arXiv subjects
Publications and source records attributed to Paul Fendley.
Interesting physical results can be obtained from sigma models by taking the number of fields N to zero. I discuss how one can make sense of this limit by using exact S matrix techniques. I review how this can be done for the case of self-avoiding polymers, and speculate on the application to the replica limit of disordered systems.
We show that two-dimensional sigma models are equivalent to certain perturbed conformal field theories. When the fields in the sigma model take values in a space G/H for a group G and a maximal subgroup H, the corresponding conformal field theory is the $k\to\infty$ limit of the coset model $(G/H)_k$, and the perturbation is related to the current of G. This correspondence allows us for example to find the free energy for the "O(n)" (=O(n)/O(n-1)) sigma model at non-zero temperature. It also results in a new approach to the CP^{n} model.
A fundamental result relevant to spin chains and two-dimensional disordered systems is that the sphere sigma model with instanton coupling theta=pi has a non-trivial low-energy fixed point and a gapless spectrum. This result is extended to two series of sigma models with theta=pi: the SU(N)/SO(N) sigma models flow to the SU(N)_1 WZW theory, while the O(2N)/O(N)\times O(N) models flow to O(2N)_1 (2N free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when theta=pi. I compute the energy in a background field, and verify that the perturbative expansions for θ=0 and pi are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with Sp(2N) symmetry, the Sp(2N)/U(N) models, flow to Sp(2N)_1.
We find the exact spectrum and degeneracies for the Gross-Neveu model in two dimensions. This model describes N interacting Majorana fermions; it is asymptotically free, and has dynamical mass generation and spontaneous chiral symmetry breaking. We show here that the spectrum contains 2^{N/2} kinks for any $N$. The unusual \sqrt{2} in the number of kinks for odd $N$ comes from restrictions on the allowed multi-kink states. These kinks are the BPS states for a generalized supersymmetry where the conserved current is of dimension N/2; the N=3 case is the {\cal N}=1 supersymmetric sine-Gordon model, for which the spectrum consists of 2\sqrt{2} kinks. We find the exact S matrix for these kinks, and the exact free energy for the model.
Sigma models arise frequently in particle physics and condensed-matter physics as low-energy effective theories. In this paper I compute the exact free energy at any temperature in two hierarchies of integrable sigma models in two dimensions. These theories, the SU(N)/SO(N) and O(2P)/O(P) x O(P) models, are asymptotically free and exhibit charge fractionalization. When the instanton coupling theta=pi, they flow to the SU(N)_1 and O(2P)_1 conformal field theories, respectively. I also generalize the free energy computation to massive and massless perturbations of the coset conformal field theories SU(N)_k/SO(N)_{2k} and O(2P)_k/O(P)_k x O(P)_k.
We consider the problem of tunneling between spinless 1D Luttinger liquid chains. We show how to map the problem onto the 4-state chiral clock model together with a free boson for the total charge mode. We use a variety of results, some of them exact, from the integrable chiral Potts model and the study of commensurate/incommensurate transitions to deduce the physics of coupled Luttinger liquids. For those intrachain interaction strengths for which interchain tunneling is relevant, we find that it can lead to the formation of symmetric and antisymmetric bands with split Fermi surfaces, depending on the relative strengths of interchain tunneling and interchain interactions. With split Fermi surfaces, the tunneling is coherent. It is not possible to have two gapless Fermi surfaces with the same Fermi momentum when interchain tunneling is relevant. However, interchain interactions can drive the formation of a gap, in which case interchain transport will be incoherent. We comment on the possible relevance of our results to c-axis transport in high-T_c superconductors.
I survey the kinds of critical behavior believed to be exhibited in two-dimensional disordered systems. I review the different replica sigma models used to describe the low-energy physics, and discuss how critical points appear because of WZW and theta terms.
The low-energy limits of models with disorder are frequently described by sigma models. In two dimensions, most sigma models admit either a Wess-Zumino-Witten or a theta term. When such a term is present the model can have a stable critical point with gapless excitations. We describe how such a critical point appears, in particular in two-dimensional superconductors with disorder. The presence of such terms is required by the underlying (anomalous) symmetries of the original electron model. This indicates that the usual symmetry classes of disordered systems in two dimensions can be further refined. Conversely, our results also indicate that models previously thought to be in different universality classes are in fact the same once the appropriate extra terms are included.
Thermodynamic Bethe ansatz equations are coupled non-linear integral equations which appear frequently when solving integrable models. Those associated with models with N=2 supersymmetry can be related to differential equations, among them Painleve III and the Toda hierarchy. In the simplest such case the massless limit of these non-linear integral equations can be solved in terms of the Airy function. This is the only known closed-form solution of thermodynamic Bethe ansatz equations, outside of free or classical models. This turns out to give the spectral determinant of the Schrodinger equation in a linear potential.
I show that physical quantities in several two-dimensional condensed-matter models are related to the Seiberg-Witten calculation of exact quantities in supersymmetric gauge theory. In particular, the magnetization in the Kondo problem and the current in the boundary sine-Gordon model can each be expressed in the form $\int dx/y$, where for example in the latter $y^2 = x + x^g - u^2$ with u related to the boundary mass scale (the analog of Λ_{QCD}) and g proportional to the radius of the boson squared. Thus for irrational g, the curve y(x) is of infinite genus, while for rational g it is of finite genus. The models are integrable and possess a quantum-group symmetry for any g, but are supersymmetric only at g=2/3. Both models also possess unique forms of g to 1/g duality.
We study multi-soliton states in two-dimensional N=2 supersymmetric theories. We calculate their energy exactly as a function of mass and volume in the simplest integrable N=2 theory, the sine-Gordon model at a particular coupling. These energies are related to the expectation value $I= tr [\exp(inπF) \exp(-H/T)]$, where F is the fermion number. For n=1, this is Witten's index; for n an odd integer, we argue that $I$ is an index in the sense that it is independent of all D-term variations.
We find exactly solvable N=2-supersymmetric flows whose infrared fixed points are the N=2 minimal models. The exact S-matrices and the Casimir energy (a c-function) are determined along the entire renormalization group trajectory. The c-function runs from c=3 (asymptotically) in the UV to the N=2 minimal model values of the central charge in the IR, leading us to interpret these theories as the Landau-Ginzburg models with superpotential $X^{k+2}$. Consideration of the elliptic genus gives further support for this interpretation. We also find an integrable model in this hierarchy which has spontaneously-broken supersymmetry and superpotential $X$, and a series of integrable models with (0,2) supersymmetry. The flows exhibit interesting behavior in the UV, including a relation to the N=2 super sine-Gordon model. We speculate about the relation between the kinetic term and the cigar target-space metric.
We show how to obtain the ultraviolet central charge from the exact S-matrix for a wide variety of models with a $U(1)$ symmetry. This is done by coupling the $U(1)$ current $J$ to a background field. In an $N$=2 superconformal theory with $J$ the fermion number current, the OPE of $J$ with itself and hence the free energy are proportional to $c$. By deforming the supersymmetry into affine $\widehat SU(2)_q$ quantum-group symmetry, this result can be generalized to many $U(1)$-invariant theories, including the N=0 and N=1 sine-Gordon models and the $SU(2)_k$ WZW models. This provides a consistency check on a conjectured S-matrix completely independent of the finite-size effects expressed in terms of dilogarithms resulting from the thermodynamic Bethe ansatz.
This is intended to be a simple discussion of work done in collaboration with S. Cecotti, K. Intriligator and C. Vafa; and with H. Saleur. I discuss how $ Tr F (-1)^F e^{-βH}$ can be computed exactly in any N=2 supersymmetric theory in two dimensions. It gives exact information on the soliton spectrum of the theory, and corresponds to the partition function of a single self-avoiding polymer looped once around a cylinder of radius $β$. It is independent of almost all deformations of the theory, and satisfies an exact differential equation as a function of $β$. For integrable theories it can also be computed from the exact S-matrix. This implies a highly non-trivial equivalence of a set of coupled integral equations with the classical sinh-Gordon and the affine Toda equations.
We find the exact quasiparticle spectrum for the continuum Kondo problem of $k$ species of electrons coupled to an impurity of spin $S$. In this description, the impurity becomes an immobile quasiparticle sitting on the boundary. The particles are ``kinks'', which can be thought of as field configurations interpolating between adjacent wells of a potential with $k+1$ degenerate minima. For the overscreened case $k>2S$, the boundary has this kink structure as well, which explains the non-integer number of boundary states previously observed. Using simple arguments along with the consistency requirements of an integrable theory, we find the exact elastic $S$-matrix for the quasiparticles scattering among themselves and off of the boundary. This allows the calculation of the exact free energy, which agrees with the known Bethe ansatz solution.
We show that ${\rm Tr}(-1)^F F e^{-βH}$ is an index for $N$=2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry's curvature) and satisfies an exact differential equation as a function of $β$. For integrable theories we can also compute the index thermodynamically, using the exact $S$-matrix. The equivalence of these two results implies a highly non-trivial equivalence of a set of coupled integral equations with these differential equations, among them Painleve III and the affine Toda equations.
In the last several years, the Casimir energy for a variety of 1+1-dimensional integrable models has been determined from the exact S-matrix. It is shown here how to modify the boundary conditions to project out the lowest-energy state, which enables one to find excited-state energies. This is done by calculating thermodynamic expectation values of operators which generate discrete symmetries. This is demonstrated with a number of perturbed conformal field theories, including the Ising model, the three-state Potts model, ${\bf Z}_n$ parafermions, Toda minimal S-matrices, and massless Goldstinos.