# Entanglement Hamiltonian of Interacting Systems: Local Temperature
Approximation and Beyond

Mahdieh Pourjafarabadi  The two first authors have contributed equally to this
work. Department of Physics, Sharif University of Technology, Tehran
14588-89694, Iran Hanieh Najafzadeh  The two first authors have contributed
equally to this work. Department of Physics, Sharif University of Technology,
Tehran 14588-89694, Iran Mohammad-Sadegh Vaezi  Pasargad Institute for
Advanced Innovative Solutions (PIAIS) , Tehran 19916-33361, Iran Abolhassan
Vaezi  Corresponding author. Email address<EMAIL_ADDRESS>Department of
Physics, Sharif University of Technology, Tehran 14588-89694, Iran

###### Abstract

We investigate the second quantization form of the entanglement Hamiltonian
(EH) of various subregions for the ground-state of several interacting lattice
fermions and spin models. The relation between the EH and the model
Hamiltonian itself is an unsolved problem for the ground-state of generic
local Hamiltonians. In this letter, we demonstrate that the EH is practically
local and its dominant components are related to the terms present in the
model Hamiltonian up to a smooth spatially varying temperature even for (a)
discrete lattice systems, (b) systems with no emergent conformal or Lorentz
symmetry, and (c) for subsystems with non-flat boundaries, up to relatively
strong interactions. We show that the mentioned local temperature at a given
point decays inversely proportional to its distance from the boundary between
the subsystem and the environment. We find the subdominant terms in the EH as
well and show that they are severely suppressed away from the boundaries of
subsystem and are relatively small near them.

Introduction.— Entanglement is a unique feature of quantum mechanics and
serves as an essential tool in quantum information, quantum gravity,
identification of topological order, quantum phase transition, etc
susskind2005introduction; nielsen2002quantum; susskind2005introduction;
ryu2006holographic; kitaev2006topological; levin2006detecting;
li2008entanglement; sterdyniak2012real; lauchli2010disentangling; liu2015non;
cian2020engineering; vaezi2017numerical; zaletel2013topological;
qi2012general; pollmann2010entanglement; yarloo2018anyonic;
calabrese2009entanglement; pichler2016measurement. The entanglement
Hamiltonian (EH) associated with a subregion $A$ embedded in a manifold
$M=A\cup B$ is defined as $\rho_{A}=e^{-K_{A}}$. Here
$\rho_{A}=\mathrm{Tr}_{B}\rho_{M}$ denotes the reduced density matrix (RDM) of
$A$, where $\rho_{M}$ represents the total density matrix. One important
question that arises from this definition is the relation between $K_{A}$ and
$H_{A}$, the Hamiltonian terms with support only in region $A$. In fact, this
problem dates back to the 19th century. A cornerstone of the classical
statistical mechanics is that a subsystem $A$ at thermal equilibrium with its
environment ($B$) is described by a thermal ensemble with
$K_{A}={H_{A}}/{T_{0}}$ ($k_{B}=\hbar=c=1$) where $T_{0}$ is a uniform and
position independent temperature. Furthermore, the eigenstate thermalization
hypothesis conjectures that the RDM of highly excited quantum states will look
thermal, again with $K_{A}={H_{A}}/{T_{0}}$, where the uniform temperature
$T_{0}$ in this case is dictated by the energy density d2016quantum. In this
letter, we revisit this fundamental problem and using the density-matrix-
renormalization-group (DMRG) approach we obtain the second quantization form
of $K_{A}$ for a number of interacting model Hamiltonians and for a variety of
boundary shapes and conditions. Comparing the components of $K_{A}$ and
$H_{A}$, we demonstrate that the above mentioned statements are not quite
accurate and for the ground-state of local Hamiltonians, $K_{A}$ is indeed
well-approximated by a local and non-uniform temperature rather than a uniform
one.

The theoretical form of $K_{A}$ is known only for a limited class of continuum
models with conformal symmetry (or Lorentz symmetry at zero temperature) and
only for certain geometries of $A$ (e.g., half-space or ball geometry). It is
known that under these conditions: (i) $K_{A}$ is local, (ii) the EH density
is related to the Hamiltonian density via a smooth local temperature, namely
$K_{A}=\int_{x\in A}d^{d}x~{}\mathcal{K}_{A}(x)=\int_{x\in
A}d^{d}x\frac{\mathcal{H}\left(x\right)}{T\left(x\right)}$, and (iii) $T(x)$
approaches $T_{0}$, the equilibrium temperature of the entire system, far away
from $\partial A$ (the boundary of A) and grows as $\frac{v}{2\pi r(x)}$ at
distance $r$ near $\partial A$. Here, $v$ is the group velocity of low energy
excitations bisognano1975duality; bisognano1976duality;
susskind2005introduction; casini2011towards; cardy2016entanglement;
arias2017local; bousso2015entropy; vaezi2018locala; vaezi2018localb. We refer
to these findings as the local temperature approximation (LTA)
vaezi2018locala; vaezi2018localb.

The LTA can be justified using the following intuitive argument. In thermal
systems, the entropy density is proportional to their temperatures. On the
other hand, for ground-states, instead of the thermal entropy, we deal with
the entanglement entropy which is not precisely an extensive property.
Nevertheless, we can still consider and gauge the contribution of individual
degrees of freedom residing inside $A$ to the overall entanglement entropy
between $A$ and $B$, $S_{A}$. Indeed, quantum mutual information can be one
candidate to quantify such local contributions. Due to the decay of quantum
mutual information with distance for the ground-state of local Hamiltonians,
the degrees of freedom that live near $\partial A$, are more entangled with
those residing at $B$ than more distant ones. Accordingly, we can assign an
effective quantum local temperature to different subregions of $A$
proportional to their contributions to $S_{A}$, which as just discussed must
diminish away from $\partial A$.

Figure 1: Various geometries of system and subsystem that we consider in this
letter for computing the EH. The blue (red) sites define the subsystem $A$
($B$). The dotted lines indicate the boundary between $A$ and $B$, $\partial
A$. In (c), $\partial A$ comprises two disjoint surfaces since we have
considered torus geometry. The green and orange lines in (b) illustrate the
rows at which the couplings in Fig. 3 are plotted respectively.

Numerically, the EH of free fermions and free bosons can be evaluated easily
chung2001density; cheong2004many; peschel2009reduced. However, for interacting
models, it becomes highly nontrivial and challenging. Recently, several
studies have analyzed the EH of quasi-one-dimensional conformal invariant or
integrable models toldin2018entanglement; dalmonte2018quantum;
giudici2018entanglement; eisler2019continuum; mendes2019entanglement;
wong2013entanglement; zhu2019reconstructing; furukawa2011entanglement;
lauchli2012entanglement; chen2013quantum; wen2018entanglement;
koeller2018local; lashkari2016modular; di2020entanglement; zhang2020lattice;
zhu2020entanglement; kokail2020entanglement. Nonetheless, we are still lacking
a systematic derivation of the EH for larger and generic interacting systems
and for various boundary geometries. In this letter, we address this problem
and introduce a DMRG-based algorithm that enables us to extract the EH for a
broader spectrum of problems.

${\bf J_{1}-J_{2}}$ Heisenberg model.– Let us first discuss the form of
$K_{A}$ for the $J_{1}-J_{2}$ Heisenberg model ($J_{1}=1$) with the following
Hamiltonian on the square lattice:

$\displaystyle H=J_{1}\sum_{\left<{\bf ij}\right>\in M}{\bf S}_{\bf i}.{\bf
S}_{\bf j}+J_{2}\sum_{\left<\left<{\bf ij}\right>\right>\in M}{\bf S}_{\bf
i}.{\bf S}_{\bf j}.$ (1)

The above Hamiltonian respects a SU(2) symmetry. Hence, $K_{A}$ must respect
SU(2) symmetry as well and thus expanded as follows:

$\displaystyle K_{A}=\sum_{\bf i,j\in A}g_{J,\bf ij}\,{\bf S}_{\bf i}.{\bf
S}_{\bf j}+\cdots~{}.$ (2)

Note that there is no restriction on ${\bf i}=\left(i_{x},i_{y}\right)$ and
${\bf j}=\left(j_{x},j_{y}\right)$ except that both must belong to $A$. In our
study, we have dropped higher order terms since the retained terms already
yield satisfactory results.

The DMRG technique is based on identifying the most relevant basis states of
the Hilbert space white1992density. Then we truncate the Hilbert space and
discard the less relevant states. The number of kept states which controls the
accuracy of DMRG is called the bond dimension, $\chi$, and its default value
equals $2^{10}$ throughout this letter. The procedure of finding the
truncation operators consecutively involves the computation and
diagonalization of the RDM at every step of DMRG and for different subsystem
sizes. Hence, $\rho_{A}$ is a natural byproduct of DMRG method and is
available at every step. Moreover, every operator component of $K_{A}$ (e.g.,
${\bf S_{i}.S_{j}}$) has a matrix representation in DMRG, albeit in the
truncated subspace. The remaining task is to adjust the EH’s couplings,
$g_{J,\bf ij}$, to bring our (simplified) guess for $K_{A}$, which we will
denote as $\widetilde{K_{A}}$, close enough to the matrix representation of
the RDM in the truncated Hilbert space achieved via DMRG:
$K_{A}=-\log\rho_{A}$. To this end, we need to define an appropriate cost
function as a measure of the distance between $\widetilde{K_{A}}$ and $K_{A}$.
In our investigations, we mainly utilized the Hilbert-Schmidt distance between
the Green’s functions, namely $\Delta_{1}:={\rm
Tr}_{A}\left(G_{A}-\widetilde{G}_{A}\right)^{2}$, where $G_{A,\bf ij}={\rm
Tr}_{A}\left({\bf S_{i}.S_{j}}{\rho}_{A}\right)$ is the Green’s function
matrix achieved by DMRG for $A$ and $\widetilde{G_{A}}$ denotes its
counterpart evaluated using the trial RDM,
$\widetilde{\rho_{A}}=\exp(-\widetilde{K_{A}})$ swingle2014reconstructing.
This cost function yields more reliable and reasonably robust results (against
changing $\chi$) in the truncated Hilbert space than other candidates, e.g.,
the quantum relative entropy between $\rho_{A}$ and $\widetilde{\rho_{A}}$. We
find the latter to overfit to numerical noises, e.g., the truncation and
computer’s roundoff errors (see the supplemental material). In the
optimization procedure, we initialized $g_{J,\bf ij}$ based on general
expectations from LTA, e.g., the locality of $g_{J,\bf ij}$ and its linear
dependence on $x_{\bf ij}$ (the minimum distance between $\overline{\bf
ij}=\frac{\bf i+j}{2}$ and $\partial A$). Then, we employed the gradient
descent algorithm and let the cost function to decide the optimum choice for
$g_{J,\bf ij}$ (see the supplemental material).

We first focus on $J_{2}=0$ Heisenberg model which is unfrustrated and is
known to host a Néel order on the square lattice sandvik1997finite;
stoudenmire2012studying. Thus, its ground-state is a symmetry broken phase
with gapless Goldstone modes and does not respect the full conformal symmetry
(e.g., the translational and rotational symmetries are broken). For this
model, the system is always subject to the PBC along $y$ axis.

  * •

As the first example, we study the manifold and subsystem $A$ depicted in Fig.
1-a, where an OBC is imposed along $x$. In this case, $\partial A$ is flat and
its locus is given by $x_{b}=6+1/2$, the line which splits columns 6 and 7.
The optimum couplings, $g_{J,\bf ij}$, which reproduce the DMRG’s Green’s
functions (with less than 0.1 $\%$ error), are plotted in Fig. 2.

Fig. 2a shows the nearest neighbor (NN) couplings along $x$ and $y$ (more
precisely, $\beta_{J,x}\left(i_{x}+1/2\right):=g_{J,\bf i,i+\hat{x}}$ and
$\beta_{J,y}\left(i_{x}\right):=g_{J,\bf i,i+\hat{y}}$) which are independent
of $i_{y}$ due to the $y$-axis translation preserving shape of $A$. Indeed,
$\beta_{J}$’s are the inverse local temperature profiles. As Fig. 2a suggests,
$\beta_{J,x}$ and $\beta_{J,y}$ follow the same profile, albeit if we shift
the argument of $\beta_{x}$ by half of the lattice spacing. This shift is due
to the fact that for $g_{J,x}$, the start and end points are located at
different positions along $x$, while for $g_{J,y}$, the two points have
identical $x$ values. In the supplemental material, we demonstrate the
robustness of $\beta_{J,x}$ and $\beta_{J,y}$ versus $\chi$. In Fig. 2b, we
have plotted $g_{J,xy}\left(i_{x}+1/2\right):=g_{J,\bf i,i+\hat{x}+\hat{y}}$,
$g_{J,yy}\left(i_{x}\right):=g_{J,\bf i,i+2\hat{y}}$, and
$g_{J,xx}\left(i_{x}+1\right):=g_{J,\bf i,i+2\hat{x}}$. Their values are
negligible everywhere and they all die off quickly away from $\partial A$.
These imply the locality of $K_{A}$ for the Heisenberg model when $\partial A$
is flat.

(a)

(b)

Figure 2: EH’s couplings for the Heisenberg model ($J_{2}=0$) for the
subsystem geometry shown in Fig 1-a. (a) The inverse temperature profiles
$\beta_{J,a}(x_{\rm avg})=J_{1}^{-1}g_{J,a}\left(\bf i,j\right)$ ($a=x,y$) for
the NN couplings versus the midpoint argument $x_{\rm
avg}:=\frac{1}{2}\left(i_{x}+j_{x}\right)$. (b) Second and third neighbor
couplings versus $x_{\rm avg}$.

  * •

We now consider the same conditions as above, but this time with a curved
$\partial A$ as shown in Fig. 1-b. In this case, $\beta_{J,x}$ and
$\beta_{J,y}$ will depend on both $i_{x}$ and $i_{y}$. In Fig. 3, we have
plotted $\beta_{J,x}$ and $\beta_{J,y}$ for two different rows marked by
orange and green lines. Interestingly, $\beta_{J,x}$ and $\beta_{J,y}$
profiles display a somewhat smooth curve satisfying our expectations from LTA.
The position dependence of the inverse temperature profile is more complicated
in this problem, since the distance between
$\bf\overline{ij}=\left(i+j\right)/2$ and $\partial A$ depends on both its $x$
and $y$ components. In Figs. 3b and 3d the second and third neighbor couplings
are plotted for the above mentioned rows. We see that for curved boundaries
between $A$ and $B$, $K_{A}$ remains local everywhere, except close to
$\partial A$ where we observe additional terms though relatively small and
subdominant.

(a)

(b)

(c)

(d)

Figure 3: EH’s couplings for the Heisenberg model ($J_{2}=0$) for the
subsystem geometry shown in Fig. 1-b. Since $A$ breaks the translational
symmetry along $y$, $g_{J,\left(\bf i,j\right)}$ depends on both $i_{x}$ and
$j_{y}$. In (a) and (b), the couplings along the green line in Fig. 1-b are
plotted and in (c) and (d), those corresponding to the orange line. The terms
beyond LTA grow substantially near $\partial A$ for non-flat boundary
geometries compared to flat boundaries (cf. Fig. 2).

  * •

Now, we consider the geometry illustrated in Fig. 1-c. Since the PBC is
imposed on $M$ along both $x$ and $y$ directions, we have chosen $N_{y}=4$ to
ensure $\chi=2^{10}$ is sufficient for DMRG’s convergence. As we see in Fig.
1-c, $\partial A$ is described by two surfaces, one of them separates columns
6 and 7 and the other one lies between the first and last columns. As a
result, LTA predicts that $\beta_{J,x}$ and $\beta_{J,y}$ must follow a
parabolic form and vanish near both boundary surfaces. Fig. 4, shows our
numerical results for the nearest as well as further neighbor couplings, both
consistent with LTA.

(a)

(b)

Figure 4: EH’s couplings for the Heisenberg model ($J_{2}=0$) subject to the
PBC along $x$, for the subsystem geometry shown in Fig. 1-c. The NN couplings
follow a parabolic curve and die off near both boundaries.

  * •

Let us now turn to the frustrated Heisenberg model with $J_{2}=0.6$ whose true
ground-state is not well-understood, though it is conjectured to be a spin
liquid phase with no classical spin order and algebraically decaying spin-spin
correlations jiang2012spin. For this model, we consider the geometry depicted
in Fig. 1-d. The ground-state is expected to be more entangled when
$J_{2}/J_{1}\sim O(1)$. Hence, we consider $N_{y}=4$ (again $N_{x}=24$) to
ensure that the ground-state is achieved reliably via $\chi=2^{10}$ in DMRG.
Since the Hamiltonian contains next nearest neighbor (NNN) couplings, we
expect significant values for the NNN in $g_{J,\bf ij}$ as well. In Fig. 5a,
$\beta_{J,x}$, $\beta_{J,y}$, and also
$\beta_{J,xy}\left(i_{x}+1/2\right):=\frac{1}{J_{2}}g_{J,\bf
i,i+\hat{x}+\hat{y}}$ are plotted. Again, we see that all these $\beta_{J}$’s
follow the same curve. Furthermore, Fig. 5b verifies the locality of $K_{A}$
everywhere except at $\partial A$ where $g_{J,yy}$ is about $29\%$ ($28\%$) of
$g_{J,y}$ ($g_{J,xy}$) at that location.

(a)

(b)

Figure 5: EH’s couplings for the $J_{1}-J_{2}$ Heisenberg model ($J_{2}=0.6$)
for the subsystem geometry shown in Fig. 1-d. LTA is valid everywhere, except
at the boundary where the second neighbor coupling along $y$ is non-
negligible.

Hubbard model.- Here, we discuss the second quantization form of $K_{A}$ for
the Hubbard model on the square lattice, whose Hamiltonian is:

$\displaystyle H=$ $\displaystyle-t_{1}\sum_{\left<{\bf ij}\right>\in
M,\sigma}c_{\bf i,\sigma}^{\dagger}c_{\bf j,\sigma}-\mu\sum_{{\bf i}\in
M,\sigma}n_{\bf i,\sigma}$ (4) $\displaystyle+U\sum_{{\bf i}\in M}\left(n_{\bf
i,\uparrow}-\frac{1}{2}\right)\left(n_{\bf
i,\downarrow}-\frac{1}{2}\right),~{}~{}~{}$

where $n_{\bf i,\sigma}=c_{\bf i,\sigma}^{\dagger}c_{\bf i,\sigma}$. In this
letter, we consider $t_{1}=1$, and $U=4$. The above Hamiltonian enjoys a
$U(1)\times SU(2)$ symmetry for generic fillings. Accordingly, $K_{A}$ must be
expanded as follows:

(a)

(b)

(c)

(d)

Figure 6: EH’s couplings for the Hubbard model at half-filling ($U=4t_{1}$)
for the subsystem geometry shown in Fig. 1-d. LTA corrections are small
(compared to the leading terms) everywhere, particularly away from $\partial
A$. The particle-hole symmetry dictates the couplings in (b) to vanish.
However, due to the finite truncation error of DMRG at $\chi=2^{10}$, we
obtain nonzero, though negligible values.

(a)

(b)

(c)

(d)

Figure 7: EH’s couplings for the Hubbard model at $p=1/8$ doping level
($U=4t_{1}$, $\mu\approx-0.92t_{1}$) for the subsystem geometry shown in Fig.
1-d.

$\displaystyle K_{A}=$ $\displaystyle-\sum_{{\bf i\neq j}\in A}g_{t,\bf
ij}\,c_{\bf i,\sigma}^{\dagger}c_{\bf j,\sigma}-\sum_{{\bf i}\in
A,\sigma}g_{\mu,\bf i}\,n_{\bf i,\sigma}$ (7) $\displaystyle+\sum_{{\bf i}\in
A}g_{U,\bf i}\left(n_{\bf i,\uparrow}-\frac{1}{2}\right)\left(n_{\bf
i,\downarrow}-\frac{1}{2}\right)$ $\displaystyle+\sum_{{\bf i\neq j}\in
A}\big{(}g_{V,\bf ij}\,\left(n_{\bf i}-1\right)\left(n_{\bf
j}-1\right)+g_{J,\bf ij}\,{\bf S_{i}.S_{j}}\big{)}+\cdots,~{}~{}~{}~{}$

where $n_{\bf i}=n_{\bf i,\uparrow}+n_{\bf i,\downarrow}$ denotes the total
electron number on site $\bf i$, and $S^{x,y,z}_{\bf
i}=\sum_{ab}\frac{1}{2}c_{{\bf i},a}^{\dagger}\sigma^{x,y,z}_{a,b}c_{{\bf
i},b}$ the three components of the spin operator at $\bf i$. Again, we have
discarded higher order terms as we attain satisfactory results with the above
structure. In the following, we consider both doped and undoped Hubbard
models. In this section, we consider the geometry shown in Fig. 1-d.

  * •

Let us start with the half-filling case. The ground-state on the square
lattice is described by a Néel anti-ferromagnetic spin order and the
charge/Mott gap opens up at moderate values of $U$ varney2009quantum;
vsimkovic2020extended. Fig. 6, summarizes our results for the optimum
couplings of the EH. Here, motivated by LTA, we define the following inverse
temperatures: $\beta_{t,x}\left(i_{x}+1/2\right):=g_{t,\bf i,i+\hat{x}}$,
$\beta_{t,y}\left(i_{x}\right):=g_{t,\bf i,i+\hat{y}}$, and
$\beta_{U}\left(i_{x}\right):=\frac{1}{U}g_{U,\bf i}$. According to Fig. 6,
they all fairly follow an identical curve, albeit by considering the
previously discussed $1/2$ shift in the argument of $\beta_{t,x}$. In Figs.
6b-d, we have shown the terms beyond LTA and again the locality of couplings
is confirmed. Only near $\partial A$, the additional terms are non-negligible.
Although, $g_{V,\bf ij}$ is insignificant everywhere, $g_{J,\bf ij}$ has
decent values near $\partial A$, yet inferior to those of
$g_{t,\bf\left<ij\right>}$ and $g_{U,\bf i}$.

  * •

We now study the Hubbard model at $p=1/8$ doping which is expected to have a
stripe order and some tendency towards superconductivity jiang2020ground;
huang2017numerical; fradkin2015colloquium; zhou2017quantum. The system is not
expected to exhibit the Lorentz invariance or conformal symmetry for these
symmetry breaking phases. At finite doping, we need to define
$\beta_{\mu}\left(i_{x}\right):=\frac{1}{\mu}g_{\mu,\bf i}$ as well (for the
current example: $\mu\approx-0.92t_{1}$). As Fig. 7 implies, various $\beta$’s
follow the expected trend and the locality of couplings is again verified,
albeit with growing corrections close to $\partial A$.

Summary.— Our DMRG-based algorithm allowed us to access the second
quantization form of the EH for several models and subsystem shapes. We showed
that the EH is local and its dominant components are related to those of the
Hamiltonian itself (more specifically the stress-energy tensor) up to a single
smooth local (inverse) temperature profile and confirmed LTA. We studied the
terms beyond LTA and demonstrated they are infinitesimal far away from
$\partial A$ and relatively small near it. In the supplemental material, we
have provided more evidences which further corroborate our main findings. To
our knowledge, the validity of LTA for the ground-state of local Hamiltonians
for generic models that do not satisfy conformal algebra or even those with
conformal symmetry but non-flat $\partial A$ is an unsolved problem despite
active research. Our results suggest that LTA is perhaps a legitimate
assumption and applicable to a broader class of problems.

Our findings pave the way for several applications of LTA. For instance, it
can be shown that LTA can practically solve the long-standing sign problem in
quantum Monte Carlo and enable us to extract the ground-state properties of
some unsolved interacting models. Furthermore, LTA can be employed to enhance
the performance and increase the accuracy of the DMRG technique. It can also
be used to recover the entire spectrum and eigenstates of an unknown
Hamiltonian by having access to its reduced density matrix (or correlation
functions) associated with a rather small subregion of that system
turkeshi2019entanglement.

Acknowledgements.— We gratefully acknowledge helpful discussions with M.
Dalmonte, A. Lucas, S. Nezami, Z. Nussinov, H. Yarloo, A. Shahbazi, E. Huang,
M. Kargarian, A. Rezakhani and S. Alipour. AV acknowledges the Gordon and
Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4302 and Stanford
Center for Topological Quantum Physics for partial financial support and
hospitality during the completion of this work. MSV acknowledges the financial
support from Pasargad Institute for Advanced Innovative Solutions (PIAIS)
under supporting Grant scheme (Project No. SG1-RCM2001-01).

## References

##  .1 SUPPLEMENTAL MATERIAL

In this supplemental material, we will delve into the details of our algorithm
and discuss the advantages and disadvantages of a number of cost function
candidates along with their pairwise comparison, and present more results on
the entanglement Hamiltonian (EH).

##  .2 A. Entanglement Hamiltonian in the truncated Hilbert space

In general, the entanglement Hamiltonian (EH) associated with subsystem $A$
which is defined as $K_{A}:=-\log\rho_{A}$, can be expanded in terms of a
complete basis of operators (not necessarily local) as follows: (the tensor
product of Pauli matrices, $\sigma_{a}$, $a=x,y,z$ with
$\sigma_{0}=\mathbb{1}$, can generate a basis for all possible operators):

$\displaystyle K_{A}=\sum_{\alpha}g_{\alpha}\hat{O}_{\alpha}$ (8)

If we are given the reduced density matrix, $\rho_{A}$, we can compute its
logarithm (which requires a lot of considerations and special care when
performed numerically) and achieve $K_{A}$ (up to computer’s round-off error).
Having $K_{A}$ available, we can easily find the expansion coefficients,
$g_{\alpha}$, via the following relation:

$\displaystyle\vec{g}=M^{-1}\vec{J}$ (9)

where

$\displaystyle J_{\beta}={\rm
Tr}_{A}\left(K_{A}\hat{O}_{\beta}^{\dagger}\right),$ (10) $\displaystyle
M_{\alpha\beta}={\rm
Tr}_{A}\left(\hat{O}_{\alpha}\hat{O}_{\beta}^{\dagger}\right)$ (11)

In the exact diagonalization (ED) method
$M_{\alpha,\beta}\propto\delta_{\alpha,\beta}$. Therefore,
$g_{\alpha}\propto{\rm Tr}_{A}\left(K_{A}\hat{O}_{\alpha}^{\dagger}\right)$.
Consequently, we do not need to consider other operators if we are interested
in reading the coefficient for a specific $\alpha$.

On the other hand, in the density-matrix-renormalization-group (DMRG)
algorithm, instead of $\rho_{A}$, and $\hat{O}_{\alpha}$, we have to deal with
$\overline{\rho_{A}}$, and $\overline{\hat{O}_{A}}$ which are their
counterparts in the truncated Hilbert space and are defined as:

$\displaystyle\overline{\rho_{A}}=T_{A}^{\dagger}\rho_{A}T_{A},$ (12)

and similarly for other operators, where $T_{A}$ denotes the truncation
(a.k.a. projection) operator. As before, we define
$\overline{K_{A}}=-\log\overline{\rho_{A}}$. Because of the numerous
truncations involved in DMRG, the situation is now more complicated for a few
reasons: (i) The matrix $M$ is not diagonal, neither sparse. Due to
consecutive truncations inherent to the DMRG method, most of operators have
non-vanishing overlaps. Therefore, we must consider all possible operators,
including highly non-local ones such as string or brane operators. (ii) $M$
can be singular and have zero eigenvalues. As a result, it might not be
invertible. (iii) The above method applied to DMRG is very sensitive to
various sources of numerical noises and errors, such as the truncation, as
well as the round-off error.

Besides the possibility of singular $M$, another main difficulty of applying
the above algorithm to DMRG is the annoying part which requires taking all
possible operators into consideration. Below, we easily demonstrate that if
$K_{A}$ contains a few relevant and dominant terms, then $\overline{K_{A}}$
contains exactly the same couplings and structure. To this end, recall that
$T_{A}$ is achieved upon concatenating the dominant eigenvectors of
$\rho_{A}$. Therefore,

$\displaystyle\overline{\rho_{A}}=T_{A}^{\dagger}\rho_{A}T_{A}=T_{A}^{\dagger}e^{-K_{A}}T_{A}=e^{-T_{A}^{\dagger}K_{A}T_{A}}.$
(13)

Therefore,

$\displaystyle\overline{K_{A}}:=-\log\overline{\rho_{A}}=T_{A}^{\dagger}K_{A}T_{A}.$
(14)

Accordingly,

$\displaystyle\overline{K_{A}}=\sum_{\alpha}g_{\alpha}T_{A}^{\dagger}\hat{O}_{\alpha}T_{A}=\sum_{\alpha}g_{\alpha}\overline{\hat{O}_{\alpha}},$
(15)

hence, assuming
$\overline{K_{A}}=\sum_{\alpha}\overline{g_{\alpha}}\overline{\hat{O}_{\alpha}}$
:

$\displaystyle\overline{g_{\alpha}}=g_{\alpha}.$ (16)

##  .3 B. Algorithm and cost function

In the above mentioned method, for ED, we can ignore insignificant couplings
since $M$ is diagonal. Nevertheless, when we apply this method to DMRG, we
have to retain all terms, no matter how infinitesimal they are due to the
complex form of $M$. Therefore, we must come up with a better algorithm to
find $g_{\alpha}$ without having to consider all irrelevant terms. For that
purpose, we must consider a valid cost function. From our physical intuitions
and expectations, we can think of the following three choices (as of now, we
drop the overline sign and keep in mind that all operators are defined in the
truncated Hilbert space):

  * •

Hilbert-Schmidt distance between the Green’s functions (GFs): $\Delta_{1}={\rm
Tr}\left(G_{A}-\widetilde{G_{A}}\right)^{2}$, where $G_{A}(\alpha,\beta)={\rm
Tr}_{A}\left(O_{\alpha}^{\dagger}O_{\beta}\rho_{A}\right)$, and
$\widetilde{G_{A}}(\alpha,\beta)={\rm
Tr}_{A}\left(O_{\alpha}^{\dagger}O_{\beta}\widetilde{\rho_{A}}\right)$. Here
$\rho_{A}$ denotes the reduced density matrix (RDM) achieved via DMRG for the
desired subsystem and $\widetilde{\rho_{A}}$ denotes the one by combining the
basis operators ($O_{\alpha}$) with $g_{\alpha}$ coefficients that are yet to
determine. The basis of this method is that the RDM contains all the
information about the equal time correlation functions within the subsystem.
Thus, if we find a RDM which recovers all the correlation functions correctly,
it must be identical to the actual one.

We would like to emphasize that in DMRG, due to finite truncation error, the
RDM yields more reliable results for the expectation value of simple operators
(e.g., two-point correlation functions for short and intermediate distances)
and becomes less reliable for more complex operators or at long distances.
Therefore, to avoid overfitting to numerical errors, instead of considering
all basis operators in the evaluation of $\Delta_{1}$, we only consider the
most physically relevant operators, i.e., simple operators motivated by
symmetry considerations, etc. For example, for the Hubbard model, we consider
the following components first:

$\displaystyle G_{A,\rm
t}\left(i,j\right)=\sum_{\sigma}\left<c_{i,\sigma}^{\dagger}c_{j,\sigma}\right>,$
(17) $\displaystyle G_{A,\rm\mu}\left(i\right)=\left<n_{i}\right>,$ (18)
$\displaystyle G_{A,\rm
U}\left(i\right)=\left<n_{i,\uparrow}n_{i,\downarrow}\right>,$ (19)
$\displaystyle G_{A,\rm J}\left(i,j\right)=\left<{\bf S}_{i}.{\bf
S}_{j}\right>,$ (20) $\displaystyle G_{A,\rm
V}\left(i,j\right)=\left<n_{i}n_{j}\right>,$

and similarly for $\tilde{G}$. We then evaluate $\eta_{a}={\rm
Tr}\left(G_{A,a}-\widetilde{G_{A,a}}\right)^{2}$ and by combining them:

$\displaystyle\Delta_{1}=w_{t}\eta_{t}+w_{\mu}\eta_{\mu}+w_{U}\eta_{U}+w_{J}\eta_{J}+w_{V}\eta_{V}.$
(21)

The exact values of $w_{t},w_{\mu},\cdots$ are not crucial as long as they all
have the same order of magnitude. Nonetheless, in most computations, we choose
$w_{t}=w_{U}=w_{\mu}=w_{J}=w_{V}=1$ for the weights.

  * •

Quantum relative entropy (QRE) of the two reduced density matrices:
$\Delta_{2}={\rm
Tr}\left(\rho_{A}\log\rho_{A}-\rho_{A}\log\widetilde{\rho}_{A}\right)$. In
this method (which is closely related to the next cost function), we try to
tune the couplings such that $\widetilde{\rho_{A}}$’s matrix form becomes very
close to $\rho_{A}$’s. This cost function converges significantly fast, in
both ED and DMRG method. In ED where we do not have to deal with truncation
errors, QRE is the superior cost function and achieves correct results.
However, for DMRG, (like $\Delta_{3}$ below), it also suffers from overfitting
to numerical errors, i.e., those parts of $\rho_{A}$ which will change upon
increasing the bond dimension of DMRG, $\chi$ (i.e., the number of retained
basis states of the Hilbert space). It is these matrix elements which are
responsible for the issues related to the expectation value or $n-$point
correlation functions of complex operators explained above. When $\chi$ is
large enough (e.g., when the truncation error becomes less than $10^{-10}$),
it yields results consistent with $\Delta_{1}$’s.

Figure 8: We study the Hubbard model on this ladder for $U=4$,
$t_{\perp}=0.5$, and $t_{\parallel}=1$ at half filling ($\mu=0$). Subsystem
$A$, whose EH is desired, is denoted by blue sites.

(a)

(b)

(c)

(d)

Figure 9: $K_{A}$’s couplings, for the geometry illustrated in Fig. 8,
obtained via applying different cost functions and for various bond
dimensions. Couplings are translational invariant due to the subsystem
geometry. (a-c) $g_{t,dx}$, $g_{J,dx}$, and $g_{V,dx}$ achieved via GF
distance ($\Delta_{1}$), QRE distance ($\Delta_{2}$), and RDM distance
($\Delta_{3}$) for $\chi=2^{11}$. Also, the corresponding renormalized onsite
couplings, $g_{U,0}$ are $16.4,14.4,14.3$ for $\Delta_{1}$, $\Delta_{2}$ and
$\Delta_{3}$, respectively. See the text for their definitions. (d) The
variation of $U_{\rm eff}:=\frac{g_{t,1}}{g_{U,0}}$ versus $\log_{2}\chi$. As
this plot clearly indicates, $\Delta_{1}$ is the most reliable cost function
for DMRG and exhibits least variations.

  * •

Hilbert-Schmidt distance between the two reduced density matrices:
$\Delta_{3}={\rm Tr}\left(\rho_{A}-\widetilde{\rho_{A}}\right)^{2}$. Similar
to $\Delta_{2}$, in this method we try to tune the couplings such that
$\widetilde{\rho_{A}}$’s matrix form becomes very close to $\rho_{A}$’s. This
cost function is slowly converging even for the ED where no truncation is
involved. Moreover, for DMRG, similar to $\Delta_{2}$, it suffers from
overfitting to numerical errors and its results are sensitive to the bond
dimension, especially for small values of $\chi$.

In section D of this supplemental material, we compare the results achieved
via all three cost functions for the two-leg ladder Heisenberg and Hubbard
models for several bond dimensions. Our results suggest that for large bond
dimensions, all three methods yield consistent outcomes. However, for
relatively small bond dimensions, it is $\Delta_{1}$ which performs better and
results in couplings which are more consistent with the results of larger bond
dimensions.

Figure 10: We study the Hubbard model on this ladder for $U=4$, $t_{\perp}=2$,
and $t_{\parallel}=1$ at half filling ($\mu=0$). Subsystem $A$, whose EH is
desired, is denoted by blue sites.

(a)

(b)

(c)

(d)

Figure 11: $K_{A}$’s couplings, for the geometry illustrated in Fig. 10,
obtained via applying GF distance ($\Delta_{1}$ cost function) and for
$\chi=2^{11}$, and $\chi=2^{9}$. (a) Various $\beta$ profiles for $\Delta_{1}$
cost function for $\chi=2^{11}$ (see the main text for their definitions). (b)
Most significant corrections to LTA corrections, $g_{J}$ for $\Delta_{1}$ cost
function for $\chi=2^{11}$. The second and third neighbors’ corrections to
$g_{t}$ are negligible due to the particle-hole symmetry. Moreover, we found
$g_{V}$ to be irrelevant as well, and that is why they are absent in this and
the following two figures. (c-d) Same as (a-b) but for $\chi=2^{9}$.

(a)

(b)

(c)

(d)

Figure 12: Same as Fig. 11 but for $\Delta_{2}$ cost function (QRE). Likewise,
(a) and (b) are achieved by considering $\chi=2^{11}$, while (c) and (d) by
$\chi=2^{9}$

(a)

(b)

(c)

(d)

Figure 13: Same as Fig. 11 but for $\Delta_{3}$ cost function (RDM distance).
Likewise, (a) and (b) are achieved by considering $\chi=2^{11}$, while (c) and
(d) by $\chi=2^{9}$.

##  .4 C. Local temperature ansatz and the initial guess for couplings

Now, let us assume we study the following Hamiltonian:

$\displaystyle H=\sum_{\alpha}J_{\alpha}\hat{O}_{\alpha},$ (22)

where, due to the locality of the Hamiltonian, only certain $J_{\alpha}$’s are
nonzero. We are interested in finding the second quantization form of the EH
expanded as follows:

$\displaystyle K_{A}=\sum_{\alpha\in A}g_{\alpha}\hat{O}_{\alpha}.$ (23)

Here, due to the renormalization procedure involved in tracing the degrees of
freedom outside $A$, $g_{\alpha}$’s can be viewed as our running coupling
constants which $J_{\alpha}$ has flown to. Thus, in principle, any
$g_{\alpha}$ consistent with symmetry considerations emerge. In practice, only
a small set of them will be relevant and non-negligible.

In our algorithm, we are trying to find $g_{\alpha}$ numerically, assuming (a
subset of relevant) correlation functions are known. In our optimization
algorithm, we initialized the coupling constants of the EH, $g_{\alpha}$,
using LTA’s ideal form. In LTA, the EH is local and its coupling constants,
$g_{\alpha}$’s, are nonzero only when the corresponding couplings of the
Hamiltonian (UV theory), $J_{\alpha}$’s, are nonzero. Another task in LTA is
to assign a position to each operator. For simple two-point operators (such as
$\bf S_{i}.S_{j}$ in the Heisenberg model, or $c_{\bf
i,\sigma}^{\dagger}c_{\bf j,\sigma}$ in the Hubbard model), the position is
defined as the average position of its components, namely $\overline{\bf
ij}=\frac{\bf i+j}{2}$. Next, we must compute the minimum distance (geodesics)
between $\overline{\bf ij}$ and the boundary separating $A$ and its
environment, $B$. Let us call this minimum distance, $x_{\bf ij}$. Finally, at
zero temperature (for ground-states) and for the open boundary condition
(OBC), LTA attributes the following form to $g_{\alpha}\left(x_{\bf
ij}\right)$ cardy2016entanglement:

Figure 14: We study the Heisenberg model on this ladder for $J_{\perp}=0.5$,
and $J_{\parallel}=1$. Subsystem $A$, whose EH is desired, is denoted by blue
sites.

(a)

(b)

Figure 15: $K_{A}$’s couplings, for the geometry illustrated in Fig. 14,
obtained via applying $\Delta_{1}$ and $\Delta_{2}$ cost functions and for
$\chi=2^{5},2^{7},2^{9},2^{11}$. The couplings are translationally invariant
due to the geometry of $A$. (a) $g_{J,dx}$ obtained by minimizing $\Delta_{1}$
cost function. (b) $g_{J,dx}$ obtained by minimizing $\Delta_{2}$ cost
function. As we see, both cost functions result in the same couplings for
$\chi=2^{11}$. Also, $\Delta_{1}$ results exhibits much less fluctuations than
$\Delta_{2}$’s and thus are more reliable for smaller bond dimensions
($\chi$’s).

$\displaystyle g_{\alpha}\left(x_{\bf
ij}\right)=J_{\alpha}\frac{4\ell}{v}\sin\left(\frac{\pi}{2\ell}x_{\bf
ij}\right).$ (24)

where $\ell$ is the maximum value of $x_{\bf ij}$ (i.e., the linear dimension
of $A$ normal to $\partial A$), and $v$ is the group velocity of low energy
excitations (quasi-particles) and is model-dependent. In our algorithm,
besides $v$, we also treated $\ell$ as a variational parameter. We first
optimized and tuned $v$, and $\ell$. Then, we took the optimized form of local
$g_{\alpha}$ (associated with $v^{*}$, and $\ell^{*}$), and using the gradient
descent algorithm we optimized our cost function. We allowed all relevant
couplings, including distant neighbors and non-local terms (which were absent
in the system’s Hamiltonian) as well as the initialized local terms to vary
and deviate from their initial point. Therefore, we have not imposed locality
in our procedure, although it finally emerged naturally as the optimum
solution (except at the boundary of $A$ with $B$, where farther neighbors
became more pronounced).

Similarly, for the periodic boundary condition (PBC) at $T=0$, LTA assigns the
following value to $g_{\alpha}\left(x_{\bf ij}\right)$ cardy2016entanglement:

$\displaystyle g_{\alpha}\left(x_{\bf
ij}\right)=J_{\alpha}\frac{2L}{v}\frac{\sin\left(\frac{\pi}{L}x_{\bf
ij}\right)\sin\left(\frac{\pi}{L}\left(\ell-x_{\bf
ij}\right)\right)}{\sin\left(\frac{\pi}{L}\ell\right)},$ (25)

where $L$ is size of the entire system ($M$) in the direction normal to
$\partial A$.

##  .5 D. A detailed comparison between the performance of $\Delta_{1}$,
$\Delta_{2}$ and $\Delta_{3}$ cost functions

Here, we compare the EH’s coefficients obtained by utilizing all three cost
functions for the ladder geometry and for the Hubbard and Heisenberg models.

Figure 16: We study the Heisenberg model on this ladder for $J_{\perp}=2$, and
$J_{\parallel}=1$. Subsystem $A$, whose EH is desired, is denoted by blue
sites.

(a)

(b)

(c)

(d)

Figure 17: $K_{A}$’s couplings, for the geometry illustrated in Fig. 14,
obtained via applying $\Delta_{1}$ cost functions and for
$\chi=2^{7},2^{11}$.(a) and (c) show the inverse local temperatures
($\beta_{y}\left(i_{x}\right):=\frac{1}{J_{\perp}}g_{J,i_{x},i_{x}}$,
$\beta_{x}\left(i_{x}+1/2\right):=\frac{1}{J_{\parallel}}g_{J,i_{x},i_{x}+1}$)
for $\chi=2^{11}$, and $2^{7}$, respectively. (b) and (d) present the second
and third neighbor couplings of the EH.

(a)

(b)

(c)

(d)

Figure 18: Same as Fig. 17 but for $\Delta_{2}$ cost function. (a) and (b) are
obtained by keeping $\chi=2^{11}$ basis states of the Hilbert space in DMRG,
while (c) and (d) by $\chi=2^{7}$.

We first consider the undoped Hubbard model on a ladder geometry (in which
$U=4$, $t_{\perp}=0.5$, $t_{\parallel}=1$) depicted in Fig. 8. This geometry
results in a highly entangled subsystem, indeed a volume law entanglement
entropy. We apply all three cost functions to this system for the following
five different bond dimensions: $\chi=2^{5},2^{7},2^{9},2^{11}$. The EH for
this case is translationally invariant, namely
$g_{\tau,i_{x},j_{x}}=g_{\tau,dx_{ij}}$ ($\tau=t,U,J,V$), where
$dx_{ij}:=j_{x}-i_{x}$. In this section, the translational symmetry is imposed
on the couplings explicitly. We first compare the EH’s couplings achieved via
applying the GF distance ($\Delta_{1}$), QRE ($\Delta_{2}$), and the RDM
distance ($\Delta_{3}$) for $\chi=2^{11}$ (see Fig. 9). With this bond
dimension, we can nearly probe the ground-state properties. The coefficients
of local terms in the EH are almost consistent in these three methods. On the
other hand, we know that ideally the ground-state must exhibit particle-hole
symmetry. Although $\chi=2^{11}$ is still insufficient for true convergence in
DMRG for such a highly entangled state ($\chi=2^{12}$ seems to be enough),
$\Delta_{1}$’s results reflect the particle-hole symmetry (e.g., the
(renormalized) second neighbor hoppings are infinitesimal), while for those of
$\Delta_{2}$ and $\Delta_{3}$ the particle-hole symmetry is visibly violated
due to overfitting issues mentioned previously. Furthermore, a previous
quantum Monte Carlo based study of a similar situation toldin2018entanglement
indicated the irrelevance of $g_{V}$ couplings which is consistent with
$\Delta_{1}$’s estimations. Additionally, perturbative studies of the EH
indicate an oscillating spin-spin couplings furukawa2011entanglement;
lauchli2012entanglement; chen2013quantum (though subdominant to the $\it
renormalized$ onsite interaction) which agrees well with our results via
minimizing GF distance ($\Delta_{1}$), while those of $\Delta_{2}$ and
$\Delta_{3}$ exhibit deviations in addition to their overestimation for the
spin-spin couplings. In Fig. 9d, we plot the normalized onsite interaction
strength $U_{\rm
eff}^{(\chi)}:=\frac{g^{(\chi)}_{U}}{g^{(\chi)}_{t}\left(1\right)}$ for all
four bond dimensions considered in our investigations. Again, as we see in
Fig. 9d, the results of the GF distance ($\Delta_{1}$) are more robust and
less sensitive to $\chi$, despite several orders of magnitude change in
$\chi$, while those of the QRE and RDM distance display stronger fluctuations.

Now, we turn to the geometry shown in Fig. 10 (where $U=4$, $t_{\perp}=2$,
$t_{\parallel}=1$) and present our results for all three cost functions in
Figs. 11, 12, and 13. Here, we have defined the following inverse local
temperatures: $\beta_{t,x}\left(i_{x}+1/2\right):=g_{t,\bf i,i+\hat{x}}$,
$\beta_{t,y}\left(i_{x}\right):=g_{t,\bf i,i+\hat{y}}$, and
$\beta_{U}\left(i_{x}\right):=\frac{1}{U}g_{U,\bf i}$. For this problem, due
to the PBC imposed along $x$ direction, we found out that even with
$\chi=2^{11}$, there is still some room for DMRG to converge to the true
ground-state. As a result, we still see some minor discrepancy among the
results of the three methods for $\chi=2^{11}$ for subdominant and correction
terms beyond LTA (though they yield highly correlated results). Nonetheless,
the local terms (i.e., dominant couplings) are reasonably consistent. We have
also plotted the results of $\chi=2^{9}$ for all three methods and again,
$\Delta_{1}$’s results proved to be more robust and $\Delta_{2}$ and
$\Delta_{3}$’s less stable. Thus, in the presence of truncation errors, we can
trust the results of the GF distance more than those of the other two
candidates for the cost function.

For the sake of completeness, we have also explored the robustness and the
accuracy of the above three cost function candidates for the Heisenberg model,
again on a ladder geometry. To this end, we first studied the geometry shown
in Fig. 14 (where $J_{\perp}=0.5$, $J_{\parallel}=1$), and presented its
results in Fig. 15. Similar to the Hubbard model case, leads to a highly
entangled ground-state. Likewise, we expect translationally invariant
couplings, namely $g_{J,i_{x},j_{x}}=g_{J,dx_{ij}}$. We have presented
$g_{J,dx}$ for $\chi=2^{5},2^{7},2^{9},2^{11}$ for the GF and QRE cost
functions (the RDM cost function yields results similar to that of the QRE).
In this case as well, the GF distance turns out to be the most stable one.

Finally, we studied the Heisenberg model on the geometry illustrated in Fig.
16 (in which $J_{\perp}=0.5$, $J_{\parallel}=1$). Their results are presented
in Figs. 17 and 18 for the GF distance and QRE, respectively. For this
problem, we indeed achieved the true ground-state using $\chi=2^{11}$.
Therefore, all cost functions must achieve the same couplings. On the other
hand, for smaller bond dimensions, e.g., $\chi=2^{7}$, $\Delta_{1}$ achieves
more accurate results (relative to $\chi=2^{11}$) than the remaining cost
functions.

## References


11institutetext:  The Private University College of Education of the Diocese
of Linz
Salesianumweg 3, A-4020 Linz, Austria
11email<EMAIL_ADDRESS>
# Another (wrong) construction of $\pi$

Zoltán Kovács

###### Abstract

A simple way is shown to construct the length $\pi$ from the unit length with
4 digits accuracy.

###### Keywords:

$\pi$, approximation, elementary geometry, automated theorem proving

##  1 Introduction

It is well-known that accurate construction of the ratio between the perimeter
and the diameter of a circle is theoretically impossible [1, 2]. Before
Lindemann’s result (and even thereafter), however, several attempts were
recorded to geometrically construct the number $\pi$ with various means, in
most cases by using a compass and a straightedge. One of the most successful
attempts is Kochański’s work [3] that produces an approximation of $\pi$ by 4
digits, $\sqrt{40/3-2\sqrt{3}}\approx 3.14153333\ldots$

Kochański’s construction is relatively easy, it requires just a little amount
of steps, and can be discussed in the school curriculum as well. In this note
the same approximation is given, by using—at least geometrically—an even
simpler approach.

##  2 The construction

The proposed way to construct $\pi$ is shown in Fig. 1 (see also [4]).

Figure 1: A new way to construct $\pi$ approximately

A proof that $|RS|=\sqrt{40/3-2\sqrt{3}}$ is as follows. We assume that
$A_{0}=(0,0)$ and $A_{1}=(1,0)$ (see Fig. 2). By using that $360^{\rm
o}/12=30^{\rm o}$, $\cos(30^{\rm o})=\sqrt{3}/2$ and $\sin(30^{\rm o})=1/2$,
the exact coordinates of the appearing vertices in the construction are
$A_{3}=(3/2+\sqrt{3}/2,1/2+\sqrt{3}/2)$, $A_{6}=(1,2+\sqrt{3})$,
$A_{7}=(0,2+\sqrt{3})$, $A_{8}=(-\sqrt{3}/2,3/2+\sqrt{3})$,
$A_{9}=(-1/2-\sqrt{3}/2,3/2+\sqrt{3}/2)$, $A_{11}=(-\sqrt{3}/2,1/2)$.

Figure 2: Explanation for the proof

To find the coordinates of $R$ we can compute the equation of the line
$A_{11}A_{6}$. By substituting the coordinates of $A_{11}$ and $A_{6}$ it can
be verified that the equation is

$(3/2+\sqrt{3})x-(1+\sqrt{3}/2)y=-2-\sqrt{3},$

and solving this for $y=0$ we obtain the exact coordinates
$R=(-2/\sqrt{3},0)$. (Alternatively, it can be shown that
$|A_{1}R|=1+2/\sqrt{3}$, because it is the shorter cathetus of the triangle
$RA_{1}A_{6}$ which is a half of an equilateral triangle—this holds because
the angle $A_{1}A_{6}A_{11}$ is an inscribed angle of the circumcircle of the
regular 12-gon, and it must be $60^{\rm o}/2=30^{\rm o}$.)

Now, $A_{3}A_{8}\perp A_{6}A_{11}$, so we are searching for the equation of
line $A_{3}A_{8}$ in form

$(1+\sqrt{3}/2)x+(3/2+\sqrt{3})y=c.$

After substituting the coordinates of $A_{3}$ in this, we obtain that
$c=(9+5\sqrt{3})/2$. Because of symmetry, it is clear that the equation of
line $A_{7}A_{9}$ is of form

$y=x+d.$

By using the coordinates of $A_{7}$, we immediately obtain that
$d=2+\sqrt{3}$.

To find the coordinates of $S$ we solve the equation system

$\displaystyle(1+\sqrt{3}/2)x+(3/2+\sqrt{3})y=$
$\displaystyle(9+5\sqrt{3})/2,$ (1) $\displaystyle y=$ $\displaystyle
x+2+\sqrt{3}$ (2)

now, which produces the coordinates $x=(\sqrt{3}-3)/2$, $y=(3\sqrt{3}+1)/2$.

Finally we compute the length of $RS$:

$|RS|=\sqrt{\left((\sqrt{3}-3)/2+(2/\sqrt{3})\right)^{2}+\left((3\sqrt{3}+1)/2\right)^{2}}=\sqrt{40/3-2\sqrt{3}}.$

##  3 Remarks

This result has been found by the software tool [5] in an automated way by
considering all possible configurations of distances between intersections of
diagonals in regular polygons. It seems very likely that the construction
described above is the simplest and most accurate one among the considered
cases. Another construction which is based on a regular star-12-gon (see [6])
can produce the same approximation.

Regular polygons with less sides have already been completely studied for the
constructible cases for $n<12$ with less accurate results (see [4] for
details). Checking cases $n=15,16,17,20$ (all are constructible) is an on-
going project.

While the software tool [5] gave a machine assisted proof by using Gröbner
bases and elimination (see [7, 8] for more details), the proof above was
compiled by the author manually.

## Acknowledgments

The author was partially supported by a grant MTM2017-88796-P from the Spanish
MINECO (Ministerio de Economia y Competitividad) and the ERDF (European
Regional Development Fund).

## References

  * [1] Wantzel, P.:  Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas.  Journal de Mathématiques Pures et Appliquées 1 (1837) 366–372
  * [2] Lindemann, F.:  Über die Zahl $\pi$.  Mathematische Annalen 20 (1882) 213–225
  * [3] Kochański, A.A.:  Observationes cyclometricae adfacilitandam praxin accomodatae.  Acta Eruditorum 4 (1685) 394–398
  * [4] Kovács, Z.:  Constructing $\pi$ from a regular 12-gon.  GeoGebra (2018) https://www.geogebra.org/m/qFvtny2G.
  * [5] Kovács, Z.:  RegularNGons.  A GitHub project (2018) https://github.com/kovzol/RegularNGons.
  * [6] Kovács, Z.:  Constructing $\pi$ from a regular star-12-gon.  GeoGebra (2018) https://www.geogebra.org/m/jnZSeBnq.
  * [7] Cox, D., Little, J., O’Shea, D.:  Ideals Varieties, and Algorithms.  Springer New York (2007)
  * [8] Recio, T., Vélez, M.P.:  Automatic discovery of theorems in elementary geometry.  Journal of Automated Reasoning 23 (1999) 63–82