Physics

Lattice models

A lattice itself may be viewed as a regular tiling of a space by a primitive cell, where in the context of condensed matter physics the primitive cell is usually called unit cell. The lattice is then constructed by repeating the unit cell in the direction of the lattice vectors and the number of linear independent lattice vectors is the dimension of the system. Note that the dimension of a system may not correspond to the coordination number of a site, which is given by the number of connections between a lattice site and its neighbors. Within a unit cell the system can be considerably more complicated compared to the global lattice structure.

So far the following site types, i.e. building blocks of the lattice, have been implemented:

• spinless fermions: consisting of an empty \(|0\rangle\) and an occupied \(|1\rangle\) with a fermionic algebra.

• spinful fermions: consisting of an empty \(|0\rangle\), a spin up \(|\uparrow\rangle\), a spin down \(|\downarrow\rangle\), and a double occupied site \(|2\rangle\) with a fermionic algebra.

• spin-S models: consisting of \(2S+1\) states \(|S^z\rangle\) with \(Sz = -S, -S+1, \cdots, S\) with a SU(2) algebra. For spin-1/2 models the sites consist of \(|\uparrow\rangle\) and \(|\downarrow\rangle\) only.

• tJ model: corresponds to the spinful fermions without the double occupied site: \(|0\rangle\), \(|\uparrow\rangle\), \(|\downarrow\rangle\).

Hamiltonians

In the following we provide an overview of the parameters defining the Hamiltonian for each site type. Please note that depending on user feedback the specific form of the Hamiltonian may change during the beta release cycles. Specifically we may switch to a fully charge compensated version.

spinless fermions

\[{\cal H} = \sum_{j} \epsilon_{j} \hat{n}_j \; + \; \sum_{j \not= k} t^{}_{jk} \, \hat{c}^\dagger_j \hat{c}^{}_k \;+\; \sum_{j < k} U^{}_{jk} \, \left(\hat{n}^{}_j - 1/2 \right) \, \left(\hat{n}^{}_k - 1/2 \right) \;+\; \sum_{j < k} D^{}_{jk} \, \hat{c}^{}_j \hat{c}^{}_k \,+\, D^{*}_{jk} \, \hat{c}^{\dagger}_k \hat{c}^{\dagger}_j\; ,\]

where \(j, k\) are site indices and we define the site occupation \(\hat{n}_j = \hat{c}^\dagger_j \hat{c}^{}_j\). Furthermore, we define \(\epsilon_j = t_{jj}\). The first term corresponds to the energy cost of putting an electron on site \(j\) , the second term denotes the energy associated with an electron hopping from site \(j\) to site \(k\) and the third terms represents the interaction energy for having an electron on site \(j\) and another electron on site \(k\). Finally, the fourth term corresponds to the anomalous pairing terms as in BCS theory.

spinful fermions

\[\begin{split}{\cal H} & = \sum_{j} \Big[ \epsilon_j \hat{n}_j - \vec{B}_j \cdot \hat{\vec{S}}_j \Big] \; + \; \sum_{j \sigma, k\tau} t_{j \sigma, k \tau} \,\hat{c}^\dagger_{j \sigma} \hat{c}^{}_{k \tau} \; + \; \sum_{j} U_{jj} \, \left(\hat{n}{}_{j \uparrow} - 1/2 \right) \, \left(\hat{n}{}_{j \downarrow} - 1/2 \right) \nonumber \\ & \phantom{=} {} \; + \; \sum_{j < k} \bigg[ U_{jk} \, \hat{n}^{}_j \hat{n}^{}_k \; - \; J_{z, jk} \hat{S}^z_{j} \hat{S}^z_{k} \; - \; \frac{1}{2} \Big( J_{jk} \hat{S}^+_{j} \hat{S}^-_{k} \; + \; J_{jk}^\star \hat{S}^-_{j} \hat{S}^+_{k} \; + \; J_{d,jk} \hat{S}^+_{j} \hat{S}^+_{k} \; + \; J_{d,jk}^\star \hat{S}^-_{j} \hat{S}^-_{k} \Big) \bigg] \nonumber \\ & \phantom{=} {} \;+\; \sum_{jk} D^{\uparrow\downarrow}_{jk} \, \hat{c}^{}_{j \uparrow} \hat{c}^{}_{k \downarrow} \,+\, D^{\uparrow\downarrow*}_{jk} \, \hat{c}^{\dagger}_{k \downarrow} \hat{c}^{\dagger}_{j \uparrow} \;+\; \sum_{j<k,\sigma,\tau} D^{\sigma\tau}_{jk} \, \hat{c}^{}_{j \sigma} \hat{c}^{}_{k \tau} \,+\, D^{\sigma\tau*}_{jk} \, \hat{c}^{\dagger}_{k \sigma} \hat{c}^{\dagger}_{j \tau}\end{split}\]

where \(\sigma, \tau\) are spin indices running over \(\uparrow, \downarrow\) and the on-site occupation is given by \(\hat{n}_j = \hat{n}_{j \uparrow} + \hat{n}_{j \downarrow}\). Similarly, the local spin magnetization is defined by

\[\begin{split}\hat{\vec{S}}_j = \frac{1}{2} \begin{pmatrix} \hat{c}^\dagger_{j \uparrow} & \hat{c}^\dagger_{j \downarrow} \end{pmatrix} \cdot \vec{\sigma} \cdot \begin{pmatrix} \hat{c}^{}_{j \uparrow} \\ \hat{c}^{}_{j \downarrow} \end{pmatrix} \;,\end{split}\]

with \(\vec{\sigma}\) being the vector composed of the Pauli matrices. Note the explicit minus sign in front of the Zeeman coupling. This implies that the spin magnetization prefers to align (as oposed to to anti-align) with the external magnetic field \(\vec{B}\). The spin raising and lowering operators are defined as \(S_{+, j} = S_{x, j} + i S_{y, j}\) and \(S_{-, j} = S_{x, j} - i S_{y, j}\), respectively. The spin-dependent inter-site interaction is specified in terms of \(J_{z, jk}\) and the complex-valued \(J_{jk} = J_{\perp, jk} + i J_{\times, jk}\). Accordingly, it can also be written as

\[\begin{split}& \sum_{j < k} \bigg[ J_{z, jk} \hat{S}^z_{j} \hat{S}^z_{k} \; + \; \frac{1}{2} \Big( J_{jk} \hat{S}^+_{j} \hat{S}^-_{k} \; + \; J_{jk}^\star \hat{S}^-_{j} \hat{S}^+_{k} \Big) \bigg] \nonumber \\ = & \sum_{j < k} \bigg[ J_{z, jk} \hat{S}^z_{j} \hat{S}_{z, k} \; + \; J_{\perp, jk} \Big(\hat{S}^x_{j} \hat{S}^x_{k} \; + \; \hat{S}^y_{j} \hat{S}^y_{k} \Big) \; + \; J_{\times, jk} \boldsymbol{e}_z \cdot \Big( \hat{\boldsymbol{S}}_j \times \hat{\boldsymbol{S}}_k \Big) \bigg] \; ,\end{split}\]

which highlights that \(\mathrm{Re}[J_{jk}] = J_{\perp, jk}\) encodes anisotropy and \(\mathrm{Im}[J_{jk}] = J_{\times, jk}\) represents a Dzyaloshinskii-Moriya interaction. The explicit minus sign in front of the inter-site spin-spin interactions implies that for isotropic spin-spin interactions (\(J_{z, jk} = J_{\perp, jk}\) and \(J_{\times, jk} = 0\)) neighboring spins align.

NOTE: In the presence of transverse magnetic fields (\(B_x \neq 0\) and/or \(B_y \neq 0\)) \(S_z\) is no longer a good quantum number. In order to allow for states to break this symmetry please specify the input parameter mod_Sz to be non-zero. mod_Sz has to be a positive even integer. For example, mod_Sz: 2 allows eigenstates to be a superposition of states differing by single or multiple spin flips.

spins

\[\begin{split}{\cal H} = {} - \sum_{i} \vec{B}_j \cdot \hat{\vec{S}}_j \nonumber \\ &\; - \; \sum_{j < k} \bigg[ J_{z, jk} \hat{S}^z_{j} \hat{S}^z_{k} \; + \; J_{\perp, jk} \Big(\hat{S}^x_{j} \hat{S}^x_{k} \; + \; \hat{S}^y_{j} \hat{S}^y_{y, k} \Big) \; + \; J_{\times, jk} \boldsymbol{e}_z \cdot \Big( \hat{\boldsymbol{S}}_j \times \hat{\boldsymbol{S}}_k \Big) \; + \; J_{d,jk} \hat{S}^+_{j} \hat{S}^+_{k} \; + \; J_{d,jk}^\star \hat{S}^-_{j} \hat{S}^-_{k} \Big) \bigg]\end{split}\]

For the site type spins, the representation of the spin operators can be controlled using the spin_representation input variable. spin_representation has to be a positive integer \(n\). \(n\) determines the spin quantum number, \(s = n/2\).

NOTE: In the presence of transverse magnetic fields (\(B_x \neq 0\) and/or \(B_y \neq 0\)) \(S_z\) is no longer a good quantum number. In order to allow for states to break this symmetry please specify the input parameter mod_Sz to be non-zero. mod_Sz has to be a positive even integer. For example, mod_Sz: 2 allows eigenstates to be a superposition of states differing by single or multiple spin flips. Since \(\hat{S}^-_{j} \hat{S}^-_{k}\) and \(\hat{S}^+_{j} \hat{S}^+_{k}\) changes \(S_z\) by 2, i.e. n by 4, we need the mod_Sz: 4 if \(B_x = 0\) or \(B_y = 0\) in case \(J_d = 0\) has finite entries.

Reduced density matrices

We define the label ordering of the single particle reduced density matrix \(\rho_{x,y}\) as

\[\begin{split}\rho_{y,x} &= \langle \hat{c}^+_{x} \hat{c}^{}_{y}\rangle \\ \rho_{y\tau,x\sigma} &= \langle \hat{c}^+_{x \sigma} \hat{c}^{}_{y \tau}\rangle\end{split}\]

with \(\sigma, \tau\) the spin and \(x, y\) the orbital / site indices. Note that this definition is consistent with most of the text book conventions, e.g. Landau IX. It is also consistent with the usual definition of the lesser and greater single particle Greensfunctions.

Rationale

The reason for exchanging the index order in the definition is given the usual convention of a density matrix \(E = \text{Tr} \,h \rho\) with \(h\) the single particle Hamiltonian matrix.

\[\begin{split}E &= \langle {\cal H} \rangle \\ &= \sum_{x\sigma,y\tau} \langle \hat{c}^+_{x, \sigma} h_{x\sigma,y\tau} \hat{c}^{}_{y \tau}\rangle \\ &= \sum_{x\sigma,y\tau} h_{x\sigma,y\tau} \langle \hat{c}^+_{x, \sigma} \hat{c}^{}_{y \tau}\rangle \\ &= \sum_{x\sigma,y\tau} h_{x\sigma,y\tau} \rho_{y\tau, x\sigma} \\ &= \text{Tr}\, h \rho\end{split}\]

Corollary

Since we should use the same convention for the \(\langle \hat{S}^+_{x} \hat{S}^{-}_{y}\rangle\) correlator we decided to use the transposed index notation for all correlators, structure and one particle reduced density matrices.

\[S_{y,x} = \langle \hat{O}_{x} \hat{Q}^{}_{y}\rangle\]

with \(\hat O\) and \(\hat Q\) being arbitrary on-site operators.

Cluster Embedding

The primary difficulty in simulating interacting quantum systems involves the exponential growth of the Hilbert space with the number of particles and the size of the system. The idea behind cluster embedding schemes is to divide a system into a cluster and its complement, referred to as the bath. One then attempts to treat the interaction within the cluster in a rigorous manner, while performing approximations on the bath to simplify the computations.

There are many ways to implement cluster embedding schemes. In the following we provide an overview of the most common embedding schemes.

Currently we are implementing the a mean field based self-consistent cluster embedding (SCCE/iMF), the cluster perturbation theory (CPT), and dynamical mean field theory.

SCCE/iMF is currently enabled in the SCCE solver for a subset of the available lattice models. The CPT is enabled within the backend.

SCCE/iMF

See also

SCCE/iMF

The scheme we have implemented is a self-consistent cluster embedding (SCCE), where the bath is obtained using an inverse mean-field (iMF) approach. Specifically, we extract a mean-field description from the cluster, which we then deploy by a periodic continuation on the complete system to obtain a mean field description. We then build a hybrid system, consisting of the cluster, which is described by the fully interacting system, the bath, and the cluster bath coupling, both described by the terms obtained from the inverse mean-field procedure. This hybrid system is currently solved via density matrix renormalization group (DMRG) calculations using PRG++.

The main advantage of this approach is that the description of the bath is physically well defined. In addition, one avoids the expensive calculation of Green's functions in order to construct the bath. Moreover, one obtains a mean-field description of the system of interest. Of course, one should keep in mind that the bath is described in mean-field only and can not account for all interaction effects.

CPT

See also

Cluster perturbation theory (CPT)

Cluster perturbation theory. In cluster perturbation theory, one calculates the local Green's functions of an isolated cluster. Subsequently, one builds a larger system by populating a larger lattice with the cluster and adding the missing hopping elements via a Dyson equation.

The main advantage of CPT is its simple structure. As one calculates the Green's function in the absence of a bath, one is able to study larger cluster sizes as compared with CDMF, which allows for the capture of reasonable momentum dependence.

DMFT

The Dynamical Mean-Field Theory is a non-perturbative, self-consistent embedding method suitable for the treatment of many-body problems which feature local electron-electron interaction. In DMFT, a many-body lattice problem is systematically mapped onto an impurity model, usually an Anderson impurity type model. Electrons on the impurity interact via a local interaction term and the impurity is coupled to a non-interacting effective bath. The mapping process of DMFT contains the approximation that the lattice self-energy \(\Sigma(\mathbf{k},i\nu_n)\) is a purely local, momentum-independent quantity. This approximation can be motivated by the fact that it becomes exact in the limit of vanishing hopping integrals \(t\rightarrow 0\), vanishing interaction \(U\rightarrow 0\), and most notably in the limit of infinite coordination number \(Z\rightarrow \infty\). In cluster dynamical mean field theory, the lattice model is mapped onto a cluster Anderson impurity model instead, to gain some insight into static fluctuations.

CDMFT

Cluster dynamical mean-field theory is the cluster extension of the DMFT.

DMET

Density matrix embedding theory. In density matrix embedding theory a system is divided into small fragments. Orbitals from the environment, consisting of the other fragments that are entangled with a specific fragment, are identified in the reduced density matrix of the environment. An embedded Hamiltonian in the subspace of the fragment, combined with its entangled environment states, can then be studied using more precise approaches. In the self-consistency loop, an effective non-interacting Hamiltonian is derived from the density matrix to identify the entangled orbitals for the next embedding step.

The advantage of DMET lies in its independence from dynamical observables. One only needs to calculate the ground state density matrices of a given system.

CASSCF

Complete active space self-consistent field. Within CASSCF one typically selects a cluster in by eigenstates of a single particle Hamiltonian. However, one can also choose an arbitrary real-space cluster as active space. The main difference lies in the treatment of the bath. CASSCF performs a self-consistent Hartree-Fock calculation for the bath, while within SCCE we extract the bath mean-field directly from the cluster.