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Square lattice Hubbard models with tunable hopping ratio $t'/t$ are highly promising for realizing a variety of quantum phases and for shedding light on key puzzles in correlated quantum materials, including higher-temperature superconductivity. We show that twisted square lattice homo-bilayers generically offer such tunability when the flat bands originate from the corner of the Brillouin zone. We reveal an emergent symmetry at low twist-angles, absent in single layers, that necessitates the vanishing of nearest neighbor hopping ($t=0$). This symmetry can be lifted by an inter-layer displacement field or by an in-plane magnetic field, introducing tunable $t$ and anisotropy, allowing access to a wide range of $t'/t$ ratios for correlated electrons on a moir\'e square lattice., Comment: 5 pages + Supplementary

We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the $\pi$-flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background $\pi$-flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal $d$-wave superconductor, and states with N\'eel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases which are very likely described by 2+1 dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and N\'eel order, and a conventional $d$-wave superconductor with gapless Bogoliubov quasiparticles at 4 nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of N\'eel, valence bond solid (Kekul\'e), and Dirac semi-metal phases., Comment: 47 pages, 20 figures

Symmetry is a powerful tool for understanding phases of matter in equilibrium. Quantum circuits with measurements have recently emerged as a platform for novel states of matter intrinsically out of equilibrium. Can symmetry be used as an organizing principle for these novel states, their phases and phase transitions? In this work, we give an affirmative answer to this question in a simple adaptive monitored circuit, which hosts an ordering transition in addition to a separate entanglement transition, upon tuning a single parameter. Starting from a symmetry-breaking initial state, depending on the tuning parameter, the steady state could (i) remain symmetry-broken, (ii) exhibit the average symmetry in the ensemble of trajectories, or (iii) exhibit the exact symmetry for each trajectory. The ordering transition is mapped to the transition in a classical majority vote model, described by the Ising universality class, while the entanglement transition lies in the percolation class. Numerical simulations are further presented to support the analytical understandings.

Symmetries and their anomalies are powerful tools for understanding quantum systems. However, realistic systems are often subject to disorders, dissipation and decoherence. In many circumstances, symmetries are not exact but only on average. This work investigates the constraints on mixed states resulting from non-commuting average symmetries. We will focus on the cases where the commutation relations of the average symmetry generators are violated by nontrivial phases, and call such average symmetry anomalous. We show that anomalous average symmetry implies degeneracy in the density matrix eigenvalues, and present several lattice examples with average symmetries, including XY chain, Heisenberg chain, and deformed toric code models. In certain cases, the results can be further extended to reduced density matrices, leading to a new lower bound on the entanglement entropy. We discuss several applications in the contexts of many body localization, quantum channels, entanglement phase transitions and also derive new constraints on the Lindbladian evolution of open quantum systems., Comment: 37 pages, 1 figure

This work investigates the gapped interfaces of 3+1d fracton phases of matter using foliated gauge theories and lattice models. We analyze the gapped boundaries and gapped interfaces in X cube model, and the gapped interfaces between the X-cube model and the toric code. The gapped interfaces are either "undecorated" or "decorated", where the "decorated" interfaces have additional Chern-Simons like actions for foliated gauge fields. We discover many new gapped boundaries and interfaces, such as (1) a gapped boundary for X-cube model where the electric lineons orthogonal to the interface become the magnetic lineons, the latter are the composite of magnetic planons; (2) a Kramers-Wannier-duality type gapped interface between the X-cube model and the toric code model from gauging planar subsystem one-form symmetry; and (3) an electromagnetic duality interface in the X-cube model that exchanges the electric and magnetic lineons., Comment: 36 pages, 10 figures

In this work, we study a slave-rotor mean-field theory of an extended Hubbard model, applicable to transition metal dichalcogenide moir\'e systems, that captures both the formation of Wigner crystals as well as exotic spin states on top of these charge backgrounds. Phase diagrams are mapped out for different choices of long-range Coulomb repulsion strength, reproducing several experimentally found Wigner crystal states. Assuming unbroken time reversal symmetry, we find several spin liquid states as well as dimer states at fractional fillings. While spin dimer states are always found to have the lowest mean field energy, several spin liquid states are energetically competitive and may be stabilized by including gauge fluctuations or further interaction terms. We further discuss possible experimental signatures of these states pertinent to two-dimensional moir\'e heterostructures., Comment: 17 pages, 12 figures

Early studies proposed a connection between cuprate superconductivity and fractionalized spin liquid states. But the low temperature phase diagram is dominated by states without fractionalization, with a competition between superconductivity and charge-ordered states which break translational symmetry. Our theory uncovers novel features associated with a particular spin-liquid presumed to underlie the pseudogap metal, and shows that it has multiple nearly-degenerate instabilities to confinement of fractionalized excitations, leading to antiferromagnetism, $d$-wave superconductivity, and/or charge order. Our theory provides routes to resolving a number of open puzzles on the cuprate phase diagram. The spin liquid is described by a SU(2) gauge theory of $N_f=2$ massless fundamental Dirac fermions, has an emergent SO(5)$_f$ global symmetry, and is presumed to confine at low energies to the N\'eel state. At non-zero doping (or smaller Hubbard repulsion at half-filling) we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges moving in $\pi$ flux. At half-filling, the low energy Higgs sector has $N_b=2$ relativistic bosons with a possible emergent SO(5)$_b$ global symmetry describing rotations between a $d$-wave superconductor, period-2 charge stripes, and the time-reversal breaking `$d$-density wave' state. We propose a deconfined quantum critical point between a confining state which breaks SO(5)$_f$ and a confining state which breaks SO(5)$_b$. The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between N\'eel order and $d$-wave superconductivity. A similar theory applies at non-zero doping and large $U$, with longer-range couplings of the chargons leading to charge order with longer periods., Comment: 10+9 pages, 10+5 figures; v4 Added Ya-Hui Zhang as co-author

Given a gapped boundary of a (3+1)d topological order (TO), one can stack on it a decoupled (2+1)d TO to get another boundary theory. Should one view these two boundaries as "different"? A natural choice would be no. Different classes of gapped boundaries of (3+1)d TO should be defined modulo these decoupled (2+1)d TOs. But is this enough? We examine the possibility of coupling the boundary of a (3+1)d TO to additional (2+1)d TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk point-like excitations, when touching the boundary, become excitations in the added (2+1)d phase, while the string-like excitations in the bulk may end on the boundary but with endpoints dressed by some other excitations in the (2+1)d phase. For a good definition of "class" for gapped boundaries of (3+1)d TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the string-like excitations that can end on the boundary, whatever their endpoints are. A concrete example is the (3+1)d bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the $m$-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a (3+1)d TO characterized by a finite group $G$, different classes correspond to different subgroups of $G$. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model and field theory., Comment: v2 with minor edits; references added

Quantum circuit dynamics with local projective measurements can realize a rich spectrum of entangled states of quantum matter. Motivated by the physics of the Kitaev quantum spin liquid [1], we study quantum circuit dynamics in (2+1)-dimensions involving local projective measurements, in which the monitored trajectories realize (i) a phase with topological quantum order or (ii) a "critical" phase with a logarithmic violation of area-law-scaling of the entanglement entropy along with long range tripartite entanglement. A Majorana parton description of these dynamics, which provides an out-of-equilibrium generalization of the parton description of the Kitaev honeycomb model, permits an analytic understanding of the universal properties of these two phases, including the entanglement properties of the steady-state, the dynamics of the system on the approach to equilibrium, and the phase transition between these states. In the topologically-ordered phase, two logical qubits can be encoded in an initial state and protected for a time which scales exponentially in the linear dimension of the system, while no robust encoding of quantum information persists in the critical phase. Extensive numerical simulations of these monitored dynamics confirm our analytic predictions., Comment: Main Text (18 pages, 20 figures), Appendices (11 pages, 17 figures)

We study the boundary theory of the $\mathbb{Z}_N$ X-cube model using a continuum perspective, from which the exchange statistics of a subset of bulk excitations can be recovered. We discuss various gapped boundary conditions that either preserve or break the translation/rotation symmetries on the boundary, and further present the corresponding ground state degeneracies on $T^2\times I$. The low-energy physics is highly sensitive to the boundary conditions: even the extensive part of the ground state degeneracy can vary when different sets of boundary conditions are chosen on the two boundaries. We also examine the anomaly inflow of the boundary theory and find that the X-cube model is not the unique (3+1)d theory that cancels the 't Hooft anomaly of the boundary., Comment: v2

When two layers of two-dimensional materials are assembled with a relative twist, moir\'e patterns arise, inducing a tremendous wealth of exotic phenomena. In this work, we consider twisting two triangular lattices hosting Dirac quantum spin liquids. A single decoupled layer is described by compact quantum electrodynamics in 2+1 dimensions (QED$_3$) with an emergent $\mathrm{U}(1)$ gauge field, which is assumed to flow to a strongly interacting fixed point in the IR with conformal symmetry. We use recent results for the quantum numbers of monopole operators, which tunnel $2 \pi$ fluxes of the compact gauge field. It is found that, in the bilayer system, interlayer monopole tunneling is a symmetry-allowed relevant perturbation which induces an (ordering) instability. We show using perturbation theory that upon twisting the two layers the system remains unstable under the interlayer interaction, but any finite twist angle softens this instability compared to the untwisted case. To analyse the resulting phase induced by the (twisted) interlayer tunneling, we use "conformal mean field theory", which reduces the interacting bilayer system to two copies of QED$_3$ coupled to background fields which are to be determined self-consistently. In the weak-coupling regime, where the interlayer coupling is weak compared to the energy scale set by the moir\'e lattice constant, we solve the self-consistency equations perturbatively. In the limit of strongly coupled layers, a local scaling approximation is used, and we find that the magnetically ordered state exhibits a lattice of magnetic vortices, with the lattice constant tunable through the twisting angle., Comment: 25 pages, 5 figures; v2 (added new Section IV on conformal perturbation theory, and several additional remarks)

We discuss examples of two dimensional metallic states with charge fractionalization, and we will demonstrate that the mechanism of charge fractionalization leads to exotic metallic behaviors at low and intermediate temperature. The simplest example of such state is constructed by fermionic partons at finite density coupled to a $Z_N$ gauge field, whose properties can be studied through rudimentary methods. This simple state has the following exotic features: (1) at low temperature this state is a "bad metal" whose resistivity can exceed the Mott-Ioffe-Regel limit; (2) while increasing temperature $T$ the resistivity $\rho(T)$ is a nonmonotonic function, and it crosses over from a bad metal at low $T$ to a good metal at relatively high $T$; (3) the optical conductivity $\sigma(\omega)$ has a small Drude weight at low $T$, and a larger Drude weight at intermediate $T$; (4) at low temperature the metallic state has a large Lorenz number, which strongly violates the Wiedemann-Franz law. A more complex example with fermionic partons at finite density coupled to a SU(N) gauge field will also be constructed., Comment: 8 pages, dedicated to Prof. Chen Ning Yang

We study the entanglement entropy in 1+1 dimensional conformal field theories in the presence of interfaces from a holographic perspective. Compared with the well-known case of boundary conformal field theories, interfaces allow for several interesting new observables. Depending on how the interface is located within the entangling region, the entanglement entropies differ and exhibit surprising new patterns and universal relations. While our analysis is performed within the framework of holography, we expect our results to hold more generally., Comment: v4, corrected formulas about entanglement entropy in the RS model

It has been proposed that an extended version of the Hubbard model which potentially hosts rich correlated physics may be well simulated by the transition metal dichalcogenide (TMD) moir\'{e} heterostructures. Motivated by recent reports of continuous metal-insulator transition (MIT) at half filling, as well as correlated insulators at various fractional fillings in TMD moir\'{e} heterostructures, we propose a theory for the potentially continuous MIT with fractionalized electric charges. The charge fractionalization at the MIT will lead to various experimental observable effects, such as a large critical resistivity as well as large universal resistivity jump at the continuous MIT. These predictions are different from previously proposed theory for interaction-driven continuous MIT. Physics in phases near the MIT will also be discussed., Comment: 24 pages, 4 figures

We derive the general continuum model for a bilayer system of staggered-flux square lattices, with arbitrary elastic deformation in each layer. Applying this general continuum model to the case where the two layers are rigidly rotated relative to each other by a small angle, we obtain the band structure of the twisted bilayer staggered-flux square lattice. We show that this band structure exhibits a "magic continuum" in the sense that an exponential reduction of the Dirac velocity and bandwidths occurs in a large parameter regime. We show that the continuum model of the twisted bilayer system effectively describes a massless Dirac fermion in a spatially modulating magnetic field, whose renormalized Dirac velocity can be exactly calculated. We further give an intuitive argument for the emergence of flattened bands near half filling in the magic continuum and provide an estimation of the large number of associated nearly-zero-energy states. We also show that the entire band structure of the twisted bilayer system is free of band gaps due to symmetry constraints., Comment: 14+1 pages, 8 figures

We extend the holographic duality between 3d pure gravity and the 2d Ising CFT proposed in [Phys. Rev. D 85 (2012) 024032] to CFTs with boundaries. Besides the usual asymptotic boundary, the dual bulk spacetime now has a real cutoff, on which live branes with finite tension, giving Neumann boundary condition on the metric tensor. The strongly coupled bulk theory requires that we dress the well-known semiclassical AdS/BCFT answer with boundary gravitons, turning the partition function into the form of Virasoro characters. Using this duality, we relate the brane tensions to the modular S-matrix elements of the dual BCFT and derive the transformation between gravitational solutions with different brane tensions under modular S action., Comment: 16+8 pages, v3

We study untwisted heterobilayers of ferromagnetic and antiferromagnetic van der Waals materials, with in particular a Dzyaloshinskii-Moriya interaction in the ferromagnetic layer. A continuum low energy field theory is utilized to study such systems. We develop a phase diagram as a function of the strength of inter-layer exchange and Dzyaloshinskii-Moriya interactions, combining perturbative and strong coupling analyses with numerical simulations using Landau-Lifshitz-Gilbert equations. Various moir\'e-periodic commensurate phases are found, and the commensurate-incommensurate transition is discussed. Among the commensurate phases, we observe an interesting skyrmion lattice phase wherein each moir\'e unit cell hosts one skyrmion., Comment: v3

We introduce a general framework to study moir\'e structures of two-dimensional Van der Waals magnets using continuum field theory. The formalism eliminates quasiperiodicity and allows a full understanding of magnetic structures and their excitations. In particular, we analyze in detail twisted bilayers of N\'eel antiferromagnets on the honeycomb lattice. A rich phase diagram with non-collinear twisted phases is obtained, and spin waves are further calculated. Direct extensions to zig-zag antiferromagnets and ferromagnets are also presented. We anticipate the results and formalism presented to lead to a broad range of applications to both fundamental research and experiments.

We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius $l$ is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge $c=3l/2G_N$ ($G_N$ is the 3D Newton constant) equals $c=1/2$, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D {\bf 85}, 024032 (2012)]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the $c=1/2$ theory among all those with $c<1$, extending the previous results of Castro et al.. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a "vacuum seed". But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge $c<1$, the sum is finite and unique {\it only} when $c=1/2$, corresponding to a dual Ising conformal field theory on the asymptotic boundary., Comment: 54 pages, 12 figures

We study the rare-earth magnets on a honeycomb lattice, and are particularly interested in the experimental consequences of the highly anisotropic spin interaction due to the spin-orbit entanglement. We perform a high-temperature series expansion using a generic nearest-neighbor Hamiltonian with anisotropic interactions, and obtain the heat capacity, the parallel and perpendicular spin susceptibilities, and the magnetic torque coefficients. We further examine the electron spin resonance linewidth as an important signature of the anisotropic spin interactions. Due to the small interaction energy scale of the rare-earth moments, it is experimentally feasible to realize the strong-field regime. Therefore, we perform the spin-wave analysis and study the possibility of topological magnons when a strong field is applied to the system. The application and relevance to the rare-earth Kitaev materials are discussed., Comment: version 3