^{3}.

We analyze two-dimensional nonlinear sigma models at nonzero chemical potentials, which are governed by a complex action. In the spirit of contour deformations (thimbles), we extend the fields into the complex plane, which allows us to incorporate the chemical potentials

Two-dimensional nonlinear sigma models have been known for a long time to share nontrivial properties—such as asymptotic freedom, dynamical mass generation, topology, supersymmetric extensions, etc.—with four-dimensional non-Abelian gauge theories; see, e.g.,

Indeed, we will analyze (purely bosonic) sigma models at nonzero chemical potentials

When treating these systems by stochastic quantization (Langevin dynamics, in the original field representation), the drift from the complex action immediately drives the field configurations into the complex plane. The thimble method

The thimble and Langevin method not only share the same saddles, but were empirically found to concentrate around similar complex configurations

Our work is motivated by the continuum trans-series description, for which sigma models serve as a showcase

For bosons, twisted boundary conditions in space and time are interchangeable (when exchanging length and inverse temperature); only fermions can distinguish between space and time.

The objects presented here are to our knowledge the first complex solutions of a field theory complexified in this way. In the sine-Gordon-like quantum mechanics that results from dimensionally reducing (supersymmetric) sigma models (at

The obtained solutions consist again of (anti)holomorphic functions; however, we have to specify twice as many functions for the doubled degrees of freedom (d.o.f.). These objects are (anti-)self-dual in the sense that their action density equals (minus) the topological density. The latter is still a total derivative and given in terms of the Laplacian of a logarithm, but the meaning of the complex total topological charge as a winding number is more intricate, as are the symmetries of the complexified system.

As examples we analyze analogues of fractional constituents and unit charge instantons in the

This work is organized into two main parts, one about the O(3) model and its specific realizations and one about the more general CP(N-1) models. In both parts, we first discuss (conventional) BPS solutions, the global symmetries to which chemical potentials couple, the method of pushing the latter into twisted boundary conditions, and the resulting complexity issue. Then we perform the field complexification doubling the d.o.f. and write down the complex field equations and their solutions in general. At the end of each part we discuss basic examples, fractional constituents, and unit charge instantons. Section

There are (at least) three ways to parametrize the O(3) field, and for reasons of illustration we will discuss them in parallel. The defining parametrization uses a real three-vector field

A famous tool to obtain classical solutions is the Bogomol’nyi bound, in which the Lagrangian is split into an (absolute) square plus (or minus) a topological term. This is most transparent in the

Consequently, configurations with

For BPS solutions, a compact formula for the topological charge density applies,

From now on we will consider a nonzero temperature

One of the three global O(3) symmetry rotations shifts

In all three cases one could try to revert the new Lagrangian

For generic

As will be shown there, Eqs.

With the help of twisted fields, BPS solutions can be extended to the case of purely imaginary chemical potential, say,

We have already argued that the action

Keeping the equivalence of the three parametrizations, e.g.,

To obtain

These complexified fields are required to obey the twisted boundary conditions

Since the stereographic coordinate

An equivalent

The transformation between the two complexifications is linear, namely,

The reverse transformation is

Again, due to the analogue of Eq.

Complex coordinates and derivatives will again be useful to write the Lagrangian with complexified fields as

Although still second order differential equations, they can in fact easily be solved in a BPS-like manner:

In the limit of purely imaginary

It is therefore tempting to see what became of the topological properties for these complex solutions. The formal equivalent of the topological density,

The topological density

The (real) O(3) model can be trivially embedded into higher O(N) models, through

As an example we consider the simplest solutions of Eq.

The product

For the action and topological density we use that

Figure

Logarithmic plot of

In expression

Evaluation of the density

Real (top) and imaginary part (bottom) of the total charge

Naively extending the notion of covering to complex target spaces, we can give an interpretation of the complex fractional charge: as

For a purely real

Other constituent solutions can easily be shown to contain jumps in

See also the discussion of Fourier components below Eq.

Finally, note that in our solution

For the family of CP(N-1) models with chemical potential, we can repeat the steps done in the O(3) model: incorporating the chemical potential into twisted boundary conditions, complexifying the twisted fields, and looking for BPS-like complex solutions.

The CP(N-1) field

The Lagrangian (at

Starting with the quartic CP(1) action and using

Searching for classical solutions we can again make use of the BPS formalism. Defining complex gauge fields and covariant derivatives,

The global U(N) symmetry

We collect the chemical potentials into a diagonal matrix,

The same relation between Lagrangians with and without

However, the problem encountered in the O(3) model also occurs here: for generic

Looking for saddles at nonzero chemical potential, we first have to complexify the fields. As in the O(3) model we do this by treating the field

For the new Lagrangian generalizing

In the same way the global symmetry is extended to

The complex nature of the gauge field also requires a modified definition of the complexified covariant derivative [completing

The operator

The topological density is also

Note that in these equations one may commute

Since the commutators are scalars and drop out of the particular projectors in the equations of motion.

). These equations are again easily solved by vanishing complex covariant derivatives,Furthermore, the variational principle implies that, for stationary points,

For the real model, this follows from a finite action requirement for

We can again construct unconstrained fields as

Again, the equations of motion

A generic (anti)holomorphic solution satisfying the boundary condition

Any solution for

The term

The topological charge, as defined in Eq.

The topological interpretation of a complex charge can be illustrated nicely by virtue of the Polyakov line

Top: Polyakov line

As an example we will now discuss a full instanton at generic

They are of the BPS-like form of Eq.

Using formula

Up to a common prefactor

In this example, we focus on the effect of a varying chemical potential

The slope

Real (top) and imaginary part (bottom) of the total charge

For

For purely imaginary

This constituent picture will essentially be validated now, looking at the behavior of the topological density. In particular, the jumps in the total charge coincide with constituents (dis)appearing at spatial infinity.

We analyze the topological density

Intersection picture for the topological charge

Logarithmic plot of

Logarithmic plot of

We have analyzed the equations of motion from the complex action of two-dimensional sigma models at nonzero chemical potential. By complexifying the fields, we were able to push the chemical potential into properly twisted boundary conditions and to solve these equations.

In a BPS manner, the solutions are provided by holomorphic and antiholomorphic solutions. Due to the doubled d.o.f., each solution is given by a pair of functions in O(3) [Eqs.

Since these (anti)baryons consist of just two (anti)quarks, the corresponding gauge symmetry would be SU(2), which shows the limitation of this analogy. Another difference is that our vectors constitute just classical solutions.

Moreover, the Fourier series of these functions are tightly linked: they must span an equal range of Fourier summands and, therefore, their moduli spaces must be of equal dimension (see the discussion at the end of Sec.One might think of looking for these objects numerically on the lattice, as was done for real twisted solutions through cooling in

It would be interesting to understand the role of these paired functions in approximate superpositions, which are needed for trans-series in (close to) neutral sectors. In some parameter space of the full instanton, the solutions seem to be extremely fine-tuned, giving strong action density peaks, whose imaginary parts fluctuate and cancel.

The coupling of fermions to these objects seems straightforward. Other desirable quantities of these solutions that are important for physical applications are the moduli space metric and the fluctuation operator eigenmodes, the latter giving a first hint at the thimbles surrounding these saddles. For both, a certain amount of technicalities from purely imaginary

F. B. thanks Tin Sulejmanpasic for discussions in an early stage of the study, as well as Muneto Nitta and his group for discussions and their hospitality during a visit. F. B. also acknowledges support by a Heisenberg Grant of Deutsche Forschungsgemeinschaft (DFG) BR 2872/6-2, BR 2872/9-1.