Invariant Ground State Research Articles

Massless Lifshitz field theory for arbitrary z

By using the notion of fractional derivatives, we introduce a class of massless Lifshitz scalar field theory in (1+1)-dimension with an arbitrary anisotropy index z. The Lifshitz scale invariant ground state of the theory is constructed explicitly and takes the form of Rokhsar-Kivelson (RK). We show that there is a continuous family of ground states with degeneracy parameterized by the choice of solution to the equation of motion of an auxiliary classical system. The quantum mechanical path integral establishes a 2d/1d correspondence with the equal time correlation functions of the Lifshitz scalar field theory. We study the entanglement properties of the Lifshitz theory for arbitrary z using the path integral representation. The entanglement measures are expressed in terms of certain cross ratio functions we specify, and satisfy the c-function monotonicity theorems. We also consider the holographic description of the Lifshitz theory. In order to match with the field theory result for the entanglement entropy, we propose a z-dependent radius scale for the Lifshitz background. This relation is consistent with the z-dependent scaling symmetry respected by the Lifshitz vacuum. Furthermore, the time-like entanglement entropy is determined using holography. Our result suggests that there should exist a fundamental definition of time-like entanglement other than employing analytic continuation as performed in relativistic field theory.

Comments on single-trace Toverline{T} holography

We explore the holographic duality between string theory in backgrounds that interpolate between asymptotically linear dilaton spacetime in the UV and AdS3 in the IR, and single-trace Toverline{T} deformed CFT. In particular, we explain how the deformation of states in the boundary theory is reflected in the bulk geometry, and show that the coupling above which the deformed energy of the SL(2, ℝ) invariant ground state of the IR CFT becomes complex is a maximal coupling.

Global phase diagram of charge-neutral graphene in the quantum Hall regime for generic interactions

Monolayer graphene at charge neutrality in a quantizing magnetic field is a quantum Hall ferromagnet. Due to the spin and valley (near) degeneracies, there is a plethora of possible ground states. Previous theoretical work, based on a stringent ultra short-range assumption on the symmetry-allowed interactions, predicts a phase diagram with distinct regions of spin-polarized, canted antiferromagnetic, inter-valley coherent, and charge density wave order. While early experiments suggested that the system was in the canted antiferromagnetic phase at a perpendicular field, recent scanning tunneling studies universally find Kekul\'e bond order, and sometimes also charge density wave order. Recently, it was found that if one relaxes the stringent assumption mentioned above, a phase with coexisting canted antiferromagnetic and Kekul\'e order exists in the region of the phase diagram believed to correspond to real samples. In this work, starting from the continuum limit appropriate for experiments, we present the complete phase diagram of $\nu=0$ graphene in the Hartree-Fock approximation, using generic symmetry-allowed interactions, assuming translation invariant ground states up to an intervalley coherence. Allowing for a sublattice potential (valley Zeeman coupling), we find numerous phases with different types of coexisting order. We conclude with a discussion of the physical signatures of the various states.

Novel semi-circle law and Hall sliding in a strongly interacting electron liquid

We study a strongly interacting, fermionic fluid in the presence of an applied magnetic field using a holographic framework. At low temperatures, translation symmetry is spontaneously broken and the resulting phase is a striped Hall fluid. Due to the magnetic field, an electric field applied parallel to the stripes causes the stripes to slide, a phenomenon we coin “Hall sliding.” We also investigate the magneto-transport of the system in the presence of an explicit translation symmetry-breaking lattice which pins the stripes. Electrical properties are well represented by a hydrodynamical model, which gives us further insight into particle-like cyclotron and pseudo-Goldstone excitations we observe. The DC conductivities obey a novel semi-circle law, which we derive analytically in the translationally invariant ground state at low temperature.

Entanglement renormalization for gauge invariant quantum fields

The continuous multiscale entaglement renormalization ansatz (cMERA) [Haegeman et al., Phys. Rev. Lett., 110, 100402 (2013)] is a variational wavefunctional for ground states of quantum field theories. So far, only scalar bosons and fermions have been considered. In this paper we explain how to generalize the cMERA framework to gauge invariant quantum fields. The fundamental difficulty to be addressed is how to make the gauge constraints (local linear constraints in the Hilbert space) compatible with the UV structure of the cMERA wavefunctional (which is generated by a quasi-local entangler). For simplicity, we consider $U(1)$ gauge theory in $d+1$ spacetime dimensions, a non-interacting theory with massless Hamiltonian $H_{U(1)}$ and Gaussian scale invariant ground state $|\Psi_{U(1)}\rangle$. We propose a gauge invariant cMERA wavefunctional $|\Psi^{\Lambda}_{U(1)}\rangle$ that, by construction, accurately reproduces the long distance properties of $|\Psi_{U(1)}\rangle$ while remaining somewhat unentangled at short distances. Moreover, $|\Psi^{\Lambda}_{U(1)}\rangle$ is the exact ground state of a gauge invariant, local Hamiltonian $H^{\Lambda}_{U(1)}$ whose low energy properties coincide with those of $H_{U(1)}$. Our construction also extends the cMERA formalism to massive (non-gauge invariant) vector boson quantum fields.

AdS2 holography: mind the cap

AdS2 plays an extremely important role in black-hole physics. We construct several infinite families of supergravity solutions that are asymptotically AdS2 in the UV, and terminate in the IR with a cap that is singular in two dimensions but smooth in ten dimensions. These solutions break conformal invariance, and should correspond to supersymmetric ground states of a holographically dual CFT1. We solve the free massless scalar wave equation on a family of these solutions, finding towers of finite-energy normalizable bound-state excitations. We discuss the intriguing possibility that these excitations correspond to time-dependent excitations of the dual CFT1, which would imply that this CFT1 is dynamical rather than topological, and hence cannot have a conformally invariant ground state.

Dynamics for QCD on an Infinite Lattice

We prove the existence of the dynamics automorphism group for Hamiltonian QCD on an infinite lattice in R^3, and this is done in a C*-algebraic context. The existence of ground states is also obtained. Starting with the finite lattice model for Hamiltonian QCD developed by Kijowski & Rudolph, we state its field algebra and a natural representation. We then generalize this representation to the infinite lattice, and construct a Hilbert space which has represented on it all the local algebras (i.e. kinematics algebras associated with finite connected sublattices) equipped with the correct graded commutation relations. On a suitably large C*-algebra acting on this Hilbert space, and containing all the local algebras, we prove that there is a one parameter automorphism group, which is the pointwise norm limit of the local time evolutions along a sequence of finite sublattices, increasing to the full lattice. This is our global time evolution. We then take as our field algebra the C*-algebra generated by all the orbits of the local algebras w.r.t. the global time evolution. Thus the time evolution creates the field algebra. The time evolution is strongly continuous on this choice of field algebra, though not on the original larger C*-algebra. We define the gauge transformations, explain how to enforce the Gauss law constraint, show that the dynamics automorphism group descends to the algebra of physical observables and prove that gauge invariant ground states exist.

Hohenberg-Kohn theorems in electrostatic and uniform magnetostatic fields.

The Hohenberg-Kohn (HK) theorems of bijectivity between the external scalar potential and the gauge invariant nondegenerate ground state density, and the consequent Euler variational principle for the density, are proved for arbitrary electrostatic field and the constraint of fixed electron number. The HK theorems are generalized for spinless electrons to the added presence of an external uniform magnetostatic field by introducing the new constraint of fixed canonical orbital angular momentum. Thereby, a bijective relationship between the external scalar and vector potentials, and the gauge invariant nondegenerate ground state density and physical current density, is proved. A corresponding Euler variational principle in terms of these densities is also developed. These theorems are further generalized to electrons with spin by imposing the added constraint of fixed canonical orbital and spin angular momenta. The proofs differ from the original HK proof and explicitly account for the many-to-one relationship between the potentials and the nondegenerate ground state wave function. A Percus-Levy-Lieb constrained-search proof expanding the domain of validity to N-representable functions, and to degenerate states, again for fixed electron number and angular momentum, is also provided.

Haag duality for Kitaev’s quantum double model for abelian groups

We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

3D field theories with Chern-Simons term for large N in the Weyl gauge

Three dimensional, $U(N)$ symmetric, field theory with fermion matter coupled to a topological Chern--Simons term, in the large $N$ limit is analyzed in details. We determine the conditions for the existence of a massless conformal invariant ground state as well as the conditions for a massive phase. We analyze the phase structure and calculate gauge invariant corelators comparing them in several cases to existing results. In addition to the non-critical explicitly broken scale invariance massive case we consider also a massive ground state where the scale symmetry is spontaneously broken. We show that such a phase appears only in the presence of a marginal deformation that is introduced by adding a certain scalar auxiliary field and discuss the fermion-boson dual mapping. The ground state contains in this case a massless $U(N)$ singlet bound state goldstone boson- the dilaton whose properties are determined. We employ here the temporal gauge which is at variance with respect to past calculations using the light-cone gauge and thus, a check (though limited) of gauge independence is at hand. The large $N$ properties are determined by using a field integral formalism and the steepest descent method. The saddle point equations, which take here the form of integral equations for non-local fields, determine the mass gap and the dressed fermion propagator. Vertex functions are calculated at leading order in $1/N$ as exact solutions of integral equations. From the vertex functions, we infer gauge invariant two-point correlation functions for scalar operators and a current. Indications about the consistency of the method are obtained by verifying that gauge-invariant quantities have a natural $O(3)$ covariant form. As a further verification, in several occasions, a few terms of the perturbative expansion are calculated and compared with the exact results in the appropriate order.

BOUNDEDNESS OF ENTANGLEMENT ENTROPY AND SPLIT PROPERTY OF QUANTUM SPIN CHAINS

We show that boundedness of entanglement entropy for pure states of bipartite quantum spin systems implies split property of subsystems. As a corollary, in one-dimensional quantum spin chains, we show that the split property with respect to left and right semi-infinite subsystems is valid for the translationally invariant pure ground states with spectral gap.

THE 1D HEISENBERG ANTIFERROMAGNET MODEL BY THE VARIATION AFTER PROJECTION METHOD

The four site and eight site 1D anti-ferromagnetic Heisenberg chains in the Jordan–Wigner representation are investigated within the standard Hartree–Fock and random phase approximation (RPA) approaches, both in the symmetry unbroken and in the symmetry broken phases. A translation invariant groundstate, obtained by the projection method as a linear combination of a symmetry-broken HF state and its image under reflection, is also considered, for each chain type. It is found that the projection method considerably improves the HF treatment for instance as far as the groundstate energy is concerned, but also with respect to the RPA energies. The results are furthermore confronted with the ones obtained within so-called SCRPA scheme.

Infinite boundary conditions for matrix product state calculations

We propose a formalism to study dynamical properties of a quantum many-body system in the thermodynamic limit by studying a finite system with infinite boundary conditions (IBC) where both finite size effects and boundary effects have been eliminated. For one-dimensional systems, infinite boundary conditions are obtained by attaching two boundary sites to a finite system, where each of these two sites effectively represents a semi-infinite extension of the system. One can then use standard finite-size matrix product state techniques to study a region of the system while avoiding many of the complications normally associated with finite-size calculations such as boundary Friedel oscillations. We illustrate the technique with an example of time evolution of a local perturbation applied to an infinite (translationally invariant) ground state, and use this to calculate the spectral function of the S=1 Heisenberg spin chain. This approach is more efficient and more accurate than conventional simulations based on finite-size matrix product state and density-matrix renormalization-group approaches.

Symmetry classification of spin-orbit-coupled spinor Bose-Einstein condensates

We develop a symmetry classification scheme to find ground states of pseudo spin-1/2, spin-1, and spin-2 spin-orbit coupled spinor Bose-Einstein condensates, and show that as the SO(2) symmetry of simultaneous spin and space rotations is broken into discrete cyclic groups, various types of lattice structures emerge in the absence of a lattice potential, examples include two different kagaome lattices for pseudo spin-1/2 condensates and a nematic vortex lattice in which uniaxial and biaxial spin textures align alternatively for spin-2 condensates. For the pseudo spin-1/2 system, although mean-field states always break time-reversal symmetry, there exists a time-reversal invariant many-body ground state, which is fragmented and expected to be observed in a micro-condensate.

THE PRINCIPLE OF LOCAL ROTATIONAL INVARIANCE AND THE COEXISTENCE OF MAGNETISM, CHARGE AND SUPERCONDUCTIVITY

We propose a macroscopic description of the superconducting state in presence of an applied external magnetic field in terms of first order differential equations. They describe a corrugated two-component order parameter intertwined with a spin-charged background, caused by spin correlations and charged dislocations. The first order differential equations are a consequence of a Weitzenböck–Liechnorowitz identity which renders a SUL(2) ⊗ UL(1) invariant ground state, based on (L) local rotational and electromagnetic gauge symmetry. The proposal is based on a long ago developed formalism by Élie Cartan to investigate curved spaces, viewed as a collection of small Euclidean granules that are translated and rotated with respect to each other. Élie Cartan's formalism unveils the principle of local rotational invariance as a gauge symmetry because the global SU(2) invariance of the order parameter is turned local by the interlacement of spin and charge to pairing.

Density and physical current density functional theory

AbstractFor a system of N electrons in an external scalar potential v(r) and external vector potential A(r), we prove that the wave function ψ is a functional of the gauge invariant ground state density ρ(r) and ground state physical current density j(r), and a gauge function α(R) (with R = r1,…,rN):ψ=ψ [ρ,j,α]. It is the presence of the gauge function α(R) that ensures the wave function functional is gauge variant. We prove this via a unitary transformation and by a proof of the bijectivity between the potentials {v(r),A(r)} and the ground state properties {ρ(r),j(r)}. Thus, the natural basic variables for the system are the gauge invariant ρ(r) and j(r). Because each choice of gauge function corresponds to the same physical system, the choice of α(R) = 0 is equally valid. As such, we construct a {ρ(r),j(r)} functional theory with the corresponding Euler equations for the density ρ(r) and physical current density j(r) , together with the constraints of charge conservation and continuity of the current. With the assumption of existence of a system of noninteracting fermions with the same ρ(r) and j(r) as that of the electrons, we provide the equations describing this model system, the definitions being within the framework of Kohn–Sham theory in terms of energy functionals of {ρ(r),j(r)} and their functional derivatives. A special case of the {ρ(r),j(r)} functional theory is the magnetic‐field density‐functional theory of Grayce and Harris. We discuss and contrast our work with the paramagnetic current‐ and density‐functional theory of Vignale and Rasolt in which the variables are the gauge invariant ground state density ρ(r), and vorticity ν(r) = ∇× (jp(r)/ρ(r)) , where jp(r) is the paramagnetic current density. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

On contour arguments for the three state Potts model with competing interactions on a semi-infinite Cayley tree

We consider the Potts model with competing two-step interactions and spin values 1, 2, 3 on a semi-infinite Cayley tree of order 2. We describe ground states and verify the Peierls condition for the model. Using a contour argument we show the existence of three different Gibbs measures associated with translation invariant ground states.

Entangling power of permutation-invariant quantum states

We investigate the von Neumann entanglement entropy as function of the size of a subsystem for permutation invariant ground states in models with finite number of states per site, e.g., in quantum spin models. We demonstrate that the entanglement entropy of $n$ sites in a system of length $L$ generically grows as $\sigma\log_{2}[2\pi en(L-n)/L]+C$, where $\sigma$ is the on-site spin and $C$ is a function depending only on magnetization.

The infinite Brownian loop on a symmetric space

The infinite Brownian loop \{B_t^0,t\ge 0\} on a Riemannian manifold \mathbb{M} is the limit in distribution of the Brownian bridge of length T around a fixed origin 0 , when T\to+\infty . It has no spectral gap. When \mathbb{M} has nonnegative Ricci curvature, B^0 is the Brownian motion itself. When \mathbb{M}=G/K is a noncompact symmetric space, B^0 is the relativized \Phi_0 -process of the Brownian motion, where \Phi_0 denotes the basic spherical function of Harish-Chandra, i.e. the K -invariant ground state of the Laplacian. In this case, we consider the polar decomposition B_t^0=(K_t,X_t) , where K_t\in K/M and X_t\in\bar{\mathfrak a}_+ , the positive Weyl chamber. Then, as t\to+\infty , K_t converges and d(0,X_t)/t\to 0 almost surely. Moreover the processes \{X_{tT}/\sqrt{T},t\ge 0\} converge in distribution, as T\to+\infty , to the intrinsic Brownian motion of the Weyl chamber. This implies in particular that d(0,X_{tT})/\sqrt{T} converges to a Bessel process of dimension D=\mathrm{rank} \mathbb{M}+2j , where j denotes the number of positive indivisible roots. An ingredient of the proof is a new estimate on \Phi_0 .

Running couplings and triviality of field theories on noncommutative spaces

We examine the issue of renormalizability of asymptotically free field theories on non-commutative spaces. As an example, we solve the non-commutative O(N) invariant Gross-Neveu model at large N. On commutative space this is a renormalizable model with non-trivial interactions. On the noncommutative space, if we take the translation invariant ground state, we find that the model is non-renormalizable. Removing the ultraviolet cutoff yields a trivial non-interacting theory.