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We show how to apply renormalization group algorithms incorporating entanglement filtering methods and a loop optimization to a tensor network which includes Grassmann variables which represent fermions in an underlying lattice field theory. As a numerical test a variety of quantities are calculated for two dimensional Wilson--Majorana fermions and for the two flavor Gross--Neveu model. The improved algorithms show much better accuracy for quantities such as the free energy and the determination of Fisher's zeros., 21 pages, 21 figures

We discuss the use of quantum simulation to study an N-flavor theory of interacting relativistic fermions in (1+1) dimensions on noisy intermediate-scale quantum (NISQ)-era machines. The case of two flavors is particularly interesting as it can be mapped to the Hubbard model. We derive the appropriate qubit Hamiltonians and associated quantum circuits. We compare classical simulation and density matrix renormalization group/time-evolving blocked decimation calculations with the results of quantum simulation on various platforms for N=2 and 4-flavors. We demonstrate that the four steps needed for calculations of real-time scattering can be implemented using current NISQ devices.

Motivated by attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we consider an extended-O(2) model that differs from the ordinary O(2) model by an explicit symmetry breaking term. Its coupling allows to smoothly interpolate between the O(2) model (zero coupling) and a $q$-state clock model (infinite coupling). In the latter case, a $q$-state clock model can also be defined for non-integer values of $q$. Thus, such a limit can also be considered as an analytic continuation of an ordinary $q$-state clock model to non-integer $q$. The phase diagram of the extended-O(2) model in the infinite coupling limit was established in our previous work, where it was shown that for non-integer $q$, there is a second-order phase transition at low temperature and a crossover at high temperature. In this work, we investigate the model at finite values of the coupling using Monte Carlo and tensor methods. The results may be relevant for configurable Rydberg-atom arrays., Comment: 7 pages, 4 Figures; Talk given at the 39th International Symposium on Lattice Field theory (LATTICE2022) in Bonn, Germany; August 8-13 2022

We present a method to extract the phase shift of a scattering process using the real-time evolution in the early and intermediate stages of the collision in order to estimate the time delay of a wave packet. This procedure is convenient when using noisy quantum computers for which the asymptotic out-state behavior is unreachable. We demonstrate that the challenging Fourier transforms involved in the state preparation and measurements can be implemented in $1+1$ dimensions with current trapped ion devices and IBM quantum computers. We compare quantum computation of the time delays obtained in the one-particle quantum mechanics limit and the scalable quantum field theory formulation with accurate numerical results. We discuss the finite volume effects in the Wigner formula connecting time delays to phase shifts. The results reported involve two- and four-qubit calculations, and we discuss the possibility of larger scale computations in the near future., 10 pages, 5 figures

The O(2) model in Euclidean space-time is the zero-gauge-coupling limit of the compact scalar quantum electrodynamics. We obtain a dual representation of it called the charge representation. We study the quantum phase transition in the charge representation with a truncation to ``spin $S$,'' where the quantum numbers have an absolute value less than or equal to $S$. The charge representation preserves the gapless-to-gapped phase transition even for the smallest spin truncation $S=1$. The phase transition for $S=1$ is an infinite-order Gaussian transition with the same critical exponents $\ensuremath{\delta}$ and $\ensuremath{\eta}$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition, while there are true BKT transitions for $S\ensuremath{\ge}2$. The essential singularity in the correlation length for $S=1$ is different from that for $S\ensuremath{\ge}2$. The exponential convergence of the phase-transition point is studied in both Lagrangian and Hamiltonian formulations. We discuss the effects of replacing the truncated ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{U}}^{\ifmmode\pm\else\textpm\fi{}}=exp(\ifmmode\pm\else\textpm\fi{}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\theta}})$ operators by the spin ladder operators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{\ifmmode\pm\else\textpm\fi{}}$ in the Hamiltonian. The marginal operators vanish at the Gaussian transition point for $S=1$, which allows us to extract the $\ensuremath{\eta}$ exponent with high accuracy.

The lattice compact Abelian Higgs model is a non-perturbative regularized formulation of low-energy scalar quantum electrodynamics. In 1+1 dimensions, this model can be quantum simulated using a ladder-shaped optical lattice with Rydberg-dressed atoms (Zhang et al., Phys. Rev. Lett. 121, 223201). In this setup, one spatial dimension is used to carry the angular momentum of the quantum rotors. One can use truncations corresponding to spin-2 and spin-1 to build local Hilbert spaces associated with the links of the lattice. We argue that ladder-shaped configurable arrays of Rydberg atoms can be used for the same purpose. We make concrete proposals involving two and three Rydberg atoms to build one local spin-1 space (a qutrit). We show that the building blocks of the Hamiltonian calculations are models with one and two spins. We compare target and simulators using perturbative and numerical methods. The two-atom setup provides an easily controllable simulator of the one-spin model while the three-atom setup involves solving nonlinear equations. This could be tested with current technology. We argue that near-term technology could be used to quantum simulate models with two or more spins., Comment: Updated version for Phys. Rev. D; Section on digital implementations and figs. added; eq. (57) corrected

Motivated by recent attempts to quantum simulate lattice models with continuous Abelian symmetries using discrete approximations, we define an extended-O(2) model by adding a $\gamma \cos(q\varphi)$ term to the ordinary O(2) model with angular values restricted to a $2\pi$ interval. In the $\gamma \rightarrow \infty$ limit, the model becomes an extended $q$-state clock model that reduces to the ordinary $q$-state clock model when $q$ is an integer and otherwise is a continuation of the clock model for noninteger $q$. By shifting the $2\pi$ integration interval, the number of angles selected can change discontinuously and two cases need to be considered. What we call case $1$ has one more angle than what we call case $2$. We investigate this class of clock models in two space-time dimensions using Monte Carlo and tensor renormalization group methods. Both the specific heat and the magnetic susceptibility show a double-peak structure for fractional $q$. In case $1$, the small-$\beta$ peak is associated with a crossover, and the large-$\beta$ peak is associated with an Ising critical point, while both peaks are crossovers in case $2$. When $q$ is close to an integer by an amount $\Delta q$ and the system is close to the small-$\beta$ Berezinskii-Kosterlitz-Thouless transition, the system has a magnetic susceptibility that scales as $\sim 1 / (\Delta q)^{1 - 1/\delta'}$ with $\delta'$ estimates consistent with the magnetic critical exponent $\delta = 15$. The crossover peak and the Ising critical point move to Berezinskii-Kosterlitz-Thouless transition points with the same power-law scaling. A phase diagram for this model in the $(\beta, q)$ plane is sketched. These results are possibly relevant for configurable Rydberg-atom arrays where the interpolations among phases with discrete symmetries can be achieved by varying continuously the distances among atoms and the detuning frequency., Comment: 23 pages, 38 figures, 2 tables

We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of motion and the discrete selection rules of the tensors. We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalence with its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss's law is always enforced. We propose a noise-robust way to implement Gauss's law in arbitrary dimensions. We reformulate Noether's theorem for global, local, continuous or discrete Abelian symmetries: for each given symmetry, there is one corresponding tensor redundancy. We discuss semi-classical approximations for classical solutions with periodic boundary conditions in two solvable cases. We show the correspondence of their weak coupling limit with the tensor formulation after Poisson summation. We briefly discuss connections with other approaches and implications for quantum computing., 12 pages, 6 figures, uses revtex, references added, extended discussion on noise-robust implementation of Gauss's law

We construct a tensor network representation of the partition function for the massless Schwinger model on a two-dimensional lattice using staggered fermions. The tensor network representation allows us to include a topological term. Using a particular implementation of the tensor renormalization group we calculate the average plaquette and topological charge density for the theory. For a range of values of the coupling constant for the topological term, $\mathrm{\ensuremath{\Theta}}$, and the gauge coupling, $\ensuremath{\beta}$, we compare with results from hybrid Monte Carlo simulations when possible and find good agreement. In order to further understand the role of fermions and topology in this model, we compare calculations of the same observables in a similar model, the Abelian Higgs model.

We report updates to an ongoing lattice-QCD calculation of the form factors for the semileptonic decays $B \to \pi \ell \nu$, $B_s \to K \ell \nu$, $B \to \pi \ell^+ \ell^-$, and $B \to K \ell^+ \ell^-$. The tree-level decays $B_{(s)} \to \pi (K) \ell \nu$ enable precise determinations of the CKM matrix element $|V_{ub}|$, while the flavor-changing neutral-current interactions $B \to \pi (K) \ell^+ \ell^-$ are sensitive to contributions from new physics. This work uses MILC's (2+1+1)-flavor HISQ ensembles at approximate lattice spacings between $0.057$ and $0.15$ fm, with physical sea-quark masses on four out of the seven ensembles. The valence sector is comprised of a clover $b$ quark (in the Fermilab interpretation) and HISQ light and $s$ quarks. We present preliminary results for the form factors $f_0$, $f_+$, and $f_T$, including studies of systematic errors.

We use lattice QCD to calculate the form factors ${f}_{+}({q}^{2})$ and ${f}_{0}({q}^{2})$ for the semileptonic decay ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\ell}\ensuremath{\nu}$. Our calculation uses six MILC asqtad $2+1$ flavor gauge-field ensembles with three lattice spacings. At the smallest and largest lattice spacing the light-quark sea mass is set to $1/10$ the strange-quark mass. At the intermediate lattice spacing, we use four values for the light-quark sea mass ranging from $1/5$ to $1/20$ of the strange-quark mass. We use the asqtad improved staggered action for the light valence quarks, and the clover action with the Fermilab interpolation for the heavy valence bottom quark. We use SU(2) hard-kaon heavy-meson rooted staggered chiral perturbation theory to take the chiral-continuum limit. A functional $z$ expansion is used to extend the form factors to the full kinematic range. We present predictions for the differential decay rate for both ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\mu}\ensuremath{\nu}$ and ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\tau}\ensuremath{\nu}$. We also present results for the forward-backward asymmetry, the lepton polarization asymmetry, ratios of the scalar and vector form factors for the decays ${B}_{s}\ensuremath{\rightarrow}K\ensuremath{\ell}\ensuremath{\nu}$ and ${B}_{s}\ensuremath{\rightarrow}{D}_{s}\ensuremath{\ell}\ensuremath{\nu}$. Our results, together with future experimental measurements, can be used to determine the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{ub}|$.

We consider the tensor formulation of the non-linear O(2) sigma model and its gauged version (the compact Abelian Higgs model), on a $D$-dimensional cubic lattice, and show that tensorial truncations are compatible with the general identities derived from the symmetries of these models. This means that the universal properties of these models can be reproduced with highly simplified formulations desirable for implementations with quantum computers or for quantum simulations experiments. We briefly discuss the extensions to non-Abelian symmetries and models with fermions., Comment: Comments and new section on non-Abelian symmetries and pure gauge models added, 8 pages, uses Revtex

In most lattice simulations, the variables of integration are compact and character expansion (for instance Fourier analysis for U(1) models) can be used to rewrite the partition function and average observables as discrete sums of contracted tensors. These reformulations have been used for RG blocking but they also naturally fit the needs of quantum computing. We discuss FAQ about tensorial reformulations: effects of truncations on symmetries, boundary conditions, Grassmann variables, and other recent aspects of quantum computing in the same context.

Starting with the 2D Abelian Higgs model with the quartic self-coupling taken infinitely large we study the finite-size scaling of the Polyakov loop. We find an exponential decay for large temporal extents which is dictated by the energy gap between the ground states of a system with the Polyakov loop inserted, and one without. We study this system using the tensor renormalization group, and we take the continuous-time limit to obtain a quantum Hamiltonian where gauge invariance has been maintained exactly. Comparing with numerical results from the density matrix renormalization group we find universal features of the finite-size scaling of the energy gap survive this continuous-time limit. We propose an optical-lattice ladder to quantum simulate this model, and observe the universal features of the energy gap scaling.

A SU(3) gauge theory with 12 flavors is a model of great interest for beyond the standard model physics. Running RHMC simulations for different masses and betas we study the Fisher zeroes in the vicinity of the endpoint of a line of first order phase transitions. The pinching of these zeros with respect to increasing volume provides information about a possible unconventional continuum limit. We also study the mass spectrum of a multiflavor linear sigma model with a splitting of fermion masses.

We derived the tree level spectrum to an extension to the linear sigma model, proposed by [1], describing an EFT for an $SU(3)_c$ gauge theory with $N_f$ flavors of fermions and $N_1$ fermions have a mass $m_l$ and $N_2$ fermions have a mass $m_h$. We examined the effects of a small mass splitting on single mass data for 8 and 12 flavors of fermions corresponding to the unperturbed case. Our intention is to encourage more simulations of split mass theories with 8, 10 and 12 flavors of fermions.

We discuss real time evolution for the quantum Ising model in one spatial dimension with $N_s$ sites. In the limit where the nearest neighbor interactions $J$ in the spatial directions are small, there is a simple physical picture where qubit states can be interpreted as approximate particle occupations. Using exact diagonalization, for initial states with one or two particles, we show that for small $J$, discrete Bessel functions provide very accurate expressions for the evolution of the occupancies corresponding to initial states with one and two particles. Boundary conditions play an important role when the evolution time is long enough. We discuss a Trotter procedure to implement the evolution on existing quantum computers and discuss the error associated with the Trotter step size. We discuss the effects of gate and measurement errors on the evolution of one and two particle states using 4 and 8 qubits circuits approximately corresponding to existing or near term quantum computers., 13 pages, 12 figures

We investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. A barrier initially confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time $t = 0$. We investigate the properties of the time-evolved density matrix, the diagonal ensemble density matrix and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation using exact diagonalization. The weights for the diagonal ensemble and the canonical ensemble match well for high initial temperatures that correspond to negative effective final temperatures after the expansion. At long times after the barrier is removed, the time-evolved R\'enyi entropy of subsystems bigger than half can equilibrate to the thermal entropy with exponentially small fluctuations. The time-evolved reduced density matrix at long times can be approximated by a thermal density matrix for small subsystems. Few-body observables, like the momentum distribution function, can be approximated by a thermal expectation of the canonical ensemble with strongly suppressed fluctuations. The negative effective temperatures for finite systems go to nonnegative temperatures in the thermodynamic limit for bosons, but is a true thermodynamic effect for fermions, which is confirmed by finite temperature density matrix renormalization group calculations. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions., Comment: 19 pages, 19 figures

Lattice gauge theories are fundamental to our understanding of high-energy physics. Nevertheless, the search for suitable platforms for their quantum simulation has proven difficult. We show that the Abelian Higgs model in 1+1 dimensions is a prime candidate for an experimental quantum simulation of a lattice gauge theory. To this end, we use a discrete tensor reformulation to smoothly connect the space-time isotropic version used in most numerical lattice simulations to the continuous-time limit corresponding to the Hamiltonian formulation. The eigenstates of the Hamiltonian are neutral for periodic boundary conditions, but we probe the nonzero charge sectors by either introducing a Polyakov loop or an external electric field. In both cases we obtain universal functions relating the mass gap, the gauge coupling, and the spatial size which are invariant under the deformation of the temporal lattice spacing. We propose to use a physical multi-leg ladder of atoms trapped in optical lattices and interacting with Rydberg-dressed interactions to quantum simulate the model and check the universal features. Our results provide a path to the analog quantum simulation of lattice gauge theories with atoms in optical lattices., Comment: 9 pages, 6 figures, experimental content, supplementary material and coauthor (JZ) added

We show that the Polyakov loop of the two-dimensional lattice Abelian Higgs model can be calculated using the tensor renormalization group approach. We check the accuracy of the results using standard Monte Carlo simulations. We show that the energy gap produced by the insertion of the Polyakov loop obeys universal finite-size scaling which persists in the time continuum limit. We briefly discuss the relevance of these results for quantum simulations., Comment: 10 pages, 11 figures, 1 table

We calculate the tree-level mass spectrum for a linear sigma model describing the scalar and pseudoscalar mesons of a $SU(3)$ local gauge theory with Dirac fermions in the fundamental representation. $N_1$ fermions have a mass $m_1$ and $N_2$ a mass $m_2$. Using recent lattice data with $m_1=m_2$ and $N_1+N_2$= 8 or 12, we predict the mass splittings for $m_2=m_1+\delta m$. At first order in $\delta m$, an interesting inverted pattern appears in the $0^{++}$ sector, where mesons with lighter fermions are heavier. This feature could be tested in ongoing calculations provided that $m_1$ and $\delta m$ are sufficiently small. We discuss possible improvements of the approach., Comment: 7 pages, 3 figures, 3 tables

We develop a methodology to test the accuracy of lattice extrapolations of the form factors in B decays only using experimental data. We test this methodology by comparing the BGL parameterization proposed by Boyd, Grinstein, and Lebed (1996) and the BCL parameterization proposed by Bourrely Caprini and Lellouch (2008) in the context of $B \rightarrow \pi \ell \nu$. Our results suggest that the BCL parameterization is a slightly better predictor of the low $q^2$ region than the BGL parameterization and that this methodology can be extended to other decays such as $B\rightarrow D\ell \nu$., Comment: 7 pages, 3 figures, 1 table. Preliminary proceedings for Lattice 2018, speaker Erik Gustafson

Using the example of configurations generated with the worm algorithm for the two-dimensional Ising model, we propose renormalization group (RG) transformations, inspired by the tensor RG, that can be applied to sets of images. We relate criticality to the logarithmic divergence of the largest principal component. We discuss the changes in link occupation under the RG transformation, suggest ways to obtain data collapse, and compare with the two state tensor RG approximation near the fixed point.