Continuum limit of fishnet graphs and AdS sigma model

Continuum limit of fishnet graphs and AdS sigma model

Benjamin Basso and De-liang Zhong
Abstract

We consider the continuum limit of 4d planar fishnet diagrams using integrable spin chain methods borrowed from the Super-Yang-Mills theory. These techniques give us control on the scaling dimensions of single-trace operators for all values of the coupling constant in the fishnet theory. We use them to study the thermodynamical limit of the BMN operator corresponding to the spin chain ferromagnetic vacuum. We find that its scaling dimension exhibits a critical behaviour when the coupling constant approaches Zamolodchikov’s critical coupling. Analysis close to that point suggests that the continuum limit of the fishnet graphs is controlled by the two-dimensional non-linear sigma model. More generally, we present evidence that the fishnet diagrams define an integrable lattice regularization of the model. A system of massless TBA equations is derived for the tachyon energy by dualizing the TBA equations of the weakly coupled planar SYM theory.

institutetext: Laboratoire de Physique Théorique de l’École Normale Supérieure, CNRS,
Université PSL, Sorbonne Universités, Université Pierre et Marie Curie,
24 rue Lhomond, 75005 Paris, France

1 Introduction

String sigma models are believed to provide a general solution for the sum over planar diagrams in gauge theories tHooft:1973alw. The AdS-CFT correspondence Aharony:1999ti shed light on this old idea and suggested new embodiments for conformally invariant gauge theories. The most famous example is the 4d SYM theory which is conjectured to be dual to string theory in AdS Maldacena:1997re. This supersymmetric gauge theory is also special in that it is believed to be integrable at large Beisert:2010jr. The latter property gives us a handle on the AdS-CFT dictionary, enabling the development of new techniques for carrying out the large re-summation of the field theory diagrams, at both planar Beisert:2010jr; Basso:2013vsa; Basso:2015zoa; Fleury:2016ykk; Eden:2016xvg and non-planar level Bargheer:2017nne; Eden:2017ozn; Ben-Israel:2018ckc. Furthermore, these methods allow us to explore a larger chunk of the correspondence between planar diagrams and sigma models by means of partial or twisted re-summations, which are naturally associated with some integrable deformations of SYM.

Figure 1: A planar fishnet diagram of order . Every intersection point is integrated over spacetime and every connecting line stands for a massless scalar propagator . In the fishnet theory, the horizontal and vertical lines correspond to trajectories of the and particles, respectively, and the diagram is weighted by powers of the coupling .

In this paper we consider such an integrable daughter of SYM, which comes with no gauge fields nor any clear-cut stringy interpretation. The theory to be studied was introduced recently by Gurdogan and Kazakov Gurdogan:2015csr and is known as the fishnet theory, see also Zamolodchikov:1980mb for earlier work and Caetano:2016ydc; Grabner:2017pgm; Kazakov:2018qbr; Kazakov:2018ugh for further developments. It consists of two complex matrix scalar fields interacting by means of a single quartic coupling111A proper definition requires introducing double-trace couplings, which for the sake of conformality must be tuned to their critical values Grabner:2017pgm; Sieg:2016vap, see also Pomoni:2008de. We will not need to worry about them here.

(1)

with the trace taken over the matrix indices, which here are just flavour indices. It can be viewed as a truncation of weakly coupled SYM in which gluons and gauginos are forcefully decoupled and only two of the three complex scalar fields are retained. The proper procedure goes through the extremal twisting Gurdogan:2015csr of the -deformed SYM theory Leigh:1995ep; Lunin:2005jy; Frolov:2005dj; Beisert:2005if which involves sending the YM coupling to zero and the deformation parameter to , while keeping the suitably rescaled coupling fixed. Owing to this “embedding”, the theory is expected to be conformally invariant and integrable for any Gromov:2017cja; Grabner:2017pgm, at least in the planar regime. In fact, the integrability of the fishnet vertex was recognized by Zamolodchikov more than 40 years ago Zamolodchikov:1980mb.

One appealing feature of the fishnet theory is that it produces many fewer graphs than SYM. Quite often only a single graph contributes at a given order in perturbation theory, in the planar limit. The price to pay for this massive cut is the loss of unitarity, as the strict ordering of the fields in the potential clashes with the reality of the action. Another drawback is that the duality with string theory is uncertain, for the AdS radius is naively small. Still, the fishnet theory proves to be a remarkable testing ground for integrability, which, in turn, sheds light on families of conformal Feynman integrals Chicherin:2017frs; Chicherin:2017cns and suggests new ways of evaluating them Gurdogan:2015csr; Caetano:2016ydc; Gromov:2017cja; Basso:2017jwq; Grabner:2017pgm; Kazakov:2018qbr.

The planar diagrams of the theory, the fishnet graphs, are special in that they all look locally like the square lattice shown in figure 1. Accordingly, every diagram can be viewed as a partition function for a 2d vertex model Zamolodchikov:1980mb, with the bulk spacetime points acting as classical “spins”, the propagators as nearest neighbour couplings and with the graph’s external lines setting up the boundary conditions. Different observables of the planar fishnet theory correspond to different boundary conditions and all the graphs obeying the same boundary conditions are summed over.

An important observation concerning the large order behaviour of the fishnet diagrams was made by Zamolodchikov Zamolodchikov:1980mb, see also Bazhanov:2016ajm for a recent discussion, who computed, using integrable vertex model techniques, the free energy per site in the thermodynamical limit

(2)

for graphs subject to periodic boundary conditions, and found that

(3)

This constant determines a critical coupling for thermodynamically large observables in the fishnet theory and one might expect, in analogy with matrix models Klebanov:1991qa; DiFrancesco:1993cyw; Ginsparg:1993is; Nakayama:2004vk, that a “dual” continuum description is taking over at that point.222Note that the critical coupling does not refer to a point at which the 4d theory is becoming critical; the fishnet theory is conformal for any . It is a point at which the planar diagrams become dense.

In this paper we examine the thermodynamical limit of fishnet graphs using integrable methods borrowed from the SYM theory and argue that the continuum description is given by the 2d (bosonic) sigma model.

Figure 2: Schematic plot of the scaling dimension of the BMN operator as function of the coupling constant , at large . The scaling dimension is nearly independent of the coupling constant, , until where it starts developing a non-trivial thermodynamical scaling, . At , the scaling function vanishes and its derivative is infinite. The physics close to this point is controlled by the 2d low energy dynamics of a large fishnet graph. The overall shape of the thermodynamical curve agrees with the findings of Gromov:2017cja for . The value of the scaling dimension at the branch point appears to be at both small and large .

Our discussion will center around the scaling dimension of the BMN operator which maps to the ground-state energy of a ferromagnetic non-compact spin chain Gromov:2017cja. Integrability will allow us to study the thermodynamical limit of this energy for generic coupling by means of a linear integral equation. It will confirm the existence of a non-trivial thermodynamical scaling , for sufficiently “strong” coupling, and the emergence of a critical behaviour close to the critical point, in line with the results of Gromov:2017cja; Grabner:2017pgm for .333The location of the branch point is function of the length ; in particular Grabner:2017pgm, . A sketch of the thermodynamical behaviour of the scaling dimension is shown in figure 2.

Dualizing our equation, by means of a particle-hole transformation, will reveal the nature of the critical point and suggest the interpretation of the BMN operator as describing the “tachyon” ground-state of the AdS sigma model,

(4)

labelled by the global time energy of the BMN operator and implicitly by the size of the worldsheet. Though we will actually never cross the line where the AdS mass squared turns negative, we will stick to the name of tachyon for the dual object.444The tachyonic domain maps to , where the scaling dimension has an imaginary part, ; see Gromov:2017cja; Grabner:2017pgm; Pomoni:2008de for discussions.

The correspondence (4) is best summarized by the formula

(5)

which relates the 4d coupling to the sigma model energy and shows that the vicinity of the critical point maps to low energies on the worldsheet. In this regime the sigma model is weakly coupled and we will be able to verify our claims directly.

We shall also test the correspondence at the level of the corrections and obtain a system of TBA equations for the tachyon ground state, valid in principle for any and . Its form will support the more general conjecture that the fishnet diagrams define an integrable lattice regularization of the sigma model.

The plan is as follows. In Section 2 we introduce the integrability set up and derive the integral equation for the large limit of the scaling dimension . With its help we reproduce Zamolodchikov’s prediction for the critical coupling. In Section 3 we obtain the dual system of equations, give arguments for its interpretation as describing the tachyon in the AdS sigma model and carry out some perturbative tests. In Section 4 we dualize the full system of TBA equations and compute the IR central charge. We conclude in Section 5. The appendices contain a detailed analysis of the dual linear equation, a discussion of spinning operators and a brief study of the thermodynamical limit of 3d triangular fishnet diagrams.

2 Thermodynamics of fishnet graphs

We start with a light review of the integrability methods for computing the scaling dimension of interest, emphasizing the correspondence with the Feynman diagrams. Readers familiar with these techniques may jump directly to Subsections 2.2 and 2.3 where we restrict our attention to the thermodynamical limit and re-derive Zamolodchikov’s critical coupling.

2.1 Grand canonical ensemble

In the planar SYM integrable framework the BMN operator is identified with the ferromagnetic ground state of a periodic spin chain of length and its scaling dimension maps to the spin chain vacuum energy. The same can be said in the fishnet theory. The main difference is that in the fishnet theory the vacuum is not protected and its energy is a very complicated function of the length and coupling . It was studied extensively in Gromov:2017cja; Grabner:2017pgm for small values of and perturbatively at weak coupling for any in Gurdogan:2015csr.

For any length , otherwise see Grabner:2017pgm, the diagrams contributing to the scaling dimension in the planar limit are the wheel diagrams Gurdogan:2015csr where virtual particles loop around the operator as shown in figure 3. A general wheel diagram is obtained by pinching the end points of the horizontal lines in figure 1 and periodically identifying the vertical ones. Obviously, each wheel costs powers of the coupling and, consequently, the scaling dimension admits an expansion in powers of at weak coupling, , referred to as the wheel expansion in the following.

Figure 3: On the left panel we show a two-wheel diagram contributing to the anomalous dimension of the BMN operator , represented in the middle by a spin chain with sites. The operator is local but magnified here for the sake of illustration. Each wheel corresponds to the trajectory of a particle. On the right panel a wheel diagram is wrapped on a cylinder of size along the global time direction and length along the spin chain one. Taking projects on the ground state at the boundary. Each vertical wheel gives rise to a tower of mirror magnons which are integrated over. Each magnon is labelled by a rapidity , or momentum along the time direction, and a Lorentz index , which comes from the partial wave decomposition of the scalar field on .

In the spin chain picture, the wheels map to long-range interactions mediated by virtual magnons travelling around the chain Ambjorn:2005wa; Bajnok:2008bm; Janik:2010kd, also known as wrapping or Lüscher corrections. The magnons circulating around the chain are not the familiar ones parameterizing spin waves on top of the ferromagnetic vacuum Minahan:2002ve. Instead they live in the orthogonal, so called mirror, kinematics where the time is interpreted as a space direction and the spin-chain length as an inverse temperature, see figure 3. The mirror picture, which in the general context is inherited from the dual string worldsheet theory, see Bajnok:2010ke; Arutyunov:2014cra for further discussion, can be motivated by considering the problem on the 4d euclidean cylinder , with the unit 3-sphere surrounding the operator and with the global time corresponding to the spin chain time . The relevant dimensional picture is obtained by dropping the 3-sphere, which becomes internal, and replacing it by the spin-chain circle, .555 would be more appropriate, but the difference is immaterial here. The partition function on this geometry returns the ground state energy, in the large volume limit ,

(6)

where the dots stand for heavier single-trace operators, with charge , and where is the length of the cylinder along the direction. The wheel expansion of the partition function, and thus of ,

(7)

is more naturally interpreted as the decomposition in the orthogonal, open string, channel, with the sum running over a complete basis of states along the direction, labelled by the total number of mirror magnons . To be more precise, each wheel gives rise to a semi-infinite family of mirror magnons, stemming from the partial wave decomposition of the scalar field on . Suppressing the Lorentz indices, each mirror magnon is tagged with an integer , for its Lorentz spins and, after smearing over the direction, with a momentum , with the so called Bethe rapidity. The micro-canonical contribution follows then from integration and summation over the magnon phase space, with proper thermodynamical weights and measures; see Kostov:2018ckg for a recent discussion of the relation between the micro-canonical partition functions and the canonical one.

The Thermodynamical Bethe Ansatz allows one to take advantage of the factorized scattering between mirror magnons and compute the bulk free energy (6) to all orders in the wheel expansion and for any temperature , see Ahn:2011xq; Bajnok:2010ke; vanTongeren:2016hhc and references therein. The latter free energy takes the usual form

(8)

with in the case at hand and with the Y functions describing the thermal distribution of the energy among magnons. In the weak coupling regime , the gas is rarefied and the TBA equations can be expanded around the Fermi-Dirac distribution, characterized by the Boltzmann weights ,

(9)

where is the dimension of the -th Lorentz representation and where is the mechanical energy of the associated magnon,

(10)

The departure from the free distribution is controlled by the interaction among magnons. To leading order, it takes the rather universal form

(11)

where is the dynamical factor of the magnon S-matrix, which specifies the model and reads

(12)

with the Euler Gamma function. The dots in (11) capture the effect of the degrees of freedom, or scattering of Lorentz indices, and are simply absent for the lightest magnons ( or ), which are Lorentz singlets. (This isotopic component of the scattering is controlled by a rational matrix, which is function of the difference of rapidities and is coupling independent.)

Formula (11) is a particular case of the NLO Lüscher formula for the ground state of the twisted SYM spin chain Ahn:2011xq, obtained by truncating the SYM magnon super-multiplet to its component and by sending the coupling constant of the gauge theory to zero in all spectral and scattering data. Once plugged into (8), it yields the wheel expansion of the scaling dimension with the obvious map “ wheel”, such that the first term is the free wheel, the next one the double wheel, etc. It was used for comparison with the direct integration of the Feynman integrals in Gurdogan:2015csr.

All the information for moving to higher orders is in principle contained in the TBA equations, which are pretty simple

(13)

if not for the dots, which accommodate for the matrix degrees of freedom and cannot be spelled out without introducing an auxiliary set of variables. They will be given in their full form in Section 4. Nonetheless, the general term in the wheel expansion (11) has not been worked out explicitly. Also, the integration over the magnons’ phase space is nearly impossible in general. A more powerful analytical treatment relies on the Baxter equation Gromov:2017cja, which relates to the (twisted) SYM Quantum Spectral Curve Gromov:2013pga; Kazakov:2015efa and enables higher loop computations of the scaling dimension at finite length . We will not need so much improvement in this paper however, since in the thermodynamical limit the TBA equations (13) simplify drastically.

A distinguished feature of the fishnet TBA equations is that all the dependence on the coupling constant comes along with the energy, like in (13) or equivalently (9). Put differently, the 4d coupling of the fishnet theory is a fugacity for the number of mirror magnons and we can think of the scaling dimension as defining the free energy of a grand canonical ensemble at chemical potential . This parallels the fact that the mirror magnons are in one-to-one correspondence with the wheels. Hence, one can easily probe fishnet graphs of arbitrarily large orders and obtain information about their continuum limit by playing with .

2.2 Thermodynamical limit

The thermodynamical limit is uninteresting at weak coupling, since the wheels are heavily suppressed, see equations (9) and (11), and the scaling dimension up to exponentially small effects. The situation changes drastically as soon the chemical potential gets bigger than the “mass gap” . Above this threshold the s-wave magnons, with , start filling a Fermi sea, see figure 4. The filling is strict in the limit with all the modes outside the sea being unoccupied,

(14)

with the step function and with the pseudo energy

(15)

defined here in such a way that inside/outside the sea. The Fermi rapidity fixes the edges of the sea and is determined by the condition . Moreover, as long as is smaller than the masses of the higher magnons, that is, naively, , the gas is mono-atomic and consists solely of fundamental magnons with . As we shall see later on we will never reach the next threshold, so in the following we drop the Lorentz index and assume that .

An immediate consequence of having a Fermi sea, obtained by plugging (14) into (8), is that the scaling dimension develops a non-trivial thermodynamical scaling

(16)

where is the free energy density of the gas at chemical potential . The same linearization is observed at the level of the TBA equations (13), see also Section 4, which can be written concisely as

(17)

with . Here, for convenience, we split the scattering kernel,

(18)

into its “boost” invariant component, , and the rest, , which depends on a single rapidity. Their explicit forms follow from equation (12), with ,

(19)

and with the digamma function. Note that both are even functions. The part of the kernel merely renormalizes the chemical potential and was absorbed into the constant ,

(20)

This system of equations, with the boundary condition , admits a unique (even) solution, which can be constructed iteratively for finite value of . The numerical solution for the free energy density is shown in figure 6.

Finally, notice that the free energy density (16) can also be read out from the large behaviour of the pseudo energy, which, according to (17) and (19), scales like

(21)

This way of obtaining will turn out to be useful later on.

2.3 Critical coupling

Let us now look for Zamoldchdikov’s scaling (2). First we note that at non zero the typical number of magnons in the ensemble is always large, since the gas has a finite particle density,

(22)

This is necessary for matching with the scaling (2), but the condition is not enough to get an actual match. The continuum limit also requires that a low-energy approximation be taken w.r.t. the fishnet Hamiltonian . Since, heuristically, , we expect that the fishnet dynamics will freeze at large magnon densities, that is when . Another way of seeing it is that the scaling (2) is a statement about the micro-canonical energy density

(23)

which, in our thermodynamical variables, translates into the requirement that

(24)

for . This behaviour cannot be observed at small , that is for a dilute gas, , since then . On the contrary, the critical regime appears for , when all the energy levels are filled, which again means that the density is infinite, .

We can verify it explicitly using the integral equation. Denoting by etc, the limiting values at , we get to solve

(25)

where denotes the convolution over the full real axis,

(26)

The solution to (25) can be found by going to Fourier space. Acting on both sides of (25) with

(27)

and using the Fourier integrals

(28)

immediately tell us that

(29)

Hence, the free energy density (16) vanishes

(30)

in conformity with our previous discussion. The transformation back to rapidity space yields the critical pseudo energy

(31)

which is positive definite and decays exponentially quickly at infinity. Plugging it back into (25) and taking a large rapidity limit fix the constant of integration, . The critical coupling follows from that condition, after recalling (20),

(32)

and it agrees with Zamolodchikov’s result (3).

Figure 4: Schematic plots of the Fermi sea and its dual. On the left, the original sea, . Low energies mean small rapidities and all the energy levels inbetween are occupied. The sea level rises upon increase of the chemical potential . On the right, the dual Fermi sea, . Low energies mean large rapidities, the dual excitations accumulate at infinity and all the energy levels beyond the Fermi rapidity are filled. There is no chemical potential here. Instead, the filling is controlled by the charge density , which pilots the large asymptotics . When , this density vanishes, the dual sea is empty and the fishnet freezes. This happens at the critical coupling .

3 Dualization and AdS sigma model

Having reached the critical point, we want now to study its neighbourhood, corresponding to a large but finite Fermi rapidity . Since “almost all” of the energy levels are filled, it is convenient to analyze this regime by means of a particle-hole transformation, which flips the notions of filled and empty states. As we shall see, the dual equation, the one for the holes, is of a totally different nature and lends itself the interpretation of a thermodynamical equation for the tachyon of the hyperbolic sigma model.

3.1 Particle-hole transformation

Formally, the particle-hole transformation amounts to introducing the dual kernel

(33)

and acting on both sides of (17) with . Straightforward algebra gives then

(34)

where the convolution is now supported on the complementary support and where is the dual of the driving term . The procedure is a bit formal since scales logarithmically at large rapidity and thus the self-convolutions in the RHS of (33) are not well defined. Fortunately, one reaches the same point by defining the dual kernel more implicitly, as the solution to

(35)

Taking derivative of this equation, going to Fourier space and fixing the constant of integration yield

(36)

where

(37)

coincides with the S-matrix of the sigma model in the symmetric channel Zamolodchikov:1977nu; Zamolodchikov:1978xm. It is written here in terms of the Bethe rapidity , which relates to the sigma model hyperbolic rapidity by . Formula (36) hints at a connection between the dual model and the sigma model. However, this cannot be the full story, as we shall discuss shortly.

The next essential piece of information comes from . Using the normalization of the dual kernel,

(38)

one concludes that the constant , and hence the chemical potential, drops out of the dual equation, , leaving us with

(39)

where we used (25) and (35). The dual driving term is thus simply given by the critical pseudo energy (31). It acquires here the meaning of a dual energy, . As noticed earlier, this one decays exponentially at large rapidity. Therefore, the dual energy describes a gapless particle, since one can lower arbitrarily the energy of a dual excitation, by sending it to larger and larger rapidities,

(40)

where sets a reference energy scale. This is in line with the fact that the dual Fermi sea has a non-compact support, see figure 4. The dual low energy modes accumulate at infinity, which is typical for gapless systems, see e.g. Fateev:1992tk; Zamolodchikov:1992zr; Fendley:1993wq; Fendley:1993xa; Fendley:2000bw; Mann:2004jr.

It is also natural in the dual picture to exchange the roles of the thermodynamical quantities. The lack of a dual chemical potential, for instance, invites us to view the free energy density as being part of the specification of the state. Namely, we can simply think of as the charge density that pilots the large behaviour,

(41)

One can also say that it triggers the formation of the dual Fermi sea, as and play interchangeable roles.666This is easily seen at the level of the equation (113) for the derivative of . Its solution is uniquely fixed at any given and so is the relation , with being defined by at large . The constant , which appears in the subleading large behaviour of , see eq. (21), is then determined by integrating and imposing that . In particular, the approach to the critical point corresponds to the low density regime .

The last important quantity is the 4d coupling , which, at the moment, is buried inside the constant . Fortunately, one can substitute to (20) the more transparent relation

(42)

where

(43)

is the critical micro-canonical distribution density, which is Legendre conjugated to , and solution to the integral equation

(44)

Formula (42) is obtained by integrating both sides of the original equation (17) against ,

(45)

Here, in the first line, we used

(46)

and transferred the action of from to , then applied (44) in the second line and finally used (20).

Introducing then a dual momentum , by means of a rotation of the energy,

(47)

and noting that , one concludes that in the dual description the coupling of the 4d theory is simply related to the 2d energy of the state through the formula (5) with, in the thermodynamical limit,

(48)

As we shall see later on, formula (5) is quite general, and not restricted to the thermodynamical limit.

Finally, let us add that putting energy and momentum together yields the dual dispersion relation, which reads

(49)

after eliminating the parametric dependence on the rapidity in (47) and (40). This formula makes the square lattice and its symmetries manifest: it preserves a subgroup of euclidean rotations, generated by , and shows a maximum momentum, , and a maximum energy, . Both features disappear at low momentum where one recovers the dispersion relation for a massless relativistic particle. The dispersion relation is depicted in figure 5, in the 1st Brillouin zone. It is analogous to the energy of a spinon above the anti-ferromagnetic vacuum of the XXX spin chain, and, following this analogy, we could say that at the critical point the spin chain settles down in its symmetric vacuum.

Figure 5: Plot of the dual energy for . At low momentum, we have a massless spectrum . The energy reaches a maximum at .

3.2 Sigma model interpretation

Let us come to the interpretation of the dual equations. They are very similar to the thermodynamical equation for a (zero temperature) finite density gas of particles, carrying maximal charge, in the non-linear sigma model Hasenfratz:1990ab,

(50)

where . The difference only comes from the dispersion relation which, in the relativistic low momentum approximation, amounts to substituting

(51)

and reversing the support of the distribution. It makes a big difference for the interpretation. In the model, the particles are massive and though the scattering kernel is repulsive the particles remain confined on a compact support. The kernel has the same effect in our case but the energy is not bounded from below and the potential runs away. Therefore, at finite charge density , the excitations start filling the energy levels around and spread in the opposite directions, towards smaller rapidities. Also, in the compact case, the charge density matches with the particle density, obtained by integrating the distribution over its support. In our case, the support is non-compact and the distribution is not normalizable, suggesting that the gapless excitations in the condensate cannot be counted.

Given the symmetries of our problem, the most natural guess is that we are dealing with the sigma model. This non-compact model is known for not developing a mass gap and for having a continuous spectrum in finite volume. Furthermore, the change (51) in the energy for a given scattering kernel embodies the “inversion of the RG flow”, which is the formal perturbative way of relating the sphere and the hyperbolic sigma model. We discuss it in more detail below.

Finally, note that there are similarities with the equations obtained for massless factorized scattering theories Fateev:1992tk; Zamolodchikov:1992zr; Fendley:1993wq; Fendley:1993xa, see also Fendley:2000bw; Mann:2004jr. In our case, since we cannot enumerate the particles in the condensate, it is not clear whether we can talk about an S matrix. Put differently, we do not think of our equation as describing the continuum limit of a dense but fundamentally discrete distribution of Bethe roots. On the contrary, the distribution is fundamentally continuous, and will remain continuous after introducing finite size corrections. It defines a one-parameter family of ground states, labelled by , or better .

3.3 Perturbative analysis

We can verify the interpetation of the dual equation by comparing its predictions against a direct finite density calculation in the sigma model. This is standard analysis for sigma models. It was carried out through two loops in Hasenfratz:1990ab; Bajnok:2008it for the sphere. We can follow the same lines for the hyperboloid. In fact, the results for the sphere carry over to the hyperboloid, since compact and non-compact models only differ perturbatively in the sign of the coupling constant, as expected on geometrical grounds Polyakov:2001af; Friess:2005be; Duncan:2007vs. We recall how this comes about below.

We consider the non-linear sigma model in Minkowskian , where in our case. Its 2d action is given by

(52)

where the embedding coordinates take values on the hyperboloid

(53)

and where are transverse directions. The worldsheet metric is taken to be flat, with signature , and we assume periodic boundary conditions in , . The coupling sets the curvature of the hyperboloid. In stringy notation and the model is weakly coupled when . The theory has exact symmetry, with associated conserved Noether currents

(54)

and .

The model is classically integrable and presumably quantum integrable, perturbatively, for the arguments supporting the integrability of the sphere Goldschmidt:1980wq also work for the hyperboloid. The integrability of the non-compact model remains puzzling at the quantum level however. The model’s spectrum has no good particle interpretation and thus cannot be handled by the conventional factorized scattering methods. On top of that the model has well known problems in the UV and might have to be completed. However, none of these complications really is a limitation here, as we do not need particles to make use of our equation and we have a lattice to make sense of the model at short distances. Nonetheless, it is interesting to note that both difficulties somehow relate to the running of the coupling , which, owing to the negative curvature of the AdS space, is governed by a positive beta function, see Friess:2005be for a recent discussion,

(55)

It says that perturbation theory can be trusted at low momentum and this is more than enough for what we intend to do here.

Getting back to our problem, we seek a state with a charge , along the global time direction, which is uniformly distributed along ,

(56)

with the Noether current (54), and which corresponds to a local extremum of the sigma model energy,

(57)

The most natural candidate is the “tachyon”, that is, a point-like and time-like geodesics at the center of AdS. It is given classically by the independent solution

(58)

or, equivalently, , where is the global time coordinate. Owing to the signature of the AdS space, “excitations” along the time-like direction contribute negatively to the sigma model energy. This applies to our state that has classical energy and charge density

(59)

with a frequency which is negative for positive. Eliminating , we obtain that the energy is quadratic in ,

(60)

up to the running of the coupling. Both the sign and the shape are in agreement with what we observe numerically from the solution to the linear integral equation shown in figure 6. As we shall see, the agreement gets even better when the running of the coupling and the perturbative corrections are included.

Figure 6: Plot of the scaling dimension as function of the coupling in the thermodynamical window . Black dots are numerical values obtained by solving the integral equation using Mathematica. The black line is a numerical interpolation. The green, orange and red lines are perturbative expressions truncated at 1st, 2nd and 3rd order, respectively. They correspond to the classical, one- and two-loop approximation in the sigma model, with the relation .

Loop corrections are more efficiently computed by exploiting the thermodynamical nature of the state under consideration. Like for the sphere Hasenfratz:1990ab, one can access to the energy density by coupling the model to a constant electric field and extremizing the path integral over an Euclidean worldsheet. Unlike the sphere and owing to the indefinite signature of the target space, one must be careful with the Wick rotation. The kinetic term of the global time coordinate comes out with the wrong sign. We evade the problem by rotating the global time coordinate along with the worldsheet one,777This is automatic, classically, . We assume that we can also do it for the fluctuations, .

(61)

It brings us to a doubly euclideanized partition function, which is perturbatively well defined to any order in . It remains then to covariantize the derivatives,

(62)

with the Euclidean coordinate, and expand at weak coupling around the center of the space, a.k.a. Goldstone vacuum,

(63)

where the fields and are canonically normalized. It gives

(64)

where is the Euclidean Lagrangian density and where the dots stand for cubic and higher couplings. The first term in (64) is the classical free energy density . The next one shows that the transverse excitations acquire a mass in the tachyon background, as in the compact case Hasenfratz:1990ab. In fact, everything is as for the sphere model up to : this substution flips the signs below the square root in (63), resulting in the compactification of , and turns the derivatives (62) into

(65)

which is the canonical way of boosting the system along a big circle , see Hasenfratz:1990ab; Bajnok:2008it. Therefore, in agreement with the discussions in Polyakov:2001af; Friess:2005be; Duncan:2007vs, the compact and non-compact problems are the same, perturbatively, if not for the sign of the coupling, .

At this stage, we could import the result directly from the perturbative studies of the sphere sigma model Hasenfratz:1990ab; Bajnok:2008it by doing the continuation to negative coupling. In particular the one loop free energy is the same in the two cases and comes directly from the determinants for the quadratic actions in (64). Their evaluation using dimensional regularization gives, in the scheme,

(66)

where solves (55),

(67)

with the scale. Taking the running of the coupling into account, the free energy (66) is verified to be independent of the subtraction scale . The energy and charge of the state are obtained by a Legendre transformation,

(68)

which, after plugging (66), eliminating the coupling (67) and fixing , yields

(69)

The two-loop calculation Bajnok:2008it would also give us the next contribution . Importantly, since the coupling has disappeared, the same formula (69) applies to both the sphere and the hyperboloid. The sole difference is that the expansion is valid for in the case the hyperboloid and for in the case of the sphere. This is the “inversion of the RG flow” alluded to before.

Equation (69) can be compared with the formula (138) obtained from the perturbative analysis of the integral equation done in Appendix A. They agree for if

(70)

which matches with the -to- ratio of the model Hasenfratz:1990ab. We recall that in our case is not a mass gap.

More generally, as shown in Appendix A, our integral equation turns out to be identical at large with the equation (50) for the compact sigma model, after continuing to . Since the Fermi rapidity plays the role of the radius of the AdS space, changing its sign has the same effect as changing the sign of the coupling . Hence, we can bypass the direct comparison with the perturbative sigma model and, assuming the validity of the equation (50) for the sigma model, conclude that our equation describes the tachyon of the model to all orders in perturbation theory.

4 TBA equations and central charge

The finite density equation only probes a diagonal subsector of the 2d theory and as such might miss some features of the model. More compelling evidence for our proposal can be found by looking at the finite size corrections. We will see that the observations made earlier at the linearized level uplift to the full set of TBA equations. We will then discuss briefly the finite size corrections to the tachyon energy level.

4.1 Massive TBA

First recall the original form of the TBA equations for the ferromagnetic vacuum . It is obtained by taking the fishnet limit of the system of TBA equations for the ground state of twisted SYM spin chain Ahn:2011xq. The relevant symmetry group is the Lorentz group and the relevant excitations describe the Lorentz harmonics of the scalar field , introduced in Section 2. These modes appear on an equal footing in the TBA with each mode mapping to a massive (non-relativistic) magnon with bare energy , see (10), and thermodynamical weight , see (11). These Y functions are subject to the equations

(71)

where and denotes the convolution, defined in (26). The kernel controls the part of the interaction that depends on the difference of rapidity, see (12),

(72)

and , with the kernel used before, see (19). The driving term in (71) is given in terms of the bare energy (10) and of a constant , which does not depend on the mode number . The latter constant captures the dependence on the coupling constant of the 4d theory and absorbs the part of the kernel that depends on a single rapidity. It reads

(73)

where

(74)

generalizes (19) to .

The last sum in the RHS of (71) describes the couplings to the matrix degrees of freedom, represented in the form of dispersion-less magnons, with wave functions . They are labelled by the dimensions of the representations. The interactions between momentum carrying magnons and isotopic ones are controlled by the scattering kernels of the XXX spin chain,

(75)

Note in particular that , meaning that there is no coupling between the s-wave magnons and the isotopic ones, as expected.

The TBA equations for the isotopic magnons are entirely controlled by the kernels (75) and read

(76)

with . For the state we are interested in, there is a left-right symmetry resulting in .

The scaling dimension of the BMN operator relates to the free energy of this polyatomic gas and was given in (8). Alternatively, it can be read out from the large asymptotics of the massive Y functions,

(77)

which follows from the universal logarithmic scaling of the kernels and energies . The non-universal part of the sub-leading behaviour comes from the isotopic Y functions . The latter tend at large rapidity to the constant solution of the Y system, which will be given later on, see equation (107). Plugging this solution into (71) and using the normalization of the XXX kernels,

(78)

one recovers that gives the dimension of the -th Lorentz representation, see (9). The relation between global quantum numbers and large rapidity asymptotics is well known in the spin chain context, see e.g. BS05, and plays an essential role in the Quantum Spectral Curve Gromov:2013pga.

Equipped with the full set of TBA equations we can verify the claims made earlier about the thermodynamical limit for . In particular, we can check that the higher modes stay under control in the presence of the condensate , all the way to the critical point where the back reaction is maximal. Plugging the critical pseudo energy, , inside (71) and using the identity (88) reveal that the functions become independent at the critical point, see (108). This behaviour is indicative of a symmetry enhancement and is further discussed below. Nonetheless, it does not change the fact that the higher modes are negligible thermodynamically. Also, the isotopic Y functions do not couple to the length , nor to , and though their actual values depend on they remain of order all the time, see (107) and (108). Therefore, in the thermodynamical limit one can legitimately substitute and inside (71) and this way recover the linear integral equation (17).888Due to an unfortunate choice of notations, in the thermodynamical limit where refers to the constant (20).

4.2 Massless TBA

The dualization introduced earlier can be applied to the TBA equations. Let us start with the dual singlet equation, which extends (34) beyond the thermodynamical limit. As before, it follows from acting with on both sides of the TBA equation (71). Then, combining