Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula

TL;DR

This paper introduces automated methods to extract conformal data from critical quantum spin chains using the Koo-Saleur formula, enabling characterization of the underlying conformal field theory without relying on integrability.

Contribution

The authors develop and demonstrate a systematic, automated procedure to identify conformal towers and extract conformal data from lattice Hamiltonians using Koo-Saleur operators.

Findings

Successfully applied to nonintegrable quantum spin chains

Identified low-energy eigenstates as conformal primary and quasiprimary operators

Extracted conformal dimensions and spins from lattice models

Abstract

We study the emergence of two-dimensional conformal symmetry in critical quantum spin chains on the finite circle. Our goal is to characterize the conformal field theory (CFT) describing the universality class of the corresponding quantum phase transition. As a means to this end, we propose and demonstrate automated procedures which, using only the lattice Hamiltonian $H = \sum_j h_j$ as an input, systematically identify the low-energy eigenstates corresponding to Virasoro primary and quasiprimary operators, and assign the remaining low-energy eigenstates to conformal towers. The energies and momenta of the primary operator states are needed to determine the primary operator scaling dimensions and conformal spins -- an essential part of the conformal data that specifies the CFT. Our techniques use the action, on the low-energy eigenstates of $H$, of the Fourier modes $H_n$ of the Hamiltonian density $h_j$. The $H_n$ were introduced as lattice representations of the Virasoro generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we demonstrate that these operators can be used to extract conformal data in a nonintegrable quantum spin chain.

Equations

$$z$$

$$\begin{array}{r}{E}_{\alpha }^{\text{textsl}CFT}=\frac{2\pi }{L}\left({\Delta }_{\alpha }-\frac{c}{12}\right),P_{\alpha }^{\text{textsl}CFT}=\frac{2\pi }{L}{S}_{\alpha },\end{array}$$

$$E_{\alpha }=\frac{2\pi }{N}({\Delta }_{\alpha }-\frac{c}{12})+\mathcal{O}(N^{-x}),P_{\alpha }=\frac{2\pi }{N}{S}_{\alpha }.$$

$$\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}H=\sum _{j=1}^N{h}_j\sim H^{\text{textsl}CFT}={\int }_0^L\mathrm{d}x{h}^{\text{textsl}CFT}(x),$$

$$h_j\text{leavevmode}\text{leavevmode}\sim \text{leavevmode}\text{leavevmode}{h}^{\text{textsl}CFT}(x).$$

$$H=\sum _{j=1}^N{h}_j,$$

$$H^{\text{textsl}Ising}(\lambda )\equiv -\sum _{j=1}^N\left[{\sigma }_j^X{\sigma }_{j+1}^X+\lambda {\sigma }_j^Z\right],$$

$$H^{\text{textsl}CFT}={\int }_0^L\mathrm{d}x{h}^{\text{textsl}CFT}(x),$$

$$h^{\text{textsl}CFT}(x)\equiv \frac{1}{2\pi }\left(T^{\text{textsl}CFT}(x)+\overline{T}{}^{\text{textsl}CFT}(x)\right).$$

$$P^{\text{textsl}CFT}={\int }_0^L\mathrm{d}x{p}^{\text{textsl}CFT}(x),$$

$$\begin{array}{rl}{L}_n^{\text{textsl}CFT}& \equiv \frac{L}{(2\pi {)}^2}{\int }_0^L\mathrm{d}x{e}^{+\mathrm{i}nx\frac{2\pi }{L}}{T}^{\text{textsl}CFT}(x)+\frac{c}{24}{\delta }_{n,0},\\ {\overline{L}}_n^{\text{textsl}CFT}& \equiv \frac{L}{(2\pi {)}^2}{\int }_0^L\mathrm{d}x{e}^{-\mathrm{i}nx\frac{2\pi }{L}}{\overline{T}}^{\text{textsl}CFT}(x)+\frac{c}{24}{\delta }_{n,0},\end{array}$$

$$\begin{array}{rl}[L_n^{\text{textsl}CFT},L_m^{\text{textsl}CFT}]& =(n-m)L_{n+m}^{\text{textsl}CFT}+\frac{c}{12}n(n^2-1){\delta }_{n+m,0}\\ [L_n^{\text{textsl}CFT},{\overline{L}}_m^{\text{textsl}CFT}]& =0\\ [{\overline{L}}_n^{\text{textsl}CFT},{\overline{L}}_m^{\text{textsl}CFT}]& =(n-m){\overline{L}}_{n+m}^{\text{textsl}CFT}+\frac{c}{12}n(n^2-1){\delta }_{n+m,0}\end{array}$$

$$H_n^{\text{textsl}CFT}$$

$$=L_n^{\text{textsl}CFT}+{\overline{L}}_{-n}^{\text{textsl}CFT}-\frac{c}{12}{\delta }_{n,0}.$$

$$H_0^{\text{textsl}CFT}=L_0^{\text{textsl}CFT}+\overline{L}{}_0^{\text{textsl}CFT}-\frac{c}{12}=\frac{L}{2\pi }{H}^{\text{textsl}CFT}.$$

$$H_n\equiv \frac{N}{2\pi }\sum _{j=1}^N{e}^{+\mathrm{i}jn\frac{2\pi }{N}}{h}_j,\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}{H}_0=\frac{N}{2\pi }H,$$

$$h_j=\frac{2\pi }{N^2}\sum _{n=-N/2}^{+N/2}{e}^{-\mathrm{i}jn\frac{2\pi }{N}}{H}_n.$$

$$\left(L_0^{\text{textsl}CFT}+\overline{L}{}_0^{\text{textsl}CFT}\right)|{\phi }_{\alpha }⟩$$

$$\left(L_0^{\text{textsl}CFT}-\overline{L}{}_0^{\text{textsl}CFT}\right)|{\phi }_{\alpha }⟩$$

$$\begin{array}{rl}{H}^{\text{textsl}CFT}& =\frac{2\pi }{L}\left(L_0^{\text{textsl}CFT}+{\overline{L}}_0^{\text{textsl}CFT}-\frac{c}{12}\right)\\ P^{\text{textsl}CFT}& =\frac{2\pi }{L}\left(L_0^{\text{textsl}CFT}-{\overline{L}}_0^{\text{textsl}CFT}\right),\end{array}$$

$$\begin{array}{r}{\Delta }_{\alpha }=\frac{L}{2\pi }{E}_{\alpha }^{\text{textsl}CFT}+\frac{c}{12},S_{\alpha }=\frac{L}{2\pi }{P}_{\alpha }^{\text{textsl}CFT}.\end{array}$$

$$\begin{array}{rl}{L}_n^{\text{textsl}CFT}|{\Delta }_{\alpha },S_{\alpha }⟩& \propto |{\Delta }_{\alpha }-n,S_{\alpha }-n⟩,\\ \overline{L}{}_n^{\text{textsl}CFT}|{\Delta }_{\alpha },S_{\alpha }⟩& \propto |{\Delta }_{\alpha }-n,S_{\alpha }+n⟩,\end{array}$$

$$\displaystyle|\varphi\rangle\text{ primary }\Leftrightarrow\leavevmode\nobreak\ L^{\textsl{\tiny CFT}}_{n}|\varphi\rangle=\overline{L}{}^{\textsl{\tiny CFT}}_{n}|\varphi\rangle=0,\leavevmode\nobreak\ \leavevmode\nobreak\ n=1,2.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\$$

$$\Delta_{\varphi^{\prime}}=\Delta_{\varphi}+n,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ S_{\varphi^{\prime}}=S_{\varphi}\pm m,\quad\text{for }n\geq m,\leavevmode\nobreak\ \leavevmode\nobreak\$$

$$|\phi ⟩\text{ quasiprimary}\text{leavevmode}\text{leavevmode}\iff \text{leavevmode}\text{leavevmode}L{}_1^{\text{textsl}CFT}|\phi ⟩=\overline{L}{}_1^{\text{textsl}CFT}|\phi ⟩=0.$$

$$\sqrt{\frac{c}{2}}|T⟩=L{}_{-2}^{\text{textsl}CFT}|I⟩\text{ and }\sqrt{\frac{c}{2}}|\overline{T}⟩=\overline{L}{}_{-2}^{\text{textsl}CFT}|I⟩,$$

$$H_n^{\text{textsl}CFT}|{\Delta }_{\alpha },S_{\alpha }⟩=$$

$$b|{\Delta }_{\alpha }+n,S_{\alpha }-n⟩,$$

$${\Gamma }_{\phi }\equiv \sum _{{\phi }_{\alpha }:E_{\alpha }<E_{\phi }}|{\phi }_{\alpha }⟩⟨{\phi }_{\alpha }|.$$

$${\Gamma }_{\phi }{H}_n^{\text{textsl}CFT}|\phi ⟩=\{\begin{array}{ll}{L}_n^{\text{textsl}CFT}|\phi ⟩& \text{if }n<0,\\ \overline{L}{}_{-n}^{\text{textsl}CFT}|\phi ⟩& \text{if }n>0.\end{array}$$

$$\displaystyle|\varphi\rangle\text{ primary }\Leftrightarrow\leavevmode\nobreak\ \Gamma_{\varphi}\;H^{\textsl{\tiny CFT}}_{n}|\varphi\rangle=0,\leavevmode\nobreak\ \leavevmode\nobreak\ n=\pm 1,\pm 2,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\$$

Full text

I Introduction

Conformal field theory (CFT) Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999) is ubiquitous in modern theoretical physics. It describes fixed points of the renormalization group flow Cardy (1996), making it central to our understanding of quantum field theory Wilson and Kogut (1974). It is also a core component both of string theory Tong (2009) and of the AdS/CFT correspondence of quantum gravity Maldacena (1999). In condensed matter, as well as in statistical mechanics, continuous phase transitions can often be understood in terms of an underlying CFT that describes their universal, long-distance/low-energy physics Belavin et al. (1984); Cardy (1996); Di Francesco et al. (2012); Henkel (1999). Based on a previous proposal by Koo and Saleur Koo and Saleur (1994), in this paper we develop tools to investigate the emergence of conformal symmetry in generic quantum spin chains at criticality.

In order to present our results, we first need to recall two well-known facts about CFTs in two spacetime dimensions Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999). (i) On the plane, parameterized by a complex coordinate $z=x+iy$, a CFT contains infinitely many scaling operators $\varphi_{\alpha}(z)$. These are fields that transform covariantly under a rescaling of the plane by a factor $\lambda>0$ or a rotation by an angle $\theta\in[0,2\pi)$:

[TABLE]

where $\Delta_{\alpha}$ and $S_{\alpha}$ are the scaling dimension and conformal spin of $\varphi_{\alpha}(z)$. Scaling operators are organized into conformal towers, each consisting of a Virasoro primary operator and its descendants (see, e.g., Fig. 1). (ii) The operator-state correspondence establishes that for each scaling operator $\varphi_{\alpha}$ there is an eigenstate $|\varphi_{\alpha}\rangle$ of the CFT Hamiltonian $H^{\textsl{\tiny CFT}}$ on the circle, with energy and momentum given by

[TABLE]

where $L$ is the length of the circle and $c$ is the central charge of the CFT, which determines the Casimir energy. The scaling dimensions, conformal spins and operator product expansion (OPE) coefficients (three-point correlators) of the primary operators, together with the central charge, fully characterize the CFT Belavin et al. (1984) and are referred to as conformal data.

I.1 Extraction of conformal data

The extraction of conformal data from lattice models has a long history. Following the landmark 1984 publication by Belavin, Polyakov and Zamolodchikov of Belavin et al. (1984), which revealed the intricate structure of 2D CFT, Cardy, Blöte, Nightingale and Affleck Cardy (1984); Blöte et al. (1986); Cardy (1986a, b); Affleck (1986) discovered that, at low energies and after suitably normalizing the lattice Hamiltonian $H$, the energies and momenta of a critical quantum spin chain made of $N$ spins must read

[TABLE]

This matches the CFT spectrum (2) up to subleading, non-universal corrections $\mathcal{O}(N^{-x})$, where $x>1$ is also model-specific 111In models with marginal operators, there may also be logarithmic finite-size corrections Cardy (1986b). We have successfully tested the proposals of this paper also on such models, including the four-state Potts quantum spin chain.. One can therefore estimate the scaling dimensions $\Delta_{\alpha}$ and conformal spins $S_{\alpha}$ of the CFT from the energies $E_{\alpha}$ and momenta $P_{\alpha}$ computed on the lattice, see e.g. Fig. 5. This result has proved extremely useful in understanding critical lattice systems e.g. von Gehlen et al. (1986); von Gehlen and Rittenberg (1986); Henkel (1987); Alcaraz et al. (1987); Baake et al. (1987); Balbao and de Felicio (1987); Allton and Hamer (1988); Affleck et al. (1989); Alcaraz et al. (1989); Frahm and Korepin (1990); Voit (1995); Boschi et al. (2003); Feiguin et al. (2007); Fuehringer et al. (2008); Xavier (2010); Gendiar et al. (2011); Katsura (2012); Wen et al. (2016).

One can think of (3) as demonstrating a low-energy correspondence between the critical lattice Hamiltonian $H$ and the CFT Hamiltonian $H^{\textsl{\tiny CFT}}$

[TABLE]

where $h_{j}$ and $h^{\textsl{\tiny CFT}}(x)$ denote the lattice and continuum Hamiltonian densities. It is then natural to ask whether this global correspondence extends to the local densities

[TABLE]

An example of this local correspondence was already found in 1971 by Kadanoff and Ceva Kadanoff and Ceva (1971), who showed that a lattice analogue of the energy-momentum tensor exists in the Ising model. Later, Koo and Saleur Koo and Saleur (1994) demonstrated the principle more generally by showing that, in some integrable models, the Fourier modes $H_{n}$ of $h_{j}$, defined so that $h_{j}=(2\pi/N^{2})\sum_{n}e^{-\mathrm{i}jn\frac{2\pi}{N}}H_{n}$, behave as lattice representations of certain linear combinations of the Virasoro generators of conformal symmetry: $H_{n}^{\textsl{\tiny CFT}}\equiv L^{\textsl{\tiny CFT}}_{n}+\overline{L}{}^{\textsl{\tiny CFT}}_{-n}-\delta_{n,0}c/12$. Other work had previously established the existence of lattice representations (or deformations) of (parts of) the Virasoro algebra in certain integrable systems Itoyama and Thacker (1987); Volkov (1988); Faddeev and Takhtajan (1988); Babelon (1990); Johannesson (1992); Volkov (1992); Belov and Chaltikian (1993); Kellendonk and Recknagel (1993), but the proposal of Koo and Saleur (1994) is of particular importance because it provides a prescription for constructing lattice analogues of all the Virasoro generators. This provides access to a wealth of information about the CFT, including the central charge. Indeed, a number of authors have used the so-called Koo-Saleur formula to extract conformal data in various models, especially logarithmic CFTs, which are nontrivial nonunitary CFTs with $c=0$, see e.g. Read and Saleur (2007a); Dubail et al. (2010); Vasseur et al. (2012); Gainutdinov et al. (2013); Gainutdinov and Vasseur (2013); Bondesan et al. (2015). However, as yet the Koo-Saleur formula has not enjoyed the same widespread use as (3), having been applied only to integrable systems.

I.2 Our results

In this paper we propose and test methods which apply the Hamiltonian-density Fourier modes $H_{n}$ Koo and Saleur (1994) to systematically identify low-energy eigenstates of a critical spin chain Hamiltonian $H$ (with local interactions) with CFT scaling operators. In particular, we present automated procedures for finding the eigenstates corresponding to primary and quasiprimary operators, as well as for assigning all remaining low-energy eigenstates to their respective (Virasoro or global) conformal towers. A key feature of these methods is that they provide a general means for determining which scaling dimensions and conformal spins derived from (3) belong to primary fields in the CFT, thus delivering a crucial piece of the conformal data. They also deliver an improved way of identifying the energy-momentum-tensor state, often used to determine the correct normalization for $H$. Furthermore, our construction sets the stage for a systematic determination of the OPE coefficients for generic critical spin chains, which involves additionally determining scaling operators on the lattice and will be discussed in Zou et al. (2017a).

Finally, we establish that our methods, and hence the Koo-Saleur formula, are applicable away from integrability by demonstrating them in the self-dual ANNNI model: a nonintegrable perturbation of the Ising model.

We stress that, although for this paper we used exact diagonalization to obtain the low-energy eigenstates of $H$, our core proposal is independent of the method used to obtain these eigenstates. Indeed, we can also apply operators $H_{n}$ to energy eigenstates obtained with more sophisticated techniques, such as periodic matrix product states Zou et al. (2017b), and in this way analyze larger systems, which carry smaller finite-size errors.

Note: Throughout the paper we differentiate between lattice objects, such as $H$, $P$, and $H_{n}$, and their CFT counterparts $H^{\textsl{\tiny CFT}}$, $P^{\textsl{\tiny CFT}}$, and $H_{n}^{\textsl{\tiny CFT}}$, by means of the superscript ${}^{\textsl{\tiny CFT}}$. On the other hand, states denoted as $|\varphi\rangle$, $|\varphi_{\alpha}\rangle$, etc. belong either to the lattice or the CFT, as can be determined from the context.

II Low-energy correspondence for Hamiltonian densities

II.1 Critical quantum spin chains and CFTs

We consider a periodic 1D lattice made of $N$ sites with a translation invariant quantum Hamiltonian

[TABLE]

that decomposes as a sum of local Hamiltonian terms, where the term $h_{j}$ is located about site $j$ and will be referred to as the Hamiltonian density on that site. A canonical example is the transverse field Ising model

[TABLE]

which is critical at $\lambda=1$. We assume that, at criticality, there is a corresponding quantum CFT Hamiltonian

[TABLE]

where $x\in(0,L]$ parameterizes a circle of radius $L/2\pi$ and the Hamiltonian-density field operator $h^{\textsl{\tiny CFT}}(x)$ can be written Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999) in terms of the chiral and anti-chiral components $T^{\textsl{\tiny CFT}}(x)$ and $\overline{T}{}^{\textsl{\tiny CFT}}(x)$ of the traceless energy-momentum tensor of the CFT on the circle,

[TABLE]

Similarly, to the lattice momentum operator $P$ (defined such that $e^{\mathrm{i}P\frac{2\pi}{N}}$ is a translation by one lattice site) we associate the CFT momentum operator

[TABLE]

where $p^{\textsl{\tiny CFT}}(x)\equiv\left(T^{\textsl{\tiny CFT}}(x)-\overline{T}^{\textsl{\tiny CFT}}(x)\right)/(2\pi)$ is the momentum density.

II.2 Fourier mode expansions

The Fourier modes $L{}^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ of the chiral and anti-chiral energy-momentum tensor operators Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999)

[TABLE]

where $n\in\mathbb{Z}$, furnish chiral and anti-chiral instances of the Virasoro algebra Virasoro (1970); Belavin et al. (1984)

[TABLE]

and are the canonical choice of generators of conformal symmetry on the CFT Hilbert space.

Importantly for our purposes, the Fourier modes $H^{\textsl{\tiny CFT}}_{n}$ of the Hamiltonian density operator $h^{\textsl{\tiny CFT}}(x)$ correspond to certain linear combinations of the above Virasoro generators,

[TABLE]

where we note that, for $n=0$

[TABLE]

In direct analogy, following the proposal of Koo and Saluer Koo and Saleur (1994), we introduce the Fourier modes $H_{n}$ of the lattice Hamiltonian density $h_{j}$

[TABLE]

in terms of which the lattice Hamiltonian density $h_{j}$ at site $j$ reads

[TABLE]

II.3 General strategy

Our goal is to use the Fourier modes $H_{n}$ of the lattice Hamiltonian density $h_{j}$ to systematically extract conformal data from the low-energy subspace of the critical lattice Hamiltonian $H$. This will be discussed in Sect. IV and then numerically demonstrated in Sect. V.

The central assumption of is that, at low energies and up to finite-size corrections, each $H_{n}$ should act on the simultaneous eigenstates $|\varphi_{\alpha}\rangle$ of $H$ and $P$ on the lattice as its CFT counterpart $H_{n}^{\textsl{\tiny CFT}}$ does on the simultaneous eigenstates of $H^{\textsl{\tiny CFT}}$ and $P^{\textsl{\tiny CFT}}$ in the continuum. Strong evidence for this was provided in Koo and Saleur (1994) and subsequent work (for integrable systems), but we will need more details for our purposes. We therefore begin in Sect. III by explaining how the Fourier modes $H_{n}^{\textsl{\tiny CFT}}$ act in the continuum. This is best understood in terms of the Fourier modes $L{}^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$, which act simply as ladder operators on the eigenbasis $|\varphi_{\alpha}\rangle$.

At this point, a natural question to ask is whether it would be more convenient to construct, and directly work with, lattice versions $L_{n}$ and $\overline{L}_{n}$ of the Virasoro generators ${}L^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$, as was done in Koo and Saleur (1994), instead of using the lattice Fourier modes $H_{n}$. After all, most CFT practitioners are already familiar with the Virasoro generators $L{}^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$, which explicitly discriminate between chiral and anti-chiral CFT modes, and not so much with the Fourier modes $H^{\textsl{\tiny CFT}}_{n}$. As explained in App. A, doing so is possible in principle but far from optimal in practice. Next we briefly summarize why.

Given the lattice Hamiltonian density $h_{j}$ as the only input, it is indeed possible to use energy conservation to obtain a lattice momentum density $p_{j}\equiv i\left[h_{j},h_{j-1}\right]$, and thus produce chiral and anti-chiral energy-momentum operators $T_{j}=\frac{1}{2}(h_{j}+p_{j})$ and $\overline{T}_{j}=\frac{1}{2}(h_{j}-p_{j})$, whose Fourier mode expansion leads to lattice Virasoro generators $L_{n}$ and $\overline{L}_{n}$ that act as $L{}^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ at low energies and up to finite size corrections. However, by construction there are additional finite-size corrections in $L_{n}$ and $\overline{L}_{n}$, compared to $H_{n}$, which can be traced back to finite-size corrections to the eigenstate energies of $H$ (see App. A). Therefore, from a numerical perspective, it is preferable to work with the lattice Fourier modes $H_{n}$, as we do in this paper.

III Conformal towers in the continuum

III.1 The Virasoro generators as ladder operators

Recall that in a 2D CFT, the combinations $L^{\textsl{\tiny CFT}}_{0}\pm\overline{L}{}^{\textsl{\tiny CFT}}_{0}$ generate the dilations and rotations in (1) Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999). Therefore, by the operator-state correspondence Fubini et al. (1973); Belavin et al. (1984), these operators act on the state $|\varphi_{\alpha}\rangle$ as

[TABLE]

which, given that $H^{\textsl{\tiny CFT}}$ and $P^{\textsl{\tiny CFT}}$ can be written in terms of $L^{\textsl{\tiny CFT}}_{0}\pm\overline{L}{}^{\textsl{\tiny CFT}}_{0}$ as

[TABLE]

automatically implies (2) or, equivalently,

[TABLE]

Let us temporarily denote $|\varphi_{\alpha}\rangle$ as $|\Delta_{\alpha},S_{\alpha}\rangle$. From (20) and the Virasoro algebra (12) it can be seen that the Virasoro generators are ladder operators of $H^{\textsl{\tiny CFT}}$ and $P^{\textsl{\tiny CFT}}$. They indeed act on an eigenstate $|\Delta_{\alpha},S_{\alpha}\rangle$ as

[TABLE]

raising $\Delta$ for $n<0$ and lowering it for $n>0$. Note also that $L^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ change $S$ in opposite directions. This is illustrated in Fig. 2.

The Virasoro operators $L^{\textsl{\tiny CFT}}_{n}$, $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ generate multiple (generally an infinite number, but not always Friedan et al. (1984)) distinct towers of eigenstates of $H^{\textsl{\tiny CFT}}$ and $P^{\textsl{\tiny CFT}}$ called conformal towers. Each tower has, at its base, a primary state (corresponding to a primary operator). Primary states are therefore those states annihilated by all ladder operators that reduce the energy Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999): $L^{\textsl{\tiny CFT}}_{n}|\varphi\rangle=\overline{L}{}^{\textsl{\tiny CFT}}_{n}|\varphi\rangle=0$ for all $n>0$. By (12), $L^{\textsl{\tiny CFT}}_{-1},L^{\textsl{\tiny CFT}}_{-2}$ generate the subalgebra $L^{\textsl{\tiny CFT}}_{n}$ for $n<0$ (and similar for $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$) so that this condition is equivalent to:

[TABLE]

By acting with products of powers of $L^{\textsl{\tiny CFT}}_{n}$, $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ with $n<0$ on a primary $|\varphi\rangle$, all descendant states in its tower can be reached. From (22), descendants $|\varphi^{\prime}\rangle$ of a primary $|\varphi\rangle$ must have scaling dimension $\Delta_{\varphi^{\prime}}$ and conformal spin $S_{\varphi^{\prime}}$ given by

[TABLE]

where $n\in\mathbb{N}$ and $m\in\mathbb{Z}$. Furthermore, it follows from (23) that all descendants can be reached from the primary using only $L^{\textsl{\tiny CFT}}_{-n}$, $\overline{L}{}^{\textsl{\tiny CFT}}_{-n}$ with $n=1,2$.

Let us pause here and briefly consider a simple example to which we will return later: The Ising CFT only has three primary operators Friedan et al. (1984):

[TABLE]

Therefore it has just three conformal towers. From this data we can infer information about the spectrum of $H^{\textsl{\tiny CFT}}$, $P^{\textsl{\tiny CFT}}$ using (2) and (24). For example, all eigenstates have either $\Delta_{\alpha}\in\mathbb{N}$ (descendants of $|I\rangle$ and $|\varepsilon\rangle$) or $\Delta_{\alpha}\in\mathbb{N}+\frac{1}{8}$ (descendants of $|\sigma\rangle$). The low-energy spectrum of the Ising CFT is shown in Fig. 1. In Fig. 2 we illustrate how the ladder operators can be used to connect states within a particular conformal tower.

III.2 Identity, energy-momentum, and central charge

Returning to a generic 2D CFT, a particularly important primary state that is always present is the “identity state” $|I\rangle$. In a unitary CFT, which is the main focus of this work, the state $|I\rangle$ corresponds to the ground state of the Hamiltonian $H^{\textsl{\tiny CFT}}$. This state is unique in having a vanishing scaling dimension $\Delta_{I}=0$ and in being annihilated by all $L^{\textsl{\tiny CFT}}_{n},\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ with $n=0,\pm 1$, which are the generators of global conformal transformations (those that are well-defined throughout the 2D plane) Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999).

Another relevant notion is that of a quasiprimary state Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999), defined as a state that is annihilated by both $L^{\textsl{\tiny CFT}}_{1}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{1}$:

[TABLE]

This includes all primary states, but also certain descendant states. Two important quasiprimary states that are present in any CFT are those corresponding to the CFT energy-momentum operators $T{}^{\textsl{\tiny CFT}}(x)$ and $\overline{T}{}^{\textsl{\tiny CFT}}(x)$. They are descended from the ground state $|I\rangle$ as

[TABLE]

where $c$ is the central charge, and thus have scaling dimensions $\Delta_{T}=\Delta_{\overline{T}}=2$ and conformal spins $S_{T}=2$, $S_{\overline{T}}=-2$. For the Ising CFT, states $|I\rangle$, $|T\rangle$, and $|\overline{T}\rangle$ can be seen in Figs. 1 and 2.

III.3 Characterization in terms of $H_{n}$

Finally, we have to translate the above statements for the Virasoro generators $L^{\textsl{\tiny CFT}}_{n}$, $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$ into statements for the Fourier modes $H_{n}^{\textsl{\tiny CFT}}$ of the Hamiltonian density defined in (13). Recalling that the Fourier modes $H_{n}$ for $n\not=0$ are linear combinations of the Virasoro generators, $H_{n}^{\textsl{\tiny CFT}}=L{}^{\textsl{\tiny CFT}}_{n}+\overline{L}{}^{\textsl{\tiny CFT}}_{-n}$, we can infer their behavior from (22):

[TABLE]

where $a$ and $b$ are determined by conformal symmetry and may equal zero Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999). The following simple observation will also prove very useful. Given an energy eigenstate $|\varphi\rangle$ with energy $E_{\varphi}$, let $\Gamma_{\varphi}$ be a projector onto all the eigenstates with energy smaller than $E_{\varphi}$,

[TABLE]

Then we have that the product $\Gamma_{\varphi}\,H_{n}^{\textsl{\tiny CFT}}$ acts on $|\varphi\rangle$ as would either just $L_{n}^{\textsl{\tiny CFT}}$ or $\overline{L}{}^{\textsl{\tiny CFT}}_{-n}$ according to

[TABLE]

It follows that we can recast the characterization (23) of a primary state as

[TABLE]

Similarly, the characterization (25) of a quasiprimary state reads

[TABLE]

More generally, by using either Eq. (29) or a similar expression with a complementary projector $\mathbb{I}-\Gamma_{\varphi}$, we can use the Fourier modes $H_{n}$ of the Hamiltonian density $h(x)$ to reproduce the action of the Virasoro generators $L_{n}^{\textsl{\tiny CFT}}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{n}$. Finally, an expression such as (26) translates directly into

[TABLE]

without the need of projectors, given that there are no states with energy below that of $|I\rangle$.

IV Extracting conformal data from the lattice

In this section we discuss how to extract conformal data by computing matrix elements of the operators $H_{n}$ of (16) between low-energy states $|\varphi_{\alpha}\rangle$. Here, each state $|\varphi_{\alpha}\rangle$ is a simultaneous eigenstate of the (normalized) critical lattice Hamiltonian $H$ and of the lattice momentum operator $P$ or, more precisely, of the lattice translation operator $e^{i\frac{2\pi}{N}P}$ that implements a translation by one lattice site,

[TABLE]

We assume that, on these low-energy states, $H_{n}$ acts analogously to $H^{\textsl{\tiny CFT}}_{n}$ of (13), up to finite-size corrections that decrease with the size $N$ of the lattice.

IV.1 Normalization of $H$ and central charge $c$

So far we have assumed that the critical lattice Hamiltonian $H$ was already normalized so that its spectrum is given by (3) (or, equivalently, so that the speed of light equals $1$ in the large-$N$ limit). However, in general the input data may be an unnormalized critical Hamiltonian $\tilde{H}$ or, equivalently, an unnormalized Hamiltonian density $\tilde{h}_{j}$, which relate to the normalized $H$ and $h_{j}$ through

[TABLE]

where $a$ and $b$ are two model-dependent constants. The constant $b$ can be computed by requiring that the extensive part of the ground state energy vanish in the limit of large $N$ (via a large-$N$ extrapolation), but in the following we will be able to simply ignore it, mostly because $b$ does not affect operators $H_{n}$ for $n\not=0$.

For a given system size $N$, the constant $a$ can be determined using states that are present in, and relations that are valid for, any CFT (see Sect. III). First we identify the states $|I\rangle$ and $|T\rangle$ as eigenstates of $\tilde{H}$

[TABLE]

such that $|I\rangle$ is the unique ground state of $\tilde{H}$ and $|T\rangle$ is the eigenstate with momentum $P_{T}=2\times\frac{2\pi}{N}$ that has maximal overlap with $\tilde{H}_{-2}|I\rangle$ (where $\tilde{H}_{-2}$ is defined as $H_{-2}$ in (16) after replacing $h_{j}$ with $\tilde{h}_{j}$). This last identification is motivated by the CFT relation (32). Then, recalling that the scaling dimension of $T$ is $\Delta_{T}=2$, and therefore $E^{\textsl{\tiny CFT}}_{T}-E^{\textsl{\tiny CFT}}_{I}=\Delta_{T}\times\frac{2\pi}{N}=2\times\frac{2\pi}{N}$, we set $a=\frac{4\pi}{N}/(\tilde{E}_{T}-\tilde{E}_{I})$, since this guarantees that the (normalized) lattice energies also fulfill $E_{T}-E_{I}=2\times\frac{2\pi}{N}$.

With this normalization of $H$ the energies and momenta on the lattice read

[TABLE]

as we wanted. We can now estimate the scaling dimensions and conformal spins. Note: In the remainder (particularly Sect. V), we slightly abuse notation, writing $H$ and $H_{n}$ for both the unnormalized and normalized operators. All results presented are obtained using the properly normalized versions.

Once we have normalized $h_{j}$, we can build the normalized Fourier modes $H_{n}$ using (16). Through the relation (32), the central charge $c$ of the emergent CFT can then be estimated by the simple expectation value Koo and Saleur (1994)

[TABLE]

Alternatively, in order to eliminate finite-size corrections of $H_{2}$ that connect $|I\rangle$ to states other than $|T\rangle$, we can use

[TABLE]

which often produces more accurate results. In either case, an extrapolation to large $N$ increases the accuracy of the lattice estimate of the central charge $c$.

The above procedures to normalize $H$ and estimate $c$ differ from previous proposals in that here we use $H_{2}$. The usual procedure to normalize $H$ is to identify $|T\rangle$ as the lowest-energy state with $P_{\alpha}=2\times\frac{2\pi}{N}$  Henkel (1999). However, this fails if finite-size corrections shift the energy of another state with $P_{\alpha}=2\times\frac{2\pi}{N}$ below that of $|T\rangle$, as happens e.g. in the ANNNI model discussed in Sect. V.3. Finally, an important advantage of estimating $c$ using $H_{2}$, compared to an extrapolation using the ground state energy alone Henkel (1999), is that the latter also requires an extrapolation of the nonzero extensive contribution to the ground state energy, represented by $b$ in (34), which must be subtracted before attempting to extrapolate $c$.

IV.2 Primary states and conformal towers

We now propose a criterion to identify candidates for primary states. In the CFT, primary states obey (30). In words, they are the states that cannot be descended to lower energies by $H_{n}^{\textsl{\tiny CFT}}$ or $\overline{H}{}^{\textsl{\tiny CFT}}_{n}$. On the lattice at finite $N$ we have corrections to the energies (3) and to the $H_{n}$, both of which must be allowed for in defining a criterion to identify candidates for a primary state. That is, on the lattice we need an approximate version of (30).

To this end, we define $\epsilon^{(n)}$ to be the norm of the matrix elements of $\frac{1}{2}(H_{+n}+H_{-n})$ that connect an energy eigenstate $|\varphi\rangle$ with states of lower energy:

[TABLE]

We then define a primary candidate as a state with small $\epsilon^{(1)}$ and $\epsilon^{(2)}$:

[TABLE]

which is analogous to (30) for $\epsilon_{\max}=0$.

Having identified primary candidate states, we can build their conformal towers by applying sequences of $H_{n}$ to them. By matching such sequences with sequences of $L^{\textsl{\tiny CFT}}_{n},\overline{L}{}^{\textsl{\tiny CFT}}_{n}$, taking finite-size corrections into account, we can then identify each nonprimary lattice eigenstate with a particular descendant state of the CFT.

However, if we only want to know which conformal tower each nonprimary state belongs to, it suffices to examine the matrix elements of a single operator – one that connects each primary state with all its descendants. We saw in Sect. III that sequences of the ladder operators $L^{\textsl{\tiny CFT}}_{-1}$, $L^{\textsl{\tiny CFT}}_{-2}$, $\overline{L}{}^{\textsl{\tiny CFT}}_{-1}$, and $\overline{L}{}^{\textsl{\tiny CFT}}_{-2}$ acting on the primary are enough to reach any descendant in the CFT. On the lattice we can therefore use the matrix elements

[TABLE]

where $H_{n}^{\diamond}$ is the projection of $H_{n}$ onto the numerically obtained low-energy subspace and the exponential generates all sequences of $H_{\pm 1,\pm 2}$ (note that $H_{n}^{\dagger}=H_{-n}$). We then assign a nonprimary state $|\varphi^{\prime}\rangle$ to the tower of the primary candidate $|\varphi\rangle$ that maximizes $\tau_{\varphi^{\prime}}^{\varphi}$. Note that this procedure is suboptimal in the sense that finite-size corrections accumulate when we take products of $H^{\diamond}_{n}$. More sophisticated schemes avoiding this issue are possible Zou et al. (2017b), but this simpler scheme is already sufficient for our purpose of illustrating the usefulness of $H_{n}$.

Armed with an identification of each eigenstate of $H$ at fixed $N$, we may examine data from a range of sizes to determine if the assignment is robust. To check that the identification of primary states is robust we note that, using (40), we can verify statements such as “With $\epsilon_{\max}=10^{-6}$ there is a primary candidate at $\Delta\approx 3$ and $S=3$ for all tested system sizes $N\geq 6$”. Since finite-size corrections typically obey power-law or logarithmic scaling in the system size Cardy (1986a, b), we rely on them varying smoothly with $N$ and assume that primary candidate states $|\varphi\rangle_{N}$ at different $N$, but with similar energy and the same momentum, represent the same primary operator in the CFT. For such sequences of primary candidate states we should find that both $\epsilon^{(1)}_{\varphi}(N)$ and $\epsilon^{(2)}_{\varphi}(N)$ go to zero in the limit of large $N$.

IV.3 Quasiprimaries and global conformal towers

The identification of primary states on the lattice, as discussed above, is a central application of the correspondence between the CFT Fourier modes $H^{\textsl{\tiny CFT}}_{n}$ and their lattice analogues $H_{n}$ Koo and Saleur (1994), because of its direct impact on our ability to compute the conformal data of the underlying CFT, which requires such an identification. However, a more refined characterization within each conformal tower is also possible on the lattice, as we discuss next.

A conformal tower (or Virasoro tower) decomposes into infinitely many global conformal towers, each consisting of a quasiprimary operator and its global descendants. To identify quasiprimary states on the lattice, we resort to an approximate version of (31) in terms of the error $\epsilon^{(1)}_{\varphi}$ defined in (39), namely

[TABLE]

which indeed is analogous to (31) for $\epsilon_{\max}=0$. Then, once a quasiprimary state $|\varphi\rangle$ has been identified, its global conformal tower (generated in the CFT by acting on $|\varphi\rangle$ with powers of $L^{\textsl{\tiny CFT}}_{-1}$ and $\overline{L}{}^{\textsl{\tiny CFT}}_{-1}$ or, equivalently, powers of $H_{1}^{\textsl{\tiny CFT}}$ and $H_{-1}^{\textsl{\tiny CFT}}$) can be produced by studying the matrix elements

[TABLE]

where $H_{1}^{\diamond},H_{-1}^{\diamond}$ are defined above and similar considerations to (41) apply.

V Results

V.1 The Ising model

As a first test of the methods introduced in Sect. IV, we examine the behavior of the Hamiltonian density modes $H_{n}$ for the integrable transverse field Ising model of (7), for which some conformal data was extracted in Koo and Saleur (1994). The Hamiltonian is invariant under a global spin flip $\prod_{j=1}^{N}\sigma_{j}^{Z}$, and is critical at its self-dual point $\lambda=1$ Henkel (1999).

We construct $H_{n}$ for the critical Ising model as

[TABLE]

where we have chosen different phases for the onsite terms $\sigma^{Z}$ and the bond terms $\sigma^{X}\sigma^{X}$ to reflect that the bonds are centered between two sites. We propose in general that terms with support on sites $j$ and $j+r$, and optionally the sites in between, be given phases consistent with the midpoint $x=j+r/2$. For the Ising model, this ensures that $H^{\textsl{\tiny Ising}}_{n}$ remains consistent with Kramers-Wannier duality, which exchanges the $\sigma^{X}_{j}\sigma^{X}_{j+1}$ and $\sigma^{Z}_{j}$ terms.

For a given finite system size $N$, we simultaneously diagonalize the Hamiltonian and the translation operator, with periodic boundary conditions, using the Arnoldi algorithm – a Krylov-subspace method for finding eigenvalue/eigenvector pairs of nonhermitian matrices Arnoldi (1951) – to find a set of low-energy eigenstates $|\varphi_{\alpha}\rangle$, with energies $E_{\alpha}$ and momenta $P_{\alpha}$. In this case, we compute the 41 lowest-energy eigenvalues and corresponding eigenvectors. With these we compute the matrix-elements $\langle\varphi_{\beta}|H_{n}^{\textsl{\tiny Ising}}|\varphi_{\alpha}\rangle$ in the low-energy eigenbasis of $H$, which we normalize according to the discussion in Sect. IV.1.

For our first test of the behavior of $H^{\textsl{\tiny Ising}}_{n}$, we examine a selection of matrix elements for $n=\pm 1,2,3$. We find that the action of these $H^{\textsl{\tiny Ising}}_{n}$ within the computed basis of 41 low-energy states is indeed consistent with that of their CFT counterparts (13), described in Sect. III, as expected from Koo and Saleur (1994). In particular, despite noticeable finite-size corrections to the energies, states $H^{\textsl{\tiny Ising}}_{n}|\varphi_{\alpha}\rangle$ have nonzero overlap only with energy eigenstates of scaling dimension $\Delta_{\alpha}\pm n+\mathcal{O}(\epsilon)$ (where $\epsilon\ll 1$ represents finite-size corrections to the energies) and spin $S_{\alpha}-n$, as expected from the CFT result of (27). Overlaps with states of incompatible scaling dimension are zero to numerical precision (within the 41 low-energy states under consideration). We plot a few examples in Fig. 3.

Applying (40) to determine the primary candidate states, we find that, even at $N=14$, we can correctly identify all three primary states using a tolerance close to machine precision, $\epsilon_{\max}=10^{-14}$. Although it is trivial that the primary states in the Ising model cannot be lowered in energy (there are no states at compatible momenta that they could be lowered to), it is nontrivial, if unsurprising, that no descendant states (again, within the 41 low-energy states under consideration) are misidentified as primary. That said, later we will see that the Potts model provides a much better proving ground for the identification of primary states.

We further observe that $\tau_{\varphi^{\prime}}^{\varphi}$ of (41) delivers a completely unambiguous tower assignment to the remaining states, consistent with the observed perfect ladder behavior of $H^{\textsl{\tiny Ising}}_{n}$. In other words, there are no significant finite-size corrections that mix conformal towers. Indeed, in this case such corrections are disallowed by the symmetries of $H_{n}^{\textsl{\tiny Ising}}$ (this is not the case for the Potts model – see below).

Corrections are present, however, which affect the size of the nonzero matrix elements of $H^{\textsl{\tiny Ising}}_{n}$, as evidenced by the central charge estimates obtained from (37) shown in Fig. 4. Nevertheless, we obtain excellent agreement with $c=\frac{1}{2}$ after extrapolation to large $N$, in concordance with the results of Koo and Saleur (1994).

Fig. 5 shows the identification of eigenstates with primary operators and their descendants at system size $N=14$. Comparing with the Ising CFT spectrum of Fig. 1 we observe that, even in cases of very significant finite-size corrections to the energies, preventing an identification of the tower using the spectrum alone, we are able to use $H^{\textsl{\tiny Ising}}_{n}$ to make an unambiguous identification.

The identification of global conformal towers using $\kappa_{\varphi^{\prime}}^{\varphi}$ of (43) was equally successful, as demonstrated in Fig. 6.

V.2 Three-state Potts model

We now test our algorithms with the three-state Potts model, which has a more complicated emergent CFT hosting more primary operators than the Ising CFT, including ones with significantly larger scaling dimensions. These are much harder to characterize numerically, partly because finite-size corrections to the $H_{n}$ operators mix conformal towers, as detailed below.

The three-state Potts model Wu (1982) may be thought of as a generalization of the Ising model in which spins have not two positions (up and down), but three. Unlike the Ising model it is not equivalent to a theory of free particles. It is, however, integrable at criticality Fateev and Zamolodchikov (1982). The Hamiltonian

[TABLE]

has a critical point at $\lambda=1$, determined by self-duality, and may be represented in terms of matrices

[TABLE]

which obey the exchange relations

[TABLE]

The Hamiltonian is manifestly invariant under the global shift $\prod_{j=1}^{N}V_{j}$, which implies that eigenstates fall into one of three $\mathbb{Z}_{3}$ charge sectors. At criticality its low-energy physics is described by the three-state Potts CFT, which has $c=4/5$ and twelve primary operators, including some with nonzero spin and four with scaling dimension $\Delta>2$ Cardy (1986a); von Gehlen and Rittenberg (1986), making their identification nontrivial. The eight primary operators of the $\mathbb{Z}_{3}$ zero-charge sector are:

[TABLE]

Here, we have largely followed the notation of Mong et al. (2014).

We first define the Hamiltonian density modes

[TABLE]

using them with the algorithms of Sect. IV to determine primary candidates and tower assignments.

At system size $N=14$ we are able to use (40) to identify all eight primary states of the charge-zero sector, as shown in Fig. 7, albeit at a relatively high tolerance $\epsilon_{\max}=0.2$. This is needed because, although we find $\epsilon^{(1)}$ to be negligible for all primary candidate states (marking them unambiguously as quasiprimary states), $\epsilon^{(2)}$ is significant for the $X$ and $Y$ primary candidates due to matrix elements of $H^{\textsl{\tiny Potts}}_{2}$ connecting those states to lower-energy states. To justify setting $\epsilon_{\max}=0.2$ to suppress these matrix elements, we must examine their scaling with $N$. In Fig. 8 we show that $\epsilon^{(2)}_{X}(N)$ and $\epsilon^{(2)}_{Y}(N)$ both appear to go to zero in the large $N$ limit, confirming the assignment of these lattice states to the $X$ and $Y$ primary operators. The scaling exponent $4/5$ used in Fig. 8 is that of the known leading finite-size correction of the Potts model von Gehlen et al. (1987); Reinicke (1987).

We note that identification of primaries is generally not possible using only the spectral data since there may be lower-energy states which, from their energies and momenta at finite size alone, cannot be excluded from being in the same tower as the primary state. That we can confidently identify all primaries in the Potts model, including at large $\Delta$, thus demonstrates a key benefit of using $H_{n}$ to extract conformal data.

Finite-size corrections to $H^{\textsl{\tiny Potts}}_{n}$ at $N=14$ also affect identification of conformal towers using (41). Comparing with the CFT spectrum in Fig. 7 we find that, although most assignments are plausible, some of the higher-energy states are clearly misidentified. For example, the erroneous matrix elements of $H^{\textsl{\tiny Potts}}_{2}$ affecting the $Y$ primary lead to the misidentification of $\varepsilon$ descendants as belonging to the $Y$ tower. Furthermore, we find that elements of the identity tower are sometimes misidentified as $X$ descendants. Although the former could easily be eliminated if, when assigning towers to descendants, we only considered primaries with lower energies than the descendant, the latter could not. For more precision, tower assignment should be based on a finite-size scaling analysis similar to that of Fig. 8.

The tower-mixing errors we observe here are consistent with the known finite-size corrections to the eigenstate energies (3) of the Potts model. These can be understood as coming from perturbations of the uncorrected CFT Hamiltonian density $h^{\textsl{\tiny CFT}}(x)$ by irrelevant operators (those with $\Delta>2$) Cardy (1986a). Of course, such perturbations must also affect the Hamiltonian density Fourier modes $H_{n}$ and we can understand the nature of these corrections in terms of the operator algebra Belavin et al. (1984); Ginsparg (1988); Di Francesco et al. (2012); Henkel (1999) of the CFT. In this case, perturbation of $h^{\textsl{\tiny CFT}}(x)$ by the primary field operator $X(x)$ von Gehlen et al. (1987); Reinicke (1987) explains the mixing of the $X$ and $Y$ towers with the $I$ and $\varepsilon$ towers, respectively, in terms of the fusion rules $X\times X=I+X$ and $X\times Y=\epsilon$ of the Potts CFT operator algebra Dotsenko (1984). As an aside for the interested reader, we also remark that the observed mixing connects different representations of the $\mathcal{W}_{3}$ algebra Fateev and Zamolodchikov (1987), a symmetry of the three-state Potts CFT which includes the Virasoro algebra.

Finally, as for the Ising model, we obtain an accurate estimate of the central charge as shown in Fig. 4.

V.3 The self-dual ANNNI model

We are now ready to test the Koo-Saleur formula, as well as our conformal data extraction procedures using the Hamiltonian density Fourier modes $H_{n}$, for a nonintegrable system. We consider the Axial Next-Nearest-Neighbor Ising (ANNNI) model Selke (1988); Milsted et al. (2015); Rahmani et al. (2015), an extension of the Ising model (7) by a next-nearest-neighbor interaction term and its counterpart under duality, resulting in the Hamiltonian

[TABLE]

which with this parameterization is self-dual for all $\gamma$. Under a Jordan-Wigner transformation it becomes a translation-invariant chain of interacting Majorana fermion modes and in this context its phase diagram has recently been numerically examined Milsted et al. (2015); Rahmani et al. (2015). It was found to have two distinct gapless regimes within the (approximate) parameter range $-5<\gamma<250$, with an emergent Ising CFT for $-0.3<\gamma<250$. We choose $\gamma=0.5$, which is far from the critical Ising integrable point, but in a regime where the universality class is well understood, making the results easier to analyze. We first compute the 71 lowest-energy eigenvectors of $H^{\textsl{\tiny ANNNI}}(\gamma=0.5)$, before evaluating the matrix elements in the low-energy eigenbasis of the Hamiltonian density Fourier modes, which we construct as

[TABLE]

in the same way as we did for the Ising model in (44).

Although the model is not integrable, we obtain similar results to those of Sec. V.1. In particular we find that (40) and (41) deliver completely unambiguous identifications of primary states and conformal towers, which we plot in Fig. 9. This is despite strong finite-size corrections to the energy eigenvalues compared to the Ising case of Fig. 5. We are also able to identify quasiprimary states and global descendants using (42) and (43), as we show in Fig. 10.

However, corrections show up in the matrix elements of $H^{\textsl{\tiny ANNNI}}_{1}$ and $H^{\textsl{\tiny ANNNI}}_{2}$ that were not present in $H^{\textsl{\tiny Ising}}_{1}$ and $H^{\textsl{\tiny Ising}}_{2}$, for example we observe that $H^{\textsl{\tiny ANNNI}}_{1}|I\rangle$ has overlap with a state corresponding to a descendant of the $I$ operator with $\Delta=5$, despite the CFT result $H^{\textsl{\tiny CFT}}_{1}|I\rangle=0$. Similarly, $H^{\textsl{\tiny ANNNI}}_{1}|\sigma\rangle$ has overlap with a state corresponding to a descendant of $\sigma$ with $\Delta=3\frac{1}{8}$, despite only one state with $\Delta=1\frac{1}{8}$ occurring as an overlap of $H^{\textsl{\tiny CFT}}_{1}|\sigma\rangle$ in the CFT, and $H^{\textsl{\tiny ANNNI}}_{1}|\varepsilon\rangle$ has overlap with a state corresponding to an $\varepsilon$-descendant with $\Delta=4$, in addition to the expected $\Delta=2$. In order to justify calling these overlaps finite-size corrections, we must of course demonstrate that they disappear as $N\rightarrow\infty$. Using the examples from the $\sigma$ and $\varepsilon$ conformal towers mentioned above, we show in Fig. 11 that this is indeed the case. We note that, as with the Ising model, there is no mixing of different conformal towers (again due to the symmetries of $H_{n}^{\textsl{\tiny ANNNI}}$), explaining why we are still able to make tower assignments unambiguously.

Unlike in the Potts model, the observed corrections to $H^{\textsl{\tiny ANNNI}}_{n}$ can only come from perturbation of $h^{\textsl{\tiny CFT}}(x)$ by irrelevant operators from the identity conformal tower, since operators from any other conformal tower would lead to tower mixing. Furthermore, there must be a perturbation that is not present in the Ising model, which did not exhibit the corrections we see here. One allowed identity-tower perturbation of $h^{\textsl{\tiny CFT}}(x)$ is the field operator $T\overline{T}(x)$, corresponding to the state $L^{\textsl{\tiny CFT}}_{-2}\overline{L}{}^{\textsl{\tiny CFT}}_{-2}|I\rangle$, which is suppressed in the Ising model Reinicke (1987), but is allowed in general. It seems a likely candidate to cause the observed correction to $H^{\textsl{\tiny ANNNI}}_{1}|I\rangle$ since adding it to $h^{\textsl{\tiny CFT}}(x)$ in (13) for $n=1$ would result in the usual $H^{\textsl{\tiny CFT}}_{1}$ plus a Fourier mode of $T\overline{T}(x)$ which, applied to $|I\rangle$, would produce global descendant states of the (quasiprimary) state $L^{\textsl{\tiny CFT}}_{-2}\overline{L}{}^{\textsl{\tiny CFT}}_{-2}|I\rangle$, including one at level $\Delta=5$. Indeed, in Zou et al. (2017b) we confirm that this perturbation is present in the ANNNI model.

Regarding finite-size corrections to the energies, we note that they are severe enough so that, at $N=14$, the states $|T\rangle$ and $|\overline{T}\rangle$ are not the lowest-energy states with $|S|=2$, as is often assumed when normalizing the Hamiltonian density (see Sect. IV.1). Where this occurs, identifying $|T\rangle$ using $H^{\textsl{\tiny ANNNI}}_{2}$ is clearly advantageous. Indeed, we observe in Fig. 12 that the difference in the normalization factors obtained is significant for affected system sizes.

Finally, in Fig. 13 we demonstrate that the central charge estimated using (38) remains accurate away from integrability. Furthermore, we compare the estimate to that obtained from the scaling of the ground state energy $E_{I}$ Henkel (1999), finding the estimates to be comparable as long as the Hamiltonian is properly normalized, which requires the use of $H^{\textsl{\tiny ANNNI}}_{2}$ at small system sizes.

VI Discussion

In this paper we have proposed and demonstrated automated procedures for extracting conformal data from generic local quantum spin chains using the Hamiltonian density Fourier modes $H_{n}$, first introduced as lattice representations of conformal generators by Koo and Saleur Koo and Saleur (1994). In particular, we explained how to use the $H_{n}$ to systematically identify the lattice energy eigenstates corresponding to Virasoro primary and quasiprimary operators of the CFT, as well as how to assign the remaining eigenstates to conformal towers. Furthermore, our demonstration included a nonintegrable model (the ANNNI model), confirming that the so-called Koo-Saleur formula continues to behave as expected away from integrability.

To extract accurate conformal data, one must examine systems of sufficient size, such that non-universal finite-size corrections (e.g. due to irrelevant perturbations) are manageable. This is often impossible using exact diagonalization techniques, which we applied to obtain spectra and low-energy eigenstates for the present work, since the computational cost scales exponentially in the system size. Fortunately, our proposals for extracting conformal data using the Hamiltonian density Fourier modes $H_{n}$ are independent of the method used to diagonalize $H$ and can also be implemented using more sophisticated tools, such as periodic matrix product states, allowing the analysis of critical quantum spin chains with hundreds of spins Zou et al. (2017b).

This work contributes toward the overarching goal of, given a generic critical quantum spin chain Hamiltonian $H$, determining the conformal data that specifies the emergent CFT. Indeed, the identification of the Virasoro primary states within the low-energy spectrum is an essential part of this task, one that cannot be accomplished in general using only the spectral information in (3), but which is made possible by using the lattice operators $H_{n}$. In order to complete this long-standing research program, a systematic way of determining the OPE coefficients relating the primary operators to each other is still missing (although progress can be made in particular cases – see for example Read and Saleur (2007b, a); Gainutdinov and Vasseur (2013)). As it turns out, however, the methods discussed in this paper can be combined with other techniques in order to also estimate the OPE coefficients on the lattice Zou et al. (2017a).

Finally, we remark that the action of lattice Virasoro generators in the low-energy subspace of quantum spin chains has found applications beyond the extraction of conformal data. For example, these techniques are used in Zou et al. (2017b) to study the RG flow between two CFTs, and in Milsted and Vidal (2017) to attach a geometric meaning to tensor networks that discretize a path integral.

Acknowledgements.

We are grateful to John Cardy, Qi Hu, Vaughan Jones, Tobias J. Osborne, Frank Verstraete, Yuan Wan and Yijian Zou for helpful and stimulating discussions. We also thank Jerôme Dubail, Hosho Katsura, Hubert Michel Saleur, and Romain Vasseur for kindly pointing out Ref. Koo and Saleur (1994), of which we were not aware while preparing the first version of this paper. Finally, we thank the referees who reviewed this paper for providing useful comments that led to improvements in our presentation. The authors acknowledge financial support from the Simons Foundation (Many Electron Collaboration). This research was supported in part by Calcul Québec and Compute Canada, as well as by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

Appendix A Lattice momentum density

The Virasoro algebra (12) fulfilled by the operators (11) together with (13), implies

[TABLE]

so that we may construct lattice analogues of $L^{\textsl{\tiny CFT}}_{n}$ and $\overline{L}^{\textsl{\tiny CFT}}_{m}$ as Koo and Saleur (1994)

[TABLE]

This is equivalent to defining a momentum density

[TABLE]

which satisfies the lattice energy-momentum conservation law

[TABLE]

and constructing $L_{n}$ and $\overline{L}_{m}$ as

[TABLE]

with

[TABLE]

in analogy with the CFT definition of the Virasoro generators (11).

We find in practice that $L_{n}$ and $\overline{L}_{m}$ defined for the Ising model have more severe finite-size corrections than $H^{\textsl{\tiny Ising}}_{n}$ (see Sect. V.1). In particular, they connect states with the wrong descendants, although they still do not mix conformal towers.

There is an obvious reason for these additional corrections, which come from finite-size corrections to the energy. Consider the action of $L_{n}$ on an energy eigenstate $|\Delta\rangle$ of a lattice Hamiltonian $H$. We first assume that $H_{n}|\Delta\rangle=a|\Delta\!-n\rangle+b|\Delta\!+n\rangle$ such that

[TABLE]

where $\epsilon$, $\epsilon^{\prime}$, $\epsilon^{\prime\prime}$ represent finite-size corrections to the energy, which will generally be different for each energy eigenstate. This scenario is consistent with $a|\Delta-n\rangle$ and $b|\Delta+n\rangle$ being the lattice counterparts of the CFT states $L^{\textsl{\tiny CFT}}_{n}|\Delta\rangle^{\textsl{\tiny CFT}}$ and $\overline{L}^{\textsl{\tiny CFT}}_{-n}|\Delta\rangle^{\textsl{\tiny CFT}}$, respectively. We then find

[TABLE]

where in case $\epsilon=\epsilon^{\prime}=\epsilon^{\prime\prime}$ almost all terms cancel and we are left with

[TABLE]

as expected. As noted above, however, generally $\epsilon\neq\epsilon^{\prime}\neq\epsilon^{\prime\prime}$ and the cancellation is prevented, leading to an erroneous matrix element of $L_{n}$ connecting $|\Delta\rangle$ and $|\Delta\!+n\rangle$.

Appendix B Degeneracies and quasiprimary states

In Figs. 6 and 10 we plot the spectra of the Ising and ANNNI models, respectively, at size $N=14$, showing global conformal towers of the quasiprimary states $|T\rangle$ and $|\overline{T}\rangle$. We find in both cases that a linear combination $|\varphi_{Q}\rangle\equiv a|\varphi_{1}\rangle+b|\varphi_{2}\rangle$ of lattice energy eigenstates $|\varphi_{1}\rangle,|\varphi_{2}\rangle$ belonging to the (Virasoro) conformal tower of $I$ at level $\Delta\approx 4$, $S=4$, fulfills the quasiprimary condition (42) to numerical precision:

[TABLE]

where $\Gamma_{\varphi_{Q}}$ projects onto states with energy lower than the energy expectation value of $|\varphi_{Q}\rangle$. The situation is analogous for the $|\overline{T}\rangle$ descendants.

In the CFT, where the states of the $I$ conformal tower at $\Delta=4$, $S=4$ are degenerate in energy and momentum (see Fig. 1), there is also a quasiprimary state in the corresponding degenerate subspace. We wish to confirm that the lattice state $|\varphi_{Q}\rangle$ corresponds to the quasiprimary in the CFT. First, we note that, from (12) and (25), the CFT quasiprimary may be built as

[TABLE]

which can be seen to be annihilated by $L^{\textsl{\tiny CFT}}_{1}$. We may construct an analogous state on the lattice as

[TABLE]

Doing so we find that, to high precision,

[TABLE]

and that furthermore $|\tilde{\varphi}_{Q}\rangle$ is approximately equal to $|\varphi_{Q}\rangle$ of (65), with appropriate normalization. This confirms that the criterion (42) for quasiprimary states on the lattice correctly distinguishes linear combinations of lattice eigenstates that correspond to CFT quasiprimary operators.

We remark here on the observation that degenerate quasiprimary and global secondary states are mixed by finite-size corrections to the energy (even when Virasoro conformal towers are not mixed) so that the quasiprimary lattice state is formed by a linear combination of energy eigenstates with different energies (which become degenerate in the limit $N\rightarrow\infty$). In the presence of finite-size effects that mix Virasoro conformal towers, it could also happen that primary states are mixed with Virasoro descendant states, although we did not observe this in the models tested in this work.