| Correlation functions in 1D Quantum gases (coll.: J.-S. Caux, N.A. Slavnov) |
|
The recent observations of 1D trapped atomic gases
provide a unique opportunity for extending our understanding of the
physics of these many-body systems.
The 1D regime can be investigated theoretically by making use of the
integrability of the model. However, although the Bethe Anstaz
solution of the delta-interacting 1D Bose gas is known since 40
years, dynamical correlation functions resisted until recently all
efforts toward an exact calculation. The most promising approach for an analytic calculation of correlation functions makes use of the so-called Algebraic Bethe Ansatz. In this framework the correlation functions are written as sums of matrix elements (called also form-factors) over the full set of excited states. Matrix elements of the most relevant observables are known since the early ninety, but their expression is a determinant of matrix whose entries are rational functions of the analytical unknown Bethe rapidities. These so cumbersome expressions blocked any further development toward a full analytical calculation. For this reason we (myself and J.-S Caux at University of Amsterdam) developed a new method that mixes integrability and numerics to calculate very precisely correlation functions for finite lenght/particles Bose gases. The overall strategy is very simple: the form-factors are calculated for each excited state inserting the numerical solutions to the Bethe equations in the determinant representation. However for number of particles that compares well with experimental data (i.e. of the order of hundred) the number of excited states we have to consider is approximately hundred millions. This large number of states requires a very smart scanning of the Fock space of the model, in order to not loose days of computation only for tiny contributions. The method we developed is applicable to any integrable model as long as the determinant representation of the matrix elements is known, and we have coined it the ABACUS method (the method has been developed also for Heisenberg spin chains by JS and collaborators). We applied successfully the ABACUS method for the calculation of the zero-temperature dynamical density-density [A1] and one-particle correlation functions [A2] in the 1D repulsive Bose gas. Their static limits (that are obtained as a subset of our results) corresponds to the experimental measured structure factor and the one-body density matrix. Another major application of the Algebraic Bethe Ansatz approach has been the understanding of the 1D attractive Bose gas. In fact, this model was usually considered as pathological because a proper thermodynamic limit is not readily defined. However we showed [A3] that, in spite of the bad thermodynamic limit, zero-temperature static and dynamical correlation functions can be exactly calculated directly from the integrability of the model and very simple closed analytical expressions have been obtained [A3]. We are currently extending these methods to understand other physical situations of 1D gases, as for example finite temperature, out-of-equilibrium correlations, different statistics (Bose-Fermi mixtures, anyons) etc... |
| [A1] J.-S. Caux and P. Calabrese ''Dynamical density-density correlations in the one-dimensional Bose gas'' Phys. Rev. A 74, 031605R (2006) [cond-mat/0603654] |
[A2] J.-S. Caux, P. Calabrese, and N. A. Slavnov, ''One-particle dynamical correlations in the one-dimensional Bose gas'' JSTAT XXXX, PXXX (2006) [cond-mat/0611321] |
[A3] P. Calabrese and J.-S. Caux, ''Correlation functions of the one-dimensional attractive Bose gas'' [cond-mat/0612192] |
| Non-equilibrium dynamics of quantum systems (coll.: J. Cardy, J.-S. Caux) |
|
Hamiltonian non-equilibrium evolution in condensed matter systems is
normally hidden by unavoidable dissipation, even at zero temperature.
The abrupt change of a Hamiltonian parameter (quantum quench)
recently realized in cold atoms experiments, offers the possibility of
studying these phenomena in an optical system.
We (in collaboration with J. Cardy) showed [B1] that the
time-dependence of correlation functions following a quantum quench may
be extracted using methods of boundary critical phenomena.
For d=1 particularly powerful results are available using CFT.
We checked our predictions against those available from some simple
solvable models. I'm currently trying to understand if and how similar calculations can be performed in systems solvable by means of Bethe Ansatz. |
| [B1] P. Calabrese and J. Cardy ''Time-dependence of correlation functions following a quantum quench'' Phys. Rev. Lett. 96, 136801 (2006) [cond-mat/0601225] |
| Quantum entanglement (coll.: J. Cardy, J.-S. Caux) |
|
Quantum entanglement is considered a basic tool to develop quantum information
and communication protocols.
However there is now also a growing interest in using entanglement measures
to characterizes extended quantum systems, in particular to understand the
interplay between quantum entanglement and quantum criticality. In this field, my main interest is the study of the so called Entanglement Entropy . Suppose a whole system is in a pure quantum state |Ψ>, and an observer A measures only a subset A of a complete commuting observables, while another observer B measures the reminder. A's reduced density matrix is ρA=TrB |Ψ><Ψ|. The Entanglement entropy SA is just the von Neumann entropy SA=-Tr ρA log ρA associated to this reduced density matrix. In collaboration with J. Cardy [C1], we carried out a systematic study of the entanglement entropy in relativistic quantum field theory describing the scaling limit of extended quantum systems close to a quantum critical point. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derived the result SA=c/3 log L when A is a finite interval of length L in an infinite system, and extended it to many other cases: finite systems, finite temperatures, systems with boundaries, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we showed that SA= NA c/6 log ξ, where NA is the number of boundary points of A. To check these prediction we developed a Corner-Transfer-Matrix approach for the calculation of SA in simple integrable models such as the Ising model in a transverse field. A non-technical introduction to these results appears in [C3]. We also studied the unitary time evolution of the entropy of entanglement of a one-dimensional system, starting from a pure state which is not an eigenstate of the Hamiltonian as following from a quantum quench [C2]. Using field-theoretical path integral methods as well as explicit computations for the transverse Ising spin chain, we showed that the entanglement entropy increases linearly with time t up to t=L/2v (v being the maximum speed of propagation of signals), after which it saturates at a value proportional to L. We interpreted this behavior as a consequence of causality in a quasi-classical approach. Several of these finding has been later confirmed in Heisenberg chains with time dependent density matrix RG [C4] (written in collaboration with G. De Chiara, S. Montangero, and R. Fazio). We are currently investigating the time evolution of the entanglement entropy from an initial state that is only locally different from the ground state and extending these findings to systems solved by means of Bethe ansatz. |
| [C1] P. Calabrese and J. Cardy ''Entanglement entropy and quantum field theory'' JSTAT 0406, P002 (2004) [hep-th/0405152] |
| [C2] P. Calabrese and J. Cardy ''Evolution of Entanglement Entropy in One-Dimensional Systems'' JSTAT 0504, P010 (2005) [cond-mat/0503393] |
| [C3] P. Calabrese and J. Cardy ''Entanglement entropy and quantum field theory: a non-technical introduction" Int. J. Quantum Inf. 4, 429 (2006) [quant-ph/0505193] |
| [C4] G. De Chiara, S. Montangero, P. Calabrese, R. Fazio ''Entanglement Entropy dynamics in Heisenberg chains'' JSTAT 0601, P001 (2006) [cond-mat/0512586] |
| Critical Aging (coll.: L. Cugliandolo, A. Gambassi) |
|
Systems with slow dynamics are currently under intensive experimental and
theoretical investigation. In some circumstances their dynamics becomes so
slow that they can not equilibrate in finite time, evolving always
in non-equilibrium conditions and displaying novel dynamic phenomena as aging.
Quite recently it has been realized that this may also be the case in
non-complex systems.
Therefore, simple models such as classical non-disordered ferromagnets
quenched at or below their critical point have become very useful toy models
to test some of the key ideas put forward in the context of complex systems.
The fluctuation-dissipation ratio (FDR) has been introduced as a sort of
measure of the "distance" from an equilibrium evolution.
It has also been shown that, at least for some mean-field glass models, the
FDR can be used to define a non-equilibrium temperature.
The interesting question is whether this allows one to devise a
non-equilibrium thermodynamics, and whether this is true for non mean-field
models as well.
In view of these open questions, there has been recently a lot of activity
around the aging dynamics of ferromagnetic systems, with particular efforts
to determine the FDR at criticality, by means of MCs and exact solutions of
specific models.
Universality of critical phenomena, on the other hand, can be exploited to
provide field-theoretical predictions for the quantities (such as two-time
response and correlation functions of the order parameter) which characterize
the non-equilibrium dynamics following a quench to the critical point. In collaboration with A. Gambassi we adopted the field-theoretical RG approach [D1] to investigate the relevant aging properties of the non-equilibrium dynamics following a quench from the high-temperature phase to the critical point. We considered (among the others) the non-equilibrium relaxational dynamics (Model A) of the d-dimensional O(N) vector model in ε-expansion [D1,D2,D5] and the weakly dilute Ising model [D3] We studied also the conserved dynamics (Model B) for the O(N) [D6] model and the dynamics of a non-conserved order parameter coupled to a conserved density (Model C) [D4]. For all the models just mentioned we computed and analyzed the scaling behaviors of the two-time response and correlation functions of the order parameter, the associated universal scaling functions, and the long-time limit of the FDR. This analysis allowed us to provide the first numerical predictions for the FDR of non-exactly solvable models (the only ones considered in the preceding literature). Some of our results have been nicely confirmed by MCs. Furthermore we provided also new estimates for quantities which have not yet been measured, but that can be determined by experiments or MCs. All these findings are summarized in our recent review article on the aging properties of critical systems [D6]. We addressed also the problem of the definition of a non-equilibrium effective temperature Teff at the critical point, on the basis of the FDR. Some numerical findings seem to indicate that this quantity is indeed independent of the observable used to define it. Motivated by these results we have studied the general properties of response and correlation functions for a generic observable of the O(N) vector-model. It turned out that within the MF approximation the FDR is actually independent of the specific observable, whereas this is no longer the case beyond MF, as we explicitly checked in a one-loop computation [D5]. In a collaboration with A. Gambassi and F. Krzakala we investigated within the RG approach and by means of MCs the general scaling properties of the two-time response and correlation functions for the Model A dynamics of the Ising universality class when considering a sudden heating from the low-temperature phase to the critical point [D7]. In collaboration with A. Gambassi and L. Cugliandolo are currently trying to extend these concepts to Quantum Phase Transitions. |
| [D1] P. Calabrese and A. Gambassi ''Aging in ferromagnetic systems at criticality near four dimensions'' Phys. Rev. E 65, 066120 (2002) [cond-mat/0203096] |
| [D2] P. Calabrese and A. Gambassi '' Two-loop Critical Fluctuation-Dissipation Ratio for the Relaxational Dynamics of the O(N) Landau-Ginzburg Hamiltonian '' Phys. Rev. E 66, 066101 (2002) [cond-mat/0207452] |
| [D3] P. Calabrese and A. Gambassi ''Aging and fluctuation-dissipation ratio for the dilute Ising Model'' Phys. Rev. B 66, 212407 (2002) [cond-mat/0207487] |
| [D4] P. Calabrese and A. Gambassi ''Aging at Criticality in Model C Dynamics'' Phys. Rev. E 67, 036111 (2003) [cond-mat/0211062] |
| [D5] P. Calabrese and A. Gambassi ''On the definition of a unique effective temperature for non-equilibrium critical systems'' JSTAT 0407, P013 (2004) [cond-mat/0406289] |
| [D6] P. Calabrese and A. Gambassi ''Ageing Properties of Critical Systems'' J. Phys. A 38, R133 (2005) [cond-mat/0410357] |
| [D7] P. Calabrese, A. Gambassi, and F. Krzakala ''Critical aging of Ising ferromagnets relaxing from an ordered state'' JSTAT 0606, P016 (2006) [cond-mat/0604412] |
| [D8] P. Calabrese and A. Gambassi, ''Slow dynamics in critical ferromagnetic vector models relaxing from a magnetized initial state'' JSTAT XXXX, PXXX (2006) [cond-mat/0610266] |
| Critical phenomena (coll.: P. Parruccini, A. Pelissetto, A. Sokolov, E. Vicari) |
| Finally, I am interested in an accurate determination of critical quantities (exponents, amplitude ratios, equation of state...) of systems described by Landau-Ginzburg Hamiltonians with complex symmetries, such as models with randomness, disorder, anisotropies, frustration and/or competing order parameters. In collaborations with P. Parruccini, A. Pelissetto, A. Sokolov, and E. Vicari, I was able to give precise theoretical estimates of universal quantities calculating and analyzing, by means of powerful resummation techniques, higher-order perturbative RG series both in fixed dimension d=3 (up to six loop), d=2 (up to five), and ε=4-d expansion (up to five loop). We found that, beside improving the accuracy, in some cases higher-order calculations turned out to be necessary to determine the correct physical picture in physical dimensions d=2,3. A short account of these works can be found in the two brief reviews [E1,E2]. |
| [E1] P. Calabrese, A. Pelissetto and E. Vicari ''The critical behavior of magnetic systems described by Landau-Ginzburg-Wilson field theories'' in Frontiers in Superconductivity Research ed B P Martins (Nova Science, 2004 Hauppage (NY)) [cond-mat/0306273] |
| [E2] P. Calabrese, A. Pelissetto, P. Rossi and E. Vicari ''Field-theory results for three-dimensional transitions with complex symmetries'' Int. J. Mod. Phys. B 17, 5829 (2003) [hep-th/0212161] |
| Last updated : | Present time: |