sigma model on a finite interval revisited
A. Milekhin^{†}^{†}footnotetext: email:
Department of Physics, Princeton University, 08540, Princeton, NJ 
Institute for Information Transmission Problems of Russian Academy of Science, 
B. Karetnyi 19, Moscow 127051, Russia 
Institute of Theoretical and Experimental Physics, B. Cheryomushkinskaya 25, 
Moscow 117218, Russia 
Abstract
In this short note we will revisit the large solution of sigma model on a finite interval of length . We will find a family of boundary conditions for which the large saddle point can be found analytically. For a certain choice of the boundary conditions the theory has only one phase for all values of . Also, we will provide an example when there are two phases: for large there is a standard phase with an unbroken gauge symmetry and for small there is Higgs phase with a broken gauge symmetry.
1 Introduction
Two dimensional sigma model in the large limit was first solved in [adda] and [witten]. The theory exhibits a plethora of nontrivial properties: asymptotic freedom, confinement and dynamical scale generation via the dimensional transmutation:
(1) 
where is the coupling constant.
Physically, 2D model naturally arises as a lowenergy effective action of nonAbelian strings in QCDlike models, see [solitons] for a review. Therefore, a finite interval geometry corresponds to a string stretched between two branes or a monopole–antimonopole pair. Such configuration was studied in [boundary].
Recently sigma model on a finite interval of length with Dirichlet boundary conditions(BC) was investigated in [old] and [kon] using large expansion. In the earlier work [old] the large saddle point equations were solved only approximately and two distinct phases were found. In [kon] saddlepoint equations were solved numerically and it was argued that there is only one phase. In this paper we will find a set boundary conditions for which the saddle point equations can be solved analytically. Strictly speaking, we will study sigma model. We will consider two different boundary conditions:

Mixed DirichletNeumann(DN) boundary conditions which will break global to . We will show that the system has at least two phases: for there is a standard ”Coulomb” phase with an unbroken gauge symmetry. This phase takes place for the model on usual . For there is ”Higgs” phase with broken . Global stays unbroken in both phases.

DirichletDirichlet and NeumannNeumann(DD and NN) boundary conditions which will break to . In this case, for all values of there is a standard phase with an unbroken gauge symmetry. Higgs phase is prohibited in this case, because it will break global to .
In case of simple Dirichlet boundary conditions studied in [old, kon], Higgs phase does not break any global symmetries, so we expect that the system will have two phases as was predicted in [old]. Let us note that the large model on a cylinder also possesses multiple phases [finite].
2 Generalized saddle point equations
Let us study model in the large limit. The field content consists of fields , real vector field and real scalar . In the Euclidian space the Lagrangian reads as:
(2) 
where and . Time coordinate takes values from to and .
Nondynamical Lagrangian multipliers and forces to lie on space: integration over yields and is responsible for invariance .
We will proceed in a standard fashion: we will integrate out 2N fields fields and then find the large N saddle point values of , and the remaining which we will denote by . After integrating out fields we have:
(3) 
So far we do not have a factor of in front of the determinant because we will impose different boundary conditions for these fields.
We will study this model on a finite interval of length with various boundary conditions. Note that the translational symmetry in direction is explicitly broken. However, we still have the time translations so we will consider only time translation invariant saddle points. By the choice of gauge we can always set . This allows us to rewrite eq. (3) as:
(4) 
Note that we have already integrated out time frequencies, so we have energies instead of of the usual . The sum over is the sum over the eigenvalues of the following equation:
(5) 
are required to be normalized.
Varying effective action (4) with respect to we get the first saddle point equation:
(6) 
To obtain this equation we have used the standard quantum mechanical first order perturbation theory for (5).
The second saddlepoint equation coincides with the equation of motion:
(7) 
Finally, we have to vary with respect to :
(8) 
Below we will study the case with real and and so this equation will be trivially satisfied.
3 DN boundary conditions: two phases
Now it is time to choose boundary conditions. Let us consider the following: For fields we will use DirichletNeumann (DN):
(9) 
And for fields we will use NeumannDirichlet(ND):
(10) 
And for we will impose NeumannNeumann(NN):
(11) 
This choice breaks global to .
Then in the DN sector we have:
(12) 
In the ND sector:
(13) 
If we plug this into the first saddle point equation (6) we will notice that and will sum up to and the dependence will disappear! So we can consider to be constant. Let us first study the phase with nonzero . From the second saddlepoint equation (7) we see that we have to put . We will call this phase ”Coulomb” phase because has zero VEV which leaves the unbroken.
The first saddlepoint equation now reads as:
(14) 
We need to separate the divergent part:
(15) 
Introducing the cutoff:
(16) 
Renormalizing using eq. (1) we will have:
(17) 
Now it is easy to see the presence of two phases: the maximum of the LHS is reached when , the corresponding value is
(18) 
It means that if the saddlepoint equations do not have a solution with nonzero .
Let consider the limit . We can expand the LHS in power series in :
(19) 
Using the following identity:
(20) 
we have:
(21) 
We see that the Coulomb phase does not exist for .
Let us now show that the ”Higgs” phase exists only for . We call this phase ”Higgs” because nonzero breaks gauge symmetry. In this case the second saddlepoint equation is satisfied. The first one reads as:
(22) 
Again using eq. (20) we have:
(23) 
4 DD and NN boundary conditions: one phase
Instead of the DN and ND boundary conditions let us investigate the case with DirichletDirichlet(DD) and NeumannNeumann(NN) boundary conditions. As we will see shortly Coulomb phase is possible for all values of . For the DD case we have the following set of eigenfunctions:
(24) 
And for NN:
(25) 
Note that now we can have which corresponds to a constant mode. Note that if we have a genuine zero mode. It means that the phase with can not exist for this choice of boundary conditions. In the saddlepoint equations and again sum to 1, so we can have a saddlepoint with constant and . From now on, we will assume that . Then from the second saddlepoint equation it follows that . The first saddlepoint equation now reads as:
(26) 
After renormalization we have:
(27) 
Unlike the DN and ND case now the LHS is not bounded from above because of the
5 Conclusion
In this paper we studied the large model on a finite interval. We have shown that for a specific choice of boundary conditions the saddlepoint equations admit a simple analytical solution. Under the DirichletDirichlet and NeumannNeumann boundary condition the system possesses a Coulomb phase with the uniform VEV, usual for the in the infinite space. This phase exists for all values of the interval length . However, under the mixed DirichletNeumann boundary conditions the system has two phases: Coulomb phase which exists for and unusual Higgs phase for with the uniform VEV. Strictly speaking, it is possible to have additional phases with nonconstant VEVs, similar to the FFLO[ff, lo] phase in superconductivity. It is even possible that the Coulomb and Higgs phases in the ND case are not adjacent on the phase diagram because of the presence of additional phases. We will postpone this analysis for future work.
Acknowledgment
We are thankful to A. Gorsky for numerous discussions and F. Popov for reading the manuscript. A.M. is grateful to RFBR grant 150202092 for travel support. \printbibliography