Application of the density matrix renormalization group method to finite temperatures and two-dimensional systems

The density matrix renormalization group (DMRG) method and its applications to finite temperatures and two-dimensional systems are reviewed. The basic idea of the original DMRG method, which allows precise study of the ground state properties and low-energy excitations, is presented for models which include long-range interactions. The DMRG scheme is then applied to the diagonalization of the quantum transfer matrix for one-dimensional systems, and a reliable algorithm at finite temperatures is formulated. Dynamic correlation functions at finite temperatures are calculated from the eigenvectors of the quantum transfer matrix with analytical continuation to the real frequency axis. An application of the DMRG method to two-dimensional quantum systems in a magnetic field is demonstrated and reliable results for quantum Hall systems are presented.Comment: 33 pages, 18 figures; corrected Eq.(117

The Primary Spin-4 Casimir Operators in the Holographic SO(N) Coset Minimal Models

Starting from SO(N) current algebra, we construct two lowest primary higher spin-4 Casimir operators which are quartic in spin-1 fields. For N is odd, one of them corresponds to the current in the WB_{\frac{N-1}{2}} minimal model. For N is even, the other corresponds to the current in the WD_{\frac{N}{2}} minimal model. These primary higher spin currents, the generators of wedge subalgebra, are obtained from the operator product expansion of fermionic (or bosonic) primary spin-N/2 field with itself in each minimal model respectively. We obtain, indirectly, the three-point functions with two real scalars, in the large N 't Hooft limit, for all values of the 't Hooft coupling which should be dual to the three-point functions in the higher spin AdS_3 gravity with matter.Comment: 65 pages; present the main results only and to appear in JHEP where one can see the Appendi

d+idd+id Holographic Superconductors

A holographic model of $d+id$ superconductors based on the action proposed by Benini, Herzog, and Yarom [arXiv:1006.0731] is studied. This model has a charged spin two field in an AdS black hole spacetime. Working in the probe limit, the normalizable solution of the spin two field in the bulk gives rise to a $d+id$ superconducting order parameter at the boundary of the AdS. We calculate the fermion spectral function in this\ superconducting background and confirm the existence of fermi arcs for non-vanishing Majorana couplings. By changing the relative strength $\gamma$ of the $d$ and $id$ condensations, the position and the size of the fermi arcs are changed. When $\gamma =1$, the spectrum becomes isotropic and the spectral function is s-wave like. By changing the fermion mass, the fermi momentum is changed. We also calculate the conductivity for these holographic $d+id$ superconductors where time reversal symmetry has been broken spontaneously. A non-vanishing Hall conductivity is obtained even without an external magnetic field.Comment: 24 pages,17 figures, Add more discussions on hall conductivity, two new figures, Matched with published versio

Thermodynamics and Crossover Phenomena in the Correlation Lengths of the One-Dimensional t-J Model

We investigate the thermodynamics of the one-dimensional t-J model using transfer matrix renormalization group (TMRG) algorithms and present results for quantities like particle number, specific heat, spin susceptibility and compressibility. Based on these results we confirm a phase diagram consisting of a Tomonaga-Luttinger liquid (TLL) phase for small J/t and a phase separated state for J/t large. Close to phase separation we find a spin-gap (Luther-Emery) phase at low densities consistent with predictions by other studies. At the supersymmetric point we compare our results with exact results from the Bethe ansatz and find excellent agreement. In particular we focus on the calculation of correlation lengths and static correlation functions and study the crossover from the non-universal high T lattice into the quantum critical regime. At the supersymmetric point we compare in detail with predictions by conformal field theory (CFT) and TLL theory and show the importance of logarithmic corrections.Comment: 14 pages, 20 figure

Holographic RG flow of the shear viscosity to entropy density ratio in strongly coupled anisotropic plasma

We study holographic RG flow of the shear viscosity tensor of anisotropic, strongly coupled N=4 super-Yang-Mills plasma by using its type IIB supergravity dual in anisotropic bulk spacetime. We find that the shear viscosity tensor has three independent components in the anisotropic bulk spacetime away from the boundary, and one of the components has a non-trivial RG flow while the other two have a trivial one. For the component of the shear viscosity tensor with non-trivial RG flow, we derive its RG flow equation, and solve the equation analytically to second order in the anisotropy parameter 'a'. We derive the RG equation using the equation of motion, holographic Wilsonian RG method, and Kubo's formula. All methods give the same result. Solving the equation, we find that the ratio of the component of the shear viscosity tensor to entropy density 'eta/s' flows from above '1/4pi' at the horizon (IR) to below '1/4pi' at the boundary (UV) where it violates the holographic shear viscosity (Kovtun-Son-Starinets) bound and where it agrees with the other longitudinal component.Comment: 17 pages, 2 figures, slight change on the title, more background material added, references added, accepted for publication in JHE

Anomalous Zero Sound

We show that the anomalous term in the current, recently suggested by Son and Yamamoto, modifies the structure of the zero sound mode in the Fermi liquid in a magnetic field.Comment: 14 pages, 2 figure

The Many Phases of Holographic Superfluids

We investigate holographic superfluids in AdS_{d+1} with d=3,4 in the non-backreacted approximation for various masses of the scalar field. In d=3 the phase structure is universal for all the masses that we consider: the critical temperature decreases as the superfluid velocity increases, and as it is cranked high enough, the order of the phase transition changes from second to first. Surprisingly, in d=4 we find that the phase structure is more intricate. For sufficiently high mass, there is always a second order phase transition to the normal phase, no matter how high the superfluid velocity. For some parameters, as we lower the temperature, this transition happens before a first order transition to a new superconducting phase. Across this first order transition, the gap in the transverse conductivity jumps from almost zero to about half its maximum value. We also introduce a double scaling limit where we can study the phase transitions (semi-)analytically in the large velocity limit. The results corroborate and complement our numerical results. In d=4, this approach has the virtue of being fully analytically tractable.Comment: 31 pages, 19 figure

The Operator Product Expansion of the Lowest Higher Spin Current at Finite N

For the N=2 Kazama-Suzuki(KS) model on CP^3, the lowest higher spin current with spins (2, 5/2, 5/2,3) is obtained from the generalized GKO coset construction. By computing the operator product expansion of this current and itself, the next higher spin current with spins (3, 7/2, 7/2, 4) is also derived. This is a realization of the N=2 W_{N+1} algebra with N=3 in the supersymmetric WZW model. By incorporating the self-coupling constant of lowest higher spin current which is known for the general (N,k), we present the complete nonlinear operator product expansion of the lowest higher spin current with spins (2, 5/2, 5/2, 3) in the N=2 KS model on CP^N space. This should coincide with the asymptotic symmetry of the higher spin AdS_3 supergravity at the quantum level. The large (N,k) 't Hooft limit and the corresponding classical nonlinear algebra are also discussed.Comment: 62 pages; the footnotes added, some redundant appendices removed, the presentations in the whole paper improved and to appear in JHE

Moduli Spaces of Cold Holographic Matter

We use holography to study (3+1)-dimensional N=4 supersymmetric Yang-Mills theory with gauge group SU(Nc), in the large-Nc and large-coupling limits, coupled to a single massless (n+1)-dimensional hypermultiplet in the fundamental representation of SU(Nc), with n=3,2,1. In particular, we study zero-temperature states with a nonzero baryon number charge density, which we call holographic matter. We demonstrate that a moduli space of such states exists in these theories, specifically a Higgs branch parameterized by the expectation values of scalar operators bilinear in the hypermultiplet scalars. At a generic point on the Higgs branch, the R-symmetry and gauge group are spontaneously broken to subgroups. Our holographic calculation consists of introducing a single probe Dp-brane into AdS5 times S^5, with p=2n+1=7,5,3, introducing an electric flux of the Dp-brane worldvolume U(1) gauge field, and then obtaining explicit solutions for the worldvolume fields dual to the scalar operators that parameterize the Higgs branch. In all three cases, we can express these solutions as non-singular self-dual U(1) instantons in a four-dimensional space with a metric determined by the electric flux. We speculate on the possibility that the existence of Higgs branches may point the way to a counting of the microstates producing a nonzero entropy in holographic matter. Additionally, we speculate on the possible classification of zero-temperature, nonzero-density states described holographically by probe D-branes with worldvolume electric flux.Comment: 56 pages, 8 PDF images, 4 figure

The Large N 't Hooft Limit of Kazama-Suzuki Model

We consider N=2 Kazama-Suzuki model on CP^N=SU(N+1)/SU(N)xU(1). It is known that the N=2 current algebra for the supersymmetric WZW model, at level k, is a nonlinear algebra. The N=2 W_3 algebra corresponding to N=2 was recovered from the generalized GKO coset construction previously. For N=4, we construct one of the higher spin currents, in N=2 W_5 algebra, with spins (2, 5/2, 5/2, 3). The self-coupling constant in the operator product expansion of this current and itself depends on N as well as k explicitly. We also observe a new higher spin primary current of spins (3, 7/2, 7/2, 4). From the behaviors of N=2, 4 cases, we expect the operator product expansion of the lowest higher spin current and itself in N=2 W_{N+1} algebra. By taking the large (N, k) limit on the various operator product expansions in components, we reproduce, at the linear order, the corresponding operator product expansions in N=2 classical W_{\infty}^{cl}[\lambda] algebra which is the asymptotic symmetry of the higher spin AdS_3 supergravity found recently.Comment: 44 pages; the two typos in the first paragraph of page 23 corrected and to appear in JHE