Estimating the central charge from the R\'enyi entanglement entropy

TL;DR

This paper demonstrates how to estimate the central charge of a conformal field theory from entanglement entropy calculations in the 1+1 dimensional O(2) model, confirming the theoretical value c=1.

Contribution

It provides a method to accurately determine the central charge from entanglement entropy in lattice models, bridging microscopic and continuum descriptions.

Findings

Entanglement entropy scales logarithmically with system size as predicted by conformal field theory.

The central charge c=1 is confirmed within 2% accuracy for the O(2) model.

The approach remains valid under continuum limits and truncations, facilitating connections to quantum Hamiltonians.

Abstract

We calculate the von Neumann and R\'enyi bipartite entanglement entropy of the $O(2)$ model with a chemical potential on a 1+1 dimensional Euclidean lattice with open and periodic boundary conditions. We show that the Calabrese-Cardy conformal field theory predictions for the leading logarithmic scaling with the spatial size of these entropies are consistent with a central charge $c=1$. This scaling survives the time continuum limit and truncations of the microscopic degrees of freedom, modifications which allow us to connect the Lagrangian formulation to quantum Hamiltonians. At half-filling, the forms of the subleading corrections imposed by conformal field theory allow the determination of the central charge with an accuracy better than two percent for moderately sized lattices. We briefly discuss the possibility of estimating the central charge using quantum simulators.

Tables

Table 1. Table 1: Slopes of the Rényi entropies using the leading-order linear fit. The fits were done using the same volumes used in the TRG calculations: N s = 2 ℓ subscript 𝑁 𝑠 superscript 2 ℓ N_{s}=2^{\ell} . For data at these volumes oscillations do not appear; oscillations occur between volumes: N s ​  mod  ​ 4 = 0 subscript 𝑁 𝑠  mod  4 0 N_{s}\text{ mod }4=0 , and N s ​  mod  ​ 4 = 2 subscript 𝑁 𝑠  mod  4 2 N_{s}\text{ mod }4=2 .

case 1	isotropic	anisotropic	DMRG	$c=1$ CFT
$S_1$ PBC	0.319	0.311	0.327	$0.\overline{3}$
$S_2$ PBC	0.273	0.265	0.267	0.25
$S_1$ OBC	0.207	0.208	0.195	$0.1\overline{6}$
$S_2$ OBC	0.182	0.152	0.168	0.125
case 2	isotropic	anisotropic	DMRG	$c=1$ CFT
$S_1$ PBC	0.328	0.296	0.329	$0.\overline{3}$
$S_2$ PBC	0.262	0.229	0.250	0.25
$S_1$ OBC	0.179	0.152	0.159	$0.1\overline{6}$
$S_2$ OBC	0.165	0.148	0.140	0.125

Table 2. Table 2: Values for A n subscript 𝐴 𝑛 A_{n} , and p n subscript 𝑝 𝑛 p_{n} for β s ​ β τ = 0.01 subscript 𝛽 𝑠 subscript 𝛽 𝜏 0.01 \beta_{s}\beta_{\tau}=0.01 , μ ​ β τ = 0.5 𝜇 subscript 𝛽 𝜏 0.5 \mu\beta_{\tau}=0.5 , from the least-squares fits to the DMRG data up to N s = 64 subscript 𝑁 𝑠 64 N_{s}=64 corresponding to Fig. 4 .

$S_n$	$p_n$	$A_n$ from fit	$A_n$ with $c=1$
$S_1$ PBC	2.315	0.3338	$0.\overline{3}$
$S_2$ PBC	0.981	0.2525	0.25
$S_1$ OBC	0.901	0.1663	$0.1\overline{6}$
$S_2$ OBC	0.443	0.1246	0.125

Table 3. Table 3: Values for A n subscript 𝐴 𝑛 A_{n} for β s ​ β τ = 4 subscript 𝛽 𝑠 subscript 𝛽 𝜏 4 \beta_{s}\beta_{\tau}=4 , μ ​ β τ = 0 𝜇 subscript 𝛽 𝜏 0 \mu\beta_{\tau}=0 , from the least-squares fits to the DMRG data up to N s = 64 subscript 𝑁 𝑠 64 N_{s}=64 corresponding to Fig. 6 .

$S_n$	$A_n$ from fit	$A_n$ with $c=1$
$S_1$ PBC	0.3337	$0.\overline{3}$
$S_2$ PBC	0.2500	0.25
$S_1$ OBC	0.1654	$0.1\overline{6}$
$S_2$ OBC	0.1278	0.125

Equations

$${\hat{\rho }}_A={\text{Tr}}_{\text{env}}[\hat{\rho }]$$

$$S_n(A)\equiv \frac{1}{1-n}\ln (\text{Tr}[{\hat{\rho }}_A^n]).$$

$$S_n(N_s)=\{\begin{array}{ll}{K}_n+\frac{c(n+1)}{6n}\ln (N_s)& \text{for PBC}\\ K_n^{\prime }+\frac{c(n+1)}{12n}\ln (N_s)& \text{for OBC}.\end{array}$$

$$Z=\frac{1}{2\pi }\int \prod _{(x,t)}d{\theta }_{(x,t)}{\mathrm{e}}^{-S}$$

$$S=$$

$$-$$

$$\lambda \equiv \frac{1}{(N_s\times N_{\tau })}\frac{\partial \ln (Z)}{\partial \mu }.$$

$${\mathbb{T}}_{(n_1,n_2,\dots ,n_{N_s})(n_1^{\prime },n_2^{\prime },\dots ,n_{N_s}^{\prime })}=$$

$$\sum _{{\tilde{n}}_1{\tilde{n}}_2\dots {\tilde{n}}_{N_s}}{T}_{{\tilde{n}}_{N_s}{\tilde{n}}_1{n}_1{n}_1^{\prime }}^{(1,t)}{T}_{{\tilde{n}}_1{\tilde{n}}_2{n}_2{n}_2^{\prime }\dots }^{(2,t)}$$

$$\dots T_{{\tilde{n}}_{N_s-1}{\tilde{n}}_{N_s}{n}_{N_s}{n}_{N_s}^{\prime }}^{(N_s,t)},$$

$$T_{{\tilde{n}}_{x-1}{\tilde{n}}_x{n}_x{n}_x^{\prime }}^{(x,t)}$$

$$\hat{H}=\frac{U}{2}\sum _x{\hat{L}}_x^2-\tilde{\mu }\sum _x{\hat{L}}_x-2J\sum _{⟨xy⟩}\cos ({\hat{\theta }}_x-{\hat{\theta }}_y),$$

$$H=\frac{U}{2}\sum _x{n}_x(n_x-1)-J\sum _x(a_x^{†}{a}_{x+1}+h.c.).$$

$$(\text{Relative Error}{)}^2=\frac{1}{N}\sum _{i=1}^N{\left(\frac{y_i-f(x_i)}{y_i}\right)}^2$$

$$S_n(N_s,l)$$

$$+\frac{C}{N_s^{p_n}}\cos (\pi l){\left|\sin \left[\frac{\pi l}{N_s}\right]\right|}^{-p_n}$$

$$+f_n(N_s,l)$$

$$\frac{1}{T}=N_{\tau }a$$

$$\frac{1}{T}=\frac{N_{\tau }}{{\beta }_{\tau }U}=\frac{{\beta }_s{N}_{\tau }}{2J}.$$

$$S_n(N_s)$$

$$+\frac{C}{N_s^{p_n}}\cos (\pi N_s/2)+E{N}_s+f(N_s),$$

$$S_n(N_s,l)$$

$$+\frac{C}{N_s^{p_n}}\sin \left[\frac{\pi (2l+1)}{2}\right]{\left|\sin \left[\frac{\pi (2l+1)}{2(N_s+1)}\right]\right|}^{-p_n}$$

$$+Dl+f_n(N_s,l).$$

Full text

I Introduction

Conformal symmetry has been a major source of inspiration for theoretical physics during the last few decades Polyakov (1987); Di Francesco et al. (1997). In two dimensions the conformal algebra is infinite dimensional and can be identified with the Virasoro algebra generating the reparameterization of the world sheet in string theory. This algebra admits central extensions labeled by the central charge, denoted $c$ hereafter. Known unitary representations with $c=1-6/(m(m+1))$ and $m=3,\dots,6$ describe the critical behavior of the two-dimensional Ising and 3-states Potts models and their tricritical versions Belavin et al. (1984); Friedan et al. (1984); Dotsenko (1984).

In three dimensions, the possibility that a cusp on the boundary of the region of anomalous dimensions allowed by the conformal bootstrap corresponds to the Ising universality class has triggered very interesting new developments El-Showk et al. (2012, 2014); Kos et al. (2014). In four dimensions, the idea that electroweak symmetry breaking could result from new strong interactions at a multi-TeV scale with an approximate conformal symmetry Appelquist et al. (1996); Miransky and Yamawaki (1997); Sekhar Chivukula (1997); Appelquist and Sannino (1999); Luty and Okui (2006); Dietrich et al. (2005) protecting a light Higgs-Brout-Englert boson has motivated numerous lattice studies DeGrand (2011); Blum et al. (2013); Kuti (2015).

QCD-like systems with various numbers of fundamental fermions and also fermions in different representations are being explored on the lattice (the latest results for various models are presented, for instance, in Refs. DeGrand et al. (2015); Brower et al. (2016); Arthur et al. (2016); Appelquist et al. (2016); Fodor et al. (2016a)). Based on the Banks-Zaks argument, systems with a large number of fermion flavors, $N_{f}$, feature a conformal phase Banks and Zaks (1982). However, precisely at what value of $N_{f}$ this happens for a particular gauge group and fermion representation remains a subject of controversy. For the $SU(3)$ gauge group and fermions in the fundamental representation, some studies claim observing conformal behavior at $N_{f}=12$ (see Cheng et al. (2014) for instance) while others (e.g. Ref. Fodor et al. (2016b)) argue that $N_{f}=12$ is not conformal.

To probe the conformality, various lattice methods, designed and well-tested for QCD, have been employed. The major obstacle, however, is that large-$N_{f}$ theories are very different from QCD. While in QCD the running of the coupling is fast enough so that one can probe both the ultraviolet perturbative and the infrared confining phenomena, as manifested, for instance, in one of the basic and extensively studied quantities, the static quark anti-quark potential, large-$N_{f}$ theories require fine lattices and large volumes to disentangle the physics from the lattice cutoff effects. This makes identifying conformal theories from the massless extrapolations of massive lattice simulations a nontrivial task Hasenfratz (2012); Fodor et al. (2011); Aoki et al. (2012); Gelzer et al. (2014); Cheng et al. (2014); Fodor et al. (2016b).

In the given examples, conformal symmetry is explicitly broken by the lattice regularization and only re-emerges in a suitable continuum limit and infinite volume limit. Given the predictive power of conformal symmetries, it is important to identify the restoration of these symmetries in practical calculations at finite volumes. The entanglement entropy may offer a promising direction in understanding the conformal behavior of systems with finite-size scaling and could be a more sensitive tool, especially for small-size systems. How the entanglement entropy of a subsystem scales with the its spatial volume provides useful information about the symmetries present and the conformality of the phases of a model Schuch et al. (2008). This is very well understood in two (one space and one Euclidean time) dimensions where Calabrese and Cardy (CC) Calabrese and Cardy (2004) have shown that various entanglement entropies scale like the logarithm of the size of the subsystem with coefficients proportional to the central charge.

Calculations of the entanglement entropy in lattice gauge theory with Monte Carlo methods have so far been performed in pure gauge theory Velytsky (2008); Buividovich and Polikarpov (2008); Itou et al. (2016) and two dimensional critical spin systems Caraglio and Gliozzi (2008). Those calculations use the “replica trick” where $n$ sheeted Riemannian surfaces are glued together over an interval, but may require extra developments for theories with fermions. The entanglement spectrum for lower order Rényi entanglement entropies has been analyzed for fermion systems in three dimensions using determinantal Monte Carlo methods, with universal behavior found that could potentially be observed in cold-atom experiments Porter and Drut (2016); Drut and Porter (2016, 2015). In the long run finite-size scaling of the entanglement entropy may provide a cleaner way to study conformal systems than just increasing lattice volumes and decreasing lattice spacings in the hope of suppressing lattice artifacts.

In the following, we use renormalization group (RG) based methods Liu et al. (2013); Denbleyker et al. (2014); Yu et al. (2014); White (1992); Östlund and Rommer (1995) to calculate the von Neumann and second order Rényi entanglement entropy of the classical $O(2)$ nonlinear sigma model with a chemical potential in 1+1 dimensions on a space-time lattice. This model is often used as an effective theory for the Bose-Hubbard model Fisher et al. (1989) and is good as a toy model for gauge theories in higher dimensions. The model has vortex solutions, no long range order, and demonstrates a confinement-deconfienment transition of vortex-anti-vortex pairs. This model Zou et al. (2014) has a superfluid (SF) phase where we expect to observe the CC scaling and multiple Mott insulator phase lobes lacking the CC scaling.

By using rectangular lattices of increasing spatial size and very large (Euclidean) temporal sizes we probe the zero temperature entanglement entropy. We focus on half-integer charge density where the entropies considered are extremal. We then take the time continuum limit and truncate the microscopic degrees of freedom in such a way that we obtain a Hamiltonian that can be quantum simulated Zou et al. (2014); Bazavov et al. (2015). These modifications should not affect the universal parts of the scaling. Our goal is to demonstrate that the constraints imposed by conformal symmetry on the finite size scaling, as well as conjectures Fagotti and Calabrese (2011); Calabrese et al. (2010); Calabrese and Essler (2010) explaining oscillations in the scaling, allows us to identify conformal behavior for modest lattice sizes.

The motivation for relating this model to a model that can be quantum simulated on optical lattices with cold atoms is prompted in current challenges with classical computation. It would be not only valuable to have efficient calculational tools for understanding conformal behavior for more complex, higher dimensional systems, but also to completely overcome the difficulty with large volume, small lattice spacing calculations entirely. This can be done by using quantum simulation, which can already reach volumes in 3+1 dimensions on the same order as classical computation and larger volumes are expected. This idea is pursued in more detail in Unmuth-Yockey et al. (2016).

Manipulations of small one-dimensional systems of cold atoms trapped in optical lattices have allowed experimental measurements of the second order Rényi entanglement entropy Islam et al. (2015) using a beam splitter method proposed in Ref. Daley et al. (2012). These measurements have been performed for small chains of four atoms. A more recent experiment on thermalization Kaufman et al. (2016) involves six atoms. It is expected that in the near future, manipulations of chains with twelve or more atoms will be possible 111private communication, Philipp Preiss.

The paper is organized as follows: in Section II we review the Rényi entropy and the corresponding conventions used in this paper. We also discuss the currently understood asymptotic scaling in the Rényi entropy as a function of system size. In Section III we introduce the $O(2)$ nonlinear sigma model on a lattice and the tensor formulation of the model. We give explicit tensor elements and discuss the isotropic and anisotropic coupling limits used in this paper as well as some results obtained in those limits. In Section IV we give results for fits to Rényi entanglement entropy data. We consider the scaling of the entanglement entropy as a function of system size. We also go into detail about the methodology used in our fits and make comparisons with theoretical predictions. In Section V we discuss the possibility of quantum simulating the $O(2)$ nonlinear sigma model and a possible quantum Hamiltonian that could be used for simulation. We also consider finite temperature effects to the entanglement entropy. Finally in Section VI we give concluding remarks about in what possible directions work could proceed, and other possible implications of this work.

II The Rényi entanglement entropy

For the calculation of the entanglement entropy, we will restrict ourselves to 1+1 dimensional space-time, or one space and one Euclidean time dimension, where the one dimensional space has an even number of sites. For all results in this work the system was divided into two identical parts, each one half the size of the entire spatial dimension, $N_{s}$ (justification for this can be found in the supplemental material of Ref. Unmuth-Yockey et al. (2016)). Other partitions of the system were considered as checks and exploratory. Tracing over one of the halves, we obtain the reduced density matrix $\hat{\rho}_{A}$ for the other half (denoted $A$),

[TABLE]

where the trace is over the “environment” leaving only the sub-system defined as $A$. The $n$-th order Rényi entropy is defined as

[TABLE]

The limit $n\rightarrow 1^{+}$ is the von Neumann entanglement entropy, or the first order Rényi entropy. $S_{2}$ is the second order Rényi entropy, and was measured in recent cold atom experiments Islam et al. (2015). An important goal for future work is to estimate the central charge, $c$, from empirical data. Using the transformation properties of the energy-momentum tensor and the Ward identities from CFT, CC established that, to leading order, the Rényi entropy scales linearly with the logarithm of the spatial volume. The constant of proportionality is the central charge multiplied by a rational that depends on the order of the Rényi entanglement entropy and the boundary conditions:

[TABLE]

The intercept is non-universal and different in the four situations considered here.

The calculation of $S_{n}$ can be performed Yang et al. (2016) using blocking (coarse-graining) methods Liu et al. (2013); Denbleyker et al. (2014); Yu et al. (2014); White (1992). In this work we used the density matrix renormalization group (DMRG) with matrix product states (MPS), as well as exact blocking formulas Liu et al. (2013); Denbleyker et al. (2014); Yu et al. (2014); Zou et al. (2014) with the tensor renormalization group method (TRG), and the only approximation in these methods consists of truncating the number of states (called $D_{\text{bond}}$). The errors associated with this truncation will be discussed later.

III The $O(2)$ model

In the following, we consider the classical $O(2)$ model on a $N_{s}\times N_{\tau}$ rectangular lattice with sites labeled $(x,t)$. This is a generalization of the Ising model where the local spin is allowed to take values on a circle, making an angle $\theta$ with respect to some direction of reference. This angle can be interpreted as the phase of a complex field and the model has an exact charge conjugation symmetry, $\theta\rightarrow-\theta$, interchanging particles and anti-particles. This symmetry can be broken by adding a chemical potential $\mu$ to the angle gradient Hasenfratz and Karsch (1983). The partition function reads:

[TABLE]

with

[TABLE]

We use periodic boundary conditions (PBC) or open boundary conditions (OBC) in space and always PBC in time. In the following, we define the charge density as

[TABLE]

We will start with the situation where the relativistic interchangeability between space and time is present ($\beta_{s}=\beta_{\tau}$), as is typical in lattice gauge theory simulations. Later, we will take the time continuum limit and switch to the Hamiltonian formulation.

For numerical purposes, and in order to connect the Hamiltonian formulation to quantum simulators, it is convenient to introduce discrete degrees of freedom on the links (bonds) of the lattice. Using Fourier expansions Savit (1980); Banerjee and Chandrasekharan (2010); Liu et al. (2013), one can show Zou et al. (2014); Yang et al. (2016) that the partition function can be expressed in terms of a transfer matrix $Z=\text{Tr}[\mathbb{T}^{N_{\tau}}]$ where the matrix elements of $\mathbb{T}$ have the explicit form

[TABLE]

with

[TABLE]

When $N_{s}$ is a power of 2, the traces in the spatial directions in Eq. (11) can be performed recursively and combined with a truncation of the number of states kept in the time direction Zou et al. (2014); Yang et al. (2016). The accuracy of this tensor renormalization group method has been tested against sampling methods Yang et al. (2016).

We can interpret $\mathbb{T}^{N_{\tau}}$ as a density matrix $\hat{\rho}$ if we normalize by the trace of the matrix. It is important to understand that the classical spin model described above can be taken in a limiting form as a quantum model in one spatial dimension. In the following, we always take $N_{\tau}\gg N_{s}$ and extrapolate to infinite $N_{\tau}$. This corresponds to the zero temperature limit in the quantum terminology. Finite temperature effects will be discussed in Sec. V and were considered in Ref. Yang et al. (2016).

The SF phase is characterized by a response of the charge density, $\lambda$, to a change in the chemical potential. This is illustrated in Refs. Banerjee and Chandrasekharan (2010); Zou et al. (2014). In contrast, in the Mott phases, the charge density keeps a fixed integer value as we increase $\mu$. This lack of response is somewhat puzzling in the functional integral formulation and is often called the “Silver blaze” phenomenon Cohen (2003) in the context of finite temperature QCD. Another characterization of the two phases is by the scaling of the Rényi entropy as a function of the volume of space.

In the following we focus on two cases for two different relationships between the spatial and temporal couplings. We consider $\beta=0.1$, $\mu\simeq 3$ (case 1), and $\beta=2$, $\mu=0$ (case 2). Both of these situations are considered in the limits of isotropic coupling ($\beta_{s}=\beta_{\tau}$), and anisotropic coupling ($\beta_{s}\beta_{\tau}=\text{const.}$ with $\beta_{\tau}\rightarrow\infty$ and $a\rightarrow 0$, where $a$ is the temporal lattice spacing). In case 1 the SF transition is driven by an increase in the chemical potential at fixed $\beta$. This is the transition driven by fluctuations in density. For case 2 the transition is driven by the presence of vortices and is the Berezinskii-Kosterlitz-Thouless (BKT) transition.

III.1 Isotropic Coupling

In this section we consider the case where the coupling in space and time are the same. This is a classical statistical two-dimensional spin system. Using TRG we can block a (Euclidean) time slice of the lattice and consider it as a transfer matrix. From it we can calculate a “zero-temperature” density matrix by taking $N_{\tau}\gg N_{s}$ (typically $N_{\tau}\approx 2^{20}N_{s}$ in practice, although even larger sizes may be used). Then, from the density matrix one can make the reduced density matrix and calculate the Rényi entropy of the desired order.

The values of $S_{1}$ and $S_{2}$ are shown in Fig. 1 for $N_{s}=4,8,16$, and $32$, for PBC and OBC.

These results are compared with the leading CFT prediction of Eq. (3) by just fitting the intercept with the CFT slope fixed. For isotropic calculations the TRG calculations kept up to 250 states. The figure shows that the discrepancies are rather small and most visible for $S_{2}$ with OBC for case 1. In case 2, the discrepancies are slightly more pronounced. In all cases, the discrepancies are due to subleading corrections not taken into account in the fits, rather than the small numerical errors.

Fig. 2 gives the values of $S_{2}$ for $N_{s}=4$ across a region of the $\beta$-$\mu$ plane.

This figure shows lobes corresponding to a fixed charge density. The largest and most prominent is the $\lambda=0$ lobe, followed by much thinner $\lambda=1$ lobe above. The lobes continue as long as the truncation on the number of states used in the TRG calculation can support the charge density. Fig. 2 also shows 3 plateaus in the SF regions between each Mott lobe. These plateaus were investigated and related to the charge density in the isotropic limit in Yang et al. (2016). In the next section we consider the time-continuum limit of the classical $O(2)$ model and how the phase diagrams transforms through taking that limit.

III.2 Anisotropic Coupling

We now proceed to take the time continuum limit. This can be achieved by taking $\beta_{\tau}$ very large while keeping constant the product $\beta_{s}\beta_{\tau}$ , and keeping $\mu\beta_{\tau}$ tuned to the desired charge density. For case 1, the limit of the chemical potential must be done carefully in order to maintain a fixed charge density corresponding to half-filling. For small volumes half-filling takes place around $\mu\beta_{\tau}=0.5$ as $\beta_{\tau}\rightarrow\infty$, but not all the data collected for larger volumes was necessarily done at that parameter specification, and instead the parameters were tuned to maintain half-filling. The time continuum limit in the tensor formulation defines Zou et al. (2014) a rotor Hamiltonian Fradkin and Susskind (1978); Kogut et al. (1979):

[TABLE]

with $[\hat{L}_{x},{\rm e}^{i\hat{\theta}_{y}}]=\delta_{xy}{\rm e}^{i\hat{\theta}_{y}}$. It’s possible to truncate to finite integer spin and approximate these commutation relations Zou et al. (2014). The normalization has been chosen in such a way that the coupling constants in the Bose-Hubbard model used in Ref. Islam et al. (2015), and here in the $O(2)$ model are the same: $\beta_{s}\beta_{\tau}\equiv 2J/U$, and $\mu\beta_{\tau}\equiv\tilde{\mu}/U$.

In the following we primarily use the spin-1 approximation which can also be implemented in the original isotropic formulation by setting the tensor elements in Eq. (12) to zero for space and time tensor indices strictly larger than 1 in absolute value (so only 3 states remain). The Hamiltonian is then a spin-1 XY model with a chemical potential and an ion anisotropy. In addition, for large enough chemical potential, the $n=-1$ component decouples and we are approximately left with a spin-1/2 XY model. Furthermore, for $\tilde{\mu}=U/2\gg J$, there is an approximate connection with the Bose-Hubbard model

[TABLE]

The Hamiltonian in Eq. (13) is never explicitly used in the blocking procedure with TRG. In practice, the TRG tensors used are the same, however the coupling constants that appear in the local tensor’s definition are tuned to reflect the scenario under consideration. Again, using the TRG the same way, one calculates the Rényi entropy for an approximately zero-temperature situation. The only minor change is that due to the increased coupling in the time direction, $N_{\tau}$ needs to be adjusted to compensate the smaller lattice spacing. This means increasing $N_{\tau}$ even more, a facile task while using a blocking method. We show slices of the Rényi entropy in the region of case 1 for $N_{s}=4$ and $8$, OBC, in Fig. 3. A more extensive plot of $S_{2}$ for the $O(2)$ model in the time-continuum limit can be found in Unmuth-Yockey et al. (2016) for a large range of couplings along with a comparison to the Bose-Hubbard Model.

In order to check the TRG calculations of $S_{n}$ in the time continuum limit (Eq. (13)), we have used DMRG White (1992) which has been used to calculate the ground state entanglement entropy and Rényi entropy for similar Hamiltonians Lange et al. (2015); Dalmonte et al. (2011); Zhang et al. (2011). Calculations with MPS optimization Östlund and Rommer (1995) have been performed using the ITensor C++ library 222(version 2.7.10), http://itensor.org/. We run enough sweeps for the entropy to converge to at least $10^{-8}$, and a large number of states, up to $1500$, was kept so that the truncation error is less than $10^{-10}$. The comparison of the results with the two methods showed excellent agreement at small volume (typically 9 digits for $N_{s}=4$) but the discrepancies increased with the volume (typically 3 digits agreement for $N_{s}=32$). We believe that the DMRG results are more accurate because firstly, it can keep many more states than the TRG by using sparse linear algebra libraries. Secondly, the truncations are made step-by-step trying to maximize the entanglement entropy. Finally, DMRG uses an environment sweep method which optimizes the ground state wave-function iteratively. For these reasons, we have used the DMRG results for the fits that follow.

IV Fits to $S_{n}$

In this section we give some results for fits to the isotropic and anisotropic data, as well as for fits to the DMRG data. The primary deviations from the leading order linear behavior of the Rényi entropy come from finite volume effects and parity oscillations. These deviations were found most prominently in OBC data, and the second order Rényi entropy. Oscillations were found between sizes $N_{s}$ mod 4 = 0 and $N_{s}$ mod 4 = 2, although in case 1 $S_{1}$ with PBC had none, as well as $S_{1}$ in case 2. Corrections to the leading-order CFT behavior to account for these oscillations have been conjectured Fagotti and Calabrese (2011); Calabrese et al. (2010); Calabrese and Essler (2010), and we check their validity for the anisotropic $O(2)$ model. In addition, non-oscillatory finite volume corrections have been derived Cardy and Calabrese (2010) and in the following we attempt to take all these corrections into account in order to fit the data as best as possible.

Initially, all the data was fit to the leading order CFT prediction (Eq. (3)), with the slope and intercept as the two free parameters. This was done for spatial volumes which matched the TRG blocking volumes, e.g. $N_{s}=2^{\ell}$, and since these sizes are multiples of 4, no oscillations were present. These fits were done for both the DMRG and TRG data in both the isotropic and anisotropic limits. The results for the slope fits are reported in Table 1 for all cases considered.

For the two cases considered here, we tried various fits that attempted to incorporate subleading corrections. We attempted fits with four or five free parameters. These included corrections $\propto 1/N_{s}$, $1/N_{s}^{2}$, $1/\ln(N_{s})$ and $1/\ln^{2}(N_{s})$. To judge the quality of the fits we compared the average relative error between fits,

[TABLE]

with $y_{i}$ the dependent data and $f(x_{i})$ the fitting function evaluated at the independent data. This measure is convenient since the error is dimensionless; in addition, a $\chi^{2}$ measure of error would depend upon the unknown DMRG error bars and fitting with uncertainties in arbitrary units gives a relatively useless estimate of the fit quality. The relative errors associated with the fits were never greater than $10^{-3}$, and never less than $10^{-7}$. For systems with subsystems of size $l$, we considered a fit of the form

[TABLE]

with $f_{n}$ a function to take into account additional corrections, and $A_{n}$, $B$, $C$, and $p_{n}$ are fit parameters. However, we focused on data with $l=N_{s}/2$. We found the best fit results by excluding data with $l<N_{s}/2$ and $l>N_{s}/2$ for small $N_{s}$ and at larger $N_{s}$ we found fits preferred data near $l\approx N_{s}/2$, resulting in data which resembled a “fan with a handle” (see Unmuth-Yockey et al. (2016)).

For case 1 the best fits included corrections like $f\propto 1/N_{s}^{2}$ and $1/\ln^{2}(N_{s})$. We found almost identical relative errors between corrections $1/N_{s}^{2}$ and $1/\ln^{2}(N_{s})$. For case 2, the OBC data had the least error with corrections $\propto 1/N_{s}$ while the PBC data had the least error with corrections $\propto 1/N_{s}^{2}$. For the oscillating term, the various $p_{n}$ are expected to follow special relations Calabrese et al. (2010) (see below). For some fits, there were no oscillations present and the fits drove $p_{n}$ very large. In these cases we replaced the $\pi N_{s}/2$ in the cosine by $\pi N_{s}$, so as to set it to unity by hand, and assumed a correction $\propto N_{s}^{-p_{n}}$. The fits were done by nonlinear least-squares minimization.

The results are shown in Fig. 4 for case 1.

The values of the slopes, $A_{n}$, for both $S_{1}$ and $S_{2}$ are plotted in Fig. 5 with the slope value predicted from CFT surrounded by a band representing a 1% deviation from the CFT value.

The values for $A_{n}$, and $p_{n}$ using all the data points up to $N_{s}=64$ are shown in Table 2. Notice the good agreement with the predicted relations Calabrese et al. (2010) $p_{1}^{\text{OBC}}=2p_{2}^{\text{OBC}}$, $p_{1}^{\text{PBC}}=2p_{2}^{\text{PBC}}$ and $p_{n}^{\text{PBC}}=2p_{n}^{\text{OBC}}$.

The fit results for case 2 are shown in Fig. 6.

As can be seen, the oscillations are very small, if at all, as compared to case 1. Also in contrast, case 2 did not yield the special relationships between the $p_{n}$ exponents that did occur for case 1. The $A_{n}$ values when fitting to all the data up to $N_{s}=64$ are found in Table 3.

In both case 1 and case 2 the first order Rényi (von Neumann) entropy with PBC possesses no oscillations, which is in agreement with what is known Fagotti and Calabrese (2011). These results suggest that both of these different regions of the phase diagram are conformal and approximately $c=1$. If either of these two regimes could be quantum simulated and experimentally realized, it may be possible to measure the central charge. We will briefly discuss the feasibility of this prospect in the next section.

V Prospect for Quantum Simulations

To better understand the possibility of quantum simulating the $O(2)$ model it is important to find a suitable condensed matter model to relate to. We considered a single species Bose-Hubbard quantum Hamiltonian (Eq. (14)) in a region of the phase diagram where the two models are essentially identical: $\tilde{\mu}\approx U/2\gg J$, i.e. similar to case 1. While the hopping parameter is very small compared to the on-site repulsion, the chain is only half-filled, allowing the superfluid regime to be probed.

We considered the second order Rényi entropy for the Bose-Hubbard model for $J/U=0.005$ and $J/U=0.1$ with OBC. We did runs using DMRG across various system sizes such that $4\leq N_{s}\leq 64$, and sub-system sizes such that $1\leq l\leq N_{s}-1$. To illuminate the legitimacy of the comparison between the two models we have plotted $S_{2}$ and $A_{2}$ for both the $O(2)$ model in the time-continuum limit for case 1, and the Bose-Hubbard model with $J/U=0.005$ in Fig. 7.

As one can see in Fig. 7 the BH model in this limit is almost identical to case 1 of the $O(2)$ model. Changing $J/U$ to 0.1 increases the discrepancy, but the models continue to agree quantitatively well, especially for smaller volumes. The exploratory fits and trials can be found in Unmuth-Yockey et al. (2016). A Bose-Hubbard model with small spatial volumes and $\mu=U/2\gg J/U=0.1$ appears as a potential candidate for quantum simulating the $O(2)$ model and experimentally measuring the central charge. The possibility of measuring the central charge with cold atoms trapped in optical lattices was investigated in Ref. Unmuth-Yockey et al. (2016).

Finite temperature effects

For quantum simulation, while most of the calculations were done at $T=0$, finite temperature effects should be considered. Here we take $k$, the Boltzmann constant, equal to unity. While working on a two-dimensional Euclidean lattice, we must relate the temporal extent to the physical temperature. This is done through the relation

[TABLE]

This relation can be derived with simple quantum statistical mechanics arguments. To take the time continuum limit, we allowed $\beta_{\tau}\rightarrow\infty$ and $a\rightarrow 0$. This allowed us to set the scale with the quantity $U\equiv 1/\beta_{\tau}a$. With this definition we have

[TABLE]

This relates a number of important quantities, for instance the spatial and temporal coupling on the Euclidean lattice to the physical temperature of a quantum Hamiltonian, as well as the number of temporal sites on the lattice. In addition it relates the hopping parameter, $J$, and the on-site repulsion, $U$, to the physical temperature and lattice couplings.

To verify this relation between the classical picture of a two-dimensional lattice and its quantum counter-part in one less dimension we again compared time-continuum TRG results with DMRG results for finite-temperature. For TRG we merely considered temporal lattice sizes which were not as great as before for the zero-temperature analysis and tuned the couplings for the time-continuum limit. The DMRG analysis used a thermal density matrix (as opposed to using the ground state) to compute the Rényi and von Neumann entanglement entropies Verstraete et al. (2004a, b); Feiguin and White (2005). In Fig. 8 one can see the agreement between the DMRG calculations and the TRG ones in the case of the $O(2)$ model. This Figure also demonstrates the effect the thermal entropy has on the entanglement entropy as the temperature increases. The peaks and valleys of the entanglement entropy become smoothed out and while the boundaries remain at approximately $\ln(2)$, the half-filling valley increases.

For systems at half-filling, i.e. the central peak (valley) like in Fig. 3 and 8, the CC scaling can be well fit to a functional form

[TABLE]

that is, adding a term linear in $N_{s}$ takes into account the finite temperature effects. For fits to a general subsystem size at fixed $N_{s}$, we find a term linear in $l$ fits the data well,

[TABLE]

This linear term can be used to subtract off finite-temperature effects Kaufman et al. (2016), however we find additional corrections are necessary to maintain the original $T=0$ fit parameters. Examples of fits done with these functional forms for various temperatures can be found in Unmuth-Yockey et al. (2016).

VI Conclusion

We have argued that finite-size scaling of the entanglement entropy may provide a sensitive tool for identifying conformal behavior in a system. It may complement the techniques currently in use in the lattice gauge theory community for studying models in the context of physics Beyond the Standard Model. Such models are harder to study on the lattice than QCD, because the running of the coupling is slow and the relevant physics may be easily masked by lattice artifacts (e.g. finite lattice spacing, finite volume).

We have calculated the Rényi entropy for the classical $O(2)$ model in the isotropic coupling limit, as well as in the anisotropic coupling limit with a quantum Hamiltonian. From fits to the Rényi entropy we have estimated the central charge. We found that this model can be mapped to a single-species Bose-Hubbard model in a particular region of the phase diagram and their Rényi entropies are quantitatively similar allowing for the possibility of quantum simulating the $O(2)$ model and observing the Calabrese-Cardy scaling during simulation. In addition we have considered finite-temperature effects on the Rényi entropy, and found fitting functions which match the data for $S_{2}$ well, with scaling in $N_{s}$ and in subsystem size. These additional fits involved including a term which is linear in either the subsystem size, $l$, or the system size, $N_{s}$.

It would be interesting to study the scaling of the Rényi entropy of the $O(3)$ nonlinear sigma model with finite chemical potential in 1+1 dimensions. This model is known to have asymptotic scaling in the continuum limit leading to a non-zero mass-gap, as well as meron (instanton) solutions due to the $O(2)$ sub-group. The phase diagram in the time-continuum limit has a similar form Bruckmann et al. (2016) to the $O(2)$ model considered here, and could be investigated in a similar fashion.

In addition it would be interesting to study the effects of a weak gauge-coupling to the $O(2)$ spins (as discussed in Jones et al. (1979); Bazavov et al. (2015)). This would be scalar electrodynamics in a perturbative limit of weak gauge coupling. Monitoring the entanglement entropy as one takes the limit of zero gauge coupling would give information about the symmetries for the phases of scalar electrodynamics, and the passage between the two models.

Acknowledgements.

We thank M. C. Banuls, I. Bloch, I. Cirac, M. Greiner, A. Kaufman, G. Ortiz, J. Osborn, H. Pichler and P. Zoller for useful suggestions or comments, as well as Philipp Preiss for input on experimental aspects of quantum simulating with cold atoms on optical lattices. This research was supported in part by the Department of Energy under Award Numbers DOE grant DE-FG02-05ER41368, DE-SC0010114 and DE-FG02-91ER40664, the NSF under grant DMR-1411345 and by the Army Research Office of the Department of Defense under Award Number W911NF-13-1-0119. L.-P. Yang was supported by Natural Science Foundation for young scientists of China (Grants No.11304404) and Research Fund for the Central Universities(No. CQDXWL-2012-Z005). Parts of the numerical calculations were done at the Argonne Leadership Computational Facilities. Y.M. thanks the Focus Group Physics with Effective Field Theories of the Institute for Advanced Study, Technische Universität München, and the workshop on “Emergent properties of space-time” at CERN for hospitality while part of this work was carried out and the Institute for Nuclear Theory for motivating this work during the workshop “Frontiers in Quantum Simulation with Cold Atoms”.