N. Datta, R. Fernandez, and J. Frohlich, \Low-temperature phase diagrams of quantum lattice systems I. Stability for quantum perturbations of classical systems with nitely many ground states," J. Stat. Phys. 84, 455 (1996). 101  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... main question is: which restrictions ensue on the unperturbed Hamiltonian H and on the perturbation P It turns out that this question can be answered using the same methods that are employed for constructing low temperature phase diagrams for classical spin systems with quantum perturbations [13, 29, 64]. Thus the Hamiltonian H can be comprised by n spin interactions (for xed n) that satisfy the Peierls condition [127] the energy cost of a local perturbation of a translationally invariant ground state is on the order of the surface area of the region that encloses the part of the ....

N. Datta, R. Fernandez, and J. Frohlich, \Low-temperature phase diagrams of quantum lattice systems I. Stability for quantum perturbations of classical systems with nitely many ground states," J. Stat. Phys. 84, 455 (1996). 101

....it is easy to verify that the all plus con guration is actually a ground state for small enough. In order to conclude that for large enough, the unique phase of H t ( is a weak perturbation of the all plus con guration (uniformly in ) we can rely on the theory developed in [3] or [6] which allows exponentially decaying perturbations of a nite range interaction satisfying the Peierls condition (see e.g. equations (1.3) 2.2) of [3] Similarly, H t ( c ) has a unique phase which is a weak perturbation of the all minus con guration. This is sucient to conclude that ....

N. Datta, R. Fernandez, J. Frohlich. Low-temperature phase diagrams of quantum lattice systems I. Stability for quantum perturbations of classical systems with nitely many ground states. J. Stat. Phys. 84, 455-534, 1996.

....theory has evolved [KP, Zah, BKL, BS, BI, BK] into a very powerful tool to study the pure phases, their coexistence and the first order phase transitions in classical spin systems at low temperature. In recent years large part of the Pirogov Sinai theory has been extended to quantum systems [Pir, BKU, DFF, DFFR, KU], including quantum spin systems as well as fermionic and bosonic lattice gases and applied to a variety of models [FR, DFF2, GKU] to describe insulating phases associated with discrete symmetry breaking. Here we formulate the Pirogov Sinai theory focusing on linear functionals that are tangent to ....

....the Pirogov Sinai theory focusing on linear functionals that are tangent to the free energy. This allows both to avoid the difficulties associated with boundary conditions, and to make the link with the notion of KMS states, using well known results [Ara, Isr, Sim] We reformulate results of [BKU, DFF, DFFR, KU] in this framework, and extend the theory to a class of models where discrete symmetries are broken at intermediate temperatures. This applies in particular to some systems with continuous symmetries. For this, we consider the restricted ensembles introduced in [BKL] that are very useful to ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez and J. Frohlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Stat. Phys. 84, 455--534 (1996)

....classical model has an infinite number of ground states. In many cases one expects that this degeneracy will be lifted as a result of quantum fluctuations , that is, the effect of a small kinetic energy T . A general theory of such effects combined with the Pirogov Sinai theory can be found in [DFFR, KU]. Notice that U(1) U( p 2) meaning that at low temperature, the chessboard phase overcomes the phase with alternate rows or columns of 1 s and 0 s. Energies (3.6) provide the zerotemperature phase diagram and allow guesses for the low temperature situation. GEOMETRIC PROBABILISTIC ASPECTS ....

....to both questions is yes and is provided by the quantum Pirogov Sinai theory. It can be viewed as a considerable extension of the Peierls argument for the Ising model. It was proposed by Pirogov and Sinai for classical lattice models [PS, Sin] and 8 DANIEL UELTSCHI extended to quantum models in [BKU, DFF, DFFR, KU, FRU]. These ideas are discussed for this model in the next section. One is then led to the phase diagram of Fig. 3. Multiple phases and occurrences of first order phase transitions are proven when fi is large and t small, i.e. at low temperature and close to the classical limit of vanishing hoppings. ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez, J. Frohlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69, 752--820 (1996)

....to both questions is yes and is provided by the quantum Pirogov Sinai theory. It can be viewed as a considerable extension of the Peierls argument for the Ising model. It was proposed by Pirogov and Sinai for classical lattice models [PS, Sin] and 8 DANIEL UELTSCHI extended to quantum models in [BKU, DFF, DFFR, KU, FRU]. These ideas are discussed for this model in the next section. One is then led to the phase diagram of Fig. 3. Multiple phases and occurrences of first order phase transitions are proven when fi is large and t small, i.e. at low temperature and close to the classical limit of vanishing hoppings. ....

....functional) at high temperature, while a domain with two extremal states, and hence long range order (LRO) is present for low temperature and small hopping (darker zone) Most of the phase diagram is not rigorously understood yet. The proof of existence of phase transitions were obtained in [BKU, DFF]; it was realized in [FRU] that tangent functionals naturally fit in the context of the Pirogov Sinai theory. The zero temperature energy takes the form (see Fig. 2) e ;h = min ( Delta Delta Delta Delta ) e ;h ( Delta Delta Delta Delta ) 3.8) where the minimum is taken over ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez and J. Frohlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Stat. Phys. 84, 455--534 (1996)

....Gibbs states with a rigid interface in the principle coordinate directions, provided that the inverse temperature # is su# ciently large and ## is su#ciently small. The proof of this theorem relies on recently developed techniques to derive Pirogov Sinai type results for quantum lattice models [7, 6]. The next two results are examples of interesting situations where perturbation theory as described above, does not apply. They also illustrate two features that distinguish quantum models from their classical counterparts: 1) in a system that has classically several equivalent energy minimizing ....

N. Datta, R. Fernandez, and J. Frohlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Stat. Phys. 84 (1996), 455--534.

....strictly positive constant c independent of . We note that in the present case the uniform density theorem [10] applies, so in particular n x = 1=2 for all fi, t and V ) In fact this result is true for any flux configuration and one can also add a small chemical potential term (see [11] [12] and [13] for recent rigorous results) Although our proof does not work in D = 2, the result is expected to hold also in two dimensions. On the flux phase conjecture 16 d) One can also consider the case of spin 1 2 electrons with attractive Hubbard interaction and a nearest neighbor repulsion, ....

Datta, N., Fernandez, R., and Frohlich, J.:"Low temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states", preprint (1995)

....that a quantum perturbation itself also removes the degeneracy of the classical ground state, but leads to a different set of configurations. Again, one would expect that this conjectured phase is stable against small perturbations, this time stemming from the thermal fluctuations (the method of [DFFR] should apply here) If a coexistence surface separates the domain in the t; fi plane that is dominated by thermal fluctuations from that which is dominated by quantum fluctuations, an interesting transition occurs, driven by the competition between two different kinds of fluctuations. We intend ....

....in [BKU] and [DFF] In the latter the sign problem arising for fermions was dealt with, so that the results also apply to lattice fermionic systems. The theory was generalized to situations where the diagonal part of the Hamiltonian has degeneracies that can be removed by non diagonal terms [DFFR, FR]. In the sequel we present an extension of these ideas to the case of boson systems. New difficulties arise here since we are dealing with Hilbert spaces of infinite dimensions (also for finite systems) and with unbounded operators. For technical reasons we assume that the Hamiltonian is of finite ....

N. Datta, R. Fern'andez, J. Frohlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy., Helv. Phys. Acta 69, 752--820 (1996)

....jx Gammayj 1 6 1 S 3 x S 3 y Gamma h X x S 3 x (1.3) where h = Gamma 4U 1 Gamma 4U 2 in d = 2, h = Gamma 6U 1 Gamma 12U 2 Gamma 8U 3 in d = 3, The model (1. 3) can be at least in some regions of the phase diagram directly treated (with the help of [BKU] or [DFF]) as a quantum perturbation of the classical lattice gas model H LG = d X k=1 U k X jx Gammayj= p k n x n y Gamma X x n x ; 1.4) with n x 2 f0; 1g. For large U 0 and small =U 0 the phase diagram of (1.4) and the t = 0 limit of (1.2) are identical. Since furthermore, 1.3) is the U ....

....situations, the Pirogov Sinai theory [PS, Sin, Zah, BI] see e.g. Kot] for an introduction to these ideas) to obtain a rigorous description of the states in the thermodynamic limit, and of the phase diagram. General theory for quantum spin lattice models has been recently proposed in [BKU] and [DFF]. In the latter the sign problem arising for fermions was dealt with, so that the results also apply to lattice fermionic systems. The theory was generalized to situations where the diagonal part of the Hamiltonian has degeneracies that can be removed by non diagonal terms [DFFR, FR] In the ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez and J. Frohlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Stat. Phys. 84, 455--534 (1996)

....certainly hold for all fi 1. When 3, it is expected that a domain with antiferromagnetic phase appears when t U and fit 2 =U const. Such a phase can be proven in the asymmetric Hubbard model, where electrons of different spins are assumed to have different hopping parameters, see [DFFR] and [KU] for two general methods to study rigorously such situations. Assuming this to be true in the standard Hubbard model, we observe that the domain D 2 has correct leading order; this is illustrated in Fig. 2. temperature t uniqueness antiferromagnetic Figure 2. Phase diagram of the ....

....model, we observe that the domain D 2 has correct leading order; this is illustrated in Fig. 2. temperature t uniqueness antiferromagnetic Figure 2. Phase diagram of the Hubbard model. Antiferromagnetic phase is expected for dimension 3; it can be proven when 2 for the asymmetric model [DFFR, KU]. Proof of Theorem 3.1. The classical free energy of the Hubbard model is easily computed and worths f 0 (fi; Gamma 1 fi log Theta 1 2 e fi e Gammafi U 2fi : 3.2) 10 DANIEL UELTSCHI Let us write the kinetic operator Gammat P Aae TA where A = x; y ; oe) and the ....

N. Datta, R. Fern'andez, J. Frohlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69, 752--820 (1996)

....was proposed by Kennedy and Lieb [KL] it was extended in [LM] to situations that are not covered by the present paper, namely to cases of such mixed systems with continuous classical variables. Results very similar to ours have already been obtained by Datta, Fernandez, Frohlich and Rey Bellet [DFFR]. Their approach is di#erent, however. Starting from a Hamiltonian H(#) H (0) #V , H (0) being a diagonal operator with infinitely many ground states, and V the quantum perturbation, the idea is to choose an antisymmetric matrix S = #S (1) # 2 S (2) in such a way that the operator ....

....to be diagonal, up to terms of order # 3 or higher. If the diagonal part of H (2) has a finite number of ground states and the excitations cost strictly positive energy, it can be shown that the ground states are stable. It is possible to include higher orders in this perturbation scheme (see [DFFR]) In fact, our first intention was to study the stability of the results of [BS] with respect to a quantum perturbation, and we began the present study as a warm up and the first simple step towards this goal. This simple step turned out however to be rather involved. Even though, at the end, the ....

[Article contains additional citation context not shown here]

N. Datta, R. Fernandez, J. Frohlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69, 752--820 (1996)

.... part only) has stable low temperature phases, and if the o# diagonal terms of the Hamiltonian are small, the contours have low probability of occurrence and it is possible to extend the Peierls argument to quantum models [Gin] More generally, one can formulate a Quantum Pirogov Sinai theory [BKU, DFF1], in order to establish that (i) low temperature phases are very close to ground states of the diagonal interaction (more precisely: the density matrix 1 Z e #H is close to the 2 EFFECTIVE INTERACTIONS DUE TO QUANTUM FLUCTUATIONS projection operator g##g , where g# is the ground state ....

....over # and # accompanied with the integration, a priori over the interval [0, #] over times # i of corresponding transitions, subjected to above formulated restrictions, c.f. 4. 2) Furthermore the contribution of # # # factorizes as a contribution of # times a product of terms for # # # [BKU, DFF1] 9 , we get Z per # = Z D per # d# Z D loop # (#) d# # per (# ### ) Z D per # d## per (# # ) Z D loop # (#) d# Y ### z(#) 4.7) Here, using (A i , # i ) i = 1, m to denote the quantum transitions of # # #, we put # per (# ### ) m Y i=1 #n ### ....

[Article contains additional citation context not shown here]

N. Datta, R. Fernandez and J. Frohlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Stat. Phys. 84, 455--534 (1996)

....or translational symmetry and therefore a Peierls argument can be applied. For other ground states this is not the case and Pirogov Sinai theory [27, 29, 31, 3] has to be used y . Recent developments of this theory allowing general long range (and also quantum) interactions may be found in [5] and [4] Using these results it is expected that the phase diagram is stable at low temperatures. The present work is limited to the large U limit. For small U the phase diagram changes drastically. This is the case in one and two dimensions where a very rich structure has been uncovered [10, 6, ....

N. Datta, R. Fern'andez and J. Frohlich, Low--temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely--many ground states, preprint ETHZ (1995)

....was proposed by Kennedy and Lieb [KL] it was extended in [LM] to situations that are not covered by the present paper, namely to cases of such mixed systems with continuous classical variables. Results very similar to ours have already been obtained by Datta, Fern andez, Frohlich and Rey Bellet [DFFR]. Their approach is different, however. Starting from a Hamiltonian H( H (0) V , H (0) being a diagonal operator with infinitely many ground states, and V the quantum perturbation, the idea is to choose an antisymmetric matrix S = S (1) 2 S (2) in such a way that the operator H ....

....to be diagonal, up to terms of order 3 or higher. If the diagonal part of H (2) has a finite number of ground states and the excitations cost strictly positive energy, it can be shown that the ground states are stable. It is possible to include higher orders in this perturbation scheme (see [DFFR]) In fact, our first intention was to study the stability of the results of [BS] with respect to a quantum perturbation, and we began the present study as a warm up and the first simple step towards this goal. This simple step turned out however to be rather involved. Even though, at the end, the ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez, J. Frohlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69, 752--820 (1996)

.... part only) has stable low temperature phases, and if the off diagonal terms of the Hamiltonian are small, the contours have low probability of occurrence and it is possible to extend the Peierls argument to quantum models [Gin] More generally, one can formulate a Quantum Pirogov Sinai theory [BKU, DFF1], in order to establish that (i) low temperature phases are very close to ground states of the diagonal interaction (more precisely: the density matrix 1 Z e GammafiH is close to the 2 EFFECTIVE INTERACTIONS DUE TO QUANTUM FLUCTUATIONS projection operator jgihgj , where jgi is the ground ....

.... Xi accompanied with the integration, a priori over the interval [0; fi] over times i of corresponding transitions, subjected to above formulated restrictions, c.f. 4. 2) Furthermore the contribution of Gamma [ Xi factorizes as a contribution of Gamma times a product of terms for 2 Xi [BKU, DFF1] 9 , we get Z per = Z D per d Gamma Z D loop ( Gamma) d Xi ae per ( Gamma[ Xi ) Z D per d Gammaae per ( Gamma ) Z D loop ( Gamma) d Xi Y 2 Xi z( 4.7) Here, using f(A i ; i ) i = 1; mg to denote the quantum transitions of Gamma [ Xi, we ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez and J. Frohlich, Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many ground states, J. Stat. Phys. 84, 455--534 (1996)

....Phase Diagrams of Quantum Lattice Systems. III. Examples. By Jurg Frohlich and Luc Rey Bellet 1 Institut fur Theoretische Physik, ETH Honggerberg 8093 Zurich, Switzerland Abstract. We use the low temperature expansion and the extension of Pirogov Sinai theory developed in [1], and the perturbation theory of [2] to describe the phase diagrams of two families of fermionic lattice systems at low temperature: the balanced model and a variant of the t Gamma J model. 1 Introduction In this paper, we illustrate the methods developed in two previous papers [1, 2] to analyze ....

....developed in [1] and the perturbation theory of [2] to describe the phase diagrams of two families of fermionic lattice systems at low temperature: the balanced model and a variant of the t Gamma J model. 1 Introduction In this paper, we illustrate the methods developed in two previous papers [1, 2] to analyze the low temperature phase diagrams of quantum lattice systems in terms of two classes of models, the balanced model and the t Gamma J models (of interest in connection with high temperature superconductivity) In this paper, we describe our main results and discuss the key ideas of ....

[Article contains additional citation context not shown here]

N. Datta, R. Fernandez, J. Frohlich. Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many groundstates. To be published in J. Stat. Phys.

....theory has evolved [KP, Zah, BKL, BS, BI, BK] into a very powerful tool to study the pure phases, their coexistence and the first order phase transitions in classical spin systems at low temperature. In recent years large part of the Pirogov Sinai theory has been extended to quantum systems [Pir, BKU, DFF, DFFR, KU], including quantum spin systems as well as fermionic and bosonic lattice gases and applied to a variety of models [FR, DFF2, GKU] to describe insulating phases associated with discrete symmetry breaking. Here we formulate the Pirogov Sinai theory focusing on linear functionals that are tangent to ....

....the Pirogov Sinai theory focusing on linear functionals that are tangent to the free energy. This allows both to avoid the difficulties associated with boundary conditions, and to make the link with the notion of KMS states, using well known results [Ara, Isr, Sim] We reformulate results of [BKU, DFF, DFFR, KU] in this framework, and extend the theory to a class of models where discrete symmetries are broken at intermediate temperatures. This applies in particular to some systems with continuous symmetries. For this, we consider the restricted ensembles introduced in [BKL] that are very useful to ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez, J. Frohlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69, 752--820 (1996)

....lattice systems which, in a sense made precise below, are small quantum perturbations of classical lattice systems. Our main concern is the analysis of the structure of groundstates of such systems and of their low temperature phase diagrams. Our results extend those presented in an earlier paper [10]. In this paper, we develop a perturbative method that enables us to analyze how small quantum perturbations of classical lattice systems lift accidental (in particular infinite) degeneracies of the classical groundstates. Once such degeneracies have been recognized to be lifted by the ....

....how small quantum perturbations of classical lattice systems lift accidental (in particular infinite) degeneracies of the classical groundstates. Once such degeneracies have been recognized to be lifted by the perturbation one can hope to apply the variant of Pirogov Sinai theory developed in [10] to analyze the low temperature phase diagram. The necessary modifications of the tools developed in [10] in order to make them applicable to the systems studied in the present paper, will be explained. We consider quantum systems on a dimensional lattice ZZ . Such systems consist of the ....

[Article contains additional citation context not shown here]

N. Datta, R. Fern'andez, and J. Frohlich. Low-temperature phase diagrams of quantum lattice systems. I. Stability for quantum perturbations of classical systems with finitely-many groundstates. To be published in J. Stat. Phys.

....of Quantum Lattice Systems. III. Examples. By Jurg Frohlich and Luc Rey Bellet 1 Institut fur Theoretische Physik, ETH Honggerberg 8093 Zurich, Switzerland Abstract. We use the low temperature expansion and the extension of Pirogov Sinai theory developed in [1] and the perturbation theory of [2] to describe the phase diagrams of two families of fermionic lattice systems at low temperature: the balanced model and a variant of the t Gamma J model. 1 Introduction In this paper, we illustrate the methods developed in two previous papers [1, 2] to analyze the low temperature phase diagrams ....

....developed in [1] and the perturbation theory of [2] to describe the phase diagrams of two families of fermionic lattice systems at low temperature: the balanced model and a variant of the t Gamma J model. 1 Introduction In this paper, we illustrate the methods developed in two previous papers [1, 2] to analyze the low temperature phase diagrams of quantum lattice systems in terms of two classes of models, the balanced model and the t Gamma J models (of interest in connection with high temperature superconductivity) In this paper, we describe our main results and discuss the key ideas of ....

[Article contains additional citation context not shown here]

N. Datta, R. Fernandez, J. Frohlich, L. Rey-Bellet. Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy. ETH-preprint.

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