Bethe Ansatz Equations in Thermodynamical Limit: Algebraic Curve and Finite Size Corrections

Vladimir Kazakov (ENS, Paris) SISSA, Trieste, 22 September 2005

N.Gromov and V.K. (to appear)

• Basic problem: Heisenberg quantum chain of spins σ · σ = s(s + 1). Solvable by Bethe ansatz. Many applications for various s.

• XXX chain hamiltonian (s = 1/2): Ĥ_{1/2} = ∑_{l=1}^{J} Ĥ_{1/2}^{l,l+1} = ∑_{l=1}^{J} (1 − σ_{l} · σ_{l+1}).

• We focus on: non-compact s = −1/2 spin chain with hamiltonian

Ĥ_{−1/2}^{l,l+1}|m, m − k⟩ = ∑_{k′=0}^{m} (δ_{k=k′} (h(k) + h(m − k)) − \frac{δ_{k≠k′}}{|k − k′|}) |k′, m − k′⟩

|k_{1}, …, k_{l}, k_{l+1}, …⟩ is a state vector, h(k) = ∑_{1}^{k} \frac{1}{k} are harmonic numbers.

• We study 1/J and 1/J^{2} corrections to solutions in thermodynamical limit: J → ∞, with energies E ∼ 1/J.

• Many similarities with N → ∞ limit in matrix models: algebraic curve, double scaling at the edge, asymptotics of Backer-Akhieser functions...

• Bethe ansatz solution (BA): an eigenvalue E is characterized by Bethe roots u_{1}, u_{2}, … u_{S}:

E = ∑_{j=1}^{S} \frac{1}{u_{j}^{2} + 1/4}

• The roots u_{j} satisfy the Bethe Ansatz eqs. (BAE)

(\frac{u_{j} − i/2}{u_{j} + i/2})^{J} = ∏_{k=1(k≠j)}^{S} \frac{u_{j} − u_{k} + i}{u_{j} − u_{k} − i}

Thermodynamical, or ”classical” limit

• S, J ∼ ∞, u ∼ Jx, E ∼ 1/J [Reshetikhin,Smirnov’83];[Sutherland, early 90’s?]; [Korchemsky’97];[Beisert,Minahan,Staudacher,Zarembo’03].

• Many, rarely changing spins: | ↑↑↑↑↑↑↑↑↑↑↓↓↓↓↓↓↓↓↓↓↓↓↓↓↑↑↑↑↑↑↑↑↑↓↓↓↓↓ ⟩ • Take log of both parts and expand in 1/u ∼ 1/J

2πn − \frac{1}{u_{p}} = 2\frac{1}{J} ∑_{q=1}^{S}\frac{′ 1}{u_{p} − u_{q}} + O (\frac{1}{J} )

or 2πn_{k} − \frac{1}{x} = 2 ∫_{Ck} \frac{dy ρ(y)}{x − y}, x ∈ C_{k}, k = 1, ⋯, K

• Quasi-momentum: p(x) = ∫ \frac{dyρ(y)}{x−y} + \frac{1}{2x} satisfies the Riemann-Hilbert eq. p(x + i0) + p(x − i0) = 2πn_{k}, x ∈ C_{k}, n_{k} ∈ Z

has the known poles at x = 0 and p(x) ∼ \frac{S/J+1/2}{x} + O(1/x^{2}), x → ∞.

Next 1/J corrections

• In terms of root density ρ(x) = \frac{1}{J} ∑_{k=1}^{J} δ(x − u_{j}/J)

∑_{k}′i log \frac{u_{j} − u_{k} + i}{u_{j} − u_{k} − i} =

= \ ∫ \frac{2ρ(x)dx}{x − y} − \frac{1}{J^{2}}\ ∫ \frac{2ρ(x)dx}{3(x − y)^{3}} + \frac{1}{J^{4}}\ ∫ \frac{2ρ(x)dx}{5(x − y)^{5}} − \frac{πρ′}{J} coth(πρ)+

+ (\frac{coth(πρ)}{sinh^{2}(πρ)}(πρ′)^{3} − \frac{2π^{2}ρ′ρ′′}{sinh^{2}(πρ)} + π coth(πρ)ρ^{(3)}) \frac{1}{12J^{3}} + O (\frac{1}{J^{5}})

• Local terms come from anomaly: singularities at close roots [Beisert,V.K.,Sakai,Zarembo’05] (similar to 2D Yang-Mills theory: [Daul,V.K.’92]). Used to study 1/J corrections for one cut distribution in [Beisert,Tseytlin,Zarembo’05].

• Solution

p′(x) = \frac{1}{f(x)} ∑_{k=−1}^{K−1} a_{k}x^{k−1},

a single-valued function on hyperelliptic Riemann surface f^{2} = ∏_{j=1}^{2K}(x − x_{j})

• Single-valuedness: ∮_{Al} dp = 0 l = 1, …, K − 1

• BA eqs. become integer B-period conditions:

∮_{Bij} dp = 2π(n_{i} − n_{j}) n_{i}, n_{j} = 1, …, K − 1 - dots corresponds to the numerical solution with total number of Bethe roots S = 300 L = 600, and three equal fractions S_{1} = S_{2} = S_{3} = 100 of roots; n_{i} = {−1, 3, 1}.

- solid line is the analytical density computed from corresponding hyper-elliptic curve at S = J/2 → ∞.

Baxter equation

• It is a second order difference eq. (discrete Schroedinger eq.)

T (u)Q(u) = W (u + i/2)Q(u + i) + W (u − i/2)Q(u − i)

where Q(u) = ∏_{k=1}^{S}(u − u_{k}), W (u) = u^{J}

• BAE from the analyticity: T (u) is a polynomial of degree J ⇔ u_{k} are Bethe roots.

Thermodynamical limit J → ∞ (”WKB”)

• In notations: x = u/J,  Φ(x) = \frac{1}{J} ∑_{k=1}^{S} log(x − x_{k}),  V (x) = log x,

2t(x) = T (Jx)/(Jx)^{J} the Baxter eq. reads:

2t(x) = exp J [Φ (x + \frac{i}{J} ) − Φ(x) + V (x + \frac{i}{2J} ) − V (x)] + c.c.

• Quasi-momentum p(x) ≡ Φ′ + V′/2 = \frac{1}{J} ∑_{k=1}^{S} \frac{1}{x−xk} + \frac{1}{2x}:

t(x) = cos p(x) [1 − \frac{1}{J} (p′(x)/2 + 1/(8x^{2})) + O(1/J^{2})]

• Substitute 1/J expansion

p(x) = p_{0}(x) + \frac{1}{J} p_{1}(x) + \frac{1}{J^{2}}p_{2}(x) + O(1/J^{3})

t(x) = t_{0}(x) + \frac{1}{J} t_{1}(x) + \frac{1}{J^{2}}t_{2}(x) + O(1/J^{3})

where all t_{k}(x) - entire functions.

Zero order in 1/J

cos p_{0}(x) = t_{0}(x)  ⇒  p′_{0}(x) = \frac{2t′_{0}(x)}{\sqrt{1 − t_{0}^{2}}}

• Finite gap assumption: all branch cuts come from this root (the Bethe roots condense!) in J → ∞ limit [Dubrovin,Novikov’74];[Its,Matveev’75].

• Asymptotics:

p_{0}(x) ∼ \frac{S/J + 1/2}{x}, x → ∞

p_{0}(x) ∼ 1/(2x) + O(x^{0}), x → 0

• As expected, p′_{0}(x) is a rational function on hyperelliptic algebraic curve

p′(x) = \frac{1}{∏_{j=1}^{2K}(x − x_{j})^{1/2}} ∑_{k=−1}^{K−1} a_{k}x^{k−1}

First order 1/J

p_{1} = −\frac{1}{2}p′_{0} cot p_{0} − (\frac{t_{0}}{8x^{2}} + t_{1}) (1 − t_{0}^{2})^{−1/2}

• The first term has no cuts since cot p_{0} has the same cuts as p′_{0}. Hence p/_{1} = −p′_{0} cot p_{0}/2

• From here and from the asymptotics p_{1} ≃ O(1) (x → 0); p_{1} ≃ O(1/x^{2}) (x → ∞).

the full p_{1}(x) can be reconstructed ([Gromov,V.K.’05], similar to KdV in [Smirnov’03])

p_{1}(x) = \frac{x}{4πif(x)} ∮_{C} \frac{f(y)p′(y) cot p(y)}{y(y − x)}dy + ∑_{j=1}^{K−2} \frac{a_{j}x^{j}}{f(x)}

where f^{2}(x) = ∏_{j=1}^{2K}(x − x_{j}).

• For K < 3 the solution is unique. For K ≥ 3 we fix K − 2 parameters a_{j} by K additional conditions that insure the filling fractions α_{j} of Bethe roots on the cuts

∮_{Cl} p_{1}(x)dx = 0, l = 1, …, K − 1

In fact only K − 2 of them are linear independent since we have already fixed total fraction by the second condition.

• Another form:

p_{1}(x) = \frac{R(x)}{4πif(x)} ∮_{C} \frac{f(y)p′(y) cot p(y)}{R(y)(y − x)}dy

Where R(x) = ∑_{k=1}^{K−2} ã_{j}x^{j}.

• Blowing up the contour we find

p_{1}(x) = −p′_{0}(x) cot p_{0}(x)/2 + \frac{R(x)}{2f(x)} ∑_{n} \frac{(R(x_{n}))^{−1}}{x_{n} − x}

where the first term comes from the residue at y = x.

Second order, 1/J^{2}

• Expanding up to 1/J^{2} we have

p_{2} = \frac{1}{2}∂_{x} (p_{1} cot p_{0} + \frac{p′_{0}}{2} cot^{2} p_{0}) − \frac{1}{8x^{3}} − \frac{t̃_{2}(x)}{2\sqrt{1 − t_{0}^{2}(x)}}

• For the principle value part on the cuts:

p/_{2} = −\frac{1}{2}∂_{x} [cot(p_{0}) (p_{1} + \frac{p′_{0}}{2} cot p_{0})] − \frac{1}{8x^{3}}

• Solution:

p_{2}(x) = \frac{x}{4πif(x)} ∮_{C} \frac{f(y)}{y(y − x)}(\frac{1}{4y^{3}} + ∂_{y} [cot(p_{0})p_{1}] +

+\frac{p′′_{0}}{12} − (p_{1} + p′_{0} cotp_{0})p_{1} cotp_{0}) + ∑_{j=0}^{K−1} \frac{b_{j}x^{j}}{f^{5}(x)}

Fixing parameters:

• At x → ∞: p_{2}(x) = O(1/x^{2})

• At x → 0: from (quasi)-periodicity P_{tot} = ∑_{j=1}^{S} [log(u_{j} + i/2) − log(u_{j} − i/2)] = 2πm we deduce p_{2}(0) = \frac{1}{24}[p′′_{0}(x) + \frac{1}{8x3}]_{x=0}.

• b_{j} are fixed from zero A-periods conditions (conserved filling fractions) ∮_{Al} p_{2}(x)dx = 0, l = 1, …, K

• Two extra regular terms under integral are fixed from the behaviour near 2K branch points (2 nontrivial conditions for each branch point, see below).

Double scaling near the branch points

• Exact branch point x_{∗}(J) defined as t(x_{∗}) = 1. Expand t(x) ≃ 1 − a(x − x_{∗})/2 if x → x_{∗}. This defines a Schroedinger-like eq. with linear potential.

• In the double scaling limit J → ∞, x → x_{∗}, v = (x − x_{∗})J^{2/3} ∼ 1, in terms of

double scaled function q(v) = e^{\frac{vJ1/3}{2x∗}}Q(x_{∗}J + vJ^{1/3}), Baxter eq. reduces to Airy eq.

q′′ − avq = 0 + O (\frac{1}{J^{1/3}})

or, for the quasi-momentum near the edge

p (x_{∗} + \frac{v}{J^{2/3}}) = \frac{1}{J^{1/3}}∂_{v} log q(v, J)

where q(v) = Ai(a^{1/3}v).

• Similar to the double scaling in gaussian matrix model. The Airy type edge universality was first discovered in Brezin,V.K.’90 for density (for the kernel: Tracy,Widom’?).

Next 1/J approximations near the edge

• We can expand further as t(x) ≃ 1 − a(x − x_{∗})/2 − b(x − x_{∗})^{2}/2 for x → x_{∗}. Note that x_{∗}, a, b depend regularly on J, so that x_{∗} = x_{0} + x_{1}/J + . . ..

• Now in terms of q_{0}(v) = Ai(a^{1/3}v) we have

p (x_{∗} + \frac{v}{J^{2/3}}) = \frac{∂_{v}q(v, J)}{q(v, J)J^{1/3}} + \frac{1}{2x_{∗}} (\frac{1}{1 + _{\frac{v}{x∗J2/3}}} − 1)

where

q ∝ [1 + \frac{v^{2}}{4x_{∗}^{2}J^{1/3}} + \frac{1}{J^{2/3}} (\frac{v^{4}}{32x_{∗}^{4}} − \frac{3b − a^{2}}{15a}v)] q_{0}(v − \frac{1}{4ax_{∗}^{2}J^{1/3}} + \frac{a^{2} + 12b}{60aJ^{2/3}}v^{2})

• The zeros of q(v) coincide with positions of the Bethe roots which we know now with precision 1/J^{2/3}. Similar to asymptotics of orthogonal polynomials! In fact, the solution of Baxter eq. Q_{S}(x) is known to be a certain set of orthogonal polynomials.

Quasi-momentum near branch point as a function of the scaling variable v. Poles corresponds to the positions of the Bether roots u_{i}. Red dashed line - ”exact” numerical value, light grey - zero order approximation given by Airy function Ai(a^{1/3}x), grey - first order and black - second order approximation.

Large v asymptotic near the edge

• Using the Airy asymptotics near an edge x_{0} in the regime y = vJ^{−2/3} ≫ 1 we find

p_{0}(x_{0} + y) = πn − \sqrt{ay} − \frac{(a^{2} + 12b)y^{2}}{24\sqrt{ay}} + …

p_{1}(x_{0} + y) = −\frac{1}{4y} + \frac{A}{\sqrt{y}} + \frac{a^{2} − 4b}{16a} + …

p_{2}(x_{0} + y) = \frac{5}{32y^{2}\sqrt{ay}} − \frac{A}{2\sqrt{a}y^{2}} + (\frac{A^{2}}{2y\sqrt{ay}} − \frac{b}{64(ay)^{3/2}} − \frac{\sqrt{ay}}{768y^{2}}) + …

where we denoted x_{1} = \frac{2A}{\sqrt{a}} − _{\frac{1}{4x2 0a}}.

• Comparing with similar expansion from the solution for p_{2}(x) we fix a, b and x_{1}, and hence all the terms under the integral in p_{2}(x). Similar to the standard procedure of gluing WKB solutions near a turning point in quantum mechanics.

Corrections to the energy

E = ∑_{k=1}^{S} \frac{1}{u_{k}^{2} + 1/4} = \frac{1}{J} ∮ \frac{dx p(x)}{x^{2} + 1/(2J)^{2}}

= \frac{1}{J} [p′(x) + \frac{1}{2x^{2}}]_{x=0} − \frac{1}{4J^{3}} [p^{(3)}(x) − \frac{1}{x^{3}}]_{x=0} + O(1/J^{5})

• Substituting the expansion p(x) = p_{0}(x) + \frac{1}{J}p_{1}(x) + \frac{1}{J2}p_{2}(x) + O(1/J) we obtain:

E = \frac{1}{J} E_{0}(x) + \frac{1}{J^{2}}E_{1} + \frac{1}{J^{3}}E_{2} + O(1/J^{4})

E_{0} = [p′_{0}(x) + \frac{1}{2x^{2}}]_{x=0}

E_{1} = p′_{1}(0) = −\frac{Q′(0)}{4πif(0)} ∮_{C} \frac{f(y)p′(y) cot p(y)}{Q(y)y}dy

E_{2} = −\frac{1}{4} [p_{0}^{(3)}(x) − \frac{1}{x^{3}}]_{x=0} + p′_{2}(0)

Energy for one cut distribution

S :  10  40  7  20  5
J :  20  20  21  20  25

(m, n) : (1, 2) (2, 1) (1, 3) (2, 2) (1, 5) Relative deviation from the exact value of the energy for finite number of roots E for zero order (light gray), first order (gray) and second order (black) approximation.

Conlcusion and prospects

• We found a regular way to generate the finite volume, 1/J corrections to solutions of Bethe equations in the thermodynamical limit.

• The behaviour near the edge of distribution of Bethe roots shows a universality of Airy type. Not so surprising, since the Baxter operator defines a certain set of orthogonal polynomials.

• Many possible generalizations: multicritical points, multicut solutions, non-hyperelliptic algebraic curves for hierarchial BAE.

• These results are a by-product of our study of integrability in 4D N = 4 sypersymmetric Yang-Mills theory. Possible exciting applications to the quantisation of superstring on AdS_{5} × S^{5} space.