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Coefficients of the ground state eigenvector as a function of the conformal energy (eigenvalue of L 0 + ¯ L 0 − c/12) in Ising field theory perturbed by the spin field σ (t = 0, h > 0) at the dimensionless volume mL = 80. Discrete dots are TCSA data at level 20 with 28624 states, while the lines show the prediction of (3.2) with τ * given in table 2 (with an overall sign difference which is due to the choice of the numerics). Different colours correspond to different modules: blue1, red-σ and black-ε; the point lying on (or close to) the horizontal axis correspond to non-diagonal contributions. Inset: zooming on the conformal energy region ≈ 14 − 15 shows the deviations from the diagonal form of the Ansatz discussed in the text.
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A bstract
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 r...
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Context 1
... t = 0, h > 0 (magnetic perturbation): as already demonstrated in figure 2 (see table 2) and c a * h corresponding to |NS and |R. It is clear the overlaps are not in full quantitative agreement. ...
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Citations
... This distinction arises from the fact that we are primarily interested in the instantaneous action of the perturbation rather than the deformation of the whole spectrum in such theories. Nevertheless, if the perturbation induces a gap in the bulk, the eigenstates ofÔ can be used as a variational ansatz to approximate the ground state of the interacting theory [68][69][70], effectively neglecting the K d + P d term in the full Hamiltonian H λ . In contrast, to produce a gapless theory in TCSA, the original Hamiltonian K d + P d is important to balance the perturbation λÔ, and the connection to our pinning flow becomes more obscure. ...
... Note that in (III.2) we have not specified the conformal boundaries |B ν ⟩, which depend on the interface between T and T gap . These are known as RG interfaces and have been determined explicitly in the 2d Ising and tricritical Ising CFTs [68][69][70]81]. We will see how to bootstrap this information efficiently in Section IV A with general criteria from monotonicity theorems and the C-symmetric condition. ...
... The results are listed in Table I where we also include the bimodule categories M labeling the conformal defects in the IR. There Vec denotes symmetryabsorbing cases (III.6) and all other cases coincide with the canonical (trivial) bimodule category C. The boundary states that participate in the factorization channels also agree with the RG interfaces identified in [68][69][70]. It is straightforward to extend this analysis to other minimal models. ...
We introduce generalized pinning fields in conformal field theory that model a large class of critical impurities at large distance, enriching the familiar universality classes. We provide a rigorous definition of such defects as certain unbounded operators on the Hilbert space and prove that when inserted on codimension-one surfaces they factorize the spacetime into two halves. The factorization channels are further constrained by symmetries in the bulk. As a corollary, we solve such critical impurities in the 2d minimal models and establish the factorization phenomena previously observed for localized mass deformations in the 3d model.
... TSMs have been used extensively in the context of (1 þ 1)-dimensional field theories, where they are commonly employed to calculate the bound state spectrum [47], matrix elements [55,56], two-point correlation functions [57], elastic S-matrix phases [54,58] and even inelastic information [59], entanglement entropy [60][61][62][63] and more. Recently, they have even been extended to study QFTs on an anti-de Sitter background [64]. ...
We propose herein an extension of truncated spectrum methodologies, a nonperturbative numerical approach able to elucidate the low energy properties of quantum field theories. TSMs, in their various flavors, involve a division of a computational Hilbert space, H , into two parts, one part, H 1 that is “kept” for the numerical computations, and one part, H 2 , that is discarded or “truncated.” Even though H 2 is discarded, truncated spectrum methodologies will often try to incorporate the effects of H 2 in some effective way. In these terms, we propose to keep the dimension of H 1 small. We pair this choice of H 1 with a Krylov subspace iterative approach able to take into account the effects of H 2 . This iterative approach can be taken to arbitrarily high order and so offers the ability to compute quantities to arbitrary precision. In many cases it also offers the advantage of not needing an explicit UV cutoff. To compute the matrix elements that arise in the Krylov iterations, we employ a Feynman diagrammatic representation that is then evaluated with Monte Carlo techniques. Each order of the Krylov iteration is variational and is guaranteed to improve upon the previous iteration. The first Krylov iteration is akin to the next-to-leading order approach of Elias-Miró [NLO renormalization in the Hamiltonian truncation, ]. To demonstrate this approach, we focus on the ( 1 + 1 d )-dimensional ϕ 4 model and compute the bulk energy and mass gaps in both the Z 2 -broken and unbroken sectors. We estimate the critical ϕ 4 coupling in the broken phase to be g c = 0.2645 ± 0.002 .
Published by the American Physical Society 2024
... TSMs have been used extensively in the context of 1+1 dimensional field theories, where they are commonly employed to calculate the bound state spectrum [49], matrix elements [57,58], two-point correlation functions [59], elastic S-matrix phases [56,60] and even inelastic information [61], entanglement entropy [62][63][64][65] and more. Recently, they have even been extended to study QFTs on an anti-de Sitter background [66]. ...
We propose herein an extension of truncated spectrum methodologies (TSMs), a non-perturbative numerical approach able to elucidate the low energy properties of quantum field theories. TSMs, in their various flavors, involve a division of a computational Hilbert space, , into two parts, one part, that is `kept' for the numerical computations, and one part, , that is discarded or `truncated'. Even though is discarded, TSMs will often try to incorporate the effects of in some effective way. In these terms, we propose to keep the dimension of small. We pair this choice of with a Krylov subspace iterative approach able to take into account the effects of . This iterative approach can be taken to arbitrarily high order and so offers the ability to compute quantities to arbitrary precision. In many cases it also offers the advantage of not needing an explicit UV cutoff. To compute the matrix elements that arise in the Krylov iterations, we employ a Feynman diagrammatic representation that is then evaluated with Monte Carlo techniques. Each order of the Krylov iteration is variational and is guaranteed to improve upon the previous iteration. The first Krylov iteration is akin to the NLO approach of Elias-Mir\'o et al. \cite{Elias-Miro:2017tup}. To demonstrate this approach, we focus on the 1+1d dimensional model and compute the bulk energy and mass gaps in both the -broken and unbroken sectors. We estimate the critical coupling in the broken phase to be .
... While [11] mainly focussed on qualitative implications of the ansatz it was later analysed in [12] with precision by comparing the ansatz with numerical results obtained using TCSA. The latter is a general method applicable to any perturbed CFT. ...
... The Hamiltonian restricted to the truncated space is then diagonalised numerically. In [12] TCSA was used to compare various predictions of Cardy's ansatz, such as the vacuum energy density, the components of the vacuum vector and the chiral entanglement entropy with TCSA numerical answers for single field perturbations of the Ising and TI models. In the absence of UV divergences a very good quantitative agreement was found which is the better the smaller the dimension of the perturbing operator is. ...
... This happens when we are in a symmetry breaking region of the perturbed theory. In this case, following [12], we need to diagonalise the matrix Māb = ⟨τ,ā|H|τ,b⟩ ⟨τ,ā|τ,ā⟩⟨τ,b|τ,b⟩ . ...
We consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy's variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this paper we study the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy's ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. For the latter we find a pair of novel transition lines that correspond to pairs of RG flows. Although the mass gap remains finite at the transition lines, several one-point functions change their behaviour. We discuss how these lines fit into the standard phase diagram of the tricritical Ising model. We show that each line extends to a two-dimensional surface in a three coupling space when we add perturbations by the subleading magnetic field. Close to this surface we locate symmetry breaking critical lines leading to the critical Ising model. Near the critical lines we find first order phase transition lines describing two-phase coexistence regions as predicted in Landau theory. The surface is determined from the CFT data using Cardy's ansatz and its properties are checked using TCSA numerics.
... Then the boundary states for a single Majorana fermion with boundary conditions ψ L = +ψ R and ψ L = −ψ R would have 12 As one of the authors emphasized in [29], the linear combination of Cardy states appears naturally as zero mode of fermionic model [30,31]. This type of states has also captured attention in the condensed matter physics community [32][33][34]. different norms. This 2 factor can also be observed as the 2 fold degeneracy of Majorana chain [29][30][31]. ...
In the last few years it was realized that every fermionic theory in 1+1 dimensions is a generalized Jordan-Wigner transform of a bosonic theory with a non-anomalous \mathbb{Z}_2} ℤ 2 symmetry. In this note we determine how the boundary states are mapped under this correspondence. We also interpret this mapping as the fusion of the original boundary with the fermionization interface.
... If one knows all conformal boundary states in the UV CFT one may minimise the energies (5.5) in τ and then choose the smallest value among all conformal boundary states. The overlaps between the trial states |τ, B with different boundary conditions are suppressed exponentially when τ R so that one does not need to consider their superpositions unless the vacuum becomes degenerate (see [32] for a nice discussion of the off-diagonal terms in Cardy's ansatz). ...
A bstract
We consider RG interfaces for boundary RG flows in two-dimensional QFTs. Such interfaces are particular boundary condition changing operators linking the UV and IR conformal boundary conditions. We refer to them as RG operators. In this paper we study their general properties putting forward a number of conjectures. We conjecture that an RG operator is always a conformal primary such that the OPE of this operator with its conjugate must contain the perturbing UV operator when taken in one order and the leading irrelevant operator (when it exists) along which the flow enters the IR fixed point, when taken in the other order. We support our conjectures by perturbative calculations for flows between nearby fixed points, by a non-perturbative variational method inspired by the variational method proposed by J. Cardy for massive RG flows, and by numerical results obtained using boundary TCSA. The variational method has a merit of its own as it can be used as a first approximation in charting the global structure of the space of boundary RG flows. We also discuss the role of the RG operators in the transport of states and local operators. Some of our considerations can be generalised to two-dimensional bulk flows, clarifying some conceptual issues related to the RG interface put forward by D. Gaiotto for bulk 𝜙 1 , 3 flows.
... Spatial EE in massive field theories has also been well studied, see for example [15][16][17][18][19][20]. One can also consider the entanglement between other partitions of the Hilbert space that are not spatial ones, as, for example, between right and left moving excitations [21][22][23] or between winding modes [24]. ...
A bstract
We consider electromagnetism in a cylindrical manifold coupled to a non-relativistic charged point-particle. Through the relation between this theory and the Landau model on a torus, we study the entanglement between the particle and the electromagnetic field. In particular, we compute the entanglement entropy in the ground state, which is degenerate, obtaining how it varies in the degeneracy subspace.
... If h = 1, a state fully polarized in the positive z-direction corresponds to a free boundary state [9], and the KW transformation in Eq. (34) maps it into a linear superposition of fixed boundary states [48,50,52]. Such a result was anticipated at the end of Sec. ...
... Its logarithm defines the universal renormalized Boundary Entropy [15], which was proven to have only two possible values depending on whether the quantum state flows to a free conformal boundary state or to a linear superposition of fixed conformal boundary states. Linear superpositions of fixed conformal boundary states appear naturally also in the analysis of topological defects [49], as a result of the Kramers-Wannier duality applied to a free boundary state [48,52,50]. As already mentioned, for technical reasons, our analysis has been limited to a chain with an even number of lattice sites and the expressions for the ground state overlaps are valid outside the circle γ 2 + h 2 = 1. ...
We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal Boundary Entropy at the critical point. The latter is a benchmark of the underlying Boundary Conformal Field Theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the Boundary Entropies are computed by exploiting a generalization of the Euler-MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.
... In addition to its application to critical systems, some theoretical application of (smeared) BCFT to gapped systems is recently proposed by Cardy [7]. More recently, his conjecture was checked by using the truncated conformal space approach (TCSA) for some models [8]. Hence further analysis of boundary states may shed new light on the analysis of the renormalization group (RG) flow to the gapped system and its realization in the lattice models. ...
We study the realizations of topological defects in 1d quantum Ising model with open boundary condition at criticality. The effect of the insertion of topological defects was predicted by Graham and Watts [K. Graham and G. M. T. Watts, JHEP 2004(4), 019 (2004)] by using boundary conformal field theory. Applying the construction introduced in [M. Hauru et al., Phys. Rev. B 94, 115125 (2016)], we prove that the Ising model on an open chain with topological defects can be transformed to the same model with boundary magnetic fields. It results in the appearance of linear combination of Cardy states, which we call Graham-Watts state, and one can understand it as an edge state of the spin chain. Our formulation suggests that an appearance of edge states can be understood as a consequence of boundary interaction which is equivalent to a topological defect in some specific cases. Moreover, we will show this edge state can be even robust under bulk perturbation whereas it is fragile to a boundary perturbation.
... the Hilbert space that are not spatial ones, as, for example, between right and left moving excitations [21][22][23] or between winding modes [24]. ...
We consider electromagnetism in a cylindrical manifold coupled to a nonrelativistic charged point-particle. Through the relation between this theory and the Landau model on a torus, we study the entanglement between the particle and the electromagnetic field. In particular, we compute the entanglement entropy in the ground state, which is degenerate, obtaining how it varies in the degeneracy subspace.
