283. Quantum kink and its excitations

This is a guest post by Arttu Rajantie from Imperial College. Dmitry.

Dmitry has kindly asked me to write about my recent paper Quantum kink and its excitations, which I wrote with my PhD student David Weir. In it we show how one can obtained detailed information about kinks and other solitons in quantum field theory in a simple way from field correlation functions. What makes this useful is that these correlators can be calculated non-perturbatively using Monte Carlo techniques. We demonstrate the method with kinks in a simple 1+1-dimensional model, but it should be easy to apply it to more realistic theories, leading to a wide range of potential applications.

Solitons are essentially localised waves that maintain their shape as they move. There are lots of fascinating examples of them in many different physical systems. In field theory, we are mostly interested in topological solitons, which are stable because of the topology of the field configuration space. Common examples of these are vortices, cosmic strings, magnetic monopoles and domain walls.

The kink, which we focused on in our paper, is the simplest possible example of a soliton. It is essentially a domain wall in one spatial dimension. The Lagrangian of the theory is

where is a real scalar field. The theory has a spontaneously broken symmetry, , and two vacua . In classical theory, the kink is a solution that interpolates between two vacua, and in this theory, one can easily find its exact form,

In this solution, energy is localised around the origin, and therefore the kink is effectively a particle with size . The mass of a classical kink is simply given by the total energy of the solution

Classical calculations like this are straightforward, because everything is described by differential equations which can be always be solved at least using numerical methods. The world is, however, not classical but quantum mechanical, so it is natural to ask what happens in quantum theory. To my knowledge, this question was first studied by Dashen, Hasslacher and Neveu in 1974, who quantised linear perturbations around the classical kink background . They found a spectrum of two discrete energy levels,

corresponding to the ground state and a bound excited state, and a continuum of states

corresponding to a free unbound scalar particle with mass and momentum . By summing up the quantum zero-point energies of all these states, they obtained an expression for the leading quantum correction to the kink mass in the weak-coupling limit

The same approach can be applied to other, physically more relevant solitons, but in practice calculations are often prohibitively hard. In non-supersymmetric theories, calculations have been done only in some very specific cases. Supersymmetry simplifies calculations, and Rebhan, van Nieuwenhuizen and Wimmer have been able to calculate leading quantum corrections to kinks, vortices and monopoles in supersymmetric theories. However, these perturbative results are only valid in weakly coupled theories.

Fig.1: Untwisted boundary conditions

In contrast, we adopted a non-perturbative approach, and set out to find a way to calculate soliton properties using lattice Monte Carlo simulations. The obvious price one then has to pay is that any results are numerical and come with statistical errors, but at least the errors can be estimated and are well under control. It is not, however, immediately obvious how to do this. Usually in lattice field theory simulations, one obtains particle masses from the decay rate of field correlation functions. To calculate the soliton mass, one would have to know the corresponding creation operator, which is usually non-local and therefore inconvenient in practical calculations. One cannot simply put in a classical kink background either, because the calculation, being non-perturbative, is background independent.

Instead, it turns out to be most convenient to use twisted boundary conditions. With the usual periodic boundary conditions , which correspond to identifying two ends of the lattice in the spatial directions (see Fig.1), the classical ground state is one of the two vacua. Therefore, a Monte Carlo simulation with periodic boundary conditions describes quantum fluctuations around the vacuum. However, if we give the field a twist by imposing antiperiodic boundary conditions , essentially defining the theory on a Mobius strip (see Fig.2), then the classical ground state is a kink. Correspondingly, a simulation with twisted boundary conditions describes quantum fluctuations around a kink.

Fig.2: Twisted boundary conditions

People have used this approach in the past to measure the quantum mechanical kink mass by expressing it as the difference in the ground state energies of the twisted and untwisted theories, which can be expressed in Euclidean field theory as

where is the length of the lattice in time direction, and and are partition functions calculated for untwisted and twisted boundary conditions. Because one cannot calculate the partition function directly using Monte Carlo simulations, one measures its derivative and integrates it along some curve in the parameter space starting from a point where the kink mass is known (such as the classical limit or the symmetric phase). Unfortunately this means that in order to calculate the mass at a given point, one needs to do simulations at a range of different values, and the statistical errors accumulate.

In our paper, we show that by calculating  field correlation functions in the twisted theory, we can obtain the kink mass directly, without having to integrate over parameter values. The statistical error in the measurement is therefore much better under control, and often smaller. Furthermore, we obtain also the excitation spectrum of the kink and even approximate wave functions of the energy eigenstates.

Consider a correlator of some operators and in the (Euclidean) time direction. The operators can be nonlocal in space. The correlator has a spectral expansion in terms of energy eigenstates ,

where is the energy of the state . These energies are defined relative to the ground state energy, so although the lowest state in the expansion corresponds to the kink ground state, it has rather than . However, we can measure the kink mass by considering operators with a non-zero momentum . Because of momentum conservation, all states in the expansion have the same momentum. In particular, the lowest state corresponds to the kink moving with momentum . The energy of the ground state is therefore

assuming that . Of course, this is simply the non-relativistic kinetic energy of a moving kink. Because the momentum is a parameter that we put in, we can easily obtain the kink mass , if we measure the energy .

Fig.3: Kink mass measurements

Physically, we are simply measuring the inertial mass of the kink. We throw a particle with a known momentum at it, and we measure how fast the kink is moving afterwards.

Our results are shown in Fig.3. The red band shows the kink mass calculated in the traditional way by integrating the derivative, and the data points show the mass determined using our method. For the same simulation time, the errors are comparable, but our method has the advantages that the mass at a given value of can be calculated directly, and that the errors are uncorrelated. The solid and dashed lines show the classical and one-loop masses.

In addition, choosing a set of operators with zero net momentum, we can obtain the spectrum of excited states in the kink rest frame. This is exactly what Dashen et al calculated at linear level to obtain the one loop kink mass, and we can compare our results with them directly.

The operators we used for this purpose were products of two field operators, and therefore the physical picture at weak coupling  is that we create a pair of particles with zero total momentum. It is then possible that both particles, one of them, or neither of them is bound to or annihilated by the kink. The energy spectrum should therefore consist of two-particle states with energy greater than twice the scalar mass , one-particle states with energy greater than the scalar mass , and discrete bound states at lower energies. To measure the spectrum, we used the Luscher-Wolff method, which constructs from the given set of operators an approximate creation operator for each of states by diagonalising their correlation matrix. At weak coupling, we can even obtain the approximate quantum wavefunctions of the states.

In our simulations, we find qualitative agreement with the linearised spectrum at weak coupling: one bound state and a continuum of one-particle states. We identify the states by looking at their volume-dependence and the shapes of their wavefunctions, which also agree well with the linearised theory.

In our paper we only looked at weak coupling, so that we could compare our results with the existing perturbative calculations. However, the method is completely non-perturbative and can be used in situations where perturbation theory breaks down. Even in this simple one-dimensional theory, we could study the critical behaviour of the kink near the phase transition. It is known that the theory is in the same universality class as the two-dimensional Ising model, which is self dual. We should therefore be able to see this duality emerging asymptotically when we approach the critical point. The properties of the kink in the broken phase should become identical to those of the scalar field in the symmetric phase.

It is straightforward to apply the same method to other solitons. For example, by simulating SU(2) or SU(N) gauge field theory with an adjoint Higgs field in four dimensions with twisted C-periodic boundary conditions, one would be able to determine the mass and excitation spectrum of the ‘t Hooft-Polyakov monopole. One could also test lattice implementations of supersymmetry by computing soliton properties for which exact analytic results are available. It is also possible to use the method with spatially extended solitons, in which case bound states would correspond to particles localised on the soliton. For instance, in domain wall brane world models, the whole Standard Model of particle physics is made up of bound states like that.