Motivation
Condensed matter theory faces a significant challenge due to the vast number of degrees of freedom in condensed matter systems, making direct solutions based on known natural laws impractical This complexity is evident in the diverse phenomena observed, including various types of ordering, phase transitions, and unique states of matter such as superconductivity and fractional Quantum Hall liquids To develop a theoretical understanding of these intricate phenomena, researchers must simplify and model the systems appropriately, followed by solving the resulting many-particle model using reliable theoretical methods.
Theoretical methods for solving quantum many-particle problems can be broadly classified in three main categories:
Different methods for solving complex systems each come with unique benefits and limitations Perturbative techniques necessitate the identification of a small parameter to enable a reliable expansion Exact analytical methods, such as the Bethe ansatz, are applicable only to specific integrable Hamiltonians Additionally, numerical solutions often involve smaller system sizes than those relevant to experiments, requiring challenging extrapolations for accurate results.
Recent advancements in condensed matter systems have been significantly enhanced by the application of various solution techniques developed over the past decades Notably, renormalization has played a crucial role in this progress, contributing to a deeper understanding of these complex systems.
Stefan Kehrein: The Flow Equation Approach to Many-Body Problems
Scaling ideas have enabled the classification of vastly different microscopic systems into specific universality classes that exhibit similar universal behaviors Notably, the behavior observed at large length scales or at sufficiently low energies is largely unaffected by the intricate details of microscopic interactions Despite this progress, many condensed matter systems characterized by strong electron correlations, such as high-temperature superconductors and heavy fermion materials, remain challenging to analyze, highlighting the need for the development of new theoretical tools.
This book presents a novel analytical approach to quantum many-particle systems through the flow equation method introduced by Wegner in 1994, which has since been applied in various fields of condensed matter physics Concurrently, Glazek and Wilson developed the similarity renormalization scheme in high-energy physics, highlighting the connection between these two methods Both approaches extend traditional scaling ideas by producing a renormalized perturbative expansion, but unlike conventional methods, they do not limit their focus to low-energy physics Notably, these techniques are also effective in addressing certain strong-coupling problems where traditional perturbative scaling leads to diverging coupling constants.
In this book we will be mainly interested in condensed matter systems and therefore use the original terminologyflow equations introduced by Wegner in 1994 [2].
Flow Equations: Basic Ideas
Condensed matter systems exhibit diverse energy scales, with electronic band widths around a few eV and experimental temperatures often significantly lower This disparity necessitates theoretical calculations that provide reliable outcomes at energy scales much smaller than the model's intrinsic energy levels Consequently, it is essential to apply perturbation theory focusing on larger energy differences before addressing smaller ones, even when a small expansion parameter exists within the model.
In condensed matter theory, scaling concepts illustrate the principle of energy scale separation by systematically lowering an ultraviolet (UV) cutoff, denoted as Λ RG, from its initial value to an experimentally relevant scale This process involves conducting a perturbative calculation using a running coupling constant that varies with the energy scale.
A schematic view of this procedure is depicted in Fig 1.1a The ma- trix denotes a many-particle Hamiltonian with single-particle energies on
1This situation is similar to atomic physics where one first calculates, e.g., thefine structure of the spectrum before deriving the hyperfine splittings based on these eigenstates calculated before.
The schematic representation of various scaling approaches illustrates two key methods: (a) conventional scaling techniques that progressively lower the high-energy cutoff, denoted as Λ RG, and (b) flow equations that enhance the band-diagonal nature of the Hamiltonian, effectively narrowing the bandwidth from the lowest energy level E = 0 to the ultraviolet cutoff Λ.
In the conventional scaling approach, the shaded off-diagonal matrix elements indicate non-vanishing couplings among various modes Degrees of freedom with single-particle energies within the range [Λ−δΛ, Λ] are eliminated, often through a path integral framework, resulting in a Hamiltonian that retains the essential physics within the reduced cutoff, Λ RG = Λ−δΛ This process modifies the coupling constants and generates a flow of these constants, typically accessible only through perturbation theory, leading to scaling equations for the coupling constants as Λ RG varies For instance, in a renormalizable theory featuring a single dimensionless coupling g(Λ RG ) and a dimensionful cutoff parameter, one can derive the corresponding scaling equation.
While a many-body Hamiltonian is typically complex and cannot be easily expressed as a simple matrix, the insights gained from this simplified model are applicable to more intricate scenarios.
The equation β(g(Λ RG)) = 0, as presented in (1.1), incorporates a β-function typically derived from an expansion involving the running coupling constant The differentiation with respect to the logarithm on the left side of (1.1) arises naturally, given that the cutoff Λ RG is the sole dimensionful parameter in the context.
The scaling approach's key feature is that the Hamiltonians H[g(Λ RG), Λ RG] consistently represent the same low-energy physics within the low-energy Hilbert space when the coupling is adjusted according to the scaling equation While the β-function is generally known only perturbatively to a limited degree, the iterative nature of the differential scaling equation enables the recovery of nonperturbative energy scales, which can be proportional to fractional powers of the coupling constant or exhibit exponential behavior This non-analytic behavior complicates the use of naive perturbation theory for such energy scales A practical perspective on scaling is that it allows for a reorganization of perturbation theory into a more manageable convergent expansion by focusing on the β-function rather than the physical observable directly Additionally, investigating low-energy fixed points of the scaling flow helps identify universality classes and their universal properties, which are largely independent of the specifics of the original interactions, determined mainly by symmetries and dimensionality For a deeper understanding of these significant concepts in modern condensed matter theory, readers are encouraged to explore the extensive literature available on the subject.
The flow equation method, illustrated in Fig 1.1b, operates on the principle of energy scale separation similar to conventional scaling approaches, but it uniquely retains the complete Hilbert space This method progressively transforms the Hamiltonian to become more energy-diagonal, as shown in Fig 1.1b, by iteratively decreasing the energy-diagonality parameter, Λ feq Unlike traditional scaling techniques that lower a fixed ultraviolet cutoff, this approach focuses on reducing the cutoff, Λ feq, associated with the energy transfer of interaction matrix elements.
From the point of view of low-energy physics close to energyE = 0, we can consider both methods as effectively equivalent with Λ feq ∝Λ RG One
3 Furthermore, renormalizability ensures that the right hand side of (1.1) only depends on the coupling constant.
The Kondo temperature exemplifies a phenomenon where its behavior is described by the equation exp(−1/ρ F J), with J > 0 representing the antiferromagnetic exchange coupling As illustrated in Fig 1.1b, higher powers of the running coupling constant, typically assumed to be small, suppress excitations to elevated energy levels EΛ feq in the flow equation procedure This flow equation framework serves as a generalization of the traditional scaling approach, notably retaining the entire Hilbert space However, this advancement comes with the trade-off of a more intricate set of scaling equations The significant advantage of this methodology lies in its ability to preserve information across all energy scales within the system, which is crucial for analyses requiring comprehensive insights into various energy levels.
– correlation functions on all energy scales
– systems that contain competing energy scales
We will discuss examples for all these applications later on in this book.
Recent analysis of non-equilibrium problems has gained significant attention within the flow equation approach, highlighting its promising potential Non-equilibrium conditions, whether through the establishment of a non-equilibrium initial state or continuous energy input, introduce multiple energy scales that influence low-temperature behavior This complexity renders conventional scaling methods ineffective, as they tend to overlook certain degrees of freedom during the scaling process Examples of these applications will be explored in Sections 5.2 and 5.3.
The primary objective of implementing a scaling flow of the Hamiltonian while preserving the full Hilbert space influences the method for generating the Hamiltonians H(Λ feq) Traditional elimination of degrees of freedom in a path integral framework is not feasible To maintain the spectrum of the flowing Hamiltonian, we seek unitary transformations that connect the Hamiltonians H(Λ feq) Additionally, to ensure a stable expansion, it is essential to respect energy scale separation, leading us to use infinitesimal unitary transformations In the initial stages of the flow, characterized by large Λ feq, we focus on eliminating off-diagonal matrix elements that couple modes with significant energy differences As the flow progresses, we begin to remove couplings between states with closer energy levels.
Any one-parameter family of unitarily equivalent Hamiltonians \( H(B) \) can be derived from the differential equation \( \frac{dH(B)}{dB} = [\eta(B), H(B)] \), where the antihermitian generator \( \eta(B) \) satisfies \( \eta(B) = -\eta^{\dagger}(B) \) Although all Hamiltonians \( H(B) \) are unitarily equivalent to the initial Hamiltonian \( H(B = 0) \) if the equation can be solved exactly, such exact solutions are typically unattainable for complex many-body systems To address this, we propose a systematic expansion that allows our Hamiltonians \( H(B) \) to be approximately unitarily equivalent to \( H(B = 0) \), with the approximation error diminishing at higher orders of the expansion This systematic approach leverages energy scale separation, where the generator \( \eta(B) \) initially targets the elimination of interaction matrix elements connecting modes with significant energy differences for small \( B \), and progressively decouples more degenerate states as \( B \) increases.
Wegner has developed a canonical generator that effectively eliminates interaction matrix elements while maintaining energy scale separation This approach involves defining the Hamiltonian's diagonal component as H₀ and the interaction component to be eliminated as H_int The canonical generator is then expressed through a commutator, represented as η(B) = [H₀(B), H_int(B)].
The operator η(B) is antihermitean due to its definition as a commutator of two hermitean operators It has a dimension of (Energy)², as the Hamiltonian also has a dimension of (Energy), which implies that the flow parameter B in equation (1.2) has a dimension of (Energy)⁻² Furthermore, we will demonstrate that the canonical generator in equation (1.4) effectively achieves the desired energy scale separation in the Hamiltonian flow, with the identification Λ feq = B⁻¹/² as shown in equation (1.5).
The interplay between equations (1.2) and (1.4) is crucial for understanding the flow equation method, which aims to diagonalize the Hamiltonian H(B) through a systematic expansion Unlike exact diagonalization techniques such as the Bethe ansatz, this method is applicable to generic non-integrable Hamiltonians The flow equation method is designed not to compete with exact analytical methods but to provide a nonperturbative approach for approximating the diagonalization of quantum many-body Hamiltonians.
Outline and Scope of this Book
This book aims to equip readers with a comprehensive understanding of the flow equation method, enabling them to apply it effectively to their own challenges Chapters 2 and 3 cover the fundamental technical aspects of the method, while Chapter 4 explores various applications to significant many-body problems, establishing a solid foundation for understanding its advantages and limitations Chapter 5 introduces recent advancements, including strong-coupling models and non-equilibrium issues Throughout the book, detailed solutions to model Hamiltonians will be presented to illustrate the method, with a primary focus on the technical aspects of the flow equation method itself.
An intriguing case is presented by the Ising-coupled Kondo impurities model, which demonstrates a significant quantum phase transition For a deeper understanding of the physical motivations behind these model Hamiltonians, we recommend consulting textbooks on condensed matter theory.
Chapter 2 introduces the flow equation method as a fundamental approach for diagonalizing many-body Hamiltonians and deriving flow equations based on the energy-diagonality parameter Λ In Section 2.1.1, the potential scattering model is explored in detail, providing a clear and pedagogical solution Although this model is simple from a many-body perspective, it serves as an essential foundation for tackling more complex problems later in the text, making it highly beneficial for thorough study.
Chapter 3 focuses on the transformation of observables under the unitary flow of the flow equation method, highlighting its significance for practical applications This method offers a unique advantage over other scaling techniques by allowing the evaluation of physical quantities across all energy scales within a single framework However, the transformation of observables can be particularly confusing for those accustomed to conventional many-body techniques, as observables may undergo substantial changes during the unitary flow This unfamiliarity is a crucial aspect of the flow equation method To clarify this transformation, we provide a simple example, the resonant level model, in Section 3.3.2, which serves as an excellent foundation for tackling more complex problems later on.
Chapter 4 explores the application of the flow equation method to complex many-body problems, focusing on the intricacies of interacting many-body Hamiltonians In Section 4.2, we provide a detailed pedagogical analysis of the Kondo model, presenting its flow equation solution through a third-order expansion of the running coupling constant Subsequently, Section 4.3 examines the spin-boson model as a representative bosonic system, enhancing our understanding of how dissipative effects arise within the purely unitary framework of the flow equation method.
Fermi liquid theory serves as the fundamental framework for understanding interacting electron systems in modern physics In Section 4.4, we will explore its connection to the flow equation approach, highlighting how this method underpins Fermi liquid theory as one of its microscopic foundations Specifically, we will examine the relationship between Landau’s quasiparticles, which possess a finite lifetime, and the transformation of fermionic creation and annihilation operators through this unitary flow.
Section 4.5.1 explores the application of flow equations in deriving effective Hamiltonians, specifically re-evaluating the Fröhlich unitary transformation through the lens of the flow equation method This analysis reveals a significantly different outcome, as initially noted by Lenz and Wegner The findings highlight the importance of retardation effects in the flow equation's effective electron-electron interaction, which are crucial for accurately determining the superconducting transition temperature in various materials.
Chapter 5 explores two promising aspects of the flow equation method for future research: strong-coupling problems in Section 5.1 and non-equilibrium issues in Sections 5.2 and 5.3 Traditional scaling methods encounter significant limitations in addressing these challenges, but the flow equation approach offers solutions to some of these obstacles.
In Section 5.1, we explore the sine–Gordon model, a significant one-dimensional strong-coupling model relevant to various low-dimensional quantum systems In its strong-coupling phase, traditional scaling methods fail due to a diverging running coupling constant However, by employing flow equations, we can identify a distinct expansion parameter that facilitates a systematic controlled expansion, even within the strong-coupling phase This approach enables a comprehensive examination of the transition from weak-coupling to strong-coupling physics in the sine–Gordon model.
In Section 5.2, we explore stationary non-equilibrium problems, focusing on the Kondo model under an applied voltage bias The steady-state current in this system is influenced by various energy scales that affect its low-temperature behavior The flow equation method's ability to retain all degrees of freedom in the Hilbert space, unlike conventional scaling approaches, is crucial for analyzing these non-equilibrium models Section 5.3 further discusses the application of the flow equation method to non-equilibrium scenarios, particularly the real-time evolution of a quantum system not prepared in its ground state, emphasizing the importance of considering the entire Hilbert space rather than just low-lying excitations near the equilibrium ground state.
Section 5.4 concludes this book with an outlook into the future perspec- tives of the flow equation method and open questions.
1 P.W Anderson:Basic Notions of Condensed Matter Physics, 6th edn (Addison-
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7 H Fr¨ohlich, Proc Roy Soc A215, 291 (1952)
8 P Lenz and F Wegner, Nucl Phys B[FS]482, 693 (1996)
This chapter establishes the foundational framework for the flow equation approach in many-particle systems, emphasizing the significance of energy scale separation This principle guides the development of the flow equation method, with key concepts outlined in Section 2.2.4 To illustrate the flow equation approach, we analyze a simple model in Section 2.3, comparing the flow equation solution to the traditional scaling method.
Energy Scale Separation
Potential Scattering Model
One realization of the potential scattering model is a gas of spinless electrons interacting with an impurity potentialV( x ) The Hamiltonian of this system is
For the scattering potential we use aδ-function of strengthg,
In a system exhibiting spherical symmetry, the Hamiltonian can be simplified to a one-dimensional model focused on s-waves near the origin The electron creation and annihilation operators are represented by \( c_k \) and \( c_k^\dagger \), adhering to the fermionic anticommutation relation \( \{c_k, c_k^\dagger\} = \delta_{kk} \) By discretizing the system with \( N \) band states, the Hamiltonian is expressed in a specific mathematical form.
In the thermodynamic limit as N approaches infinity, we define the density of states ρ(ε) as k times the Dirac delta function δ(−k), based on the original problem presented in equation (2.1) For simplicity, we consider a constant density of states within an energy band of width D, represented as ρ(ε) = 1/D for the range 0 < ε < D.
The density of states resembles that of a two-dimensional tight-binding square lattice, characterized by a discontinuity at the band edge This discontinuity is crucial as it leads to nonperturbative behavior in the coupling constant g, which will be discussed further.
To begin, we derive the precise solution for the spectrum of the quadratic Hamiltonian (2.3), which presents a straightforward problem In contrast, for interacting non-quadratic systems, we typically need to apply perturbation theory The Hamiltonian (2.3) can be expressed as an N × N matrix.
H =H 0+H int (2.6) and the diagonal part
, (2.7) where i = (i−1)∆ ,∆=D/(N−1) The interaction part is given by
For an eigenvector v = (v 1 , , v N ) with eigenvalueE we can write
Summing the left hand side of this equation overiallows us to eliminate the v i ’s, and yields the following condition for the eigenvalueE:
Fig 2.1 Left hand side of (2.12) The circles denote solutions for a repulsive potential with g > 0 and the squares show solutions for an attractive potential withg 0, applying the findings from equation (2.14) In this scenario, all band energies experience a shift due to impurity contributions, which are proportional to 1/N, while maintaining a small coupling assumption, |ρg| 1, to minimize band edge effects Consequently, we conclude that all band energies are elevated, reflecting a consistent energy shift.
– that is determined by the coupling constantg at large energies (i.e., en- ergies not much smaller than the bandwidth) and
– that vanishes as 1/ln(D/) for small energies.
Energy shifts influence the impurity contribution ∆E imp to the system's total energy at absolute zero In a conduction band filled with electrons up to the chemical potential, this relationship becomes evident.
For small fillingàDexp(−1/ρg) this gives
∆E imp =à 1 ln(D/à) , (2.18) while foràD exp(−1/ρg) one finds
In the study of many-body systems, we often rely on perturbation expansions to achieve analytical understanding of complex Hamiltonians To illustrate this, we apply conventional second-order perturbation theory to the potential scattering Hamiltonian (2.3) This approach will provide valuable insights into the nature of perturbation expansions when contrasted with exact results.
Our starting point is the standard perturbative expansion for the eigenen- ergies in second order:
, (2.21) where we have replaced the summation by an integral (which is possible for
N → ∞) For eigenenergies much smaller than the bandwidthD the energy shift in second order perturbation theory is therefore given by
The expansion of the exact expression (2.14) in powers of the coupling constant is valid only when |ρ g ln(D/j)| is significantly less than one As energies approach the lower band edge, specifically at j D exp(−1/|ρg|), perturbative results fail and may even indicate a negative energy shift, which contradicts the correct outcome This reveals a low-energy regime where perturbation theory becomes unreliable, even with a small coupling constant that typically suggests a dependable expansion parameter For instance, the second-order perturbation theory outcome for the impurity's contribution to the total energy illustrates this breakdown.
We notice that this is quite wrong for small fillingsàD exp(−1/|ρg|) by comparison with the exact result (2.18).
The analysis of expression (2.14) reveals that increasing the perturbative expansion's power in the coupling constant does not resolve the issue; instead, it generates additional uncontrolled terms in the low-filling limit This conclusion applies to both repulsive and attractive potential scattering scenarios.
Perturbation theory can fail even with a simple quadratic Hamiltonian when using a small expansion parameter To address this issue, the conventional method involves reorganizing the perturbation expansion through scaling theory This approach focuses on initially analyzing the impact of high-lying states on low-energy physics, progressively working down to lower energy levels By applying the principle of energy scale separation, a more stable expansion can be achieved.
To achieve our goal, we can utilize various diagrammatic and path integral methods, as detailed in the extensive literature on the subject In this discussion, we will focus on a method that constructs low-energy effective Hamiltonians, emphasizing key concepts The fundamental approach involves partitioning the Hilbert space H of our system into two components through the use of projection operators P and Q, ensuring that P + Q = 1.
H P = PHP denotes a low-energy Hilbert space and H Q = QHQ is the complementary high-energy Hilbert space Let |Ψ be an eigenstate of the Hamiltonian:
We can split up|Ψ in a part that lives inH P and one that lives inH Q ,
The eigenvalue condition (2.24) then takes the following form:
We eliminate the component of the eigenstate in the high-energy Hilbert space,
E−QHQQHP|Ψ P , (2.27) and express the eigenvalue equation (2.24) in terms of the low-energy Hilbert
In many applications, the focus is on eigenenergies E that are close to the ground state energy E_gs, which are significantly lower than the energies found in the high-energy Hilbert space H_Q Consequently, we can simplify the left side of equation (2.28) using an effective Hamiltonian H_eff.
The eigenvalue equation (2.24) reduces to an eigenvalue equation forH eff in the low-energy Hilbert spaceH P :
While this method may not be ideal for exploring higher orders of scaling expansion, it serves as an effective educational tool for understanding the core concepts of the scaling approach.
Kondo Model
The potential scattering model, represented by a quadratic Hamiltonian, can be solved exactly using basic methods In contrast, the Kondo model presents a significant interacting many-body problem that is crucial for understanding strongly correlated electron systems Notably, its thermodynamic properties can be precisely determined through the Bethe ansatz.
The impurity contribution to the total energy, denoted as ∆E imp, is analyzed in the context of a repulsive potential scattering model with a density parameter ρ g = 0.2, illustrating its dependence on the chemical potential The solid line represents the exact results derived from scaling theory, while the dashed line indicates the findings from unrenormalized second-order perturbation theory This approach, based on integrability, has limitations as it cannot be applied universally to different conduction band densities of states or for assessing correlation functions beyond the low-energy regime.
The Kondo model, akin to the potential scattering model, addresses the potential scattering of electrons, but it uniquely incorporates a quantum mechanical spin-1/2 degree of freedom, S, which follows its own dynamics The Kondo Hamiltonian is expressed in a specific mathematical form that captures these interactions.
In correlated electron systems, the antiferromagnetic exchange coupling J (with J > 0) influences the spin of an electron in a strongly correlated singly occupied orbital, as described by the equation Sã k,k ,α,β c † k α σ αβ c kβ The system's naive twofold ground state degeneracy is effectively screened by many-particle processes, resulting in the impurity spin being screened below the Kondo temperature T K ∝D exp(−1/ρJ) As the temperature approaches zero, the impurity spin susceptibility becomes finite, and the Fermi sea responds to the formation of the screened Kondo singlet by exhibiting an increased density of states at the Fermi level This increase in density of states subsequently contributes to a higher impurity contribution to the specific heat, highlighting the significance of the Kondo problem in condensed matter physics.
The advancement of dynamical mean field theory for strongly correlated electron systems has sparked renewed interest in effective analytical and numerical methods for solving the Kondo model, particularly for various conduction band densities of states This area of research is pivotal in modern condensed matter theory and serves as a valuable test case for the flow equation method For a comprehensive understanding of the Kondo problem, readers are encouraged to explore the extensive literature available on the topic.
We analyze the Kondo model through scaling methods, assuming a symmetric conduction band around the Fermi surface at F = 0, with a bandwidth of 2D, where k ranges from -D to D The projection operator Q is defined to project onto states with energies within the range of |q| between D - δD and D.
Equation (2.29) then yields the following contribution to the low-energy in- teraction:
We use the standard relation for the spin-1/2 SU(2)-algebra,
The occupation numbers are defined by the equation n(q) = F S|c † qβ c qβ |F S, indicating the relationship to the non-interacting Fermi sea, with no summation over β Consequently, the right-hand side of the equation can be simplified, leading to a clearer contribution to the interaction term in Hamiltonian H P.
We have ignored an uninteresting constant that contributes to the ground state energy The scaling equation for the spin-spin interaction then reads dJ dD =−ρJ 2
At zero temperature the occupation number is given by n( ) =Θ(−) and
(2.46) takes the same structure as the scaling equation for the potential scat- tering model (2.36) except for a minus sign: dJ d lnD =−ρJ 2 (2.47)
Since the spin-spin coupling is generally antiferromagnetic, J >0, the solu- tion is
1−ρJln(D/D eff) (2.48) or ρJ eff (D eff ) = 1 ln(D eff /T K ) (2.49)
Here we have introduced the Kondo temperature
The scaling approach fails when the effective bandwidth reaches the Kondo temperature, leading to a divergence in the expansion parameter ρJ eff Consequently, the Kondo model demonstrates strong-coupling behavior akin to that observed in the attractive potential scattering model.
The scaling approach is a valuable tool for resuming perturbation theory as it highlights the low-energy scale, T_K, of the Kondo model Screening of the magnetic impurity moment occurs at temperatures below T_K However, identifying T_K through perturbation theory is not feasible, even with small coupling constants J, since the Taylor expansion around ρJ = 0 does not converge.
Flow Equation Approach
Motivation
The flow equation approach, originating from the work of Glazek and Wilson and Wegner in the early 1990s, offers a novel method for implementing the principle of energy scale separation Glazek and Wilson introduced this concept in high-energy physics as the similarity renormalization scheme, while Wegner applied it to condensed matter theory, coining the term flow equations This book focuses on solid-state systems, adopting Wegner's notation for clarity For additional insights into the similarity renormalization scheme, readers are encouraged to consult the relevant literature.
The similarity between the Kondo model and the potential scattering model arises from a fundamental discontinuity In the Kondo model, this discontinuity is reflected in the zero temperature occupation number at the Fermi level, while in the potential scattering model, it is characterized by a discontinuous band density of states at the lower band edge.
The key distinction between flow equations and the traditional scaling method is illustrated in Fig 1.1 While the conventional approach removes highly excited states within the energy range |E| ∈ [Λ RG − δΛ RG, Λ RG], the flow equation method focuses on eliminating matrix elements that connect states with differing energy levels.
In low-energy physics, both the flow equation approach and the conventional scaling method appear similar when considering the energy range |∆E| ∈ [Λ feq − δΛ feq, Λ feq] An excitation to a higher energy level, Λ feq, in the flow equation method necessitates multiple interactions with a small coupling constant Consequently, for low-energy excitations above the ground state, both approaches produce identical scaling flows, leading to the identification of Λ RG with Λ feq Thus, the flow equation method can be viewed as encompassing the conventional scaling approach as a special case.
However, in the flow equation approach the Hilbert space remains unchanged so that we can also get information about, for example, correlation functions at higher energies.
Since its inception in 1994, the flow equation approach has been applied to various contexts, including non-equilibrium models that do not prioritize the ground state This article will explore both the similarities and differences between the flow equation approach and traditional scaling methods, providing insights into different models By doing so, readers will be better equipped to select the most appropriate method for their specific research problems.
Infinitesimal Unitary Transformations
To achieve the flow from Λ feq to Λ feq - δΛ feq, it is essential to develop new tools, as simply eliminating part of the Hilbert space is inadequate Since our Hamiltonians must remain unitarily equivalent throughout this flow, we must explore infinitesimal unitary transformations that can effectively remove interaction matrix elements coupling states with an energy transfer |∆E| within the range of [Λ feq - δΛ feq, Λ feq] For instance, considering a diagonal Hamiltonian can be a useful approach in this context.
H 0 n n c † n c n (2.51) and a specific (hermitean) interaction matrix element M that we want to eliminate from the total interaction termH int:
We assume that all interaction terms are proportional to a small coupling constantg Then we construct a new Hamiltonian ˜H via a unitary transfor- mationU (U −1 =U † ),
H˜ =U H U † , (2.53) whereU = e η with an antihermitean generatorη In our case we choose η= g m
We have therefore eliminated the interaction term M to first order in the coupling constant from the total interaction term.
We now want to repeat this procedure iteratively by looking at smaller and smaller energy differences ∆E = m
2 − m 1 − m 2 A convenient way to do this for general interaction terms has been suggested by Wegner
[9] We label the one-parameter family of unitarily equivalent Hamiltonians with some flow parameterB 4 and consider the differential equation dH(B) dB = [η(B), H(B)], (2.57) where η(B) =−η † (B) is an antihermitean generator.
Let us first of all verify that the solution of (2.57) generates Hamiltonians
H(B) that are unitarily equivalent to the initial HamiltonianH(B= 0) We claim that the solutionH(B) of (2.57) is given by
Here T B { .}denotesB-ordering defined in the same way as the usual time ordering The generator η(B i ) with the largest B i is commuted to the left etc.,
In this article, we denote the flow parameter as B to prevent confusion with the logarithmic notation commonly used in conventional scaling equations We consider the permutation π ∈ S n, represented as B π(1) B π(2) B π(n) It can be verified that U(B) is a unitary operator, satisfying the condition U(B)U † (B) = 1 Additionally, we explore the derivative with respect to B, expressed as d/dB.
With our definition (2.59) one finds dU(B) dB U † (B) =η(B)U(B)U † (B) =η(B), (2.63) and therefore d dB
This shows thatH(B) defined by H(B) =U(B)H(B= 0)U † (B) obeys the differential equation (2.57) The correct initial condition is also fulfilled since
U(B= 0) = 1 ThereforeH(B) from (2.58) is the solution of (2.57).
After showing unitary equivalence, we next need to construct a suitable antihermitean generatorη(B) that implements (2.56) in an energy scale sep- arated way.
Choice of Generator
The selection of the generator η, as illustrated in Fig 1.1, is crucial to the flow equation method This method involves a Hamiltonian that can be divided into two components: a diagonal part and an interaction part.
H(B) =H 0(B) +H int(B), (2.65) Wegner suggested the followingcanonical generator 5 η(B) def = [H 0(B), H int(B)] (2.66)
The choice of η(B) as the commutator of two Hermitian operators ensures that it is anti-Hermitian, which is a necessary condition With a dimension of (Energy)², the flow parameter B consequently has a dimension of (Energy)⁻² This relationship is significant when analyzing a typical interacting fermion model characterized by a specific kinetic energy.
H 0 (B) n n (B)c † n c n (2.67) and a two-particle interaction term
5Similar ideas calleddouble bracket flowandisospectral flowhave independently been developed in numerical mathematics [17, 18, 19].
Equation (2.66) yields the canonical generator η(B) m 1 ,m 2 ,m 1 ,m 2 g m 1 m 2 m 1 m 2(B) ( m
2 c m 1 c m 2 (2.69) and this in turn theflow equation (2.57): dH(B) dB = [η(B), H 0(B)] + [η(B), H int(B)]
One can easily verify that the structure of these equations is generic for all interaction terms:
– In the generator the interaction coefficient is multiplied by the energy transfer of the scattering process.
– In the flow equation the interaction coefficient is multiplied by the square of the energy transfer of the scattering process, and the overall sign is negative.
Identification of the same interaction terms on the left hand and right hand side of (2.70) leads to the differential equation dg m
As long as we can neglect the coupling constant in comparison with the energy transfer, we can use the approximatelinearized solution: g m
The canonical choice of the generator effectively decouples interaction matrix elements across different energy scales, as illustrated in Fig 1.1 For dimensional consistency, we identify Λ feq as B - 1/2 Initially, with small values of B, the flow removes interaction matrix elements with significant energy differences As the flow parameter B increases, the process begins to decouple increasingly degenerate states Notably, the intriguing scaling properties emerge from the O(g²) term in the analysis.
The initial separation of the Hamiltonian into diagonal and interaction components introduces a level of arbitrariness, as the "correct" diagonal part is not predetermined Choosing an incorrect diagonal can lead to large coupling constants and expansion parameters in the flow equation approach, resulting in uncontrolled behavior during the flow process A flow equation solution is only justified a posteriori if the chosen generator maintains a small expansion parameter throughout the flow.
To enhance the effectiveness of the canonical generator in making the Hamiltonian more energy diagonal, two essential conditions must be satisfied.
Here the trace is taken over all states in the Hilbert space Notice that these conditions are always fulfilled if
The diagonal elements of a quantum state preserve its quantum numbers, which is exemplified by the Hamiltonian H₀(B) being representable as a sum of number operators, such as kinetic energy.
– and if the interaction term H int(B) contains only scattering processes which change at least one quantum number.
The product of H 0(B) and H int(B) in (2.74) and (2.75) then necessarily changes at least one quantum number of any state that it acts on This implies that the traces (2.74) and (2.75) vanish.
If (2.74) and (2.75) are fulfilled, the canonical generator reduces the in- teraction part of the Hamiltonian in the sense that d dB Tr(H int 2 (B))0 (2.76)
This is the property that we want to prove now We first use (2.75) to show d dB Tr
Next we employ the definition of the flow equation (2.57) and the possibility for cyclic exchange under the trace: d dB Tr
We use the definition of the canonical generator (2.66) and the fact that it can equivalently be expressed as η(B) = [H(B), H int(B)] (2.79) Therefore (2.78) is equivalent to d dB Tr
0, (2.80) since η † (B)η(B) is a positive semi-definite operator This implies that the
Flow enhances the diagonal nature of the Hamiltonian, provided that the generator η(B) remains non-zero If η(B) does vanish, the Hamiltonians H₀(B) and H_int(B) can commute, allowing them to be diagonalized within a shared eigenbasis According to Wegner's canonical choice of generator, it effectively generates a Hamiltonian that becomes increasingly energy diagonal as B approaches infinity.
While the property described in equation (2.80) is highly desirable, it is important to note that in many-body Hamiltonians, traces often diverge in the thermodynamic limit Additionally, approximations are typically necessary when analyzing the flow of a many-body Hamiltonian, even though (2.80) applies to the exact unitary flow Despite these challenges, the canonical generator remains an excellent choice, exhibiting the desired properties in numerous many-body problems.
Flow Equations
Let us sum up the main ingredients of the flow equation method that we have developed so far.
I The unitary flow of the Hamiltonian is generated by the flow equation dH(B) dB = [η(B), H(B)], (2.81) where H(B = 0) is the initial Hamiltonian and H(B =∞) is the final energy-diagonal Hamiltonian We want to achieve this flow in an energy scale separated way as indicated in Fig 1.1 with the identificationΛ feq B −1/2 η(B) is an antihermitean operator that generates this flow.
II The canonical generator is defined as the commutator of the diagonal part with the interaction part of the Hamiltonian: η(B) def = [H 0 (B), H int (B)] (2.82)
This choice makes the flowing Hamiltonian more energy-diagonal in an energy scale separated way as can be seen from (2.80).
III An interaction matrix elementg ∆E with energy transfer∆E(as measured with respect to H 0 ) decays like g ∆E (B) =g ∆E (B= 0) e −B(∆E) 2 +O(g 2 ) (2.83) for the canonical choice of the generator The higher order terms in this equation will turn out to be responsible for nontrivial scaling properties, i.e nontrivial β-functions The analysis of such higher order terms will comprise the main part of this book.
Some remarks are in order concerning this general methodology.
1 The main challenge of the flow equation approach lies in the genera- tion of higher and higher order interaction terms during the flow This is apparent from looking at the differential equation (2.81) If the original interaction contains two-particle interaction terms, that is terms with two fermion creation and two fermion annihilation operators, thenη has the same structure The commutator of η with the interaction Hamiltonian on the right hand side of (2.81) then produces a term with three creation and three annihilation operators This makes the generator more compli- cated and this progression of higher order terms continues indefinitely In Chap 4 we will discuss possibilities to truncate this infinite sequence to produce a systematic expansion.
2 If our initial Hamiltonian commutes with some symmetry operatorS
[H(0), S] = 0, (2.84) then the canonical generator also commutes withS 6 Consequently, the flow preserves this symmetry:
One can only destroy this symmetry by making an approximation on the right hand side of (2.81) that does not respect the symmetry.
6Notice that [H 0 , S] = 0 holds trivially in all realistic examples.
3 The canonical choice of the generator (2.82) is particularly well-behaved as we have seen in the previous chapter.As a general rule I recommend working with the canonical generator since this choice is very robust.How- ever, there can be situations where other choices of the generator turn out to be more convenient in the sense that they simplify the structure of differential equations during the flow 7 Different choices of the generator amount to different ways of doing the expansion around the noninteract- ing model A particular choice of the generator can be justified a poste- riori if the resulting expansion has a small expansion parameter during the flow since we typically have to neglect certain higher orders in our expansion parameter on the right hand side of (2.81).
The flexibility in selecting various generators within the flow equation approach is not indicative of a lack of control; rather, it highlights the existence of multiple possible expansions When higher-order expansions are achievable and convergent, the observable outcomes will remain consistent The essential strategy is to identify an expansion that captures a significant amount of information at lower orders, facilitating manageable calculations This emphasizes the importance of maintaining a small expansion parameter throughout the flow process.
In certain situations, utilizing a generator other than the canonical one can be beneficial, but it is crucial to maintain energy scale separation This principle is upheld when the coefficient associated with an interaction term in η decreases at least linearly with the energy transfer ∆E Throughout this book, we will primarily use the canonical generator, as it consistently produces stable expansions, except in specific cases such as strong-coupling models.
4 One can ask the question whether it is useful to reconstruct the full uni- tary transformationU(B=∞) from the infinitesimal steps using (2.59).
Many-body problems present significant challenges due to the non-commuting nature of generators at different B-values, where [η(B1), η(B2)] ≠ 0 This complicates the evaluation of the B-ordered exponential Additionally, the full unitary transformation U(B=∞) obscures its representation as a B-ordered exponential, indicating that it is generated by infinitesimal transformations.
7 We will discuss an example for this in Sect 4.3.
8Notice that the complexity of such calculations typically grows considerably with every order.
A generator that disregards energy scale separation does not inherently lead to an uncontrolled approach, as evidenced by the successful applications of conventional perturbation theory in various interactions However, caution is essential when using such a generator to avoid potential uncontrolled errors in higher orders To ensure accuracy, it is advisable to follow steps that maintain energy scale separation, making the infinitesimal formulation a more natural choice This approach allows for a proper justification of the necessary approximations in generic many-body problems.
Example: Potential Scattering Model
Setting up the Flow Equations
The flow equation approach aims to enhance the energy-diagonal nature of interactions To achieve this, the ansatz for the flowing Hamiltonian must be appropriately formulated In the context of the potential scattering model, this concept is expressed through the equation provided in section 2.3.
( k +g kk (B))c † k c k + k,k :k =k g k k (B)c † k c k (2.86) with the initial condition g k k (B = 0) = g
The appropriate separation of diagonal and interaction part of the Hamil- tonian is obvious:
Here we have introduced the abbreviationE k (B) def = k +g kk (B).
We begin by constructing the canonical generator as outlined in equation (2.82) It is essential to first calculate the fundamental commutators, which will then be incorporated into the various commutators responsible for generating the flow In the context of the potential scattering model, this necessitates the computation of these key elements.
Let us first think of potential scattering for fermions with the basic anticom- mutation relations
We use the following expression that holds generally for all operators A, B,
=A{B, C}D−C{D, A}B+CA{B, D} − {C, A}BD (2.93) With this expression it is straightforward to evaluate (2.91): 11
Alternatively, one can consider potential scattering of bosonic particles Then we have the basic commutation relations:
Here we can use the following expression that also holds generally for all operatorsA, B, C, D:
Remarkably, we see by comparison with (2.93) that only the anticommutators are replaced by commutators Hence we also find for bosonic particles
Our calculation is applicable to both fermionic and bosonic cases, as all commutators in the flow equation solution of the potential scattering model can be derived from this expression.
The canonical generator of the flow equation approach (2.82) is given by η(B) = [H 0 (B), H int (B)] p,p η p p c † p c p (2.98)
I strongly endorse this procedure for tackling complex many-body problems, as it effectively organizes the calculations and simplifies the process of identifying and tracing errors back to fundamental commutators.
The commutator bilinear in fermions can often be expressed through basic fermionic anticommutation relations, which may come as a surprise It is advisable to examine equation (2.93) for clarity In this context, we define η p p as (E p − E p )g p p, utilizing relation (2.94) For brevity, we will generally omit the B-dependence of the parameters, as it is understood that all couplings vary with B during the flow To derive the right-hand side of the fundamental flow equation (2.57), we need to compute the commutator of η(B) with the Hamiltonian H(B) This process is conveniently executed in two steps: first, we calculate the commutator [η(B), H 0 (B)], followed by [η(B), H int (B)] The evaluation of the second commutator is crucial for the flow equation approach in genuine many-particle models, and again, applying (2.94) will facilitate this calculation.
We are now ready to compare the coefficients of the operators on the left and and the right hand side of the fundamental flow equation dH dB = [η(B), H(B)] (2.102)
The structures of differential equations relevant to interacting many-particle systems frequently recur in this context The flow equations effectively achieve the desired diagonalization through the linear coupling constant component, represented as \( g_{kk}(B) \propto g_{kk}(0) e^{-B(E_k - E_k)^2} \) with \( B \propto \Lambda^{-2} f_{eq} \) Additionally, the shift in single-particle energies, denoted as \( \Delta_k \), is determined by the exact solution provided in equation (2.14).
∆ k =g kk (B=∞), (2.105) since all other couplings withk =kvanish forB→ ∞according to (2.104).
Methods of Solution
Equation (2.103) and the initial condition (2.87) represent the solution to the flow equation in the potential scattering model In this article, we will explore three distinct methods for solving these coupled differential equations, which are also typical of interacting many-particle systems.
1 Exact solution based on numerical implementation on a computer.
2 Diagonal parametrization of the running coupling constant.
3 IR-parametrization of the running coupling constant.
The exact solution of flow equations provides precise quantitative results without approximations, making it essential for detailed analysis In contrast, the IR-parametrization simplifies these equations to a single differential equation, enabling analytical solutions that offer valuable insights into low-energy physics, albeit with some quantitative errors The diagonal parametrization serves as a middle ground, reducing the number of differential equations in a manner that scales linearly with the number of band states, balancing accuracy and complexity.
In a genuine many-particle system, it is common to begin with the infrared (IR) parametrization before advancing to more precise solution techniques However, for educational purposes and to evaluate the accuracy of different methods, we will first examine the behavior of the exact solution.
To numerically solve the system of differential equations described in (2.103), an adaptive step-size 5th order Runge-Kutta algorithm is recommended, widely available in numerical algorithm compilations By discretizing the conduction band into N states, we generate N² coupled differential equations Standard workstations can efficiently handle systems of up to 500×500 coupled equations within a few hours For successful implementation, consider optimizing your algorithm and ensuring efficient memory management.
– Consider the differential equation for a coupling constantg k k (B) for given k , k Once the flow has proceeded to values ofBsuch thatB( k − k ) 2
5, the couplingg k k (B) has decayed to values that are typically only e − 5 ≈
For larger values of B, it is advisable to set k(B) to zero, as the accuracy of the numerical solution will be inherently limited, given that it is only 0.007 of its initial value (less than 1%) This approach significantly reduces the number of differential equations in the later stages of the flow and allows the numerical algorithm to determine the appropriate step size for large B values effectively.
The flow of coupling constants \( g_{kk}(B) \) is illustrated for a repulsive potential scattering model with \( \rho_g = 0.2 \) and band energies \( k \) ranging from 0 to 1 The diagrams depict the evolution of coupling constants at various flow parameters: \( B = 0 \) (upper left), \( B = 5 \) (upper right), \( B = 50 \) (lower left), and \( B = 500 \) (lower right) It is evident that as \( B \) approaches infinity, only the diagonal couplings \( (k = k) \) remain significant Additionally, the nonperturbative scaling effects are most pronounced at the lower band edge \( (k = k = 0) \).
When dealing with systems where the interesting low-energy scale, such as the Kondo temperature (T_K), is significantly smaller than the system's bandwidth, it is beneficial to implement a finer discretization mesh at the relevant low-energy scale This approach aids in accurately resolving low-energy phenomena while also optimizing computational efficiency One effective method is to utilize a variant of logarithmic discretization, commonly used in numerical renormalization group (NRG) techniques, where the discretization parameter is defined as i−1 / i = Λ with Λ > 1, allowing for a focus on low-energy limits as i approaches infinity Additionally, it is possible to create a finer mesh around any specific nonzero energy scale of interest.
Figure 2.3 illustrates the typical flow of coupling constants in the repulsive potential scattering model, revealing patterns akin to those observed in interacting many-body systems This resemblance underscores the importance of examining the model in greater detail, particularly as the Hamiltonian H(B) exhibits an increasing complexity.
Logarithmic discretization is a crucial approximation in Numerical Renormalization Group (NRG) methods, enabling their effective application Conversely, in the numerical implementation of flow equations, a finer mesh serves merely as a numerical enhancement to improve solution efficiency, as it is theoretically possible to approach the limit of Λ→1.
The flow of the diagonal coupling constants \( g_{kk}(B) \) is illustrated in Figure 2.4, based on a repulsive potential scattering model with parameters identical to those in Figure 2.3 (where \( \rho_g = 0.2 \) and band energies \( k \) range from 0 to 1) It is important to note that the nonperturbative scaling effects are most pronounced at the lower band edge \( k = 0 \), ultimately leading to \( g_{00}(B) \) approaching zero.
B→ ∞ energy-diagonal for larger values ofBas expected in the flow equation frame- work Only the diagonal couplings with k = k remain nonzero forB → ∞.
The banded structure of the flowing Hamiltonian highlights its advantages over the conventional scaling approach, which is characterized by a reduced size of the Hilbert space This distinction is clearly illustrated in Figure 1.1, while Figures 2.3 and 2.4 further emphasize these differences.
The renormalization of diagonal couplings demonstrates the non-perturbative scaling properties of our model Notably, the numerical solution for B=∞ aligns perfectly with the exact solution presented in Section 2.1.1 Additionally, the diagonal couplings g kk (B =∞) provide the precise shift of single-particle energy levels as outlined in equations (2.105) and (2.14).
Figures 2.3 and 2.4 motivate an approximate parametrization of the N ×
The N coupling constants can be expressed through their N diagonal entries by addressing the linear component of the differential equation (2.103) For simplified notation, we define g k (B) as g kk (B) and let p represent the arithmetic mean of k, calculated as p = (k + k)/2.
Thediagonal parametrization then amounts to the ansatz g k k (B) =g k k (B) e −B( k − k ) 2 (2.107)
Equation (2.107) serves as an approximation that aligns closely with the exact solution, which will be explored further We will now analyze its implications for the flow equations presented in (2.103), focusing specifically on the differential equations related to the diagonal couplings, expressed as dg k dB = 2p.
In the repulsive case of the impurity model, energy levels are shifted by a factor of 1/N Consequently, in the thermodynamic limit as N approaches infinity, we can simplify the expression by substituting (E k − E p) with (k − p).
Strong-Coupling Case
Our analysis has so far focussed on the case of repulsive potential scattering.
As we had noticed in the exact solution in Sect 2.1.1, for attractive scattering g