Conformal Space Approach Research Articles

Multicriticality in Yang-Lee edge singularity

In this paper we study the non-unitary deformations of the two-dimensional Tricritical Ising Model obtained by coupling its two spin ℤ2 odd operators to imaginary magnetic fields. Varying the strengths of these imaginary magnetic fields and adjusting correspondingly the coupling constants of the two spin ℤ2 even fields, we establish the presence of two universality classes of infrared fixed points on the critical surface. The first class corresponds to the familiar Yang-Lee edge singularity, while the second class to its tricritical version. We argue that these two universality classes are controlled by the conformal non-unitary minimal models mathcal{M} (2, 5) and mathcal{M} (2, 7) respectively, which is supported by considerations based on PT symmetry and the corresponding extension of Zamolodchikov’s c-theorem, and also verified numerically using the truncated conformal space approach. Our results are in agreement with a previous numerical study of the lattice version of the Tricritical Ising Model [1]. We also conjecture the classes of universality corresponding to higher non-unitary multicritical points obtained by perturbing the conformal unitary models with imaginary coupling magnetic fields.

Variations on vacuum decay: The scaling Ising and tricritical Ising field theories

We study the decay of the false vacuum in the scaling Ising and tricritical Ising field theories using the Truncated Conformal Space Approach and compare the numerical results to theoretical predictions in the thin wall limit. In the Ising case, the results are consistent with previous studies on the quantum spin chain and the $\varphi^4$ quantum field theory; in particular we confirm that while the theoretical predictions get the dependence of the bubble nucleation rate on the latent heat right, they are off by a model dependent overall coefficient. The tricritical Ising model allows us on the other hand to examine more exotic vacuum degeneracy structures, such as three vacua or two asymmetric vacua, which leads us to study several novel scenarios of false vacuum decay by lifting the vacuum degeneracy using different perturbations.

Duality and form factors in the thermally deformed two-dimensional tricritical Ising model

The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional E_7E7 Lie algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation operators, \sigma(x)σ(x) and \sigma'(x)σ′(x), in either phase are accompanied by associated disorder operators, \mu(x)μ(x) and \mu'(x)μ′(x). Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four operators. For \sigma(x)σ(x) and \sigma'(x)σ′(x), the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest \mathbb{Z}_2ℤ2 odd particles. Similarly, the equations for \mu(x)μ(x) and \mu'(x)μ′(x) have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions are eventually not independent; instead, the parameters of the solutions for \sigma(x)/\sigma'(x)σ(x)/σ′(x) are fixed in terms of those for \mu(x)/\mu'(x)μ(x)/μ′(x). We use the truncated conformal space approach to confirm numerically the derived expressions of the matrix elements as well as the validity of the \DeltaΔ-sum rule as applied to the off-critical correlators. We employ the derived form factors of the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron or Raman scattering experiments.

Chirally factorised truncated conformal space approach

Truncated Conformal Space Approach (TCSA) is a highly efficient method to compute spectra, operator matrix elements and time evolution in quantum field theories defined as relevant perturbations of 1+1-dimensional conformal field theories. However, similarly to other exact diagonalisation methods, TCSA is ridden with the “curse of dimensionality”: the dimension of the Hilbert space increases exponentially with the (square root of the) truncation level, limiting its precision by the available memory resources. Here we describe an algorithm which exploits the chiral factorisation property of conformal field theory with periodic boundary conditions to achieve a substantial improvement in the truncation level. The Chirally Factorised TCSA (CFTCSA) algorithm presented here works with inputs describing the necessary CFT data in a specified format. It makes possible much more precise calculations with given computing resources and extends the reach of the method to problems requiring large Hilbert space dimensions. In fact, it has already been used in a number of recent works ranging from determination of form factors, through studying confinement of topological excitations to non-equilibrium dynamics. Besides the description of the algorithm, a MATLAB implementation of the algorithm is also provided as an ancillary file package, supplemented with example codes computing spectra, matrix elements and time evolution, and with CFT data for three different quantum field theories. We also give a detailed how-to guide for constructing the required CFT data for Virasoro minimal models with central charge c<1, and for the massless free boson with c=1.

Periodically driven perturbed CFTs: The sine-Gordon model

We analyze a version of the sine-Gordon model in which the strength of the cosine potential has a periodic dependence on time. This model can be considered as the continuum limit of the many-body generalization of the Kapitza pendulum. Based on the perturbed CFT point of view, we develop a truncated conformal space approach (TCSA) to investigate the Floquet quasienergy spectrum. We focus on the effective behaviour for large driving frequencies, which we also derive exactly. Depending on the driving protocol, we can recover the original sine-Gordon model or its two-frequency version. The rich structure of the two-frequency model implies that the time-periodic drive can break integrability, can lead to new states in the spectrum, or can result in a phase transition. Our method is applicable for any periodically driven perturbed conformal field theories.

TCSA and the finite volume boundary state

We develop a new way to calculate the overlap of a boundary state with a finite volume bulk state in the truncated conformal space approach. We check this method in the thermally perturbed Ising model analytically, while in the scaling Lee-Yang model numerically by comparing our results to excited state g-functions, which we obtained by the analytical continuation method. We also give a simple argument for the structure of the asymptotic overlap between the finite volume boundary state and a periodic multiparticle state, which includes the ratio of Gaudin type determinants.

Conformal defects from string field theory

Unlike conformal boundary conditions, conformal defects of Virasoro minimal models lack classification. Alternatively to the defect perturbation theory and the truncated conformal space approach, we employ open string field theory (OSFT) techniques to explore the space of conformal defects. We illustrate the method by an analysis of OSFT around the background associated to the (1, 2) topological defect in diagonal unitary minimal models. Numerical analysis of OSFT equations of motion leads to an identification of a nice family of solutions, recovering the picture of infrared fixed points due to Kormos, Runkel and Watts. In particular, we find a continuum of solutions in the Ising model case and 6 solutions for other minimal models. OSFT provides us with numerical estimates of the g-function and other coefficients of the boundary state.

Kibble–Zurek mechanism in the Ising Field Theory

The Kibble--Zurek mechanism captures universality when a system is driven through a continuous phase transition. Here we study the dynamical aspect of quantum phase transitions in the Ising Field Theory where the quantum critical point can be crossed in different directions in the two-dimensional coupling space leading to different scaling laws. Using the Truncated Conformal Space Approach, we investigate the microscopic details of the Kibble--Zurek mechanism in terms of instantaneous eigenstates in a genuinely interacting field theory. For different protocols, we demonstrate dynamical scaling in the non-adiabatic time window and provide analytic and numerical evidence for specific scaling properties of various quantities. In particular, we argue that the higher cumulants of the excess heat exhibit universal scaling in generic interacting models for a slow enough ramp.

On asymptotic behaviour in truncated conformal space approach

The Truncated conformal space approach (TCSA) is a numerical technique for finding finite size spectrum of Hamiltonians in quantum field theory described as perturbations of conformal field theories. The truncation errors of the method have been systematically studied near the UV fixed point (when the characteristic energy related to the coupling is less than the truncation cutoff) where a good theoretical understanding has been achieved. However numerically the method demonstrated a good agreement with other methods for much larger values of the coupling when the RG flow approaches a new fixed point in the infrared. In the present paper we investigate this regime for a number of boundary RG flows testing the leading exponent and truncation errors. We also study the flows beyond the first fixed point which have been observed numerically but yet lack a theoretical understanding. We show that while in some models such flows approximate reversed physical RG flows, in other models the spectrum approaches a stable regime that does not correspond to any local boundary condition. Furthermore we find that in general the flows beyond the first fixed point are very sensitive to modifications of the truncation scheme.

Perturbative post-quench overlaps in quantum field theory

In analytic descriptions of quantum quenches, the overlaps between the initial pre-quench state and the eigenstates of the time evolving Hamiltonian are crucial ingredients. We construct perturbative expansions of these overlaps in quantum field theories where either the pre-quench or the post-quench Hamiltonian is integrable. Using the E8 Ising field theory for concrete computations, we give explicit expressions for the overlaps up to second order in the quench size, and verify our results against numerical results obtained using the Truncated Conformal Space Approach. We demonstrate that the expansion using the post-quench basis is very effective, but find some serious limitations for the alternative approach using the pre-quench basis.

Nonequilibrium time evolution and rephasing in the quantum sine-Gordon model

We discuss the non-equilibrium time evolution of the phase field in the sine-Gordon model using two very different approaches: the truncated Wigner approximation and the truncated conformal space approach. We demonstrate that the two approaches agree for a period covering the first few oscillations, thereby giving a solid theoretical prediction in the framework of sine-Gordon model, which is thought to describe the dynamics of two bosonic condensates in quasi-one-dimensional traps coupled via a Josephson tunneling term. We conclude, however, that the recently observed phase-locking behavior cannot be explained in terms of homogeneous sine-Gordon dynamics, which hints at the role of other degrees of freedom or inhomogeneity in the experimental system.

Chiral entanglement in massive quantum field theories in 1+1 dimensions

We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy’s Ansatz. The chiral entanglement entropy contains a universal O(1) term γ for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.

Quench dynamics of the Ising field theory in a magnetic field

We numerically simulate the time evolution of the Ising field theory after quenches starting from the $E_8$ integrable model using the Truncated Conformal Space Approach. The results are compared with two different analytic predictions based on form factor expansions in the pre-quench and post-quench basis, respectively. Our results clarify the domain of validity of these expansions and suggest directions for further improvement. We show for quenches in the $E_8$ model that the initial state is not of the integrable pair state form. We also construct quench overlap functions and show that their high-energy asymptotics are markedly different from those constructed before in the sinh/sine-Gordon theory, and argue that this is related to properties of the ultraviolet fixed point.

Correlation Functions of the Quantum Sine-Gordon Model in and out of Equilibrium.

Complete information on the equilibrium behavior and dynamics of quantum field theory (QFT) is provided by multipoint correlation functions. However, their theoretical calculation is a challenging problem, even for exactly solvable models. This has recently become an experimentally relevant problem, due to progress in cold-atom experiments simulating QFT models and directly measuring higher order correlations. Here we compute correlation functions of the quantum sine-Gordon model, a prototype integrable model of central interest from both theoretical and experimental points of view. Building upon the so-called truncated conformal space approach, we numerically construct higher order correlations in a system of finite size in various physical states of experimental relevance, both in and out of equilibrium. We measure deviations from Gaussianity due to the presence of interaction and analyze their dependence on temperature, explaining the experimentally observed crossover between Gaussian and non-Gaussian regimes. We find that correlations of excited states are markedly different from the thermal case, which can be explained by the integrability of the system. We also study dynamics after a quench, observing the effects of the interaction on the time evolution of correlation functions, their spatial dependence, and their non-Gaussianity as measured by the kurtosis.

Overlaps after quantum quenches in the sine-Gordon model

We present a numerical computation of overlaps in mass quenches in sine-Gordon quantum field theory using truncated conformal space approach (TCSA). To improve the cut-off dependence of the method, we use a novel running coupling definition which has a general applicability in free boson TCSA. The numerical results for the first breather overlaps are compared with the analytic continuation of a previously proposed analytical Ansatz for the initial state in a related sinh-Gordon quench, and good agreement is found between the numerical data and the analytical prediction in a large energy range.

Truncated Hilbert space approach to the 2d ϕ 4 theory

We apply the massive analogue of the truncated conformal space approach to study the two dimensional $\phi^{4}$ theory in finite volume. We focus on the broken phase and determine the finite size spectrum of the model numerically. We interpret the results in terms of the Bethe-Yang spectrum, from which we extract the infinite volume masses and scattering matrices for various couplings. We compare these results against semiclassical analysis and perturbation theory. We also analyze the critical point of the model and confirm that it is in the Ising universality class.

Particle formation and ordering in strongly correlated fermionic systems: Solving a model of quantum chromodynamics

In this paper we study a (1+1)-dimensional version of the famous Nambu-Jona-Lasinio model of Quantum Chromodynamics (QCD2) both at zero and finite hadron density. We use non-perturbative techniques (non-Abelian bosonization and Truncated Conformal Space Approach). At zero density we describe a formation of fermion three-quark (nucleons and $\Delta$-baryons) and boson (two-quark mesons, six-quark deuterons) bound states and also a formation of a topologically nontrivial phase. At finite hadron density, the model has a rich phase diagram which includes phases with density wave and superfluid quasi-long-range (QLR) order and also a phase of a baryon Tomonaga-Luttinger liquid (strange metal). The QLR order results as a condensation of scalar mesons (the density wave) or six-quark bound states (deuterons).

Confinement in the q-state Potts model: an RG-TCSA study

In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field induces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting "hadron" spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also the Bethe-Salpeter equation. Here we compare the available predictions to numerical results from renormalization group improved truncated conformal space approach (RG-TCSA). While mesonic states in the Ising model have already been considered in a different truncated Hamiltonian approach, this is the first time that a precision numerical study is performed for the 3-state Potts model. We find that the semiclassical approach provides a very accurate description for the mesonic spectrum in all the parameter regime for weak magnetic field, while the low-energy expansion from the Bethe-Salpeter equation is only valid for very weak fields where it gives a slight improvement over the semiclassical results. In addition, we confirm the validity of the recent predictions for the baryon spectrum obtained from solving the quantum mechanical three-body problem.

Exact finite volume expectation values of local operators in excited states

We present a conjecture for the exact expression of finite volume expectation values in excited states in integrable quantum field theories, which is an extension of an earlier conjecture to the case of general diagonal factorized scattering with bound states and a nontrivial bootstrap structure. The conjectured expression is a spectral expansion which uses the exact form factors and the excited state thermodynamic Bethe Ansatz as building blocks. The conjecture is proven for the case of the trace of the energy-moment tensor. Concerning its validity for more general operators, we provide numerical evidence using the truncated conformal space approach. It is found that the expansion fails to be well-defined for small values of the volume in cases when the singularity structure of the TBA equations undergoes a non-trivial rearrangement under some critical value of the volume. Despite these shortcomings, the conjectured expression is expected to be valid for all volumes for most of the excited states, and as an expansion above the critical volume for the rest.

Truncated conformal space approach inddimensions: A cheap alternative to lattice field theory?

We show how to perform accurate, nonperturbative and controlled calculations in quantum field theory in d dimensions. We use the Truncated Conformal Space Approach (TCSA), a Hamiltonian method which exploits the conformal structure of the UV fixed point. The theory is regulated in the IR by putting it on a sphere of a large finite radius. The QFT Hamiltonian is expressed as a matrix in the Hilbert space of CFT states. After restricting ourselves to energies below a certain UV cutoff, an approximation to the spectrum is obtained by numerical diagonalization of the resulting finite-dimensional matrix. The cutoff dependence of the results can be computed and efficiently reduced via a renormalization procedure. We work out the details of the method for the phi^4 theory in d dimensions with d not necessarily integer. A numerical analysis is then performed for the specific case d = 2.5, a value chosen in the range where UV divergences are absent. By going from weak to intermediate to strong coupling, we are able to observe the symmetry-preserving, symmetry-breaking, and conformal phases of the theory, and perform rough measurements of masses and critical exponents. As a byproduct of our investigations we find that both the free and the interacting theories in non integral d are not unitary, which however does not seem to cause much effect at low energies.