We map spin ladders with $n_l$ legs and couplings $J'$ across all rungs and $J(1 \pm \gamma)$... more We map spin ladders with $n_l$ legs and couplings $J'$ across all rungs and $J(1 \pm \gamma)$ along the legs, staggered in both directions, to a sigma model. Setting its $\theta = (2m+1)\pi$ (where it is known to be gapless), we locate the critical curves in the $\gamma$ versus ${J'\over J}$ plane at each $n_l$, and spin $S$. The phase diagram is rich and has some surprises: when two gapped chains are suitably coupled, the combination becomes gapless. With $n_l, \gamma $ and $J'/J$ to control, the prospects for experimentally observing any one of these equivalent transitions seem bright. We discuss the order parameters and the behavior of holes in the RVB description.
We propose a quantum circuit that creates a pure state corresponding to the quantum superposition... more We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.
We review the non linear sigma model approach (NLSM) to spin chains and spin ladders, presenting ... more We review the non linear sigma model approach (NLSM) to spin chains and spin ladders, presenting new results. The generalization of the Haldane's map to ladders in the Hamiltonian approach, give rise to different values of the θ parameter depending on the spin S, the number of legs n ℓ and the choice of blocks needed to built up the NLSM fields. For rectangular blocks we obtain θ = 0 or 2πS depending on wether n ℓ , is even or odd, while for diagonal blocks we obtain θ = 2πSn ℓ. Both results agree modulo 2π, and yield the same prediction, namely that even (resp. odd) ladders are gapped (resp. gapless). For even leeged ladders we show that the spin gap collapses exponentially with n ℓ and we propose a finite size correction to the gap formula recently derived by Chakravarty using the 2+1 NSLM, which gives a good fit of numerical results. We show the existence of a Haldane phase in the two legged ladder using diagonal blocks and finally we consider the phase diagram of dimerized ladders.
We propose that the valence bonds forming the ground state of the 2D-AF Heisenberg model on a squ... more We propose that the valence bonds forming the ground state of the 2D-AF Heisenberg model on a square lattice may rotate under the effect of the antiferromagnetic background. To test this idea we apply a real space renormalization group approach to construct a rotating-valence-bond ground state anstaz. Our results are analytic and valid for any value of the spin S of the AF-magnet, which allow us to perfom semiclassical expansions of the energy and the staggered magnetization.
Recently, Coulomb blockade physics was observed at room temperature in a carbon nanotube single-e... more Recently, Coulomb blockade physics was observed at room temperature in a carbon nanotube single-electron transistor (H. W. Ch. Postma, et. al., Science 293, 76 (2001)). In this work, we suggest that these devices may be promising for studying the Kondo effect. In particular, they could allow for a detailed investigation of the 2-channel Kondo fixed point. Moreover, fabricating a similar device in a short nanotube could be promising for studying the effect of a magnetic impurity in an ultrasmall metallic grain. Experimental signatures of the Kondo effect in these systems is discussed.
We study the ground state of the four-leg spin ladder using a dimer-resonating valence bond ͑RVB͒... more We study the ground state of the four-leg spin ladder using a dimer-resonating valence bond ͑RVB͒ ansatz and the Lanczos method. Besides the well-known resonance mechanism between valence bond configurations we find interference effects among nearby resonances. These effects were missed by standard factorizing approaches to the RVB states. ͓S0163-1829͑99͒08441-6͔ PHYSICAL REVIEW B
We introduce a class of models defined on ladders with a diagonal structure generated by np plaqu... more We introduce a class of models defined on ladders with a diagonal structure generated by np plaquettes. The case np = 1 corresponds to the necklace ladder and has remarkable properties which are studied using DMRG and recurrent variational ansatzes. The AF Heisenberg model on this ladder is equivalent to the alternating spin-1/spin-1 2 AFH chain which is known to have a ferrimagnetic ground state (GS). For doping 1/3 the GS is a fully doped (1,1) stripe with the holes located mostly along the principal diagonal while the minor diagonals are occupied by spin singlets. This state can be seen as a Mott insulator of localized Cooper pairs on the plaquettes. A physical picture of our results is provided by a tp − Jp − t d model of plaquettes coupled diagonally with a hopping parameter t d. In the limit t d → ∞ we recover the original t − J model on the necklace ladder while for weak hopping parameter the model is easily solvable. The GS in the strong hopping regime is essentially an "on link" Gutzwiller projection of the weak hopping GS. We generalize the tp − Jp − t d model to diagonal ladders with np > 1 and the 2D square lattice. We use in our construction concepts familiar in Statistical Mechanics such as medial graphs and Bratelli diagrams.
Using the Boltzmann weights of classical Statistical Mechanics vertex models we define a new clas... more Using the Boltzmann weights of classical Statistical Mechanics vertex models we define a new class of Tensor Product Ansatzs for 2D quantum lattice systems, characterized by a strong anisotropy, which gives rise to stripe like structures. In the case of the six vertex model we compute exactly, in the thermodynamic limit, the norm of the ansatz and other observables. Employing this ansatz we study the phase diagram of a Hamiltonian given by the sum of XXZ Hamiltonians along the legs coupled by an Ising term. Finally, we suggest a connection between the six and eight-vertex Anisotropic Tensor Product Ansatzs, and their associated Hamiltonians, with the smectic stripe phases recently discussed in the literature.
Progress in Theoretical Chemistry and Physics, 2000
In this note, first the Recurrent Variational Approach (RVA) is introduced by using as example a ... more In this note, first the Recurrent Variational Approach (RVA) is introduced by using as example a non-trivial spin-model, the spin-1/2 antiferromagnetic two-leg-ladder. Then, a first application of this scheme to the electronic structure of conjugated polymers is proposed. An analogy between the two-leg ladder and the dimerized chain described by the Extended Peierls-Hubbard Hamiltonian is underlined and comparisons are made with DMRG calculations. The proposed ansatz is useful to get some analytical insight and it is a good starting point for further improvements using the RVA.
We consider bipartitions of one-dimensional extended systems whose probability distribution funct... more We consider bipartitions of one-dimensional extended systems whose probability distribution functions describe stationary states of stochastic models. We define estimators of the shared information between the two subsystems. If the correlation length is finite, the estimators stay finite for large system sizes. If the correlation length diverges, so do the estimators. The definition of the estimators is inspired by information theory. We look at several models and compare the behavior of the estimators in the finite-size scaling limit. Analytical and numerical methods as well as Monte Carlo simulations are used. We show how the finite-size scaling functions change for various phase transitions, including the case where one has conformal invariance.
We find the classification of diagonal spin ladders depending on a characteristic integer Np in t... more We find the classification of diagonal spin ladders depending on a characteristic integer Np in terms of ferrimagnetic, gapped and critical phases. We use the finite algorithm DMRG, non-linear sigma model and bosonization techniques to prove our results. We find stoichiometric contents in cuprate CuO2 planes that allow for the existence of weakly interacting diagonal ladders.
We provide a physically meaningful picture of the nature of the ground state of the Cr8Ni compoun... more We provide a physically meaningful picture of the nature of the ground state of the Cr8Ni compound in the regime where it is a spin singlet. According to this picture, the anisotropy of the N i atom in the Cr ring induces a dimerization in the molecule that makes the ground state to stabilize in a Valence Bond Solid phase of virtual spins. We characterize rigorously this phase by means of a particular non-local order parameter denoted the generalized string order parameter. In the completely antiferromagnetic regime, the system becomes frustrated. We have performed a numerical real-time evolution study of the correlations between the spin of the N i impurity and the rest of the spins in order to show the reaction of the system under this frustration. We provide results that manifestly show the agreement with the Valence Bond Solid picture introduced.
We determine the existence of critical lines in dimerized quantum spin ladders in their phase dia... more We determine the existence of critical lines in dimerized quantum spin ladders in their phase diagram of coupling constants using the finite-size DMRG algorithm. We consider both staggered and columnar dimerization patterns, and antiferromagnetic and ferromagnetic inter-leg couplings. The existence of critical phases depends on the precise combination of these patterns. The nature of the massive phases separating the critical lines are characterized with generalized string order parameters that determine their valence bond solid (VBS) content.
In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp giv... more In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.
Journal of Statistical Mechanics: Theory and Experiment, 2011
We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantu... more We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantum spin chains and corresponding ground state wave function(s). The approach is quite general, and we illustrate it by constructing a family of Hamiltonians whose ground states are the chiral correlators of the SU (2) k WZW model for integer values of the level k. The simplest example corresponds to k = 1 and is essentially a nonuniform generalization of the Haldane-Shastry model with long-range exchange couplings. At level k = 2, we analyze the model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show, respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors to derive a set of algebraic, linear equations relating spin correlators within each model. At level k = 1, these equations allow us to compute the two-point spin correlators analytically for the finite chain uniform Haldane-Shastry model and to obtain numerical results for the nonuniform case and for higher-point spin correlators in a very simple way and without resorting to Monte Carlo techniques.
We map spin ladders with $n_l$ legs and couplings $J'$ across all rungs and $J(1 \pm \gamma)$... more We map spin ladders with $n_l$ legs and couplings $J'$ across all rungs and $J(1 \pm \gamma)$ along the legs, staggered in both directions, to a sigma model. Setting its $\theta = (2m+1)\pi$ (where it is known to be gapless), we locate the critical curves in the $\gamma$ versus ${J'\over J}$ plane at each $n_l$, and spin $S$. The phase diagram is rich and has some surprises: when two gapped chains are suitably coupled, the combination becomes gapless. With $n_l, \gamma $ and $J'/J$ to control, the prospects for experimentally observing any one of these equivalent transitions seem bright. We discuss the order parameters and the behavior of holes in the RVB description.
We propose a quantum circuit that creates a pure state corresponding to the quantum superposition... more We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than $2^n$, where $n$ is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.
We review the non linear sigma model approach (NLSM) to spin chains and spin ladders, presenting ... more We review the non linear sigma model approach (NLSM) to spin chains and spin ladders, presenting new results. The generalization of the Haldane's map to ladders in the Hamiltonian approach, give rise to different values of the θ parameter depending on the spin S, the number of legs n ℓ and the choice of blocks needed to built up the NLSM fields. For rectangular blocks we obtain θ = 0 or 2πS depending on wether n ℓ , is even or odd, while for diagonal blocks we obtain θ = 2πSn ℓ. Both results agree modulo 2π, and yield the same prediction, namely that even (resp. odd) ladders are gapped (resp. gapless). For even leeged ladders we show that the spin gap collapses exponentially with n ℓ and we propose a finite size correction to the gap formula recently derived by Chakravarty using the 2+1 NSLM, which gives a good fit of numerical results. We show the existence of a Haldane phase in the two legged ladder using diagonal blocks and finally we consider the phase diagram of dimerized ladders.
We propose that the valence bonds forming the ground state of the 2D-AF Heisenberg model on a squ... more We propose that the valence bonds forming the ground state of the 2D-AF Heisenberg model on a square lattice may rotate under the effect of the antiferromagnetic background. To test this idea we apply a real space renormalization group approach to construct a rotating-valence-bond ground state anstaz. Our results are analytic and valid for any value of the spin S of the AF-magnet, which allow us to perfom semiclassical expansions of the energy and the staggered magnetization.
Recently, Coulomb blockade physics was observed at room temperature in a carbon nanotube single-e... more Recently, Coulomb blockade physics was observed at room temperature in a carbon nanotube single-electron transistor (H. W. Ch. Postma, et. al., Science 293, 76 (2001)). In this work, we suggest that these devices may be promising for studying the Kondo effect. In particular, they could allow for a detailed investigation of the 2-channel Kondo fixed point. Moreover, fabricating a similar device in a short nanotube could be promising for studying the effect of a magnetic impurity in an ultrasmall metallic grain. Experimental signatures of the Kondo effect in these systems is discussed.
We study the ground state of the four-leg spin ladder using a dimer-resonating valence bond ͑RVB͒... more We study the ground state of the four-leg spin ladder using a dimer-resonating valence bond ͑RVB͒ ansatz and the Lanczos method. Besides the well-known resonance mechanism between valence bond configurations we find interference effects among nearby resonances. These effects were missed by standard factorizing approaches to the RVB states. ͓S0163-1829͑99͒08441-6͔ PHYSICAL REVIEW B
We introduce a class of models defined on ladders with a diagonal structure generated by np plaqu... more We introduce a class of models defined on ladders with a diagonal structure generated by np plaquettes. The case np = 1 corresponds to the necklace ladder and has remarkable properties which are studied using DMRG and recurrent variational ansatzes. The AF Heisenberg model on this ladder is equivalent to the alternating spin-1/spin-1 2 AFH chain which is known to have a ferrimagnetic ground state (GS). For doping 1/3 the GS is a fully doped (1,1) stripe with the holes located mostly along the principal diagonal while the minor diagonals are occupied by spin singlets. This state can be seen as a Mott insulator of localized Cooper pairs on the plaquettes. A physical picture of our results is provided by a tp − Jp − t d model of plaquettes coupled diagonally with a hopping parameter t d. In the limit t d → ∞ we recover the original t − J model on the necklace ladder while for weak hopping parameter the model is easily solvable. The GS in the strong hopping regime is essentially an "on link" Gutzwiller projection of the weak hopping GS. We generalize the tp − Jp − t d model to diagonal ladders with np > 1 and the 2D square lattice. We use in our construction concepts familiar in Statistical Mechanics such as medial graphs and Bratelli diagrams.
Using the Boltzmann weights of classical Statistical Mechanics vertex models we define a new clas... more Using the Boltzmann weights of classical Statistical Mechanics vertex models we define a new class of Tensor Product Ansatzs for 2D quantum lattice systems, characterized by a strong anisotropy, which gives rise to stripe like structures. In the case of the six vertex model we compute exactly, in the thermodynamic limit, the norm of the ansatz and other observables. Employing this ansatz we study the phase diagram of a Hamiltonian given by the sum of XXZ Hamiltonians along the legs coupled by an Ising term. Finally, we suggest a connection between the six and eight-vertex Anisotropic Tensor Product Ansatzs, and their associated Hamiltonians, with the smectic stripe phases recently discussed in the literature.
Progress in Theoretical Chemistry and Physics, 2000
In this note, first the Recurrent Variational Approach (RVA) is introduced by using as example a ... more In this note, first the Recurrent Variational Approach (RVA) is introduced by using as example a non-trivial spin-model, the spin-1/2 antiferromagnetic two-leg-ladder. Then, a first application of this scheme to the electronic structure of conjugated polymers is proposed. An analogy between the two-leg ladder and the dimerized chain described by the Extended Peierls-Hubbard Hamiltonian is underlined and comparisons are made with DMRG calculations. The proposed ansatz is useful to get some analytical insight and it is a good starting point for further improvements using the RVA.
We consider bipartitions of one-dimensional extended systems whose probability distribution funct... more We consider bipartitions of one-dimensional extended systems whose probability distribution functions describe stationary states of stochastic models. We define estimators of the shared information between the two subsystems. If the correlation length is finite, the estimators stay finite for large system sizes. If the correlation length diverges, so do the estimators. The definition of the estimators is inspired by information theory. We look at several models and compare the behavior of the estimators in the finite-size scaling limit. Analytical and numerical methods as well as Monte Carlo simulations are used. We show how the finite-size scaling functions change for various phase transitions, including the case where one has conformal invariance.
We find the classification of diagonal spin ladders depending on a characteristic integer Np in t... more We find the classification of diagonal spin ladders depending on a characteristic integer Np in terms of ferrimagnetic, gapped and critical phases. We use the finite algorithm DMRG, non-linear sigma model and bosonization techniques to prove our results. We find stoichiometric contents in cuprate CuO2 planes that allow for the existence of weakly interacting diagonal ladders.
We provide a physically meaningful picture of the nature of the ground state of the Cr8Ni compoun... more We provide a physically meaningful picture of the nature of the ground state of the Cr8Ni compound in the regime where it is a spin singlet. According to this picture, the anisotropy of the N i atom in the Cr ring induces a dimerization in the molecule that makes the ground state to stabilize in a Valence Bond Solid phase of virtual spins. We characterize rigorously this phase by means of a particular non-local order parameter denoted the generalized string order parameter. In the completely antiferromagnetic regime, the system becomes frustrated. We have performed a numerical real-time evolution study of the correlations between the spin of the N i impurity and the rest of the spins in order to show the reaction of the system under this frustration. We provide results that manifestly show the agreement with the Valence Bond Solid picture introduced.
We determine the existence of critical lines in dimerized quantum spin ladders in their phase dia... more We determine the existence of critical lines in dimerized quantum spin ladders in their phase diagram of coupling constants using the finite-size DMRG algorithm. We consider both staggered and columnar dimerization patterns, and antiferromagnetic and ferromagnetic inter-leg couplings. The existence of critical phases depends on the precise combination of these patterns. The nature of the massive phases separating the critical lines are characterized with generalized string order parameters that determine their valence bond solid (VBS) content.
In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp giv... more In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.
Journal of Statistical Mechanics: Theory and Experiment, 2011
We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantu... more We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantum spin chains and corresponding ground state wave function(s). The approach is quite general, and we illustrate it by constructing a family of Hamiltonians whose ground states are the chiral correlators of the SU (2) k WZW model for integer values of the level k. The simplest example corresponds to k = 1 and is essentially a nonuniform generalization of the Haldane-Shastry model with long-range exchange couplings. At level k = 2, we analyze the model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show, respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors to derive a set of algebraic, linear equations relating spin correlators within each model. At level k = 1, these equations allow us to compute the two-point spin correlators analytically for the finite chain uniform Haldane-Shastry model and to obtain numerical results for the nonuniform case and for higher-point spin correlators in a very simple way and without resorting to Monte Carlo techniques.
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Papers by Germán Sierra