Research StatementTOMMASO ROSCILDE

Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484(Dated: November 9, 2004)

My past and current research efforts focus on the field of theoretical condensedmatter physics, with particular emphasis on thermal and quantum phase tran-sitions, and macroscopic quantum coherence. Over the last five years of PhDand post-doctoral research, I have specialized in novel computational tech-niques to treat static and dynamic properties of quantum many-body systemsin extreme conditions, especially at ultra-low temperatures and in the proxim-ity of quantum critical points. Tremendous progress has recently been made inthe experimental investigation of condensed and soft matter systems exhibitingquantum effects at the macroscopic scale, such as Bose-Einstein condensates,low-dimensional quantum magnets, and superconducting devices. The model-ing of such systems represents a formidable challenge to theoreticians, and theaid of computational techniques such as Exact Diagonalization (ED) QuantumMonte Carlo (QMC) and the Density Matrix Renormalization Group (DMRG)has proven to be essential for understanding of the main phenomena underly-ing their physics.Here I present the guidelines of my current and future research.

2

Quantum percolation: a novel interplay between quantum fluctuations and geometric randomness

The interplay between geometric randomness and quantum fluctua-

FIG. 1: Quantum magnetic fragmentationof a cluster due to the formation of a localresonating valence-bond state.

tions is an exciting topic in quantum many-body physics, leading tothe emergence of novel quantum phases in strongly correlated electronsystems. Low-dimensional quantum antiferromagnets are an ideal play-ground for this study, especially since controlled non-magnetic dopingof the magnetic lattice (site dilution) and controlled bond disorder hasbeen demonstrated in real compounds. Recently, site dilution of thesquare lattice antiferromagnet has been realized by non-magnetic dop-ing of La2CuO4 [25], opening the question whether disorder can drive thesystem toward a novel quantum disordered state before the geometricpercolation threshold is reached. Making use of Stochastic Series Ex-pansion QMC, we have demonstrated [17] that in two dimensions thisis indeed possible by introducing inhomogeneous bond-dilution in thesquare-lattice quantum antiferromagnet. This kind of dilution promotesthe appearence of structures with ladder-like and dimer-like shape, thussupporting local strongly fluctuating quantum states such as resonat-ing valence bond states or dimer-singlet states. The appearence of suchstates has the effect of suppressing spin-spin correlations in the percolat-ing cluster, effectively fragmenting it into non-percolating clusters andthus hindering the onset of magnetic order with respect to geometric per-colation. A novel two-dimensional quantum spin liquid phase appearstherefore between the classically ordered phase and the non-percolatedphase. The quantum phase transition separating the ordered and the spin-liquid phase has a persistent percolativecharacter, as indicated by the estimated critical exponents, and we therefore named it quantum percolation.Our current investigations are focused on a deeper understanding of the nature of the spin liquid phase, and on thebehavior of such phase under applications of a uniform magnetic field, possibly leading to quantum glassy states.

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Finite clustersm = 0

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m ≠ 0m = 0

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zν

ν=1

ν= 4−3

z=1.9

FIG. 2: Left. Phase diagram of the inhomogeneously bond-diluted Heisenberg antiferromagnet; P and P ′ are the prob-abilities of activation of dimer bonds and ladder bonds respectively (see Ref. [17]). The intermediate orange region is anovel two-dimensional spin-liquid phase. Right. Dynamical critical exponent z and correlation length exponent ν along thequantum critical line, parametrized by the angle θ (see left panel).

3

Entanglement in quantum-critical many-body systems

Entanglement represents one of the most strik-

FIG. 3: Entanglement masks the individual spins involved in aBell state.

ing features of quantum many-particle states, whichpartially or completely prevents one from the con-ventional local description of the state of the sin-gle particles. More recently entanglement has at-tracted a renewed interest, being recognized as thekey ingredient for the efficiency of quantum infor-mation and quantum computation protocols. Thisnon-local aspect of quantum states is expected to beenhanced close to zero-temperature quantum phasetransitions, where collective quantum effects drivethe system through a global reconfiguration of itsground state. Understanding how this enhancementoccurs is very challenging, due to the often counter-intuitive nature of entanglement as a form of corre-lation, and also very intriguing, as it potentially allows to control entanglement by means of collective phenomena.Test cases of well-understood quantum critical behavior are offered by low-dimensional antiferromagnets, as forinstance quantum spin chains. In the spin-1/2 case, estimators have been recently developed to quantify theentanglement between any two spins, and the bipartite entanglement between a group of spins and the remainderof the system. We have applied these entanglement estimators to the study of the quantum phase transition inanisotropic spin chains in presence of an applied uniform field, and studied it through Stochastic Series ExpansionQMC [14]. The specific case considered (XY-like chain with field applied in the xy plane) is remarkably realized inrecent experiments on the quantum spin chain compound Cs2CoCl4 [13]. We find that the quantum critical pointis characterized by a strong feature in the entanglement dependence on the magnetic field, in the form a minimumin the pairwise-to-global entanglement ratio. This indicates that multi-spin entanglement, rather than pairwiseentanglement, is critically enhanced at the transition, so that the quantum-critical state is maximally non-local.Moreover the quantum phase transition is anticipated by the occurrence of an exactly factorized state with zeroentanglement, so that the system realizes an entanglement switch effect close to the transition.Recent investigations are devoted to the observation of the same scenario in different geometries, such as ladders[15] and square lattices [16]. To our surprise, not only the critical enhancement of multi-spin entanglement showsup as a clear signature of the transition, but also the presence of an exactly factorized state persists. This demon-strates that the study of entanglement provides us with a novel insight in the global properties of the ground statewave function of a complex quantum system.

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FIG. 4: Pairwise-to-global entanglement ratio in the easy-plane spin-1/2 Heisenberg antiferromagnet as a function of thefield applied in the easy-plane. Left. One-dimensional case. Right. Square lattice. In both cases the dip marks the quantumcritical field.

4

Phase transitions and crossovers in low-dimensional quantum antiferromagnets

Critical phenomena and ordering of quantum mag-

2D antiferromagnetism in 3D crystals

• Square-lattice layers

• Strong n.n. intra-layer AFcoupling, J = 10 ÷ 1000 K

• Long inter-layer super-exchange paths

• Body-centered magneticlattice: frustrated inter-layerAFM coupling

• → J ≈ 104 ÷ 108 J ′

• Phase trans. at TN≈ JS2

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Sr

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Copper Formate Tetradeuterate(CFTD)

nets in reduced spatial dimensions represent a long-standing problem. The unprecedented productionof compounds with quasi-low-dimensional magneticproperties has opened the possibility of experimen-tal observation of genuine low-dimensional critical-ity. The experimental situation also demands a bet-ter understanding of the role of quantum fluctua-tions, enhanced by the reduced dimensionality, aswell as the identification of specific signatures oflow-dimensional critical behavior that distinguish itfrom the conventional 3D picture. My research ac-tivity has been devoted to a detailed study of low-dimensional criticality in quasi two-dimensional quantum antiferromagnets, which represent to date some of thebest established and most intensively investigated quantum systems in low dimensions. The path-integral quantumMonte Carlo method in continuous Euclidean time has been used to study the emergence of Ising and Berezinskii-Kosterlitz-Thouless (BKT) critical behavior in realistic models of two-dimensional quantum antiferromagnets,described by the spin-1/2 Heisenberg model with extremely weak anisotropy. We have demonstrated that theobservation of the low-dimensional critical behavior is indeed possible in the realistic conditions dictated by themagnetic interactions of quasi-2D compounds [3]. In the particular case of Sr2CuO2Cl2 [4], we have identifiedthe signature for the onset of a phase with vortex-antivortex excitations, marked by a minimum in the uniformsusceptibility perpendicular to the easy plane. Such signature anticipates the occurrence of BKT criticality [5].Finally, we have described the whole phase diagram of the Heisenberg antiferromagnet in a uniform Zeeman field,and we have shown how the signatures of a vortex-antivortex phase are controlled and enhanced by the field [6],potentially leading to a unambiguous observation of the BKT scenario in layered antiferromagnets.

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FIG. 5: Left. Uniform susceptibilities χxxu χzz

u of x and z spin components of Sr2CuO2Cl2 (×’s symbols) compared tothe theoretical predictions for the Heisenberg model with easy-plane anisotropy (diamonds). Right. Phase diagram of thequantum Heisenberg antiferromagnet in a uniform field.

5

Transitions in frustrated antiferromagnets

The presence of competing (frustrated) interactions in low-dimensional antiferromagnets leads to an excitingvariety of scenarios, ranging from frustration-induced disorder in the ground state to frustration-induced finite-temperature transitions. The understanding of models with competing magnetic interactions is nonetheless hin-dered by the lack of exact results and of efficient numerical methods, both on the classical and on the quantumside.

My attention has focused on the spin-1/2 J1 − J2 antiferromagnet

J1

2J

FIG. 6: J1 − J2 lattice.

on the square lattice, very extensively investigated and debated in thelast decade, and yet relatively poorly understood. The knowledge of itsthermodynamic behavior has become particularly compelling since thesynthesis of compounds (Li2VOSi(Ge)O4, VOMoO4 [7]) most likely de-scribed by the J1 − J2 hamiltonian in their paramagnetic phase. Thisproblem is quite challenging, since quantum Monte Carlo fails becauseof the sign problem, while conventional perturbative schemes are notsuitable to treat the paramagnetic phase. Moreover, a novel non-magnetic Chandra-Coleman-Larkin (CCL) phasetransition [8] is predicted to occur in the collinear phase of the model, and its interplay with the magnetic transi-tion observed in the frustrated compounds is unclear.For the study of the thermodynamics of the J1−J2 hamiltonian, we have generalized a semiclassical method called“Pure-Quantum Self-Consistent” Harmonic Approximation which has already proven quite successful in the studyof non-frustrated quantum magnets [9]. The main outcome of such study is the finite-temperature phase diagramof the quantum J1 − J2 model, which shows the existence of the CCL transition even in the extreme quantumcase of S = 1/2 [10]. We have also considered the more general case of anisotropic frustration, in which differentfrustrating next-to-nearest neighbor couplings involve the x(y) spin component on the one side, and the z compo-nent on the other [11]. In the extreme case of axial-only frustration, the complete phase diagram of the model wasdrawn with the aid of QMC, revealing a surprisingly rich scenario of topological transitions for low frustration,and magnetic transitions for higher frustration. With the aid of exact diagonalization and mean-field theory, wecould interpolate between the case of axial frustration and the fully isotropic case, revealing the persistence of thescenario obtained by QMC.Current ongoing investigations are concentrated on the interplay between chiral and magnetic transitions in frus-trated weakly coupled planar chains, in light of recent experiments on frustrated molecular magnets [12].

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2 /J1

and α⊥ = J⊥2 /J1.

6

Classical and Quantum Monte Carlo simulations

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FIG. 8: (D+1)-dimensional path-integral representation of a D-dimensional quantum spin system. The red crosses mark the exchangeof spin configurations in the imaginary-time evolution.

body systems, the Monte Carlo method is themost versatile and suitable for the study ofcritical phenomena in any spatial dimensions.I have implemented several Monte Carlo codesfor the study of classical and quantum spinsystems with genuine phase transitions. Inthe quantum case, I have used both the Path-Integral formulation in continuous Euclideantime [1], and the Stochastic Series Expansionformulation [2]. Both simulation schemes havethe advantage of being fully quantum withoutany semiclassical approximation, and to be ex-tremely accurate even in the limit of utra-lowtemperature. Faithful monitoring of the crit-ical behavior requires the development of ad-vanced cluster algorithms which are able to“follow” critical correlations in the system. Ihave implemented cluster algorithms for bothclassical and quantum spins, and in particu-lar I have generalized classical algorithms tothe case of frustrated spins, and generalized

optimal quantum algorithms to the case of applied non-homogeneous Zeeman fields and of low-symmetry hamil-tionians. Advanced numerical schemes required for the study of first-order transition, such as parallel tempering,have also been successfully implemented.

Future research plans

My research plans for the next few years are articulated along three main lines.

Optimal design of q-bits based on strongly correlated electron systems. Goal of the project is to design atunable strongly correlated electron system, e.g. realized by a cluster of quantum dots, whose low-energyspectrum in the spin channel has a well isolated two-level structure, so that the system behaves collectively as aqubit [18]. The controlled growing of arrays of quantum dots opens the possibility of designing mesoscopic clustersof strongly correlated electrons. Quantum computation using the spin degree of freedom of electrons confined onsingle quantum dots has been recently proposed [19]. The advantage of a cluster-qubit over the single-dot one istwo-fold: it can be addressed more easily by means of, e.g., the developing techniques of magnetic resonance forcemicroscopy [20]. Moreover, the internal interactions of the cluster can lead to an energetic stabilization of some ofthe qubit states against the effect of the environment. Exact diagonalization of Hubbard clusters with spin-orbitinteractions, modeling clusters of coupled quantum dots, will be supplemented with adaptive design algorithmsto find the optimal geometry and interactions. The coupling to the environment and the qubit-qubit couplingwill then be investigated.

Quantum phase transitions of ultra-cold atomic systems in optical lattices. Bose-Einstein condensates loadedin optical lattices represent the most tunable quantum many-body system at hand, in which dimensionality,filling and interactions can be controlled with unprecedented accuracy. QMC and DMRG techniques can mimicvery closely the real conditions in which the experiments are carried, and they can single out the most relevantsignatures of the different quantum phases [21]. The goal of this project is to numerically investigate the static

7

and dynamic properties of ultra-cold atomic systems in optical lattices, modeled by the Bose-Hubbard model inD dimensions, with particular emphasis on quantum phase transitions between superfluid and insulating phases.In the case of quasi-one-dimensional condensates, recently developed DMRG techniques [23] allow for a real-timestudy of out-of-equilibrium behavior under the effect of non-adiabatic change of the system parameters. Inparticular, I plan to study the static and dynamic behavior of atoms loaded in optical superlattices and disorderedoptical lattices [24], which stand as an emerging field in the experimental investigations, and represent an idealplayground to study effects of quantum localization and glassy behavior in a controlled way.

Quantum magnets on disordered lattices. Is percolation in quantum systems the same as in classical ones?The goal of this project is to identify and classify relevant random lattice geometries leading to a strong enhance-ment of quantum effects in low-dimensional quantum Heisenberg antiferromagnets, and showing a substantialcorrection to the classical picture of percolation. The recent experimental realization of site percolation in thetwo-dimensional quantum antiferromagnet on the square lattice by non-magnetic doping of La2CuO4 [25] opensthe perspective of driving ordered magnets towards novel quantum-disordered states by a controlled disorderingof the magnetic lattice. Our recent theoretical investigations [17] have demonstrated that this is indeed possibleby inhomogeneously bond-diluting the square-lattice quantum antiferromagnet. With the aid of Stochastic SeriesExpansion QMC [2] a systematic investigation of non-frustrated realistic disordered lattices leading to quantumspin-liquid phases is definitely at hand. The application of a uniform magnetic field, partly quenching quantumfluctuations, is expected to lead to novel quantum glassy phases, such as the Bose glass phase [22]. The numericalinvestigation will be supplemented with theoretical analysis based on the bond-operator approach and withfield-theoretical techniques.

[1] B. B. Beard and U.-J. Wiese, Phys. Rev. Lett. 77, 5130 (1996).[2] O. F. Syljuasen and A. W. Sandvik, Phys. Rev. E 66, 046701 (2002).[3] A. Cuccoli, T. Roscilde, V. Tognetti, P. Verrucchi, and R. Vaia, Phys. Rev. B. 67, 104414 (2003).[4] M. Greven et al., Z. Phys. B 96, 465 (1995).[5] A. Cuccoli, T. Roscilde, P. Verrucchi, and R. Vaia, Phys. Rev. Lett. 90, 167205 (2003).[6] A. Cuccoli, T. Roscilde, P. Verrucchi, and R. Vaia, Phys. Rev. B 68, 060402(R) (2003).[7] For a recent review, see: P. Carretta et al., J. Phys.: Condens. Matter 16, S849 (2004).[8] P. Chandra, P. Coleman, and A. I. Larkin, Phys. Rev. Lett. 64, 88 (1990).[9] A. Cuccoli et al., Phys. Rev. Lett. 77, 3439 (1996); Phys. Rev. B 56, 144456 (1997).

[10] L. Capriotti, A. Fubini, T. Roscilde, and V. Tognetti, Phys. Rev. Lett. 92, 157202 (2004).[11] T. Roscilde, A. Feiguin, A. Chernyshev, S. Liu, and S. Haas, Phys. Rev. Lett. 93, 017203 (2004).[12] A. Lascialfari et al., Phys. Rev. B 67, 224408 (2003)[13] M. Kenzelmann et al., Phys. Rev. B 65, 144432 (2002).[14] T. Roscilde, A. Fubini, P. Verrucchi, S. Haas, and V. Tognetti, Phys. Rev. Lett. 93, 167203 (2004).[15] T. Roscilde, A. Fubini, P. Verrucchi, S. Haas, and V. Tognetti, In Quantum computation: Solid State Systems, edited

by P. Delsing, C. Granata, Y. Pashkin, B. Ruggiero, and P. Silvestrini (Kluwer, Dordrecht, in press, 2004).[16] T. Roscilde, A. Fubini, P. Verrucchi, S. Haas, and V. Tognetti, in preparation (2004).[17] Rong Yu, T. Roscilde, and S. Haas, cond-mat/0410041, submitted, 2004.[18] F. Meier et al., Phys. Rev. Lett. 90, 047901 (2003).[19] D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998).[20] D. Rugar et al., Nature 430, 329 (2004).[21] S. Wessel et al., cond-mat/0404552; S. R. Clark and D. Jaksch, cond-mat/0405580.[22] M. P. A. Fisher et al., Phys. Rev. B 40, 546 (1989).[23] S. R. White and A. E. Feiguin Phys. Rev. Lett. 93, 076401 (2004).[24] S. Peil et al., Phys. Rev. A 67, 051603 (2003); B. Damski et al., Phys. Rev. Lett. 91, 080403 (2003).[25] O. P. Vajk et al., Science 295, 1691 (2002).