Finite-size Spectrum Research Articles

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.

Transition from theZ2spin liquid to antiferromagnetic order: Spectrum on the torus

We describe the finite-size spectrum in the vicinity of the quantum critical point between a $\mathbb{Z}_2$ spin liquid and a coplanar antiferromagnet on the torus. We obtain the universal evolution of all low-lying states in an antiferromagnet with global SU(2) spin rotation symmetry, as it moves from the 4-fold topological degeneracy in a gapped $\mathbb{Z}_2$ spin liquid to the Anderson "tower-of-states" in the ordered antiferromagnet. Due to the existence of nontrivial order on either side of this transition, this critical point cannot be described in a conventional Landau-Ginzburg-Wilson framework. Instead it is described by a theory involving fractionalized degrees of freedom known as the O$(4)^\ast$ model, whose spectrum is altered in a significant way by its proximity to a topologically ordered phase. We compute the spectrum by relating it to the spectrum of the O$(4)$ Wilson-Fisher fixed point on the torus, modified with a selection rule on the states, and with nontrivial boundary conditions corresponding to topological sectors in the spin liquid. The spectrum of the critical O($2N$) model is calculated directly at $N=\infty$, which then allows a reconstruction of the full spectrum of the O($2N)^\ast$ model at leading order in 1/N. This spectrum is a unique characteristic of the vicinity of a fractionalized quantum critical point, as well as a universal signature of the existence of proximate $\mathbb{Z}_2$ topological and antiferromagnetically-ordered phases, and can be compared with numerical computations on quantum antiferromagnets on two dimensional lattices.

Spinon bases in supersymmetric CFTs

We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with or (nonlinear) superconformal symmetry. Generalising the spinon basis of the WZW theories, we introduce supersymmetric spinons , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection of (discrete) momenta. The state-content for given choice of is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the theories. In the simplest case, which is the superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the theory. It includes an operator H2, akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the superconformal CFTs, this finitisation precisely leads to the so-called Mk supersymmetric lattice models with characteristic order-k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.

Emergent quantum criticality from fractionalizing one-dimensional SO(5) symmetric valence-bond solid states

A common feature of topological phases of matter is the fractionalization of the quantum number in their low-energy excitations. Such information is encoded in their ground state wave functions, but emerges in the bipartite entanglement spectra. The symmetric extensive bipartition is an effective novel method to create deconfined fractionalized edge particles in the reduced subsystem, which lead to quantum critical behavior associated with the transition from the topological phase to its adjacent trivial phase. Here we report the interesting results revealed by applying this method to the one-dimensional SO(5) symmetric valence-bond solid state being a spin-2 symmetry protected topological phase. From the finite-size entanglement spectrum, we find the lowest level to be an SO(5) singlet with a logarithmic entanglement entropy. Surprisingly, the first excited level is also an SO(5) singlet and the spectral gap scales with the subsystem size as ${L}_{A}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\simeq}1.978$. In the thermodynamic limit, a novel quantum criticality emerges with SO(5) spinons and their four-body singlet bound states as elementary excitations, hence ruling out the possibility of being described by a conformal field theory. Moreover, the entanglement Hamiltonian can be determined as an SO(5) symmetric nearest neighbor spin-3/2 quadruple-quadruple interaction with a negative coupling. Our work thus demonstrates the power of this new method in the study of quantum criticality encoded in the topological ground states.

Finite size spectrum of SU(N) principal chiral field from discrete Hirota dynamics

Using recently proposed method of discrete Hirota dynamics for integrable (1+1)D quantum field theories on a finite space circle of length L we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N)×SU(N) principal chiral field model as functions of mL, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of Wronskian determinants of N×N matrices parameterized by N−1 functions of the spectral parameter θ with the known analytic properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2 case. As a demonstration of efficiency of our method, we solve these equations numerically for a few low-lying states at N=3 in a wide range of mL.

Finite-volume spectra of the Lee-Yang model

We consider the non-unitary Lee-Yang minimal model ${\cal M}(2,5)$ in three different finite geometries: (i) on the interval with integrable boundary conditions labelled by the Kac labels $(r,s)=(1,1),(1,2)$, (ii) on the circle with periodic boundary conditions and (iii) on the periodic circle including an integrable purely transmitting defect. We apply $\varphi_{1,3}$ integrable perturbations on the boundary and on the defect and describe the flow of the spectrum. Adding a $\Phi_{1,3}$ integrable perturbation to move off-criticality in the bulk, we determine the finite size spectrum of the massive scattering theory in the three geometries via Thermodynamic Bethe Ansatz (TBA) equations. We derive these integral equations for all excitations by solving, in the continuum scaling limit, the TBA functional equations satisfied by the transfer matrices of the associated $A_{4}$ RSOS lattice model of Forrester and Baxter in Regime III. The excitations are classified in terms of $(m,n)$ systems. The excited state TBA equations agree with the previously conjectured equations in the boundary and periodic cases. In the defect case, new TBA equations confirm previously conjectured transmission factors.

The exact spectrum and mirror duality of the (AdS5 × S 5) η superstring

We discuss the spectrum of a string propagating on η-deformed AdS5 × S5 by treating its worldsheet theory as an integrable quantum field theory. The exact S-matrix of this field theory is given by a qdeformation of the AdS5 × S5 worldsheet S-matrix with a real deformation parameter. By considering mirror (double Wick-rotated) versions of these worldsheet theories, we give the thermodynamic Bethe ansatz description of their exact finite-size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the string level, this seems to indicate that the light-cone worldsheet theories of strings on particular pairs of backgrounds are related by a double Wick rotation, a feature we call “mirror duality.” We provide a partial verification of these statements at the level of a sigma model by considering reduced actions and their corresponding (mirror) giant magnon solutions.

Integrable anyon chains: From fusion rules to face models to effective field theories

Starting from the fusion rules for the algebra SO(5)2 we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of ‘interactions round the face’ (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra, namely WB2 and WD5, respectively. The latter contains primaries with half and quarter spin. The modular partition function and fusion rules are derived and found to be consistent with the results for the lattice model.

Logarithmic superconformal minimal models

The higher fusion level logarithmic minimal models have recently been constructed as the diagonal GKO cosets where n ≥ 1 is an integer fusion level and k = nP/(P′− P) − 2 is a fractional level. For n = 1, these are the well-studied logarithmic minimal models . For n ≥ 2, we argue that these critical theories are realized on the lattice by n × n fusion of the n = 1 models. We study the critical fused lattice models within a lattice approach and focus our study on the n = 2 models. We call these logarithmic superconformal minimal models where P = |2p − p′|, P′ = p′ and p, p′ are coprime. These models share the central charges of the rational superconformal minimal models . Lattice realizations of these theories are constructed by fusing 2 × 2 blocks of the elementary face operators of the n = 1 logarithmic minimal models . Algebraically, this entails the fused planar Temperley–Lieb algebra which is a spin-1 Birman–Murakami–Wenzl tangle algebra with loop fugacity β2 = [x]3 = x2 + 1 + x−2 and twist ω = x4 where x = eiλ and λ = (p′− p)π/p′. The first two members of this n = 2 series are superconformal dense polymers with , β2 = 0 and superconformal percolation with c = 0, β2 = 1. We calculate the bulk and boundary free energies analytically. By numerically studying finite-size conformal spectra on the strip with appropriate boundary conditions, we argue that, in the continuum scaling limit, these lattice models are associated with the logarithmic superconformal models . For system size N, we propose finitized Kac character formulae of the form for s-type boundary conditions with r = 1, s = 1,2,3, …, ℓ = 0,1,2. The P, P′ dependence enters only in the fractional power of q in the prefactor and ℓ = 0,2 label the Neveu–Schwarz sectors (r + s even) and ℓ = 1 labels the Ramond sectors (r + s odd). Combinatorially, the finitized characters involve Motzkin and Riordan polynomials defined in terms of q-trinomial coefficients. Using the Hamiltonian limit and the finitized characters, we argue, from examples of finite-lattice calculations, that there exist reducible yet indecomposable representations for which the Virasoro dilatation operator L0 exhibits rank-2 Jordan cells, confirming that these theories are indeed logarithmic. We relate these results to the N = 1 superconformal representation theory.

Conducting fixed points for inhomogeneous quantum wires: A conformally invariant boundary theory

Inhomogeneities and junctions in wires are natural sources of scattering, and hence resistance. A conducting fixed point usually requires an adiabatically smooth system. One notable exception is "healing", which has been predicted in systems with special symmetries, where the system is driven to the homogeneous fixed point. Here we present theoretical results for a different type of conducting fixed point which occurs in inhomogeneous wires with an abrupt jump in hopping and interaction strength. We show that it is always possible to tune the system to an unstable conducting fixed point which does not correspond to translational invariance. We analyze the temperature scaling of correlation functions at and near this fixed point and show that two distinct boundary exponents appear, which correspond to different effective Luttinger liquid parameters. Even though the system consists of two separate interacting parts, the fixed point is described by a single conformally invariant boundary theory. We present details of the general effective bosonic field theory including the mode expansion and the finite size spectrum. The results are confirmed by numerical quantum Monte Carlo simulations on spinless fermions. We predict characteristic experimental signatures of the local density of states near junctions.

The staggered six-vertex model: Conformal invariance and corrections to scaling

We study the emergence of non-compact degrees of freedom in the low energy effective theory for a class of Z2-staggered six-vertex models. In the finite size spectrum of the vertex model this shows up through the appearance of a continuum of critical exponents. To analyze this part of the spectrum we derive a set of coupled nonlinear integral equations from the Bethe ansatz solution of the vertex model which allow to compute the energies of the system for a range of anisotropies and of the staggering parameter. The critical theory is found to be independent of the staggering. Its spectrum and density of states coincide with the SL(2,R)/U(1) Euclidean black hole conformal field theory which has been identified previously in the continuum limit of the vertex model for a particular ‘self-dual’ choice of the staggering. We also study the asymptotic behavior of subleading corrections to the finite size scaling and discuss our findings in the context of the conformal field theory.

The D(D3)-anyon chain: integrable boundary conditions and excitation spectra

Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group D3 are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the Bethe equations for the spin chain. The low-energy effective theories and operator content of the models (in both the spin chain and fusion path formulation) are identified from analytical and numerical studies of the finite-size spectra. For all boundary conditions considered the continuum theory is found to be a product of two conformal field theories. Depending on the coupling constants the factors can be a Z4 parafermion or a minimal model.

Asymptotic scattering and duality in the one-dimensional three-state quantum Potts model on a lattice

We determine numerically the single-particle and the two-particle spectrum of the three-state quantum Potts model on a lattice by using the density matrix renormalization group method, and extract information on the asymptotic (small momentum) S-matrix of the quasiparticles. The low energy part of the finite size spectrum can be understood in terms of a simple effective model introduced in a previous work, and is consistent with an asymptoticS-matrix of an exchange form below a momentum scale p*. This scale appears to vanish faster than the Compton scale, mc, as one approaches the critical point, suggesting that a dangerously irrelevant operator may be responsible for the behaviour observed on the lattice.

SU(3) Anderson impurity model: A numerical renormalization group approach exploiting non-Abelian symmetries

We show how the density-matrix numerical renormalization group (DM-NRG) method can be used in combination with non-Abelian symmetries such as SU(N), where the decomposition of the direct product of two irreducible representations requires the use of a so-called outer multiplicity label. We apply this scheme to the SU(3) symmetrical Anderson model, for which we analyze the finite size spectrum, determine local fermionic, spin, superconducting, and trion spectral functions, and also compute the temperature dependence of the conductance. Our calculations reveal a rich Fermi liquid structure.

A NUMERICAL TEST OF THE Y-SYSTEM IN THE SMALL SIZE LIMIT OF THE SU(2) × SU(2) PRINCIPAL CHIRAL MODEL

Recently, Kazakov, Gromov and Vieira applied the discrete Hirota dynamics to study the finite size spectra of integrable two dimensional quantum field theories. The method has been tested from large values of the size L down to moderate values using the SU (2) × SU (2) principal chiral model as a theoretical laboratory. We continue the numerical analysis of the proposed nonlinear integral equations showing that the deep ultraviolet region L → 0 is numerically accessible. To this aim, we introduce a relaxed iterative algorithm for the numerical computation of the low-lying part of the spectrum in the U (1) sector. We discuss in detail the systematic errors involved in the computation. When a comparison is possible, full agreement is found with previous thermodynamical Bethe ansatz computations.

Finite size properties of staggered [formula omitted] superspin chains

Based on the exact solution of the eigenvalue problem for the U q [ sl ( 2 | 1 ) ] vertex model built from alternating three-dimensional fundamental and dual representations by means of the algebraic Bethe ansatz we investigate the ground state and low energy excitations of the corresponding mixed superspin chain for deformation parameter q = exp ( − i γ / 2 ) . The model has a line of critical points with central charge c = 0 and continua of conformal dimensions grouped into sectors with γ-dependent lower edges for 0 ⩽ γ < π / 2 . The finite size scaling behavior is consistent with a low energy effective theory consisting of one compact and one non-compact bosonic degree of freedom. In the ‘ferromagnetic’ regime π < γ ⩽ 2 π the critical theory has c = − 1 with exponents varying continuously with the deformation parameter. Spin and charge degrees of freedom are separated in the finite size spectrum which coincides with that of the U q [ osp ( 2 | 2 ) ] spin chain. In the intermediate regime π / 2 < γ < π the finite size scaling of the ground state energy depends on the deformation parameter.

The complete conformal spectrum of asl(2|1) invariant network model and logarithmic corrections

We investigate the low temperature asymptotics and the finite-size spectrum of a class of Temperley–Liebmodels. As reference system we use the spin-1/2 Heisenberg chain with anisotropy parameterΔ and twisted boundary conditions. Special emphasis is placed onthe study of logarithmic corrections appearing in the case ofΔ = 1/2 inthe bulk susceptibility data and in the low-energy spectrum yielding the conformal dimensions. For thesl(2|1) invariant 3-state representation of the Temperley–Lieb algebra withΔ = 1/2 we give the complete set of scaling dimensions, which show huge degeneracies.

The finite size spectrum of the 2-dimensional [formula omitted] nonlinear σ-model

Nonlinear integral equations are proposed for the description of the full finite size spectrum of the 2-dimensional O ( 3 ) nonlinear σ-model in a periodic box. Numerical results for the energy eigenvalues are compared to the rotator spectrum and perturbation theory for small volumes and with the recently proposed generalized Lüscher formulas at large volumes.

Ground state and excitations of quantum dots with magnetic impurities

Received 30 December 2008; revised manuscript received 19 June 2009; published 24 July 2009 We consider an “impurity” with a spin degree of freedom coupled to a finite reservoir of noninteracting electrons, a system which may be realized by either a true impurity in a metallic nanoparticle or a small quantum dot coupled to a large one. We show how the physics of such a spin impurity is revealed in the many-body spectrum of the entire finite-size system; in particular, the evolution of the spectrum with the strength of the impurity-reservoir coupling reflects the fundamental many-body correlations present. Explicit calculation in the strong- and the weak-coupling limits shows that the spectrum and its evolution are sensitive to the nature of the impurity and the parity of electrons in the reservoir. The effect of the finite-size spectrum on two experimental observables is considered. First, we propose an experimental setup in which the spectrum may be conveniently measured using tunneling spectroscopy. A rate equation calculation of the differential conductance suggests how the many-body spectral features may be observed. Second, the finite-temperature magnetic susceptibility is presented, both the impurity and the local susceptibilities. Extensive quantum Monte Carlo calculations show that the local susceptibility deviates from its bulk scaling form. Nevertheless, for special assumptions about the reservoir—the “clean Kondo box” model—we demonstrate that finite-size scaling is recovered. Explicit numerical evaluations of these scaling functions are given, both for even and odd parities and for the canonical and the grand-canonical ensembles.

Finite-size effects for multi-magnon states

We propose the generalized Luscher formula for multi-magnon states by which one can compute the finite-size correction to the energy of multi giant magnons at classical and one-loop levels. It is shown that the F-term of our formula is consistent with the exact finite-size spectrum of the sinh-Gordon model, and the μ-term agrees with the finite-size energy of magnon boundstate of the asymptotic Bethe Ansatz in the su(2) sector at strong coupling. In an appendix, we evaluate our formula at weak coupling under some approximations, and find that the transcendental terms arise from a sum over an infinite tower of BPS boundstates.