Continuous-time quantum Monte Carlo

In second quantisation, the Hamiltonian of the Anderson impurity model reads:[1]

${\displaystyle H=\underbrace {\sum _{ij}E_{ij}c_{i}^{\dagger }c_{j}} _{H_{\mathrm {loc0} }}+\underbrace {{\frac {1}{2}}\sum _{ijkl}U_{ikjl}c_{i}^{\dagger }c_{j}^{\dagger }c_{l}c_{k}} _{H_{\mathrm {int} }}+\underbrace {\sum _{p,i}{\big (}V_{pi}f_{p}^{\dagger }c_{i}+V_{pi}^{*}c_{i}^{\dagger }f_{p}{\big )}} _{H_{\mathrm {hyb} }}+\underbrace {\sum _{p}\epsilon _{p}f_{p}^{\dagger }f_{p}} _{H_{\mathrm {bath} }}}$,

where ${\displaystyle c_{i}^{\dagger }}$ and ${\displaystyle c_{i}}$ are the creation and annihilation operators, respectively, of a fermion on the impurity. The index ${\displaystyle i}$ collects the spin index and possibly other quantum numbers such as orbital (in the case of a multi-orbital impurity) and cluster site (in the case of multi-site impurity). ${\displaystyle f_{p}^{\dagger }}$ and ${\displaystyle f_{p}}$ are the corresponding fermion operators on the non-interacting bath, where the bath quantum number ${\displaystyle p}$ will typically be continuous.

Step 1 of CT-QMC is to split the Hamiltonian into an exactly solvable term, ${\displaystyle H_{0}}$, and the rest, ${\displaystyle H_{\mathrm {I} }}$. Different choices correspond to different expansions and thus different algorithmic descriptions. Common choices are:

• Interaction expansion (CT-INT):[2] ${\displaystyle H_{\mathrm {I} }=H_{\mathrm {int} }}$
• Hybridization expansion (CT-HYB):[3][4] ${\displaystyle H_{\mathrm {I} }=H_{\mathrm {hyb} }}$
• Auxiliary field expansion (CT-AUX):[5] like CT-INT, but the interaction term is first decoupled using a discrete Hubbard-Stratonovich transformation

Step 2 is to switch to the interaction picture and expand the partition function in terms of a Dyson series:

${\displaystyle Z=\operatorname {tr} \left(\mathrm {e} ^{-\beta H}\right)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}\int _{0}^{\beta }\mathrm {d} ^{n}\tau \;\operatorname {tr} \left[\mathrm {e} ^{-\beta H_{0}}T_{\tau }H_{\mathrm {I} }(\tau _{1})H_{\mathrm {I} }(\tau _{2})\cdots H_{\mathrm {I} }(\tau _{n})\right]}$,

where ${\displaystyle \beta }$ is the inverse temperature and ${\displaystyle T_{\tau }}$ denotes imaginary time ordering. The presence of a (zero-dimensional) lattice regularises the series and the finite size and temperature of the system makes renormalisation unnecessary.[2]

The Dyson series generates a factorial number of identical diagrams per order, which makes sampling more difficult and possibly worsen the sign problem. Thus, as step 3, one uses Wick's theorem to group identical diagrams into determinants. This leads to the expressions:[1]

• Interaction expansion (CT-INT):

${\displaystyle Z=\sum _{n=0}^{\infty }\prod _{\alpha =1}^{n}\sum _{i_{\alpha }j_{\alpha }k_{\alpha }l_{\alpha }}\int \mathrm {d} \tau _{\alpha }\left(-{\frac {1}{2}}U_{i_{\alpha }k_{\alpha }j_{\alpha }l_{\alpha }}\right)\det \left[{\begin{array}{cc}\langle T_{\tau }c_{i_{\alpha }}^{\dagger }\!(\tau _{\alpha })\ c_{k_{\beta }}\!(\tau _{\beta })\rangle _{0}&\langle T_{\tau }c_{i_{\alpha }}^{\dagger }\!(\tau _{\alpha })\ c_{l_{\beta }}\!(\tau _{\beta })\rangle _{0}\\\langle T_{\tau }c_{j_{\alpha }}^{\dagger }\!(\tau _{\alpha })\ c_{k_{\beta }}\!(\tau _{\beta })\rangle _{0}&\langle T_{\tau }c_{j_{\alpha }}^{\dagger }\!(\tau _{\alpha })\ c_{l_{\beta }}\!(\tau _{\beta })\rangle _{0}\end{array}}\right]_{\alpha \beta }}$

• Hybridisation expansion (CT-HYB):

${\displaystyle Z=\sum _{n=0}^{\infty }\prod _{\alpha =1}^{n}\sum _{i_{\alpha }j_{\alpha }}\int \mathrm {d} \tau _{\alpha }\mathrm {d} \tau _{\alpha }^{\prime }\operatorname {tr} {\Big [}\mathrm {e} ^{-\beta (H_{\mathrm {loc0} }+H_{\mathrm {int} })}T_{\tau }\prod _{\alpha =1}^{n}c_{i_{\alpha }}^{\dagger }(\tau _{\alpha })c_{i_{\alpha }}(\tau _{\alpha }^{\prime }){\Big ]}\;\det {\big [}\Delta _{i_{\alpha }j_{\beta }}(\tau _{\alpha }-\tau _{\beta }^{\prime }){\big ]}_{\alpha \beta }}$

In a final step, one notes that this is nothing but an integral over a large domain and performs it using a Monte Carlo method, usually the Metropolis–Hastings algorithm.

• Metropolis–Hastings algorithm
• Quantum Monte Carlo
• Dynamical mean field theory