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Path Integral Methods for Single Band Hubbard Model

N. Heydari, S. Azakov

Published Apr 1, 1997• 0 citations
OpenAlex

Abstract

We review various ways to express the partition function of the single-band Hubard model as a path integral. The emphasis is made on the derivation of the action in the integrand of the path integral and the results obtained from this approach are discussed only briefly. Since the single-band Hubbard model is a pure fermionic model on the lattice and its Hamiltonian is a polynomial in creation and annihilation fermionic operators, with the help of the fermionic coherent states of holomorphic representions the partition function of the model can be straightforwardly expressed as a path integral with respect to Grassmann fields. Then the question arises how to calculate this path integral specially in the most interesting case of the strong coupling, which is described by the term quartic in the fermion fields. The standard way to get rid of this quartic term is to use so-called Hubbard-Stratonovich transformation by introducing auxilary boson field (this procedure is usually called bosonization). Since there are several equivalent forms of the interaction term in the single-band Hubbard model the bosonization procedure is not unique. In the first section of the article we discuss two types of bosoniztions one with two real scalar fields (charge and spin fields) and another with a real vector field. The second section is devoted to a spin-rotation-invariant bosonization procedure invented by Schulz and Weng, Ting and Lee, where an arbitary quantization axis is introduced at each point in space and time and an integration over all possible orientations enters the path integral measure. Spin rotation invariant formulation is necessary to study e.g. spiral magnetization. In the third section we present the contstruction of a path integral for the partition function when the single-band Hubbard Hamiltonian is written in terms of the Hubbard operators, which are elements of the doubly graded semi-simple algebra. In this case one should use generalized coherent states for superalgebra in order to get a path integral formulation. We follow the procedure proposed recently by T ngler and Kopp, to investigate the case of strong coupling.

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We review various ways to express the partition function of the single-band Hubard model as a path integral. The emphasis is made on the derivation of the action in the integrand of the path integral and the results obtained from this approach are discussed only briefly. Since the single-band Hubbard model is a pure fermionic model on the lattice and its Hamiltonian is a polynomial in creation and annihilation fermionic operators, with the help of the fermionic coherent states of holomorphic representions the partition function of the model can be straightforwardly expressed as a path integral with respect to Grassmann fields. Then the question arises how to calculate this path integral specially in the most interesting case of the strong coupling, which is described by the term quartic in the fermion fields. The standard way to get rid of this quartic term is to use so-called Hubbard-Stratonovich transformation by introducing auxilary boson field (this procedure is usually called bosonization). Since there are several equivalent forms of the interaction term in the single-band Hubbard model the bosonization procedure is not unique. In the first section of the article we discuss two types of bosoniztions one with two real scalar fields (charge and spin fields) and another with a real vector field. The second section is devoted to a spin-rotation-invariant bosonization procedure invented by Schulz and Weng, Ting and Lee, where an arbitary quantization axis is introduced at each point in space and time and an integration over all possible orientations enters the path integral measure. Spin rotation invariant formulation is necessary to study e.g. spiral magnetization. In the third section we present the contstruction of a path integral for the partition function when the single-band Hubbard Hamiltonian is written in terms of the Hubbard operators, which are elements of the doubly graded semi-simple algebra. In this case one should use generalized coherent states for superalgebra in order to get a path integral formulation. We follow the procedure proposed recently by T ngler and Kopp, to investigate the case of strong coupling.
Functional integral methods provide a way to define mean–field theories and to systematically improve them. For the Hubbard model and similar strong–correlation problems, methods based in particular on the Hubbard–Stratonovich transformation have however been plagued by difficulties to formulate the problem in a spin–rotation invariant way. Here a formalism circumventing this problem by using a space – and time–dependent spin reference axis is discussed. This formulation is then used to suggest a possible alternative to Nagaoka ferromagnetism in the strongly correlated Hubbard model in the vicinity of half–filling. Finally, some aspects of single–particle spectra in a simplified model for a short–range ordered antiferromagnet are discussed. PACS numbers: 71.28.+d, 75.10.Lp, 75.30.Kz 1.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
The path integral single spin partition function in the basis of boson coherent states, for a general normal ordered two-mode Schwinger boson Hamiltonian, has been computed. Within the initial phase of the calculational process, by a specific approach, the Schwinger boson constraint by means of the projection operator has been implemented.\nAfter, the case of the Zeeman Hamiltonian has been taken on. Using the expression for the partition function obtained for the general case, the appropriate single spin partition function has been produced. Additionally, partition functions have been computed for a few specific spin quantum numbers and then compared with those calculated by the straightforward means, which brought about great confidence in the projection operator implementation; both results match perfectly. \nFinally, the Heisenberg Model has been tackled, and the appropriate expressions for the partition functions have been computed. Their validity has not been verified for any specific cases, but the Zeeman Hamiltonian case dealt with earlier appears to be a strong indicator that these are indeed the correct expressions. \nMany open questions remain, however, for future research to address. All in all, it has been discovered that using the projection operator as a means to enforce the Schwinger boson constraint, veritably works for a Zeeman Hamiltonian problem for specific spin quantum numbers.
The Hubbard model is investigated using functional integrals and a Hubbard-Stratonovich decomposition of the interaction term. Using a spin-space reference frame that fluctuates in time and space, the effective action both is spin-rotation invariant and exhibits the correct Hartree-Fock saddle point. For strong correlation the effective action for free carriers and spins coupled via a gauge field is obtained as a systematic expansion in t/U. The spiral state existing for small doping is discussed. The formalism allows straightforward generalization to intermediate and weak coupling.
Functional-integral techniques are used to study correlated fermion models of popular interest: the prototypical Hubbard models. Recent progress has been made in the functional-integral approach to such correlated Fermi systems by introducing a spin-space reference frame that :fluctuates in space and time, Introducing fermions with varying quantization a.xis on the lattice allows the functional-integral representation to assume a uniform approximation over the entire range of the interaction strength U, reproducing the expected mean fields at the saddlepoints along with gapless excitation spectrums. First, the framework necessary for this study is developed: second quantization and the second quantized models of correlated fermions that are to be examined are briefly reviewed. Next, an introduction to first and second quantized functional integrals is followed by basic functional-integral techniques employed to study quantum many-particle systems. Once the background necessary to investigate this approach has been given, the :fluctuating spin-space reference frame is described and is examined in the two-site problem, where an exact diagonalization may be performed. A functional-integral over one-vertex polygonal paths is used to approximate the partition function in a fixed reference frame orientation. This functional-integral representation is studied in the low-to-intermediate and high temperature regimes. A classification of the quantization a.xis configurations, based on classical spin waves, is motivated and presented.
The present work consists of two parts. The first part deals with hyperbolic Hubbard-Stratonovich transformations. Such transformations are used to derive non-linear sigma models that describe the low energy behaviour of disordered electron systems. For a long time the mathematical status of hyperbolic Hubbard-Stratonovich transformations of Pruisken-Schäfer type remained unclear. Only recently the two special cases of pseudounitary and pseudoorthogonal symmetry were proven . In this thesis we prove the transformation for a general (essentially semisimple) symmetry group. The proof is descriptive and shows explicitly the connection to the standard Gaussian integrals. In the second part we develop a novel method to bosonise granular fermionic systems. The method is related to the method of coherent states. In particular it is not based on the well known bosonisation of (1+1)-dimensional systems. One approach is to use the colour-flavour transformation to transform the Grassmann path integral representation of a grand canonical partition function in a way that allows to eliminate the Grassmann variables. The result is a path integral in generalised coherent states with special boundary conditions.