Simulations of Discrete Quantum Systems in Continuous Euclidean Time

Beard, Bernard B.; Wiese, U.‐J.

doi:10.1103/physrevlett.77.5130

Cited by 346 publications

(442 citation statements)

References 12 publications

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“…The point of using this loop-cluster definition for the importance sampling of classical spin systems, as was treated rigorously in [17] [18], is that the sum over discrete steps in the probabilistic Markov chain, M, can then essentially be interchanged with the sum over discrete loop-clusters in the definition of the lattice partition function, which leads to more efficient numerical sampling. As was noticed in [15] and [16], however, when this loop-cluster definition of the new lattice configurations is applied to the Trotter-Suzuki formalism in (4), this definition has the advantage that closed loops on the D +1 dimensional lattice in (4) automatically preserve the definition of the trace. Therefore, as is used explicitly in [15], using this approach it is possible to interchange, t, with, M, and to define the lattice partition via the with respect to a trace operation defined over the length of the Markov chain, rather than with respect to Euclidean-time.…”

Section: B Continuous-timementioning

confidence: 99%

“…As was noticed in [15] and [16], however, when this loop-cluster definition of the new lattice configurations is applied to the Trotter-Suzuki formalism in (4), this definition has the advantage that closed loops on the D +1 dimensional lattice in (4) automatically preserve the definition of the trace. Therefore, as is used explicitly in [15], using this approach it is possible to interchange, t, with, M, and to define the lattice partition via the with respect to a trace operation defined over the length of the Markov chain, rather than with respect to Euclidean-time. A formal definition of the continuous-time properties of the general continuous-time Monte Carlo method is given in [19], and it is explained in detail how, t, and, M, can be interchanged in a general…”

Section: B Continuous-timementioning

confidence: 99%

“…This recovers the quantum Hamiltonian used for the Trotter-Suzuki decomposition of a quantum spin system with an AFM nearest-neighbour spin interaction means that the lattice partition function can be expressed via a basis of two-by-two site blocks known as plaquettes [15],…”

Section: A the Loop-cluster Lattice Partition Functionmentioning

confidence: 99%

“…To be clear, these transition probabilities, λ i , are taken from the final figure in [15], but there is no explicit expression in [15] that defines the lattice partition function as a product of these matrix elements. Therefore, to obtain (6) we have substituted these transition probabilities from the final figure into the transfer matrix definition in (4) (which is also from [15]).…”

Section: B Continuous-timementioning

confidence: 99%

“…We define the lattice partition function of the continuous-time QMC method from [15] by the following product of local transfer matrices,…”

Section: B Continuous-timementioning

confidence: 99%

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The partition function zeroes of quantum critical points

Crompton¹

2009

Nuclear Physics B

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The Lee-Yang theorem for the zeroes of the partition function is not strictly applicable to quantum systems because the zeroes are defined in units of the fugacity e h∆τ , and the Euclidean-time lattice spacing ∆τ can be divergent in the infrared (IR). We recently presented analytic arguments describing how a new space-Euclidean time zeroes expansion can be defined, which reproduces Lee and Yang's scaling but avoids the unresolved branch points associated with the breaking of nonlocal symmetries such as parity. We now present a first numerical analysis for this new zeros approach for a quantum spin chain system. We use our scheme to quantify the renormalization group flow of the physical lattice couplings to the IR fixed point of this system. We argue that the generic Finite-Size Scaling (FSS) function of our scheme is identically the entanglement entropy of the lattice partition function and, therefore, that we are able to directly extract the central charge, c, of the quantum spin chain system using conformal predictions for the scaling of the entanglement entropy.

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Section: B Continuous-timementioning

confidence: 99%

Section: B Continuous-timementioning

confidence: 99%

Section: A the Loop-cluster Lattice Partition Functionmentioning

confidence: 99%

Section: B Continuous-timementioning

confidence: 99%

“…We define the lattice partition function of the continuous-time QMC method from [15] by the following product of local transfer matrices,…”

Section: B Continuous-timementioning

confidence: 99%

See 3 more Smart Citations

The partition function zeroes of quantum critical points

Crompton¹

2009

Nuclear Physics B

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QuantumMonteCarlo Methods

Assaad

Troyer

2007

Handbook of Magnetism and Advanced Magnetic Materials

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We present a review of quantum Monte Carlo algorithms for the simulation of quantum magnets. A general introduction to Monte Carlo sampling is followed by an overview of local updates, cluster updates, and generalized ensemble techniques such as multicanonical and Wang‐Landau sampling for classical magnets. For quantum systems we present two complementary approaches: world‐line methods in (i) a path‐integral and (ii) a stochastic series expansion (SSE) representation, which are best suited for pure spin Hamiltonians. Determinental quantum Monte Carlo algorithms are the methods of choice for the simulation of fermionic Hamiltonians, such as the Hubbard model. An overview of the methods is followed by a selection of typical applications.

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Computing Quantum Phase Transitions

Vojta

2008

Reviews in Computational Chemistry

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A phase transition occurs when the thermodynamic properties of a material display a singularity as a function of the external parameters. Imagine, for instance, taking a piece of ice out of the freezer. Initially, its properties change only slowly with increasing temperature. However, at 0°C, a sudden and dramatic change occurs. The thermal motion of the water molecules becomes so strong that it destroys the crystal structure. The ice melts, and a new phase of water forms, the liquid phase. At the phase transition temperature of 0°C the solid (ice) and the liquid phases of water coexist. A finite amount of heat, the so-called latent heat, is required to transform the ice into liquid water. Phase transitions involving latent heat are called first-order transitions. Another well known example of a phase transition is the ferromagnetic transition of iron. At room temperature, iron is ferromagnetic, i.e., it displays a spontaneous magnetization. With rising temperature, the magnetization decreases continuously due to thermal fluctuations of the spins.At the transition temperature (the so-called Curie point) of 770°C, the magnetization vanishes, and iron is paramagnetic at higher temperatures. In contrast to the previous example, there is no phase coexistence at the transition temperature; the ferromagnetic and paramagnetic phases rather become indistinguishable. Consequently, there is no latent heat. This type of phase transition is called continuous (second-order) transition or critical point.

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Simulations of Discrete Quantum Systems in Continuous Euclidean Time

Cited by 346 publications

References 12 publications

The partition function zeroes of quantum critical points

The partition function zeroes of quantum critical points

QuantumMonteCarlo Methods

Computing Quantum Phase Transitions

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