonzero angular momentum term as
suggested in [10],
d ℓ d ℓ
( )( + )+x E ψ(x,E,ℓ) = 0. (29)
− dx − x dx x −
(cid:16) (cid:17)
Below we will argue that this replacement leads to the desired T, Y system
and to the TBA.
The Stokes multiplier has the form (19),
D∓(EΩ−2,ℓ) D∓(EΩ2,ℓ)
T (E,ℓ) = ξ∓ +ξ± , (30)
1 D∓(E,ℓ) D∓(E,ℓ)
ξ = eπ−3hi (31)
where h = 2ℓπ. We assume that the following limit exists,
−
1
lim D±(E,ℓ) = Q±(E). (32)
ℓ→0 √2ℓ+1
The quantum Wronskian relation is then rewritten with ξ as,
T (E,ℓ) =ξ−(j+1)D+(EΩj+1,ℓ)D−(EΩ−(j+1),ℓ)
j
ξ(j+1)D−(EΩj+1,ℓ)D+(EΩ−(j+1),ℓ). (33)
−
When q is at a root of unity, the SU(2) T-system (13) closes among finite
elements [25]. In the present case, this is due to a simple relation,
T (E,ℓ) = ξ3 +ξ−3 +T (E,ℓ). (34)
3 1
Then one ends up with
T (EΩ,ℓ)T (EΩ−1,ℓ) = 1+T (E,ℓ),
1 1 2
T (EΩ,ℓ)T (EΩ−1,ℓ) = 1+T (E,ℓ)T (E,ℓ)
2 2 1 3
= (ξ3 +T (E,ℓ))(ξ−3+T (E,ℓ)). (35)
1 1
In the following, we derive (21) from the above truncated T-system as the
first nontrivial equation in the expansion of h. Then we will show that a
similar expansion of the quantum Wronskian relation (20) yields Fendley’s
solution (22).
In this example, the T-system is identified with the Y-system. This is
achieved by introducing
Y (θ,ℓ) = T (E,ℓ), Y (θ,ℓ) = T (E,ℓ). (36)
t 1 1 2
9
Here the parameter θ is related to E by
E = E0e32θ, (37)
and the constant E will be determined later.
0
The Y-system is represented by new variables as
π π
Y (θ i,ℓ)Y (θ + i,ℓ) = 1+Y (θ,ℓ), (38)
t t 1
− 2 2
π π
Y (θ i,ℓ)Y (θ+ i,ℓ) = (ξ3 +Y (θ,ℓ))(ξ−3 +Y (θ,ℓ)). (39)
1 1 t t
− 2 2
Next we consider the expansion in h. The solution (28), strictly at h = 0,
suggests the expansions
Y (θ,ℓ) = 1+hy (θ)+O(h2), Y (θ,ℓ) = hy (θ)+O(h2). (40)
t t 1 1
The first nontrivial equation in the expansion of (38) is O(h), while it is
O(h2) for (39),
π π
y (θ + i)+y (θ i) = y (θ),
t t 1
2 − 2
π π
y (θ+ i)y (θ i) = y (θ)2 +1. (41)
1 1 t
2 − 2
We set
y (θ) = B(θ), y (θ) = e−A(θ) (42)
t 1
−
and assume that y and y are analytic and nonzero in the strip mθ
1 t
ℑ ∈
[ π/2,π/2]. Wealsoassumethattherighthandsidesof(40)areanalyticand
−
nonzero in the narrow strip including the real axis of θ. These assumptions
are justified by the solution in (49), a posteriori. One then obtains
∞ dθ′ 1
A(θ) = m eθ +C ln(1+B(θ′)2),
A A − 2π cosh(θ θ′)
Z−∞ −
∞ dθ′ 1
B(θ) = m eθ +C e−A(θ′) (43)
B B − 2π cosh(θ θ′)
Z−∞ −
where “mass terms” are introduced to take account of the zero mode in the
Fourier transformation. Without loss of generality we can always choose
m = 1 by tuning the origin of θ (or a redefinition of E ). It will be later
A 0
shown that m = 0. The integration constants C , C are found to be zero.
B A B
This can be verified from the asymptotic values e−A(−∞) = 2/√3,B( ) =
−∞
1/√3.
−
We have a remark. The quantum sine Gordon model has = 2 super-
N
symmetry at a special coupling constant. Fendley et al. [14] utilized this and
10