[a]Wolfgang Unger

# Hamiltonian Lattice QCD from Strong Coupling Expansion

Pratitee Pattanaik

###### Abstract

We present generalizations of Hamiltonian Lattice QCD as derived from the
continuous time limit of strong coupling lattice QCD: we discuss the flavor
dependence and the effect of gauge corrections. This formalism can be applied
at finite temperature and baryon density as well as isospin density and allows
both for analytic and numeric investigations that are sign problem-free.

##  1 Introduction

The Hamiltonian formulation of lattice QCD has been discussed in detail in [1]
in the strong coupling limit for ${N_{f}}=1$. In contrast to Hamiltonian
formulations in the early days of lattice QCD [2] this formulation is based on
the dual representation that is obtained when integrating out the gauge links
first, and then the Grassmann-valued fermions [3]. The resulting dual degrees
of freedom are color singlets, such as mesons and baryons. In this
representation, the finite density sign problem is much milder, as it is re-
expressed in terms of the geometry of baryonic world-lines. The dual
representation of lattice QCD with staggered fermions has been studied in the
strong coupling limit both via mean-field theory [4] and Monte Carlo [5, 6]
and has been extended in both approaches to include gauge corrections [7, 8].

The Hamiltonian formulation is based on the continuous time limit of the dual
representation: at fixed bare temperature $aT=\frac{a}{a_{t}N_{t}}$, the
limits $a_{t}\rightarrow 0$ and $N_{t}\rightarrow\infty$ are taken
simultaneously [9]. This limit requires to determine the non-perturbative
relation between the bare anisotropy $\gamma$ and the physical anisotropy
$\frac{a}{a_{t}}\equiv\xi(\gamma)$, based on pion current fluctuations. A
conservation law for the pion current can be identified: if a quantum number
$\mathfrak{m}(x)$ is raised/lowered by a spatial dimer, then at the site
connected by the spatial dimer the quantum number is lowered/raised. This is a
direct consequence of the even/odd decomposition for staggered fermions. With
interactions derived from a high temperature series, the resulting partition
sum can be expressed in terms of a Hamiltonian that is composed of mesonic
annihilation and creation operators $\hat{J}^{\pm}$:

$\displaystyle Z_{\rm CT}(\mathcal{T},\mu_{\mathcal{B}})={\rm
Tr}_{\mathfrak{h}}\left[e^{(\hat{\mathcal{H}}+\hat{\mathcal{N}}\mu_{\mathcal{B}})/\mathcal{T}}\right],\qquad\hat{\mathcal{H}}=\frac{1}{2}\sum_{\langle\vec{x},\vec{y}\rangle}\left(\hat{J}^{+}_{\vec{x}}\hat{J}^{-}_{\vec{y}}+\hat{J}^{-}_{\vec{x}}\hat{J}^{+}_{\vec{y}}\right),\qquad\hat{\mathcal{N}}=\sum_{\vec{x}}\hat{\omega}_{x},$
$\displaystyle\hat{J}^{+}={\footnotesize\left(\begin{array}[]{cccc|cc}0&0&0&0&&\\\
\hat{v}_{\hskip
0.56003pt\text{\lx@text@tweaked{yoffset=-0.5ex}{\lx@mvs@LSteel}}\hskip
0.56003pt}&0&0&0&&\\\ 0&\hat{v}_{\hskip
0.56003pt\text{\lx@text@tweaked{yoffset=0.5ex}{\lx@mvs@TSteel}}\hskip
0.56003pt}&0&0&&\\\ 0&0&\hat{v}_{\hskip
0.56003pt\text{\lx@text@tweaked{yoffset=-0.5ex}{\lx@mvs@LSteel}}\hskip
0.56003pt}&0&&\\\ \hline\cr&&&&0&0\\\ &&&&0&0\\\
\end{array}\right)},\qquad\hat{J}^{-}=(\hat{J}^{+})^{T},\qquad\hat{\omega}={\footnotesize\left(\begin{array}[]{cccc|cc}0&0&0&0&&\\\
0&0&0&0&&\\\ 0&0&0&0&&\\\ 0&0&0&0&&\\\ \hline\cr&&&&1&0\\\ &&&&0&-1\\\
\end{array}\right)}$ (13)

with the local Hilbert space given by
$|\mathfrak{h}\rangle=|\mathfrak{m};\mathfrak{b}\rangle=|0,\pi,2\pi,3\pi;B^{+},B^{-}\rangle.$
The matrix elements $\hat{v}_{\hskip
0.70004pt\text{\lx@text@tweaked{yoffset=-0.5ex}{\lx@mvs@LSteel}}\hskip
0.70004pt}=1$ and $\hat{v}_{\hskip
0.70004pt\text{\lx@text@tweaked{yoffset=0.5ex}{\lx@mvs@TSteel}}\hskip
0.70004pt}=\frac{2}{\sqrt{3}}$ are computed from the local weights of the
corresponding meson vertices. Since Pauli saturation holds on the level of the
quarks and mesons have a fermionic substructure, the meson occupation numbers
$|\mathfrak{m}\rangle$ are also bounded from above. This results in a
particle-hole symmetry, leading to an SU(2) algebra which is
$d={N_{c}}+1$-dimensional:

$\displaystyle\hat{J}_{1}$
$\displaystyle=\frac{\sqrt{{N_{c}}}}{2}\left(\hat{J}^{+}+\hat{J}^{-}\right),\quad\hat{J}_{2}=\frac{\sqrt{{N_{c}}}}{2i}\left(\hat{J}^{+}-\hat{J}^{-}\right),$
$\displaystyle\hat{J}_{3}$
$\displaystyle=i[J_{1},J_{2}]=\frac{{N_{c}}}{2}[\hat{J}^{+},\hat{J}^{-}],\quad\hat{J}^{2}=\frac{{N_{c}}({N_{c}}+2)}{4},$
$\displaystyle\hat{J}_{3}\left|\frac{{N_{c}}}{2},\mathfrak{s}\right\rangle$
$\displaystyle=\mathfrak{s}\left|\frac{{N_{c}}}{2},\mathfrak{s}\right\rangle,$
$\displaystyle\hat{J}^{2}\left|\frac{{N_{c}}}{2}\mathfrak{s}\right\rangle$
$\displaystyle=\frac{{N_{c}}\left({N_{c}}+2\right)}{4}\left|\frac{{N_{c}}}{2},\mathfrak{s}\right\rangle,$
$\displaystyle[\hat{J}^{2},\hat{J}_{3}]$ $\displaystyle=0$ (14)

with $\mathfrak{m}\mapsto\mathfrak{s}=\mathfrak{m}-\frac{{N_{c}}}{2}$. The
${N_{f}}=1$ formulation at strong coupling has many similarities with full
QCD, and via Quantum Monte Carlo the grand-canonical and canonical phase
diagram could be determined [1]. The nuclear interactions are of entropic
nature: the nucleons, which are point-like in the strong coupling limit,
attract each other due to the modification of the pion bath that surrounds
them. However, pion exchange cannot be realized in the strong coupling limit
for ${N_{f}}=1$ since the Grassmann constraint does not allow for pions and
nucleons to overlap. This is clearly a lattice artifact of the strong coupling
limit, but this is overcome by formulations with ${N_{f}}>1$ and/or by
including gauge corrections.

The aim of the generalizations discussed in this proceedings is to allow for
Quantum Monte Carlo simulations (QMC) at both non-zero baryon and isospin
chemical potential, and possibly also strangeness chemical potential. We will
argue that in the strong coupling limit, the ${N_{f}}=2$ formulation remains
sign problem-free.

##  2 Hamiltonian formulation in the strong coupling limit for any $N_{f}$

Strong coupling lattice QCD with staggered fermions for ${N_{f}}>1$ admits
more than one baryon per site and the Grassmann constraint allows for pion
exchange between them, modifying nuclear interactions substantially. It has so
far only been studied via mean-field theory [10]. A Hamiltonian formulation
for ${N_{f}}=2$ allows for QMC and provides a more realistic scenario for
nuclear interactions and the phase diagram. It also compares better to the
strong coupling regime with Wilson fermions in a world-line formulation, as
discussed in the context of the 3-dim. Polyakov effective theory (for a
review, see [11]). Also for ${N_{f}}>1$ the suppression of spatial bonds
$\gamma^{-k}$, $k>2$ applies, hence the continuous time limit is well defined.

The first step to derive the Hamiltonian is to determine the local Hilbert
space $\mathbbm{H}_{\mathfrak{h}}$ via canonical sectors
$B\in\\{-{N_{f}},\ldots,{N_{f}}\\}$, i.e. we need to consider all possible
single-site quantum states $\mathfrak{h}$ in a non-interacting theory to
establish the basis of quantum states that generalize the ${N_{f}}=1$ states
$|\mathfrak{m}\rangle$ and $|\mathfrak{b}\rangle$. The static partition sum
for any ${N_{c}}$, ${N_{f}}$ is:

$\displaystyle Z_{\rm
stat}({N_{c}},{N_{f}})\sum_{B=-{N_{f}}}^{{N_{f}}}\prod_{a=0}^{N_{c}}\frac{a!(2{N_{f}}+a)!}{({N_{f}}+a+B)!(Nf+a-B)!}e^{B\mu_{B}/T}$
(15)

where the coefficient encode the number of hadronic states, i.e. the dimension
of $\mathfrak{h}$. For ${N_{c}}=3$ (separated in canonical sectors)

  * •

for ${N_{f}}=1$: $d=[1,4,1]=6$,

  * •

for ${N_{f}}=2$: $d=[1,20,50,20,1]=92$,

  * •

for ${N_{f}}=3$: $d=[1,56,490,980,490,56,1]=2074$.

Unfortunately there is no general formula known including the isospin sectors,
required for $\mu_{I}\neq 0$. The individual states $\mathfrak{h}$ are also
required for constructing the flavored $\hat{J}^{\pm}$. To arrive at this
basis, we consider the ${\rm SU}(3)$ one-link integrals in terms of the
fermion matrix
$(\mathcal{M})_{ij}=\bar{\chi}^{\alpha}_{i}(x)\chi^{\alpha}_{i}(y)$ and
$(\mathcal{M}^{\dagger})_{kl}=\chi^{\beta}_{k}(y)\bar{\chi}^{\beta}_{l}(x)$,
valid also for ${N_{f}}\geq 1$:

$\displaystyle\mathcal{J}(\mathcal{M},\mathcal{M}^{\dagger})$
$\displaystyle=\int\limits_{{\rm SU}(3)}dUe^{{\rm
tr}[U\mathcal{M}^{\dagger}+U^{\dagger}\mathcal{M}]}=\sum_{B=-{N_{f}}}^{N_{f}}\sum_{n_{1},n_{2},n_{3}}C_{B,n_{1},n_{2},n_{3}}\frac{E^{B}}{|B|!}\prod_{i=1}^{3}\frac{X_{i}^{n_{i}}}{n_{i}!},\quad
E=\begin{cases}\det{\mathcal{M}}&B>0\\\ 1&B=0\\\
\det{\mathcal{M}^{\dagger}}&B<0\\\ \end{cases}$ $\displaystyle
C_{B,n_{1},n_{2},n_{3}}$
$\displaystyle=2\binom{n_{1}+2n_{2}+4n_{3}+2|B|+2}{n_{3}+|B|}\frac{|B|!}{(n_{1}+2n_{2}+3n_{3}+2|B|+2)!(n_{2}+2n_{3}+|B|+1)!}$
(16)

which in contrasts to [12] is now expressed in a more suitable basis involving
the baryon sectors $B$. The sum over $B$ and $n_{i}$ $(i=1,\ldots 3)$
terminates due to the Grassmann integration, depending on ${N_{f}}$. The
corresponding invariants $D$, $X_{i}$ can be evaluated as follows:

$\displaystyle X_{1}$ $\displaystyle={\rm
tr}[\mathcal{M}\mathcal{M}^{\dagger}]={\rm
Tr}[M_{x}M_{y}]=k_{U}+k_{D}+\ldots+k_{\pi^{+}}+k_{\pi^{-}}+\ldots$
$\displaystyle X_{2}$ $\displaystyle=\frac{1}{2}\left({\rm
tr}[\mathcal{M}\mathcal{M}^{\dagger}]-{\rm
tr}[(\mathcal{M}\mathcal{M}^{\dagger})^{2}]\right)=\frac{1}{2}\left({\rm
Tr}[M_{x}M_{y}]^{2}+{\rm Tr}[(M_{x}M_{y})^{2}]\right)=X_{1}^{2}-D_{2}$
$\displaystyle X_{3}$
$\displaystyle=\det[\mathcal{M}\mathcal{M}^{\dagger}]=\frac{1}{6}\left({\rm
Tr}[M_{x}M_{y}]^{3}+3{\rm Tr}[M_{x}M_{y}]{\rm Tr}[(M_{x}M_{y})^{2}]+2{\rm
Tr}[(M_{x}M_{y})^{3}]\right)=X_{1}^{3}-2X_{1}D_{2}+D_{3}$ $\displaystyle
D_{2}$ $\displaystyle=\frac{1}{2}\left({\rm Tr}[M_{x}M_{y}]^{2}-{\rm
Tr}[(M_{x}M_{y})^{2}]\right)=k_{U}k_{D}+k_{\pi^{+}}k_{\pi^{-}}+\ldots-(k^{(2)}_{\pi^{+}\pi^{-},UD}+k^{(2)}_{UD,\pi^{+}\pi^{-}}+\ldots)$
$\displaystyle D_{3}$ $\displaystyle=\frac{1}{6}\left({\rm
Tr}[M_{x}M_{y}]^{3}-3{\rm Tr}[M_{x}M_{y}]{\rm Tr}[(M_{x}M_{y})^{2}]+2{\rm
Tr}[(M_{x}M_{y})^{3}]\right)=\frac{1}{6}(3X_{1}D_{2}-X_{1}^{3})$
$\displaystyle E$ $\displaystyle=\det[\mathcal{M}]=\sum_{f\leq g\leq
h}B_{fgh}\quad(B>0)\quad\text{or}\quad
E=\det[\mathcal{M}^{\dagger}]=\sum_{f\leq g\leq h}\bar{B}_{fgh},\quad(B<0)$
(17)

where the invariants are expressed in terms of nearest neighbors
$(M_{x}M_{y})^{n}=(-1)^{n+1}(\mathcal{M}\mathcal{M}^{\dagger})^{n}$. To be
explicit, for ${N_{f}}=2$, the (anti-) baryons and flavored dimers in terms of
quarks are:

$\displaystyle B_{uud}$ $\displaystyle=\bar{u}\bar{u}\bar{d}_{x}uud_{y},$
$\displaystyle B_{udd}$ $\displaystyle=\bar{u}\bar{d}\bar{d}_{x}udd_{y},$
$\displaystyle B_{uuu}$ $\displaystyle=\bar{u}\bar{u}\bar{u}_{x}uuu_{y},$
$\displaystyle B_{ddd}$ $\displaystyle=\bar{d}\bar{d}\bar{d}_{x}ddd_{y},$
$\displaystyle\bar{B}_{uud}$ $\displaystyle=uud_{x}\bar{u}\bar{u}\bar{d}_{y},$
$\displaystyle\bar{B}_{udd}$ $\displaystyle=udd_{x}\bar{u}\bar{d}\bar{d}_{y},$
$\displaystyle\bar{B}_{uuu}$ $\displaystyle=uuu_{x}\bar{u}\bar{u}\bar{u}_{y},$
$\displaystyle\bar{B}_{ddd}$ $\displaystyle=ddd_{x}\bar{d}\bar{d}\bar{d}_{y},$
$\displaystyle k_{U}$ $\displaystyle=\bar{u}u(x)\bar{u}u(y),$ $\displaystyle
k_{D}$ $\displaystyle=\bar{d}d(x)\bar{d}d(y),$ $\displaystyle k_{\pi^{+}}$
$\displaystyle=\bar{u}d(x)\bar{d}u(y),$ $\displaystyle k_{\pi^{-}}$
$\displaystyle=\bar{d}u(x)\bar{u}d(y)$ $\displaystyle
k^{(2)}_{\pi^{+}\pi^{-},UD}$
$\displaystyle=\bar{u}d(x)\bar{d}u(x)\bar{u}u(y)\bar{d}d(y),$ $\displaystyle
k^{(2)}_{UD,\pi^{+}\pi^{-}}$
$\displaystyle=\bar{u}u(x)\bar{d}d(x)\bar{u}d(y)\bar{d}u(y).$ (18)

Every state of the local Hilbert space can be described by a set of
${N_{f}}^{2}$ charges: baryon number $B\in\\{-{N_{f}},\ldots{N_{f}}\\}$,
isospin number $I\in\\{-{N_{c}},\ldots{N_{c}}\\}$, diagonal flavor numbers
$U,D,S,\ldots\in\\{0,\ldots{N_{c}}\\}$ and for ${N_{f}}>2$ additional
generalizations of isospin
$K_{1},K_{2},\ldots\in\\{-{N_{c}},\ldots{N_{c}}\\}$. The one-link integral
$J_{k}^{(B)}$ from Eq. (16) of order $k=n_{1}+2n_{2}+3n_{3}$ can be evaluated
for various ${N_{f}}$. We show here the general result for SU(3) in terms of
$X=X_{1}$, $D_{2}$, $D_{3}$ and $E$: in black the ${N_{f}}=1$ contributions,
in blue the additional ${N_{f}}=2$ contributions and in red the additional
${N_{f}}=3$ contributions:

$\displaystyle J_{0}^{(0)}$ $\displaystyle=1,\qquad
J_{1}^{(0)}=\frac{1}{3}X,\qquad
J_{2}^{(0)}=\frac{1}{12}X^{2}-{\color[rgb]{0,0,1}{\frac{1}{24}{D_{2}}}},\qquad
J_{3}^{(0)}=\frac{1}{36}X^{3}-{\color[rgb]{0,0,1}{\frac{1}{24}X{D_{2}}}}+{\color[rgb]{1,0,0}{\frac{1}{60}{D_{3}}}},$
$\displaystyle J_{4}^{(0)}$
$\displaystyle=\frac{1}{720}\left({\color[rgb]{0,0,1}{\frac{37}{12}X^{4}-\frac{11}{2}X^{2}{D_{2}}+\frac{1}{6}{D_{2}}^{2}}}+{\color[rgb]{1,0,0}{\frac{7}{3}X{D_{3}}}}\right),$
$\displaystyle J_{5}^{(0)}$
$\displaystyle=\frac{1}{840}\left({\color[rgb]{0,0,1}{\frac{47}{120}X^{5}-\frac{31}{36}X^{3}{D_{2}}+\frac{1}{4}X{D_{2}}^{2}}}+{\color[rgb]{1,0,0}{\frac{1}{3}X^{2}{D_{3}}-\frac{1}{9}\,D_{2}D_{3}}}\right),$
$\displaystyle J_{6}^{(0)}$
$\displaystyle=\frac{1}{448}\left({\color[rgb]{0,0,1}{\frac{53}{2700}X^{6}-\frac{1}{18}X^{4}{D_{2}}+\frac{5}{144}X^{2}{D_{2}}^{2}+\frac{1}{45}{D_{3}}X^{3}-\frac{1}{6480}D_{2}^{3}}}-{\color[rgb]{1,0,0}{\frac{1}{40}XD_{2}{D_{3}}+\frac{1}{240}{D_{3}}^{2}}}\right),$
$\displaystyle J_{7}^{(0)}$
$\displaystyle=\frac{1}{25920}\left({\color[rgb]{1,0,0}{\frac{241}{2940}X^{7}-\frac{19}{70}D_{2}X^{5}+\frac{227}{1008}{D_{2}}^{2}X^{3}+\frac{1}{9}\,{D_{3}}X^{4}-\frac{13}{1008}D_{2}^{3}X-\frac{29}{168}D_{2}D_{3}X^{2}+\frac{1}{168}{D_{2}}^{2}D_{3}}}+{\color[rgb]{1,0,0}{\frac{11}{336}\,D_{3}^{2}X}}\right),$
$\displaystyle J_{8}^{(0)}$
$\displaystyle=\frac{1}{21772800}\left({\color[rgb]{1,0,0}{\frac{13259}{3360}\,X^{8}-\frac{149}{10}D_{2}X^{6}+\frac{631}{40}{D_{2}}^{2}X^{4}+\frac{121}{20}\,D_{3}X^{5}-\frac{403}{120}\,D_{2}^{3}X^{2}-\frac{143}{12}D_{2}D_{3}X^{3}+\frac{1}{240}\,D_{2}^{4}}}\right.$
$\displaystyle\left.\hskip
45.5244pt{\color[rgb]{1,0,0}{+\frac{11}{4}X{D_{2}}^{2}D_{3}+\frac{11}{5}X^{2}D_{3}^{2}-\frac{11}{20}D_{2}D_{3}^{2}}}\right),$
$\displaystyle J_{9}^{(0)}$
$\displaystyle=\frac{1}{17107200}\left({\color[rgb]{1,0,0}{\frac{63163}{423360}X^{9}-\frac{377}{588}X^{7}D_{2}+\frac{1429}{1680}X^{5}{D_{2}}^{2}+\frac{73}{280}X^{6}D_{3}-\frac{1663}{5040}X^{3}{D_{2}}^{3}-\frac{551}{840}X^{4}{D_{2}}{D_{3}}}}\right.$
$\displaystyle\hskip
45.5244pt\left.{\color[rgb]{1,0,0}{+\frac{17}{3360}X{D_{2}}^{4}+\frac{71}{210}X^{2}{D_{2}}^{2}D_{3}+\frac{13}{105}X^{3}{D_{3}}^{2}-\frac{1}{420}{D_{2}}^{3}{D_{3}}-\frac{13}{120}X{D_{2}}{D_{3}}^{2}+\frac{13}{1260}{D_{3}}^{3}}}\right),$
$\displaystyle J_{0}^{(1)}$ $\displaystyle=\frac{E}{6},\qquad
J_{1}^{(1)}=\frac{E}{6}\left({\color[rgb]{0,0,1}{\frac{1}{4}X}}\right),\qquad
J_{2}^{(1)}=\frac{E}{6}\left({\color[rgb]{0,0,1}{\frac{1}{24}X^{2}-\frac{1}{60}D_{2}}}\right),\qquad
J_{3}^{(1)}=\frac{E}{6}\left({\color[rgb]{0,0,1}{\frac{1}{144}X^{3}-\frac{1}{120}XD_{2}}}+{\color[rgb]{1,0,0}{\frac{1}{360}{D_{3}}}}\right),$
$\displaystyle J_{4}^{(1)}$
$\displaystyle=\frac{E}{6}{\color[rgb]{1,0,0}{\left(\frac{1}{1344}X^{4}-\frac{1}{840}X^{2}D_{2}+\frac{1}{20160}{D_{2}}^{2}+\frac{1}{2240}X{D_{3}}\right)}},$
$\displaystyle J_{5}^{(1)}$
$\displaystyle=\frac{E}{6}{\color[rgb]{1,0,0}{\left(\frac{143}{2419200}X^{5}-\frac{29}{241920}\,D_{2}X^{3}+\frac{1}{32256}D_{2}^{2}X+\frac{1}{23040}\,D_{3}X^{2}-\frac{1}{80640}D_{2}D_{3}\right)}},$
$\displaystyle J_{6}^{(1)}$
$\displaystyle=\frac{E}{6}{\color[rgb]{1,0,0}{\left(\frac{19}{4838400}X^{6}-\frac{221}{21772800}X^{4}{D_{2}}+\frac{241}{43545600}X^{2}{D_{2}}^{2}+\frac{11}{2903040}X^{3}D_{3}-\frac{1}{21772800}{D_{2}}^{3}-\frac{11}{3110400}X{D_{2}}{D_{3}}+\frac{11}{21772800}{D_{3}}^{2}\right)}},$
$\displaystyle J_{0}^{(2)}$
$\displaystyle={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}},\qquad
J_{1}^{(2)}={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}}\left({\color[rgb]{1,0,0}{\frac{1}{5}X}}\right),\qquad
J_{2}^{(2)}={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}}\left({\color[rgb]{1,0,0}{\frac{1}{40}X^{2}-\frac{1}{120}D_{2}}}\right),\qquad
J_{3}^{(2)}={\color[rgb]{0,0,1}{\frac{E^{2}}{144}}}\left({\color[rgb]{1,0,0}{\frac{1}{360}X^{3}-\frac{1}{360}XD_{2}+\frac{1}{1260}D_{3}}}\right),$
$\displaystyle J_{0}^{(3)}$
$\displaystyle={\color[rgb]{1,0,0}{\frac{1}{8640}E^{3}}}$ (19)

The number of hadronic states in each conserved charge sector $(B,I,S,\ldots)$
can be obtained from the one-link integral by combining them in alternating
chains $J_{k}^{(B)}J_{(N_{f}-B){N_{c}}-k}^{(B)}$, with $k$ denoting the
mesonic occupation number $\mathfrak{m}_{k}$. To obtain the Hamiltonian, we
still have to integrate out the Grassmann variables. Here we consider the
chiral limit only. The Grassmann constraint then dictates that all quarks
$u,d,s,\ldots$ and anti-quarks $\bar{u},\bar{d},\bar{s},\ldots$ within mesons
or baryons appear exactly ${N_{c}}$ times. The Grassmann integral in the
chiral limit for U(3) (i.e. SU(3) with $B=0$) for a given site $x$ is

$\displaystyle I_{G}$ $\displaystyle=\int\prod_{\alpha}\prod_{f}[{\rm
d}\bar{f}_{\alpha}{\rm
d}f_{\alpha}]\prod_{f,g}(\bar{f}g)^{k_{\bar{f}g}}(-1)^{\frac{1}{2}\sum\limits_{f\neq
g}k_{\bar{f}g}}F_{N_{f}}({N_{c}},\\{k_{\bar{f}g}\\})$ (20)

with $k_{\bar{f}g}$ the sum of flavored dimers attached to site $x$, and with
$F_{N_{f}}({N_{c}})$ some ${N_{c}}$-dependent function, given the power
$k_{fg}$ of charged fluxes ($f\neq g$). The result for non-zero baryon number
$B\neq 0$ and details on $F_{N_{f}}({N_{c}})$ will be presented in a
forthcoming publication. On a given configuration, the Grassmann integration
simplifies due to flux conservation: for the non-diagonal pseudo-scalar mesons
$k_{\bar{f}g}=k_{\bar{g}f}$ such that all minus signs from the Grassmann
integration cancel. Only the non-trivial contribution of type
$k^{(2)}_{UD,\pi^{+}\pi^{-}}$ and its generalizations for ${N_{f}}>2$ induce
minus signs. Those link states can be resummed and diagonalized, e.g. the
following states have an eigenvalue 1:

$\displaystyle\pi_{0}^{2}$
$\displaystyle=\left(k_{U}k_{D}+\frac{1}{\sqrt{{N_{c}}}}k^{(2)}_{\pi^{+}\pi^{-},UD}\right),$
$\displaystyle\bar{\pi}_{0}^{2}$
$\displaystyle=\left(k_{\pi^{+}}k_{\pi^{-}}+\frac{1}{\sqrt{{N_{c}}}}k^{(2)}_{UD,\pi^{+}\pi^{-}}\right)$
(21)

Likewise all dimer-based states are resummed, as states are only
distinguishable on the quark level. This reduces the state space drastically
to the physical Hilbert state, e.g. for ${N_{f}}=2$: $B=0:340\mapsto 50$,
$B=\pm 1:152\mapsto 20$, $B=\pm 2:4\mapsto 1$, resulting in the local Hilbert
space $\mathbbm{H}_{\mathfrak{h}}$ as given in Tab. 1, classified by baryon
number $B$, isospin number $I$ and meson occupation number $\mathfrak{m}$. The
particle-hole symmetry generalizes to
$\mathfrak{m}\mapsto\mathfrak{s}=\mathfrak{m}-\frac{{N_{c}}}{2}({N_{f}}-|B|)$
as the meson raising and lowering operators fulfill an SU(2) algebra for each
meson charge $Q_{i}$. All states $\mathfrak{h}\in\mathbbm{H}_{\mathfrak{h}}$
for ${N_{f}}=2,3$ contribute with a weight 1, although some dimer-based link
weights are negative.

$B$ | $I$ | $\mathfrak{s}=\mathfrak{m}-\frac{3}{2}(2-|B|)$ | $\Sigma$
---|---|---|---
|  | $-3$ | $-\frac{5}{2}$ | $-2$ | $-\frac{3}{2}$ | $-1$ | $-\frac{1}{2}$ | $\,0\,$ | $+\frac{1}{2}$ | $+1$ | $+\frac{3}{2}$ | $+2$ | $+\frac{5}{2}$ | $+3$ |
-2 | 0 |  |  |  |  |  |  | 1 |  |  |  |  |  |  | 1
-1 | $-\frac{3}{2}$ |  |  |  | 1 |  | 1 |  | 1 |  | 1 |  |  |  | 4
-1 | $-\frac{1}{2}$ |  |  |  | 1 |  | 2 |  | 2 |  | 1 |  |  |  | 6
-1 | $+\frac{1}{2}$ |  |  |  | 1 |  | 2 |  | 2 |  | 1 |  |  |  | 6
-1 | $+\frac{3}{2}$ |  |  |  | 1 |  | 1 |  | 1 |  | 1 |  |  |  | 4
0 | -3 |  |  |  |  |  |  | 1 |  |  |  |  |  |  | 1
0 | -2 |  |  |  |  | 1 |  | 2 |  | 1 |  |  |  |  | 4
0 | -1 |  |  | 1 |  | 2 |  | 4 |  | 2 |  | 1 |  |  | 10
0 | 0 | 1 |  | 2 |  | 4 |  | 6 |  | 4 |  | 2 |  | 1 | 20
0 | -1 |  |  | 1 |  | 2 |  | 4 |  | 2 |  | 1 |  |  | 10
0 | -2 |  |  |  |  | 1 |  | 2 |  | 1 |  |  |  |  | 4
0 | -3 |  |  |  |  |  |  | 1 |  |  |  |  |  |  | 1
1 | $-\frac{3}{2}$ |  |  |  | 1 |  | 1 |  | 1 |  | 1 |  |  |  | 4
1 | $-\frac{1}{2}$ |  |  |  | 1 |  | 2 |  | 2 |  | 1 |  |  |  | 6
1 | $+\frac{1}{2}$ |  |  |  | 1 |  | 2 |  | 2 |  | 1 |  |  |  | 6
1 | $+\frac{3}{2}$ |  |  |  | 1 |  | 1 |  | 1 |  | 1 |  |  |  | 4
2 | 0 |  |  |  |  |  |  | 1 |  |  |  |  |  |  | 1
$\Sigma$ |  | 1 | 0 | 4 | 8 | 10 | 12 | 22 | 12 | 10 | 8 | 4 | 0 | 1 | 92
Table 1:  All 92 possible quantum states for the ${N_{f}}=2$ Hamiltonian
formulation with ${\rm SU}(3)$ gauge group. The number of states are given for
the sectors specified baryon number $B$ and isospin number $I$, and
symmetrized meson occupation number
$\mathfrak{s}=\mathfrak{m}-\frac{{N_{c}}}{2}({N_{f}}-|B|)$. Note the mesonic
particle-hole symmetry $\mathfrak{s}\leftrightarrow-\mathfrak{s}$ which
corresponds to the shift symmetry by ${a_{\tau}}$.

Only single meson exchange is possible (multiple meson exchange becomes
resolved into single mesons in the continuous time limit), the resulting
interaction Hamiltonian has ${N_{f}}^{2}$ terms:

$\displaystyle\hat{\mathcal{H}}$
$\displaystyle=\frac{1}{2}\sum_{\langle\vec{x},\vec{y}\rangle}\sum_{Q_{i}}\left({\hat{J}_{Q_{i},\vec{x}}}^{+}{\hat{J}_{Q_{i},\vec{y}}}^{-}+{\hat{J}_{Q_{i},\vec{x}}}^{-}{\hat{J}_{Q_{i},\vec{y}}}^{+}\right),$
(22)

e.g. for ${N_{f}}=2$: $Q_{i}\in\\{\pi^{+},\pi^{-},\pi_{U},\pi_{D}\\}$ and for
${N_{f}}=3$:
$Q_{i}\in\\{\pi^{+},\pi^{-},K^{+},K^{-},K_{0},\bar{K_{0}},\pi_{U},\pi_{D},\pi_{S}\\}$.
For the transition $\mathfrak{h}_{1}\mapsto\mathfrak{h}_{2}$, the matrix
elements ${\langle\mathfrak{h}_{1}|Q_{i}|\mathfrak{h}_{2}\rangle}$ of
$\hat{J}_{Q_{i}}^{\pm}$ are determined from Grassmann integration and a square
root per participating link weight (in, hopping, out); only those matrix
elements are non-zero which are consistent with current conservation for all
$Q_{i}$. Some matrix elements are negative, but in the combination of a closed
charged meson loop the total weight remains positive. Some examples for
${N_{f}}=2$ are:

$\displaystyle{\langle\pi^{+},\pi^{-}|\pi^{+}|\pi_{U},\pi_{D}\pi^{-}\rangle}$
$\displaystyle=\frac{\sqrt{3}}{2},$
$\displaystyle{\langle\pi_{U},\pi^{-}|\pi^{+}|\pi_{U},2\pi^{-}\rangle}$
$\displaystyle=\frac{\sqrt{6}}{3},$ $\displaystyle{\langle
B_{uud}|\pi^{+}|B_{uud},\pi_{U},\pi_{D}\rangle}$
$\displaystyle=\frac{2}{\sqrt{5}},$ $\displaystyle{\langle
B_{uud}\pi^{+}|\pi^{+}|B_{uud},\pi_{D}\rangle}$
$\displaystyle=-\frac{\sqrt{3}}{6}$ (23)

There are 130 non-zero matrix elements for $B=0$ and 40 matrix elements for
$B=\pm 1$. The $B=\pm 2$ states do not allow for pion exchange. Only after
contraction of the matrix elements the resummation is carried out for the
internal hadronic states, resulting in positive weights. An important
application of the ${N_{f}}=2$ partition function is to determine the QCD
phase diagram with both finite baryon and isospin chemical potential [13, 14].
Our formulation is still sign-problem free in the continuous time limit. As we
have not yet performed dynamical simulations, we can only obtain analytic
results in a high temperature expansion of the partition sum $Z$, which is an
expansion around the static limit

$\displaystyle Z\left(\frac{\mu_{B}}{T},\frac{\mu_{I}}{T}\right)=$
$\displaystyle
2\cosh\frac{3\mu_{I}}{T}+8\cosh\frac{2\mu_{I}}{T}+20\cosh\frac{\mu_{I}}{T}+20$
$\displaystyle+2\cosh\frac{\mu_{B}}{T}\left(8\cosh{\frac{\frac{3}{2}\mu_{I}}{T}}+12\cosh\frac{\frac{1}{2}\mu_{I}}{T}\right)+2\cosh\frac{2\mu_{B}}{T}$
(24)

in the number of spatial mesons. This will be presented in a forthcoming
publication.

##  3 Leading order gauge corrections to the Hamiltonian formulation for
$N_{f}=1$

The QCD Partition can be expanded via the strong coupling expansion in
$\beta$:

$\displaystyle Z_{QCD}$ $\displaystyle=\int d\psi
d\bar{\psi}dUe^{S_{G}+S_{F}}=\int d\psi d\bar{\psi}Z_{F}{\langle
e^{S_{G}}\rangle}_{Z_{F}},$ $\displaystyle{\langle e^{S_{G}}\rangle}_{Z_{F}}$
$\displaystyle\simeq 1+{\langle
S_{G}\rangle}_{Z_{F}}+\mathcal{O}(\beta^{2})=1+\frac{\beta}{2{N_{c}}}\sum_{P}{\langle{\rm
tr}[U_{P}+U_{P}^{\dagger}]\rangle}_{Z_{F}}+\mathcal{O}(\beta^{2})$ (25)

Additional color singlet link states are due to plaquette excitations. For
discrete time lattices, the dual formulation has been extended beyond
$\mathcal{O}(\beta)$ in terms of a tensor network [8]. Here we want to
consider the leading order gauge corrections in a Hamiltonian formulation. On
anisotropic lattices, the anisotropy $\xi=\frac{a_{s}}{a_{t}}$ is a function
of two bare anisotropies $\gamma_{F}$ and
$\gamma_{G}=\sqrt{\frac{\beta_{t}}{\beta_{s}}}$, i.e
$\xi=\xi(\gamma_{F},\gamma_{G},\beta)$. However, in the continuous time limit
$a_{t}\rightarrow 0$ ($\xi\rightarrow\infty$) and for small $\beta$, spatial
plaquettes are suppressed over temporal plaquettes by
$(\gamma_{G}\gamma_{F})^{-2}$, hence only temporal plaquettes need to be
considered. They are of the same order as meson exchange, hence the
$\mathcal{O}(\beta)$ weights will contribute to the corresponding operators
$\hat{J}^{\pm}$. Note that $\hat{J}^{\pm}$ still has block-diagonal structure,
but they will also allows to couple to baryons. Fig. 1 shows the relevant
weights as computed in [8].

Figure 1: Tensors and their weights of $\mathcal{O}(\beta)$ that can be
incorporated into the Hamiltonian formulation. Blue: dimers, red: baryon flux,
green: plaquette flux due to gauge corrections.

##  4 Summary and Outlook

We have presented two extensions to the Hamiltonian formulation of lattice
QCD: (1) generalization from ${N_{f}}=1$ to ${N_{f}}>1$ in the strong coupling
limit, (2) $\mathcal{O}(\beta)$ gauge corrections for ${N_{f}}=1$. In a
forthcoming publication, we also address gauge corrections to the ${N_{f}}=2$
Hamiltonian. It turns out that - although not sign problem-free - the sign
problem is much milder than in the corresponding formulation at finite
${N_{\tau}}$ as for ${N_{\tau}}\rightarrow\infty$ only temporal plaquettes
contribute. Meson exchange between baryons can thus be either due to the gauge
corrections or the flavor content. All results presented here are valid in the
chiral limit. We have argued in [1] that also a Hamiltonian formulation at
finite quark mass $am_{q}$ is well-defined, which extends this approach to
staggered lattice QCD further. We are preparing Quantum Monte Carlo
simulations that will help to obtain the QCD phase diagram on the lattice in
the parameter space $(aT,a\mu_{B},a\mu_{I},am_{q},\beta)$ in lattice units or
in units of the baryon mass $am_{B}$.

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – project number 315477589 – TRR 211.“

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